Proceedings of the International Congress of Mathematicians, Madrid 2006 Volume 3

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ICM_vol3_titelei

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Proceedings of the

International Congress of Mathematicians Madrid, August 22–30, 2006

VOLUME III

Invited Lectures Marta Sanz-Solé Javier Soria Juan Luis Varona Joan Verdera Editors

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S E M E S

European Mathematical Society

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Editors: Marta Sanz-Solé Facultat de Matemàtiques Universitat de Barcelona Gran Via 585 08007 Barcelona Spain Javier Soria Departament de Matemàtica Aplicada i Anàlisi Facultat de Matemàtiques Universitat de Barcelona Gran Via 585 08007 Barcelona Spain

Juan Luis Varona Departamento de Matemáticas y Computación Universidad de La Rioja Edificio J. L. Vives Calle Luis de Ulloa s/n 26004 Logroño Spain

Joan Verdera Departament de Matemàtiques Universitat Autònoma de Barcelona 08193 Bellaterra (Barcelona) Spain

2000 Mathematics Subject Classification: 00Bxx

ISBN 978-3-03719-022-7 Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.ddb.de. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. ©2006 European Mathematical Society Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum FLI C4 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the author’s TEX files: I. Zimmermann, Freiburg Printed in Germany 987654321

Contents 10

Ordinary differential equations and dynamical systems (continued)

Robert Ghrist Braids and differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

Anton Gorodetski, Brian Hunt and Vadim Kaloshin* Newton interpolation polynomials, discretization method, and certain prevalent properties in dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

Bryna Kra From combinatorics to ergodic theory and back again . . . . . . . . . . . . . . . . . . . . . . .

57

Patrice Le Calvez From Brouwer theory to the study of homeomorphisms of surfaces . . . . . . . . . .

77

Michael Shub All, most, some differentiable dynamical systems . . . . . . . . . . . . . . . . . . . . . . . . . .

99

Anton Zorich Geodesics on flat surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

11

Partial differential equations

Stefano Bianchini Asymptotic behavior of smooth solutions for partially dissipative hyperbolic systems and relaxation approximation . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Patrick Gérard Nonlinear Schrödinger equations in inhomogeneous media: wellposedness and illposedness of the Cauchy problem . . . . . . . . . . . . . . . . . . . . . 157 François Golse The periodic Lorentz gas in the Boltzmann-Grad limit . . . . . . . . . . . . . . . . . . . . . . 183 Matthew J. Gursky Conformal invariants and nonlinear elliptic equations . . . . . . . . . . . . . . . . . . . . . . . 203 Hitoshi Ishii Asymptotic solutions for large time of Hamilton–Jacobi equations . . . . . . . . . . . 213 Mario Pulvirenti The weak-coupling limit of large classical and quantum systems . . . . . . . . . . . . . 229 Ovidiu Savin Symmetry of entire solutions for a class of semilinear elliptic equations . . . . . . 257 Sylvia Serfaty Vortices in the Ginzburg–Landau model of superconductivity . . . . . . . . . . . . . . . . 267 ∗ In case of several authors, invited speakers are marked with an asterisk.

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Contents

Neil S. Trudinger Recent developments in elliptic partial differential equations of Monge–Ampère type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 Luis Vega The initial value problem for nonlinear Schrödinger equations . . . . . . . . . . . . . . . 303 Juan J. L. Velázquez Singular solutions of partial differential equations modelling chemotactic aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

12

Mathematical physics

Alberto S. Cattaneo From topological field theory to deformation quantization and reduction . . . . . . 339 Bernard Derrida Matrix ansatz and large deviations of the density in exclusion processes . . . . . . 367 Jean-Michel Maillet Correlation functions of the XXZ Heisenberg spin chain: Bethe ansatz approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 Marcos Mariño Gromov–Witten invariants and topological strings: a progress report . . . . . . . . . 409 Igor Rodnianski The Cauchy problem in General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421 Christoph Schweigert*, Jürgen Fuchs, and Ingo Runkel Categorification and correlation functions in conformal field theory . . . . . . . . . . 443 Avy Soffer Soliton dynamics and scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 Cédric Villani Hypocoercive diffusion operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473

13

Probability and statistics

Anton Bovier Metastability: a potential theoretic approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 Raphaël Cerf On Ising droplets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519 Amir Dembo Simple random covering, disconnection, late and favorite points . . . . . . . . . . . . . 535 Peter Donnelly Modelling genes: mathematical and statistical challenges in genomics . . . . . . . 559 K. David Elworthy* and Xue-Mei Li Geometric stochastic analysis on path spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575

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vii

Jianqing Fan* and Runze Li Statistical challenges with high dimensionality: feature selection in knowledge discovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595 Alice Guionnet Random matrices and enumeration of maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623 Steven P. Lalley The weak/strong survival transition on trees and nonamenable graphs . . . . . . . . 637 Yves Le Jan New developments in stochastic dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649 Peter McCullagh* and Jie Yang Stochastic classification models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 669 Andrei Okounkov Random partitions and instanton counting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687 Dominique Picard* and Gérard Kerkyacharian Estimation in inverse problems and second-generation wavelets . . . . . . . . . . . . . 713 Wendelin Werner Conformal restriction properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 741

14

Combinatorics

Alexander Barvinok The complexity of generating functions for integer points in polyhedra and beyond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 763 Mireille Bousquet-Mélou Rational and algebraic series in combinatorial enumeration . . . . . . . . . . . . . . . . . . 789 Jim Geelen, Bert Gerards*, and Geoff Whittle Towards a structure theory for matrices and matroids . . . . . . . . . . . . . . . . . . . . . . . 827 Mark Haiman Cherednik algebras, Macdonald polynomials and combinatorics . . . . . . . . . . . . . 843 Jeong Han Kim Poisson cloning model for random graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 873 Tomasz Łuczak Randomness and regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 899 Imre Z. Ruzsa Additive combinatorics and geometry of numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 911 Francisco Santos Geometric bistellar flips: the setting, the context and a construction . . . . . . . . . . 931 Robin Thomas A survey of Pfaffian orientations of graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 963

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Mathematical aspects of computer science

Manindra Agrawal Determinant versus permanent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alexander S. Holevo The additivity problem in quantum information theory . . . . . . . . . . . . . . . . . . . . . . Jon Kleinberg Complex networks and decentralized search algorithms . . . . . . . . . . . . . . . . . . . . . Omer Reingold On expander graphs and connectivity in small space . . . . . . . . . . . . . . . . . . . . . . . . Tim Roughgarden Potential functions and the inefficiency of equilibria . . . . . . . . . . . . . . . . . . . . . . . . Ronitt Rubinfeld Sublinear time algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Luca Trevisan Pseudorandomness and combinatorial constructions . . . . . . . . . . . . . . . . . . . . . . . .

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999 1019 1045 1071 1095 1111

Numerical analysis and scientific computing

Pavel Bochev and Max Gunzburger* Least-squares finite element methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zhiming Chen A posteriori error analysis and adaptive methods for partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ricardo G. Durán Error estimates for anisotropic finite elements and applications . . . . . . . . . . . . . . Nira Dyn Linear subdivision schemes for the refinement of geometric objects . . . . . . . . . . Randall J. LeVeque Wave propagation software, computational science, and reproducible research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yvon Maday Reduced basis method for the rapid and reliable solution of partial differential equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Endre Süli Finite element algorithms for transport-diffusion problems: stability, adaptivity, tractability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

985

1137

1163 1181 1201

1227

1255

1271

Control theory and optimization

Vivek S. Borkar Ergodic control of diffusion processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1299

Contents

Stephen Boyd Convex optimization of graph Laplacian eigenvalues . . . . . . . . . . . . . . . . . . . . . . . Oleg Yu. Emanouilov (Imanuvilov) Controllability of evolution equations of fluid dynamics . . . . . . . . . . . . . . . . . . . . . Arjan van der Schaft Port-Hamiltonian systems: an introductory survey . . . . . . . . . . . . . . . . . . . . . . . . . . Olof J. Staffans Passive linear discrete time-invariant systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Enrique Zuazua Control and numerical approximation of the wave and heat equations . . . . . . . .

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1311 1321 1339 1367 1389

18 Application of mathematics in the sciences Russel E. Caflisch Multiscale modeling for epitaxial growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Emmanuel J. Candès Compressive sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vicent Caselles Total variation based image denoising and restoration . . . . . . . . . . . . . . . . . . . . . . . Michael Griebel* and Jan Hamaekers A wavelet based sparse grid method for the electronic Schrödinger equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Claude Le Bris Mathematical and numerical analysis for molecular simulation: accomplishments and challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Martin A. Nowak Evolutionary dynamics of cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . David Nualart Fractional Brownian motion: stochastic calculus and applications . . . . . . . . . . . . Anders Szepessy Atomistic and continuum models for phase change dynamics . . . . . . . . . . . . . . . .

19

1419 1433 1453

1473

1507 1523 1541 1563

Mathematics education and popularization of mathematics

Petar S. Kenderov Competitions and mathematics education . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1583 Alan Siegel Understanding and misunderstanding the Third International Mathematics and Science Study: what is at stake and why K-12 education studies matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1599 Ian Stewart Mathematics, the media, and the public . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1631

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Contents

Michèle Artigue, Ehud de Shalit, and Anthony Ralston Panel A: Controversial issues in K-12 mathematical education . . . . . . . . . . . . . . . 1645 Lee Peng Yee, Jan de Lange, and William Schmidt Panel B: What are PISA and TIMSS? What do they tell us? . . . . . . . . . . . . . . . . . 1663 Fr. Ben Nebres, Shiu-Yuen Cheng, Konrad Osterwalder, and Hung-Hsi Wu Panel C: The role of mathematicians in K-12 mathematics education . . . . . . . . . 1673

20

History of mathematics

Leo Corry On the origins of Hilbert’s sixth problem: physics and the empiricist approach to axiomatization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1697 Niccolò Guicciardini Method versus calculus in Newton’s criticisms of Descartes and Leibniz . . . . . . 1719

Special activity Sebastià Xambó Descamps, Hyman Bass, Gilda Bolaños Evia, Ruedi Seiler, and Mika Seppälä e-learning mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1743 Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1769

Braids and differential equations Robert Ghrist∗

Abstract. Forcing theorems based on topological features of invariant sets have played a fundamental role in dynamics and differential equations. This talk focuses on the recent work of Vandervorst, Van den Berg, and the author using braids to construct a forcing theory for scalar parabolic PDEs, second-order Lagrangian ODEs, and one-dimensional lattice dynamics. Mathematics Subject Classification (2000). Primary 37B30, 35K90; Secondary 34C25, 37L60, 57M25. Keywords. Braids, Conley index, dynamical systems, parabolic PDEs, second order Lagrangian.

This talk covers a particular type of forcing theory for parabolic dynamics which uses the topology of braids in an index theory.

1. Topological forcing theorems Throughout the last century of work in dynamical systems, forcing theorems have played a substantial role in establishing coarse minimal conditions for complicated dynamics. Forcing theorems in dynamics tend to take the following form: given a dynamical system of a specified class, the existence of some invariant set of one topological type implies the existence of invariant sets of other topological types. This forcing is often encoded by some sort of ordering on topological types of invariant sets. 1.1. Examples. Three canonical examples of forcing theorems frame our work. Example 1 (Morse Theory [43]). The class of systems is that of nondegenerate gradient flows on an oriented manifold M. The invariant sets of interest are the fixed points, and the topological type associated to a fixed point is its Morse index – the dimension of its unstable manifold. A suitable chain complex generated by fixed points and graded by the Morse index yields a homology which is isomorphic to that ∗ Research supported by the National Science Foundation, PECASE DMS-0337713. The author wishes to thank Rob Vandervorst, without whom the work described here would not exist.

Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Robert Ghrist

of M, allowing one to deduce the existence and indices of additional critical points based on partial knowledge of the invariant sets and the homology of M. Morse theory has blossomed into a powerful array of topological and dynamical theories. One significant extension is the theory of Conley [14] which associates to an ‘isolated’ invariant set of a finite dimensional dynamical system an index – the Conley index – which, like the Morse index, can be used to force the existence of certain invariant sets. Instead of being a number (the dimension of the unstable manifold), the Conley index is a homotopy class of spaces (roughly speaking, the homotopy type of the unstable set). See [44] and the references therein for a sampling of applications to differential equations. Following on the heels of Conley’s index theory is the extension of Floer to infinitedimensional gradient-like dynamics. This, in turn, has led to an explosion of results in topology and geometry. The recent flurry of activity in contact homology and symplectic field theory [18] is a descendent of these foundational ideas. Example 2 (The Poincaré–Birkhoff Theorem [5]). This theorem applies to orientation and area preserving homeomorphisms of the annulus whose boundaries are twisted in opposite directions. As with Morse theory, the forcing is in terms of a lower bound (two) on the number of fixed points. The Poincaré–Birkhoff Theorem is the first of many dynamical theorems to exploit the particular features of symplectic manifolds and maps which preserve this symplectic structure. The marriage of this type of theorem with the Morse-type forcing results is the Arnold Conjecture, for which Floer theory was first and most strikingly used. There is a very powerful extension of the Poincaré–Birkhoff Theorem due to Franks [25] (Gambaudo and LeCalvez [39, App.] proved a slightly different version independently at about the same time). Franks’ theorem states that if an area and orientation preserving diffeomorphism of the annulus has at least one periodic point, then it has infinitely many periodic orbits. See [26] for this and related results. Franks’ Theorem is an excellent example of how a forcing theorem in dynamics often provides a sharp threshold for complicated dynamics: one simple invariant set implies the existence of infinitely many others. This principle finds its clearest exponent in the theorem of Sharkovsky. Example 3 (Sharkovsky’s Theorem [48]). For continuous maps of the compact interval to the reals, this theorem gives a total ordering on the periods of periodic orbits. The theorem states that if a map has an orbit of minimal period P then it has periodic orbits of minimal period Q for all P Q. That the minimal element of is three has led to the popular coinage “period three implies chaos.” The Sharkovsky theorem is remarkable in that there are no assumptions on the systems beyond dimension and continuity. Yet, the topological datum assigned to a periodic orbit is merely the period and nothing more sophisticated. In general, the resolution with which a forcing theorem can act depends on two factors: (1) how narrowly one constrains the class of dynamical systems; and (2) what type of topological data one assigns to the invariant sets.

Braids and differential equations

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1.2. Overview. This paper motivates and describes a forcing theory developed by R. Vandervorst in collaboration with J.-B. Van den Berg and the author. In this context, the class of dynamics is, roughly speaking, scalar parabolic lattice dynamics. The topological data which drives the forcing theory is a relative Conley index for invariant sets based on the theory of braids. The resulting forcing theory shares features with all three of the above examples. The index we construct – the homotopy braid index – is a Conley–Morse index and leads to Morse-type inequalities. The discrete version of the forcing theory is similar in spirit to LeCalvez’work on twist maps for annuli [38], [39], which itself is an elegant descendent of the Poincaré–Birkhoff Theorem. As with the Sharkovsky Theorem, we obtain a (partial) order on invariant sets. This leads to very simple conditions on invariant sets which force an infinite collection of additional invariant sets. 1.3. Braids and braid types. The use of braids in forcing theorems is not without precedent. There are various types of topological forcing in dimensions two and three related to braids. In the two-dimensional discrete case, one considers the isotopy class of a map relative to some periodic orbit(s): these are related to braids. One definition of a topological braid on n strands is a loop with basepoint in the configuration space of n distinct unlabeled points in the disc D 2 . One usually visualizes a braid as an embedding of n intervals u = {uα (t)}n1 into D 2 × [0, 1] such that each slice D 2 × {t} is a set of n points and the initial and final configurations the same: u(0) = u(1). See Figure 1 [left]. Given a braid u, its braid class {u} is the equivalence class of braids isotopic to u, that is, homotopic to u through braids, fixing the endpoints.

Figure 1. A braid on five strands, illustrated as a collection of embedded arcs in D 2 × [0, 1] [left]. A typical generator of the braid group has all strands ‘straight’ with a pair of incident strands crossing [right].

There is an obvious algebraic structure on n-strand braid classes by passing to the fundamental group Bn of the configuration space, the group operation being concatenation of the braids in D 2 × [0, 1]. The standard presentation for Bn has n − 1 generators, where the i th generator consists of n parallel strands (the identity braid) except that the i th strand crosses over the (i + 1)st strand as in Figure 3 [right]. See [6] for more details on the topology and algebra of braids.

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There is a wonderful analogue of the Sharkovsky Theorem for forcing periodic orbits in surface homeomorphisms. In this setting, the period is not a sufficiently fine datum – one must use what Boyland [7] coined the braid type of a periodic orbit. Consider, for the sake of simplicity, an orientation preserving homeomorphism f : D 2 → D 2 of the closed disc with a periodic orbit P of period n. The braid type bt(P ) is, roughly speaking, the isotopy class of f relative to P . Using the relationship between braid groups and mapping class groups [6], it is possible to formally identify bt(P ) with a conjugacy class in the braid group Bn modulo its center. This is best seen by suspending the disc map to a flow on D 2 × S 1 . When embedded in R3 , the periodic orbit is a braid. The choice of how many meridional twists to employ in the embedding is the genesis of modding out by the center of Bn . Boyland defined the following forcing order on braid types: one says that γ ≤ β if and only if for any homeomorphism f : D 2 → D 2 with γ a braid type for some periodic orbit of f , then β must also be a braid type for some (perhaps distinct) periodic orbit of f as well. Boyland showed that this is a partial order on braid types [8], which, though weaker than the total order of the Sharkovsky theory, is nevertheless efficacious in forcing complicated dynamics. Boyland’s theory, when generalized to surfaces, entwines with the Nielsen– Thurston theory for surface homeomorphisms. This combination of braid types together with Nielsen–Thurston theory has matured to yield numerous strong results, not only in the dynamics of horseshoe and Hénon maps [11], [13], but also in problems of fluid mixing [9], [33]. 1.4. Knots and links. In the case of flows in dimension three, embedding and isotopy data is likewise crucial. Since each periodic orbit is an embedded loop, it is a knot, and the set of periodic orbits forms a (perhaps infinite) link. The relationship between the link of periodic orbits and the dynamics of the flow is very subtle. A forcing theory for flows in not straightforward. Roughly speaking, the counterexamples to the Seifert Conjecture constructed by K. Kuperberg [37] imply that there cannot be a forcing theorem for general smooth nonsingular 3-d flows – one can always insert a Kuperberg plug and destroy any isolated periodic orbit. At one extreme, Kuperberg’s work implies that there exist smooth nonsingular flows on S 3 without any periodic orbits whatsoever. At the other extreme, it is possible to have a smooth, nonsingular, structurally stable flow on S 3 which displays all possible knots and links as periodic orbits [29]. These phenomena do not bode well for a forcing theory based on knots and links. However, upon restriction to the correct subclass of flows, it is often possible to retain some vestige of forcing based on knot and link types. One principle which persists is that simple dynamics implicate simple types of knots. For example, in the class of nonsingular Morse–Smale flows on S 3 , only certain knot types and link types can appear, a complete classification being given by Wada [54]. This result has a nearly dual counterpart in the class of integrable Hamiltonian dynamics on an invariant 3-sphere, as shown by Fomenko and Nguyen [24] and explained best by Casasayas et

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al. [12]. Other instantiations of this principle appear in smooth, integrable fluid flows on Riemannian 3-spheres [20] and in gradient fields on S 3 kinematically constrained by a plane field distribution [19]. A complementary principle also holds, that complex dynamics implicate complex knot types in a flow on a 3-sphere. The best example of this type of result is the theorem of Franks and Williams [27], which states that any C 2 flow with positive topological entropy has a link of periodic orbits which has an infinite number of distinct knot types represented. Other results on knotting and linking for suspensions of Smale horseshoes have been proved by Holmes and Williams [35] and used to force bifurcations in Hénon maps. These results all employ the relationship between knots, links, and closed braids – conjugacy classes of braids in the braid group which are visualized by identifying the left and right endpoints of a braid. 1.5. Toward higher dimensions. Forcing theorems based on knots, links, or braids in higher dimensional dynamics seem hopeless at first: these objects fall apart in dimension higher than three. One possibility is to try to work with embedding data associated to higher-dimensional invariant sets, say spheres or tori, which can be knotted and linked in the appropriate codimension. At present, there is some initial work on braiding of 2-d invariant tori in 4-d flows [50] which may lead to a forcing theory. There is a great deal now known about the peculiar constraints of embedding spheres and tori in symplectic manifolds, but as yet without much in the way of dynamical implications. We now turn to a braid-theoretic forcing theory for certain types of PDEs, where the stationary equation allows us to import three-dimensional embedding constraints into an infinite-dimensional dynamical system.

2. Braids for parabolic dynamics Our motivation for using braids to force dynamics comes from a very simple observation about parabolic partial differential equations. 2.1. Motivation: parabolic PDEs. Consider the scalar parabolic PDE ut = uxx + f (x, u, ux ),

(1)

where f satisfies one’s favorite technical assumptions to guarantee no finite-time blowups of solutions. For simplicity, we assume periodic boundary conditions (x ∈ [0, 1]/0 ∼ 1). We view Equation (1) as an evolution equation on the curve u( ·, t). As t increases, the graph of u evolves in the (x, u) plane. Thus, the PDE induces a flow on a certain infinite-dimensional space of curves. It is a result of Fiedler and Mallet-Paret [21] that a type of Poincaré–Bendixson Theorem holds for these types of equations: the only bounded invariant sets are stationary solutions, time-periodic solutions, and connecting orbits.

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We augment the types of solutions under consideration as follows. First, we allow multiple graphs to evolve by the product flow. That is, if u1 = u1 (t) : [0, 1] → R and u2 = u2 (t) : [0, 1] → R are solutions to Equation (1), then we consider the union u = (u1 , u2 ) as a solution to the product flow. These two strands evolve together, independently, as a pair of graphs on the (x, u) plane. In general, we can consider an n-tuple u = (uk )n1 of strands which evolve under the dynamics. Second, we allow for strands of multiple spatial period. That is, we allow for a collection u = (uk )n1 of strands of the form uk : [0, 1] → R with the endpoints equivalent as sets: {uk (0)}n1 = {uk (1)}. Even though the endpoints do not match strandwise, the union of the endpoints of the strands do match, and thus the entire collection evolves under the PDE so as to respect the spatial periodicity. One can think of such a collection of strands as a single-strand curve on the n-fold cover [0, n]/0 ∼ n of the spatial variable x. It is a well-known fact (going back to Sturm, but revived and extended considerably by Matano [41], Brunovsky and Fiedler [10], Angenent [1], and others) that there is a comparison principle for Equation (1). Specifically, let u1 (t) and u2 (t) be solutions to Equation (1). Then the number of intersections of the graphs of u1 (t) and u2 (t) is a weak Lyapunov function for the dynamics: it is non-increasing in t. Furthermore, at those particular times t for which the graphs of u1 (t) and u2 (t) are tangent, the number of intersections decreases strictly in t, even in the case where the tangencies are of arbitrarily high order [1]. These facts are all at heart an application of classical maximum principle arguments which have a topological interpretation: parabolic dynamics separates tangencies monotonically. This monotonicity is easily seen. Assume that u1 and u2 are solutions to Equation (1) which have a simple tangency where u1 (x, t) = u2 (x, t). Then the evolution of the difference between u1 and u2 is given by ∂2 ∂ 1 u (x, t) − u2 (x, t) = 2 u1 (x, t) − u2 (x, t) . ∂t ∂x

(2)

Since the nonlinear terms cancel, the evolution is governed purely on the basis of the curvature of the graphs. Using this comparison principle (also known as lap number or zero crossing techniques), numerous authors have analyzed the dynamics of Equation (1) in varying degrees of generality. We note in particular the paper of Fiedler and Mallet-Paret [21], in which the comparison principle is used to show that the dynamics of Equation (1) is often Morse–Smale, and also the paper of Fiedler and Rocha [22], in which the global attractor for the dynamics is roughly classified. 2.2. Idea: dynamics on spaces of braids. A typical collection of strands is illustrated in Figure 2 [left], in which one notices a resemblance to the planar projection of a braid. By lifting this collection of strands in the (x, u) plane to the 1-jet extension of the strands in (x, u, ux ) space, we obtain a Legendrian braid tangent to the contact structure {dy − z dx = 0}. Such a braid is closed, due to the periodicity of

7


the strands. Being Legendrian, the braid is positive – in the standard generators for the braid group, only positive powers of generators are permitted. There is a globalization of the comparison principle using braids. For a motivating example, consider again a pair of evolving curves u1 (t) and u2 (t) in the (x, u) plane. If we lift these curves to the three-dimensional (x, u, ux ) space, we no longer have intersecting curves, unless t is such that the planar graphs of u1 and u2 intersect tangentially. The graphs of u1 and u2 in the (x, u, ux ) space are instead a closed braid on two strands. What was the intersection number of their projections is now the linking number of the pair of strands. We see therefore that the comparison principle takes on a linking number interpretation (a fact utilized in a discrete setting by LeCalvez [38]). After lifting solutions u1 and u2 to the (x, u, ux ) space, the comparison principle says that the linking number is a nonincreasing function of time which decreases strictly at those times at which the curves are tangent. This two-strand example is merely motivation for adopting a braid-theoretic perspective on multiple strands, as in Figure 2.

u

u

x

ux

x

Figure 2. Curves in the (x, u) plane lift to a braid in (x, u, ux ).

The key observation is that the comparison principle passes from a local statement (“linking number decreases at a tangency”) to a global statement (“algebraic length in the braid group decreases at a tangency”). A related globalization of the comparison principle for geodesic flows on Riemannian surfaces appears in the recent work of Angenent [2]. 2.3. Goal: forcing. Our goal is to produce a forcing theory for the dynamics of Equation (1) and more general parabolic systems. For simplicity, we focus on forcing stationary solutions, though periodic and connecting orbits are likewise accessible. Say that one has found a skeleton of stationary strands {v 1 , v 2 , . . . , v m } for a particular representative of Equation (1). How many and which types of other stationary curves are forced to be present? Since the skeleton of known fixed curves v = {v 1 }m i=1 lifts to a braid, the problem is naturally couched in braid-theoretic terms: given a braid v fixed by a particular uniform parabolic PDE, which other classes of braids u are forced to exist as stationary curves? The spirit of our forcing theory is as follows:

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Robert Ghrist

1. Given a braid of stationary solutions v, construct the configuration space of all n-strand braids u which have v as a sub-braid. 2. Use the braid-theoretic comparison principle to give a Morse-type decompose of this configuration space into dynamically isolated braid classes. 3. Define the homotopy braid index – a Conley index for relative braid classes which depends only on the topology of the braids, and not on the analytical details of the dynamics. 4. Prove Morse-type inequalities for forcing stationary and/or time-periodic solutions. To execute this requires a significant generalization to spatially discretized systems, which in turn generalizes the results far beyond parabolic PDEs.

3. Spaces of braids for parabolic dynamics 3.1. Braids, topological and discrete. The motivation of §2.1 leads one to consider spaces of braids obtained from curves in the (x, u) plane. Consider the space of all such n-strand braids u which are both closed and positive. For the sake of intuition, one should think of these topological braids as smooth braids lifted from the 1-jet extension of graphs in the plane. In reality, one completes this space to include nonsmooth braids as well. These configuration spaces of braids are infinite dimensional. By projecting to finite dimensional approximations, we avoid a great deal of analytic and topological difficulties. We briefly outline the “finite dimensional” braid theory needed. The class of discretized braids are best visualized as piecewise-linear braid diagrams, as in Figure 3 [left]. A discretized braid, u, on n strands of period p, is determined by np anchor points: u = {uαi }α=1,...,n i=0,...,p . Superscripts α = 1, . . . , n refer to strand numbers, and subscripts i = 0, . . . , p refer to spatial discretizations. One connects the anchor point uαi to uαi−1 and uαi+1 via straight lines. Since “height” is determined by slope, all crossings in the braid diagram are of the same sign (as in Figure 3 [left] but not in Figure 1 [left]). Since we employ periodic boundary conditions on the x variable, all of the braids are closed: left and right hand endpoints of strands are abstractly identified and the endpoints are free to move. This necessitates a periodicity convention for the subscript. For a single-strand component uα , we have that uαi+p = uαi for all i. For multi-strand components, one cycles between the strands according to the permutation of strands. Denote by Dpn the space of all n-strand period p discretized braids: Dpn is homeomorphic to Rnp . For topological braids, a singular braid is one for which one or more strands intersect. For braids which are lifts of graphs, the only possible intersection is that which occurs when two strands are tangent in the projection. For a discretized braid u,

9


Figure 3. A discretized braid in D64 with three components (note: left and right hand sides are identified) [left]; Two types of singular discretized braids: a simple tangency, and a high-order contact [right].

the singular braids are defined to be those braids at which anchor points on two different strands coincide in a topologically non-transverse fashion with respect to immediate neighbors. Denote by the singular braids, β

β

β

= {u : uαi = ui for some i and α = β, and (uαi−1 − ui−1 )(uαi+1 − ui+1 ) ≥ 0}. (3) The set is a discriminant that carves Dpn into components: these are the discretized braid classes, denoted [u]. Within , there is a subspace of collapsed braids, − ⊂ , consisting of those braids for which distinct components of the braid (or a single component with multiple period) collapse to yield a braid on fewer strands. More specifically, − = {u ∈ : uαi = ui for all i ∈ Z and some α = β}, β

(4)

under the convention of subscript periodicity mod p as regulated by the braid. 3.2. Parabolic dynamics on braids. A parabolic PDE of the form in Equation (1) gives rise to a flow on the space of topological braids. There is likewise a broad class of flows on spaces of discretized braids which are best described as parabolic. These come from nearest-neighbor lattice dynamics. Discretizing Equation (1) in the standard way would yield a family of nearest d neighbor equations of the form dt ui = fi (ui−1 , ui , ui+1 ) in which uniform parabolicity would manifest itself in terms of the derivatives of fi with respect to the first and third variables. Instead of explicitly discretizing the PDE itself, we use the broadest possible category of nearest neighbor equations for which a comparison principle holds: these are related to the monotone systems of, e.g., [49], [34], [21] and others. A parabolic relation of period p is a sequence of maps R = {Ri : R3 → R}, such that ∂1 Ri > 0 and ∂3 Ri ≥ 0 for every i. These include discretizations of uniform parabolic PDE’s, as well as a variety of other discrete systems [40], [42], including

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Robert Ghrist

monotone twist maps [38]. The small amount of degeneracy permitted (∂3 Ri = 0) does not prevent the manifestation of a comparison principle. Given a discretized braid u = {uαi } and a parabolic relation R, one evolves the braid according to the equation d α (u ) = Ri (uαi−1 , uαi , uαi+1 ). (5) dt i Any parabolic relation R therefore induces a flow on Dpn . Fixed points of this flow correspond to stationary braids u satisfying Ri (uαi ) = 0 for all i and α. It will be useful at certain points to work with parabolic relations which induce a gradient flow on Dpn . One calls R exact if there exist generating functions Si such that Ri (ui−1 , ui , ui+1 ) = ∂2 Si−1 (ui−1 , ui ) + ∂1 Si (ui , ui+1 ),

(6)

for all i. In the exact case, the flow of Equation (5) is given by the gradient of i Si . All parabolic relations, exact or non-exact, possess a discrete braid-theoretic comparison principle. Lemma 4 (Comparison principle for braids [32]). Let R be any parabolic relation and u ∈ − − any non-collapsed singular braid. Then the flowline u(t) of R passing through u = u(0) leaves a neighborhood of in forward and backward time so as to strictly decrease the algebraic length of u(t) in the braid group as t increases through zero. Lemma 4 implies that the flow of parabolic dynamics is gradient-like on the (noncollapsed portions of) boundaries of braid classes. This suggests a Morse-theoretic approach. For example, if the flow points in to a given braid class everywhere along the boundary, then the braid class should serve as a ‘sink’ for the dynamics and thus be assigned a Morse index of zero. At least some invariant set would have to lie within this braid class, even if the dynamics is not gradient everywhere. For more complicated behaviors on the boundary of a braid class, Conley’s version of Morse theory is the appropriate tool, with the notion of a Morse index generalizing to the Conley index, a homotopy class of spaces.

4. The homotopy braid index One significant problem with this idea is the prevalence of collapsed braids, which are invariant under the flow and foil the straightforward application of Morse theory. Clearly, any braid class [u] borders the set of collapsed braids − somewhere. One need simply collapse all the strands together as an extreme degeneracy. 4.1. Relative braids. We are therefore naturally confronted with the need for a forcing theory. Given that a particular parabolic relation possesses a stationary braid v, does it force some other braid u to also be stationary with respect to the dynamics? This necessitates understanding how the strands of u braid relative to those of v.


11

Given a discrete braid v ∈ Dpm , consider the set of nonsingular braids

u ∈ Dpn : u ∪ v ∈ Dpn+m − pn+m ,

the path components of which define the relative braid classes [u rel v]. Not only are tangencies between strands of u illegal, so are tangencies with the strands of v. In this setting, the braid v is called the skeleton. Elements within [u rel v] are equivalent as discrete braids fixing all strands of v. In this context, it is possible to define a Conley index for certain discrete relative braid classes. To do so, it must be shown that the braid classes [u rel v] are isolated in the sense that no flowlines within [u rel v] are tangent to the boundary of this set. It follows from Lemma 4 that [u rel v] is isolated for the flow of Equation (5) assuming that the braid class avoids the collapsed braids − . We therefore declare a braid class [u rel v] to be proper if no free strands of u can “collapse” onto v or onto each other: see Figure 4. Furthermore, to ensure compactness, it is convenient to assume that the braid class [u rel v] is bounded – free strands cannot wander off to ±∞.

Figure 4. A bounded but improper braid class [left]. A proper, but unbounded braid class. Solid strands form the skeleton; dashed strands are free [right].

4.2. The index: discrete version. The homotopy braid index of a proper, bounded, discrete relative braid class [u rel v] is defined as follows. Choose any parabolic relation R which fixes v (such an R exists). Define E to be the exit set: those braids on the boundary of the braid class [u rel v] along which evolution under the flow of R exits the braid class. The homotopy braid index is defined to be the pointed homotopy class h([u rel v]) = [u rel v]/E , {E } . (7) This is simply the Conley index of the closure of [u rel v] in Dpn under the flow of R. Lemma 4, combined with the basic stability properties of the Conley index yields the following:

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Robert Ghrist

Lemma 5. The index h([u rel v]) is well-defined and independent of the choice of R (so long as it is parabolic and fixes v) as well as the choice of v within its braid class [v]. Thanks to the comparison principle for braids, the computation of the index h does not require a choice of R. One can identify the exit set E purely on the basis of which singular braids will decrease algebraic length under parabolic evolution. This is the basis for an algorithm to compute the homological index H∗ (h[u rel v]) numerically [17]. Example 6. Consider the proper period-2 braid illustrated in Figure 5 [left]. There is exactly one free strand with two anchor points (via periodicity). The anchor point in the middle, u1 , is free to move vertically between the fixed points on the skeleton. At the endpoints, one has a singular braid in which is on the exit set. The end anchor point, u0 (= u2 ) can freely move vertically in between the two fixed points on the skeleton. The singular boundaries are not on the exit set since pushing u0 across the skeleton increases the number of crossings.

u0

u2 u1

u1 u0

Figure 5. The braid of Example 6 [left] and the associated configuration space with parabolic flow [middle]. Collapsing out the exit set leads to a space [right] which has the homotopy type of a circle.

Since the points u0 and u1 can be moved independently, the braid class is the product of two intervals. The exit set consists of those points on the boundary for which u1 is a boundary point. Thus, the homotopy braid index is S 1 , as seen in Figure 5 [right]. Example 7. Consider the proper relative braid presented in Figure 6 [left]. Since there is one free strand of period three, the configuration space is determined by the vector of positions (u0 , u1 , u2 ) of the anchor points. This example differs greatly from the previous example. For instance, the point u0 (as represented in the figure) may pass through the nearest strand of the skeleton above and below without changing the braid class. The points u1 and u2 may not pass through any strands of the skeleton

13


without changing the braid class unless u0 has already passed through. In this case, either u1 or u2 (depending on whether the upper or lower strand is crossed) becomes free. The skeleton induces a cubical partition of R3 by planes of singular braids. The relative braid class is the collection of cubes in R3 illustrated in Figure 6 [right]: it is homeomorphic to D 2 × S 1 . In this case, the exit set is the entire boundary and the quotient space is homotopic to the wedge-sum S 2 ∨S 3 , the space defined by abstractly gluing a point of S 2 to a point of S 3 .

u1 u0

u3 u2

Figure 6. The braid of Example 7 and the associated relative braid class.

Example 8. The braid pair of Figure 7 [right] has index h S 4 ∨ S 5 (as computed in [32, Lem. 50]); the pair on the left has trivial index, even though the linking numbers and periods of all strands are identical. This exemplifies the extra information carried by the braiding data.

Figure 7. Discretized braid pairs with trivial [left] and nontrivial [right] homotopy index.

4.3. The index: topological version. As defined, the homotopy braid index h is a function of discretized braid classes. For topological braids, one could hope that any discretization yields the same discrete index. It does, modulo two technicalities.

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Robert Ghrist

The first is simple. Given a topological relative braid pair u rel v and a discretization period p, consider the discrete braid pair whose anchor points are defined in the obvious way using xi = i/p as the spatial discretization points. Only for p sufficiently large will this discrete braid pair be isotopic as a topological braid to the pair u rel v. Thus, one must choose p so that the correct braid class is obtained by discretization. The second technicality is more subtle. Even if the discretized braid is topologically isotopic to the original, it is possible to “fracture” the homotopy type of the topological braid class via discretization. Consider the discrete braids of Figure 8: these braid pairs are equivalent as topological closed braids, but not as discrete closed braids. There is simply not enough freedom to maneuver.

Figure 8. An example of two discretized braids which are of the same topological braid class but define disjoint discretized braid classes in D41 rel v.

To overcome this difficulty, we define a modification of the homotopy braid index as follows. Given a fixed period p and a discrete proper relative braid class β = [u rel v] ∈ Dpn , let S(β) denote the set of all braid classes in Dpn rel v which are isotopic as topological braids to a representative of β. Define the index H to be H (β) = h(βi ). (8) βi ∈S(β)

This is a wedge sum of the indices of all discrete period-p representatives of the given topological braid class. The wedge sum is well-defined since each h is a pointed homotopy class. This index H is an invariant of topological braid classes. Consider the following n , which appends a trivial period-1 braid to stabilization operator, E : Dpn → Dp+1 the end of a discrete braid: uαi , i = 0, . . . , p, (Eu)αi = (9) upα , i = p + 1. The most important result about the index is the following invariance theorem:


15

Theorem 9 (Stabilization [32]). For u rel v any bounded proper discretized braid pair, the topological homotopy braid index is invariant under the extension operator: H (Eu rel Ev) = H (u rel v).

(10)

The proof of this theorem involves, surprisingly enough, a dynamical argument, utilizing a singular perturbation of a particular parabolic relation adapted to E. This is a very convenient way to prove homotopy equivalence, given the robustness of the Conley index with respect to singular perturbations [15]. This theorem allows for a proof of topological invariance. Theorem 10 (Invariance [32]). Given u rel v ∈ Dpn rel v and u˜ rel v˜ ∈ Dpn˜ rel v˜ which are topologically isotopic as bounded proper braid pairs, then H (u rel v) = H (u˜ rel v˜ ).

(11)

The key ingredients in this proof are the Stabilization Theorem combined with a braid-theoretic argument that the moduli space of discretized braids converges to that of topological braids under sufficiently many applications of E – the length of the braid in the word metric suffices.

5. Forcing theorems: parabolic lattice dynamics The dynamical consequences of the index are forcing results. A simple example: given any parabolic relation R which has as stationary solutions the skeleton of Figure 7 [right], then, since adding the dashed strand from that figure yields a nontrivial braid index, there must be some invariant set for R within this braid class. At this point, one uses Morse theory ideas: if R is exact, then there must be a stationary solution of the form of the grey strand. If the flow is not a gradient flow, then finer information can still detect stationary solutions. More specifically, let h be the homotopy braid index of a proper bounded discrete braid class [u rel v]. Let Pτ (h) denote the Poincaré polynomial of the index – the polynomial in Z[τ ] whose coefficients are the Betti numbers of the homology of the index, H∗ (h; R). The following results are consequences of degenerate Morse theory (cf. [16]). Theorem 11 ([32]). Given a parabolic relation R which fixes v and h = h([u rel v]), the following hold: 1. The number of stationary braids in this braid class is bounded below by the Euler characteristic χ(h) = P−1 (h). 2. If R is exact, then the number of stationary braids in this braid class is bounded below by the number of nonzero monomials of Pτ (h).

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Stronger results are available if it is known that the parabolic relation is nondegenerate. By iterating the process of adding free strands and computing a nontrivial index, one can go quite far. The following forcing theorem (for exact R) is very general, requiring only that the parabolic relation is exact (yielding a gradient flow) and dissipative, meaning that Ri → −∞ as |ui | → +∞. Theorem 12 ([32]). Let R be a parabolic relation which is both exact and dissipative. If R fixes a discretized braid v which is not a trivial braid class, then there exist an infinite number of distinct braid classes which arise as stationary solutions of R. This theorem is very much in the spirit of “period-three implies chaos.” The dissipative boundary condition at infinity can be replaced with a coercive condition (infinity is attracting) or with mixtures thereof with only minor adjustments to the theorem statements [32].

6. Forcing theorems: second-order Lagrangians This forcing theory gives an elegant approach to a class of fourth-order equations arising from a Lagrangian. Consider a second order Lagrangian, L(u, ux , uxx ), such as is found in the Swift–Hohenberg equation: L=

1 1 − α 2 u4 (uxx )2 − (ux )2 + u + . 2 2 4

(12)

Assume the standard convexity assumption that ∂u2xx L ≥ δ > 0. The Euler– Lagrange equations yield a fourth-order ODE. The objective is to find bounded functions u : R → R which are stationary for the action integral J [u] = L(u, ux , uxx ) dx. Due to the translation invariance x → x + c, the solutions of the Euler–Lagrange equation satisfy the energy constraint

d ∂L ∂L ∂L − ux + uxx − L(u, ux , uxx ) = E = constant, ∂ux dx ∂uxx ∂uxx

(13)

where E is the energy of a solution. To find bounded solutions for given values of E, T we employ the variational principle δu,T 0 (L(u, ux , uxx )+E) dx = 0, which forces solutions to have energy E. The Lagrangian problem can be reformulated as a two degree-of-freedom Hamiltonian system. In that context, bounded periodic solutions are closed characteristics of the corresponding energy manifold M 3 ⊂ R4 . Unlike the case of first-order Lagrangian systems, the energy hypersurface is not of contact type in general [4], and is never compact. The recent stunning results in contact homology [18] are inapplicable. 6.1. The twist condition. The homotopy braid index provides a very effective means of forcing periodic orbits. By restricting to systems which satisfy a mild variational


17

hypothesis, one can employ a “broken geodesics” construction which yields a restricted form of parabolic relation. Closed characteristics at a fixed energy level E are concatenations of monotone laps between alternating minima and maxima (ui )i∈Z , which form a periodic sequence with even period. The problem of finding closed characteristics can, in most cases, be formulated as a finite dimensional variational problem on the extrema (ui ), as realized by Vandervorst, in his definition of the twist condition. The twist condition is a weaker version of the hypothesis that assumes that the monotone laps between extrema are unique and is valid for large classes of Lagrangians L, including Equation (12). The following result of [52] is the motivation and basis for the applications of the homotopy braid index to second-order Lagrangians. Lemma 13. Extremal points {ui } for bounded solutions of second order Lagrangian twist systems are solutions of an exact parabolic relation with the constraints that (i) (−1)i ui < (−1)i ui+1 ; and (ii) the relation blows up along any sequence satisfying ui = ui+1 . 6.2. A general result. It is necessary to retool the homotopy braid index to the setting of Lemma 13 and show that the index properties with respect to this restricted class of parabolic relations are invariant. Upon so doing, one extracts very general forcing theorems, a simple example of which is the following: Theorem 14 ([32]). Let L(u, ux , uxx ) be a Lagrangian which is dissipative (infinity is repelling) and twist. Then, at any regular energy level, the existence of a single periodic orbit which traces out a self-intersecting curve in the (u, ux ) plane implies the existence of infinitely many other periodic orbits at this energy level. Additional results give lower bounds on the multiplicity of solutions in a given braid class based on the Poincaré polynomial and apply to singular energy levels, as well as to non-dissipative systems [32].

7. Forcing theorems: parabolic PDEs The homotopy braid index, being inspired by parabolic PDEs, is efficacious in this context also, thanks to Theorem 10. By performing a spatial discretization of the dynamics of Equation (1), it is possible to reduce the dynamics of the PDE to those of a parabolic relation on a finite-dimensional space of discretized braids. On account of the robustness of the homotopy index with respect to the dynamics, there is very little one needs to assume about the nonlinearity in Equation (1). The first, crucial, hypothesis is a growth condition on the ux term of f . For simplicity, let us call Equation (1) subquadratic if there exist constants C > 0 and 0 < γ < 2, such that |f (x, u, v)| ≤ C(1 + |v|γ ), uniformly in both x ∈ S 1 and on compact intervals in u. This is necessary for regularity and control of derivatives of solution

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curves, cf. [3]. This condition is sharp: one can find examples of f with quadratic growth in ux for which solutions have singularities in ux . Since our topological data are drawn from graphs of u, the bounds on u imply bounds on ux and uxx . A second gradient hypothesis will sometimes be assumed. One says Equation (1) is exact if

d uxx + f (x, u, ux ) = a(x, u, ux ) ∂ux L − ∂u L , (14) dx for a strictly positive and bounded function a = a(x, u, ux ) and some Lagrangian L satisfying a(x, u, ux ) · ∂u2x L(x, u, ux ) = 1. In this case, one has a gradient system whose stationary solutions are critical points of the action L(x, u, ux ) dx over loops of integer period in x. This condition holds for a wide variety of systems. In general, systems with Neumann or Dirichlet boundary conditions admit a gradient-like structure which precludes the existence of nonstationary time-periodic solutions. It was shown by Zelenyak [55] that this gradient-like condition holds for many nonlinear boundary conditions which are a mixture of Dirichlet and Neumann. 7.1. Stationary solutions. Assume for the following theorems that {u rel v} is a topological braid class which is both bounded and proper. Assume further that v is stationary for Equation (1). We state our existence and multiplicity results in terms of the Poincaré polynomial Pτ (H ) of the topological (as opposed to the discrete) braid index H = H {u rel v}, computed via a discretization of the topological braid. Theorem 15 ([31]). Let Equation (1) be subquadratic with v a stationary braid, and H = H ({u rel v}). 1. There exists a stationary solution in this braid class if the Euler characteristic of the index, χ(H ) = P−1 (H ), is nonvanishing. 2. If Equation (1) is furthermore exact, then there exists a stationary solution in this braid class if Pτ (H ) = 0. Additional results are available concerning multiplicity of solutions, alternate boundary conditions, and non-uniformly parabolic equations: see [31]. A version of Theorem 12 on infinite numbers of braids being forced by a single nontrivial stationary braid persists in this context. The result is simplest to state if the PDE is dissipative; that is, u f (x, u, 0) → −∞ as |u| → +∞ uniformly in x ∈ S 1 . This is a fairly benign restriction. Theorem 16 ([31]). Let Equation (1) be subquadratic, exact, and dissipative. If v is a nontrivially braided stationary skeleton, then there are infinitely many braid classes represented as stationary solutions. Moreover, the number of single-freestrand braid classes is bounded from below by ι/2 − 1, where ι is the maximal number of intersections between two strands of v.

19


7.2. Examples. The following family of spatially inhomogeneousAllen–Cahn equations was studied by Nakashima [45], [46]: ε2 ut = ε2 uxx + g(x)u(1 − u2 ),

(15)

where g : S 1 → (0, 1) is not a constant. This equation has stationary solutions u = 0, ±1 and is exact with Lagrangian L=

1 2 2 1 ε ux − g(x)u2 (2 − u2 ). 2 4 +1

1

0 0

g

−1

Figure 9. Given a function g : S 1 → (0, 1) and ε small, there exists a skeleton of stationary curves for Equation (15) which forms a nontrivial braid. This forces infinitely many other stationary braids.

According to [45], for any N > 0, there exists an εN > 0 so that for all 0 < ε < εN , there exist at least two stationary solutions which intersect u = 0 exactly N times. (The cited works impose Neumann boundary conditions: it is a simple generalization to periodic boundary conditions.) Via Theorem 16 we have that for any such g and any small ε, this equation admits an infinite collection of stationary periodic curves; furthermore, there is a lower bound of N on the number of 1-periodic solutions. As a second explicit example, consider the equation ut = uxx −

5 cos x u(u2 − 1), sin 2x ux + 8 cos x + √3

(16)

5

with x ∈ S 1 = R/2π Z. This gives an exact system with Lagrangian 5

L = e− 16 cos 2x

1 2 (u2 − 1)2 cos x ux − , 2 4 cos x + √3

(17)

5

5

and weight a(x, u, ux ) = e 16 cos 2x (cf. Equation (14)). √ One checks easily that there are stationary solutions ±1 and ± 21 5 cos x + 1 , as in Figure 10 [left]. These curves comprise a skeleton v which can be discretized to yield the skeleton of Example 6. This skeleton forces a stationary solution of the braid

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Robert Ghrist

+1

−1

Figure 10. This collection of stationary solutions for Equation (16) [left] discretizes to the braid skeleton of Example 6.

class indicated in Figure 5 [left]: of course, this is detecting the obvious stationary solution u = 0. Note, however, that since H S 1 , this solution is unstable. What is more interesting is the fact that one can take periodic extensions of the skeleton and add free strands in a manner which makes the relative braid spatially non-periodic. Let v n be the n-fold periodic extension of v on [0, n]/0 ∼ n and consider a single free strand that weaves through v n as in Figure 11. The homotopy index of such a braid is a sphere whose dimension is a function of the linking number of the free strand with the skeletal strands. The appropriate Morse inequalities imply that for each n > 0 there exist at least 3n − 2 distinct stationary solutions. This information can be used to prove that the time-2π map of the stationary equation has positive entropy, see e.g. [47], [53].

Figure 11. Taking a lift of the spatial domain allows one to weave free strands through the lifted skeleton. These project to multiply-periodic solutions downstairs. The braid pictured has index H S2.

7.3. Time-periodic solutions. A fundamental class of time-periodic solutions to Equation (1) are the so-called rotating waves. For an equation which is autonomous in x, one makes the rotating wave hypothesis that u(t, x) = U (x − ct), where c is


21

the unknown wave speed. Stationary solutions for the resulting equation on U (ξ ) yield rotating waves. In [3] it was proved that time-periodic solutions are necessarily rotating waves for an equation autonomous in x. However, in the non-autonomous case, the rotating wave assumption is highly restrictive. The homotopy braid index presents a very general technique for finding timeperiodic solutions without the rotating wave hypothesis. Theorem 17 ([31]). Let {u rel v} be a bounded proper topological braid class with u a single-component braid, v an arbitrary stationary braid, and Pτ (H ) = 0. If the braid class is not stationary for Equation (1) – the equation does not contain stationary braids in this braid class – then there exists a time-periodic solution in this braid class. It was shown in [3] that a singularly perturbed van der Pol equation, ut = εuxx + u(1 − δ 2 u2 ) + ux u2 ,

(18)

possesses an arbitrarily large number of rotating waves for ε 1 sufficiently small and fixed 0 < δ. The homotopy braid index methods extend these results dramatically. Theorem 18 ([31]). Consider the equation ut = uxx + ub(u) + ux c(x, u, ux ),

(19)

where c has sub-linear growth in ux at infinity. Moreover, b and c satisfy the following hypotheses: 1. b(0) > 0, and b has at least one positive and one negative root; 2. c(x, 0, 0) = 0, and c > 0 on {uux = 0}. Then this equation possesses time-periodic solutions spanning an infinite collection of braid classes. All of the periodic solutions implied are dynamically unstable. In the most general case (those systems with x-dependence), the periodic solutions are not rigid rotating waves and thus would seem to be very difficult to detect.

8. What does this index mean? The most important fact about the homotopy braid index H is that it is an invariant of topological braid pairs. Though it is not realistic to think that this is of interest in knot theory as a means of distinguishing braid pairs, the homotopy braid index nevertheless entwines both topological and dynamical content. Thinking in terms of braid classes gives finer information than relying merely on intersection numbers. With the braid-theoretic approach, various analytic conditions

22

Robert Ghrist

on a PDE or lattice system (dispersive, coercive, etc.) can be ‘modeled’by an auxiliary braid when computing the index. Likewise, spatial boundary conditions (Neumann, Dirichlet, periodic, etc.) can be viewed as restrictions on braids (fixed, closed, etc.). Any such restrictions which yield topologically equivalent braids have the same dynamical implications with respect to forcing. One may replace complicated analytic constraints with braids. The precise topological content to the homotopy braid index is not transparent. A few steps toward unmasking the meaning of the index are as follows. 8.1. Duality. One special feature of working with discretized braids in a fixed period is a natural duality made possible by the fact that the index pair used to compute the homotopy braid index can be chosen to be a manifold pair. n → The duality operator on discretized braids of even period is the map D : D2p n D2p given by (Du)αi = (−1)i uαi .

(20)

Clearly D induces a map on relative braid diagrams by defining D(u rel v) to be Du rel Dv. The topological action of D is to insert a half-twist at each spatial segment of the braid. This has the effect of linking unlinked strands, and, since D is an involution, linked strands are unlinked by D, as in Figure 12.

D

i=0

i=1

i=2

i=3

i=4

i=0

i=1

i=2

i=3

i=4

Figure 12. The topological action of D.

For the two duality theorems to follow, we assume that all braids considered have even periods and that all of the braid classes and their duals are proper, so that the homotopy index is well-defined. In this case, the duality map D respects braid classes: if [u] = [u ] then [D(u)] = [D(u )]. Bounded braid classes are taken to bounded braid classes by D. The effect of D on the index pair is to reverse the direction of the parabolic flow. This is the key to proving the following:

23


Theorem 19 (Duality [32]1 ). For [u rel v] having period 2p and n free strands, Hq (H (D(u rel v)); R) ∼ = H2np−q (H (u rel v); R).

(21)

This duality operator is very useful in computing the homology of the braid index: see the computations in [32]. 8.2. Twists. The duality operator yields a result on the behavior of the index under appending a full twist. Theorem 20 (Shift [32]). Appending a full twist to a braid shifts the homology of the index up by dimension equal to twice the number of free strands. We include a sketch of the proof (a more careful version of which would deal with some boundedness issues). Assume that [u rel v] is a braid of period 2p with n free strands. A period two full-twist braid can be realized as the dual of the trivial braid of period two. Thus, the effect of adding a full twist to a braid can be realized by the operator DEED. By combining Theorems 9 and 19, we obtain: Hq (H (DEED[u rel v])) ∼ = H2np+2n−q (H (DEE[u rel v])) ∼ = H2np+2n−q (H (D[u rel v])) ∼ Hq−2n (H ([u rel v])). =

(22)

A homotopy version of Equation (22) should be achievable by following a similar procedure as in the proof of Theorem 9. We suspect one obtains an iterated suspension of the homotopy index, as opposed to a shift in homology.

9. Toward arbitrary braids Given the motivation from PDEs and the comparison principle, the types of braids considered in this paper are positive braids. One naturally wonders whether an extension to arbitrary braids – those with mixed crossing types – is possible. Unfortunately, passing to discretized braids is no longer simple, as anchor points alone cannot capture crossing information for arbitrary braids. One way to define a formal index for general braid pairs is to use Garside’s Theorem [6], slightly modified. Garside’s Theorem states that any braid can be placed into a unique normal form of a positive braid times a (minimal) number of negative halftwists. Clearly, one can define a modified Garside normal form that gives a unique decomposition into a positive braid and a (minimal) number of negative full twists. By applying Theorem 20, one can define a homological braid index (with negative grading permitted) by shifting the braid index of the positive normal form down by 1 The theorem in the reference has a slight error in the statement. There, it was implicitly assumed that the braid has one free strand. The present statement is correct for arbitrary numbers of strands.

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the appropriate amount. A homotopy theoretic version could be defined in terms of spectra via suspensions. This, then, yields a formal index for arbitrary (proper) braid pairs. The real question is what dynamical meaning this generalized index entails. The passage from positive braids to arbitrary braids is akin to the passage from a Lagrangian to a Hamiltonian settings, and such an extended index appears to be a relative Floer homology for (multiply) periodic solutions to time-periodic Hamiltonian systems.

References [1] Angenent, S., The zero set of a solution of a parabolic equation. J. Reine Ang. Math. 390 (1988), 79–96. [2] Angenent, S., Curve Shortening and the topology of closed geodesics on surfaces. Ann. of Math. 162 (2005), 1187–1241. [3] Angenent, S., Fiedler, B., The dynamics of rotating waves in scalar reaction diffusion equations. Trans. Amer. Math. Soc. 307 (2) (1988), 545–568. [4] Angenent, S., Van den Berg, B., Vandervorst, R., Contact and noncontact energy hypersurfaces in second order Lagrangian systems. Preprint, 2001. [5] Birkhoff, G., Proof of Poincaré’s Geometric Theorem. Trans. Amer. Math. Soc. 14 (1913), 14–22. [6] Birman, J. S., Braids, Links and Mapping Class Groups, Ann. of Math. Stud. 82, Princeton University Press, Princeton, N.J., 1975. [7] Boyland, P., Braid types and a topological method for proving positive entropy. Preprint, Boston University, 1984. [8] Boyland, P., Topological methods in surface dynamics. Topology Appl. 58 (3) (1994), 223– 298. [9] Boyland, P., Aref, H., Stremler, M., Topological fluid mechanics of stirring. J. Fluid Mech. 403 (2000), 277–304. [10] Brunovský P., Fiedler B., Connecting orbits in scalar reaction-diffusion equations. In Dynamics Reported, Vol. 1, Dynam. Report. Ser. Dynam. Systems Appl. 1, John Wiley & Sons, Ltd., Chichester; B. G. Teubner, Stuttgart, 1988, 57–89. [11] de Carvalho, A., Hall, T., Pruning theory and Thurston’s classification of surface homeomorphisms. J. European Math. Soc. 3 (4) (2001), 287–333. [12] Casasayas, J., Martinez Alfaro, J., Nunes, A., Knots and links in integrable Hamiltonian systems. J. Knot Theory Ramifications 7 (2) (1998), 123–153. [13] Collins, P., Forcing relations for homoclinic orbits of the Smale horseshoe map. Experimental Math. 14 (1) (2005), 75–86. [14] Conley, C., Isolated Invariant Sets and the Morse Index. CBMS Reg. Conf. Ser. Math. 38, Amer. Math. Soc., Providence, R.I., 1978. [15] Conley, C., Fife, P., Critical manifolds, travelling waves, and an example from population genetics. J. Math. Biol. 14 (1982), 159–176.


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[16] Dancer, N., Degenerate critical points, homotopy indices and Morse inequalities. J. Reine Angew. Math. 350 (1984), 1–22. [17] Day, S., Van den Berg, J., Vandervorst, R., Computing the homotopy braid index. In preparation, 2005. [18] Eliashberg,Y., Givental, A., Hofer, H., Introduction to symplectic field theory. Geom. Func. Anal. Special Volume II (2000), 560–673. [19] Etnyre, J., Ghrist, R., Gradient flows within plane fields. Commun. Math. Helv. 74 (1999), 507–529. [20] Etnyre, J., Ghrist, R., Stratified integrals and unknots in inviscid flows. Contemp. Math.. 246 (1999), 99–112. [21] Fiedler, B., Mallet-Paret, J., A Poincaré-Bendixson theorem for scalar reaction diffusion equations. Arch. Rational Mech. Anal. 107 (4) (1989), 325–345. [22] Fiedler, B., Rocha, C., Orbit equivalence of global attractors of semilinear parabolic differential equations. Trans. Amer. Math. Soc. 352 (1) (2000), 257–284. [23] Floer, A., A refinement of the Conley index and an application to the stability of hyperbolic invariant sets. Ergodic Theory Dynam. Systems 7 (1987), 93–103. [24] Fomenko, A., Nguyen, T.-Z., Topological classification of integrable nondegenerate Hamiltonians on isoenergy three-dimensional spheres. In Topological classification of integrable systems, Adv. Soviet Math. 6, Amer. Math. Soc., Providence, RI, 1991, 267–296. [25] Franks, J., Geodesics on S 2 and periodic points of annulus homeomorphisms. Invent. Math. 108 (1992), 403–418. [26] Franks, J., Rotation numbers and instability sets. Bull. Amer. Math. Soc. 40 (2003), 263–279. [27] Franks, J., Williams, R., Entropy and knots. Trans. Amer. Math. Soc. 291 (1) (1985), 241–253. [28] Fusco, G., Oliva, W., Jacobi matrices and transversality. Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), 231–243. [29] R. Ghrist, Branched two-manifolds supporting all links. Topology 36 (2) (1997), 423–448. [30] Ghrist, R., Holmes, P., Sullivan, M., Knots and Links in Three-Dimensional Flows. Lecture Notes in Math. 1654, Springer-Verlag, Berlin 1997. [31] Ghrist, R., Vandervorst, R., Scalar parabolic PDE’s and braids. Preprint, 2005. [32] Ghrist, R., Van den Berg, J., Vandervorst, R., Morse theory on spaces of braids and Lagrangian dynamics. Invent. Math. 152 (2003), 369–432. [33] Gouillart, E., Thiffeault, J.-L., Finn, M., Topological mixing with ghost rods. Preprint, 2005. [34] M. Hirsch, Systems of differential equations which are competitive or cooperative, I: Limit sets. SIAM J. Math. Anal. 13 (1982), 167–179. [35] Holmes, P., Williams, R., Knotted periodic orbits in suspensions of Smale’s horseshoe: torus knots and bifurcation sequences. Arch. Rational Mech. Anal. 90 (2) (1985), 115–193. [36] Kalies, W., Vandervorst, R., Closed characteristics of second order Lagrangians. Preprint, 2002. [37] Kuperberg, K., A smooth counterexample to the Seifert conjecture. Ann. of Math. 140 (1994), 723–732.

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[38] LeCalvez, P., Propriété dynamique des diffeomorphismes de l’anneau et du tore. Astérique 204 1991. [39] LeCalvez, P., Décomposition des difféomorphismes du tore en applications déviant la verticale. Mém. Soc. Math. France (N.S.) 79 (1999). [40] Mallet-Paret, J., Smith, H., The Poincaré-Bendixson theorem for monotone cyclic feedback systems. J. Dynam. Differential Equations 2 (1990), 367–421. [41] Matano, H., Nonincrease of the lap-number of a solution for a one-dimensional semi-linear parabolic equation. J. Fac. Sci. Tokyo 1A 29 (1982), 645–673. [42] Middleton, A., Asymptotic uniqueness of the sliding state for charge-density waves. Phys. Rev. Lett. 68 (5) (1992), 670–673. [43] Milnor, J., Morse Theory. Ann. of Math. Stud. 51, Princeton University Press, Princeton, NJ, 1963. [44] Mischaikow, K., Conley index theory. In Dynamical Systems (Montecatini Terme), Lecture Notes in Math. 1609, Springer-Verlag, Berlin 1995, 119–207. [45] Nakashima, K., Stable transition layers in a balanced bistable equation. Differential Integral Equations 13 (7–9) (2000), 1025–1038. [46] Nakashima, K., Multi-layered stationary solutions for a spatially inhomogeneous AllenCahn equation. J. Differential Equations 191 (1) (2003), 234–276. [47] Séré, E., Looking for the Bernoulli shift. Ann. Inst. Henri Poincaré 10 (5) (1993), 561–590. [48] Sharkovski, A., Coexistence of cycles of a continuous map of a line to itself. Ukranian Math. J. 16 (1964), 61–71. [49] Smillie, J., Competative and cooperative tridiagonal systems of differential equations. SIAM J. Math. Anal. 15 (1984), 531–534. [50] Spears, B., Hutchings, M., Szeri, A., Topological bifurcations of attracting 2-tori of quasiperiodically driven nonlinear oscillators. J. Nonlinear Sci. 15 (6) (2005) 423–452. [51] Sturm, C., Mémoire sur une classe d’équations à différences partielles. J. Math. Pure Appl. 1 (1836), 373–444. [52] Van den Berg, J., Vandervorst, R., Fourth order conservative Twist systems: simple closed characteristics. Trans. Amer. Math. Soc. 354 (2002), 1383–1420. [53] Van den Berg, J., Vandervorst, R., Wójcik, W., Chaos in orientation preserving twist maps of the plane. Preprint, 2004. [54] Wada, M., Closed orbits of nonsingular Morse-Smale flows on S 3 . J. Math. Soc. Japan 41 (3) (1989), 405–413. [55] Zelenyak, T., Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable. Differential Equations 4 (1968), 17–22. Department of Mathematics and Coordinated Science Laboratory, University of Illinois, Urbana, IL 61801, U.S.A. E-mail: [email protected]

Newton interpolation polynomials, discretization method, and certain prevalent properties in dynamical systems Anton Gorodetski, Brian Hunt∗, and Vadim Kaloshin†

Abstract. We describe a general method of studying prevalent properties of diffeomorphisms of a compact manifold M, where by prevalent we mean true for Lebesgue almost every parameter ε in a generic finite-parameter family {fε } of diffeomorphisms on M. Usually a dynamical property P can be formulated in terms of properties Pn of trajectories of finite length n. Let P be such a dynamical property that can be expressed in terms of only periodic trajectories. The first idea of the method is to discretize M and split the set of all possible periodic trajectories of length n for the entire family {fε } into a finite number of approximating periodic pseudotrajectories. Then for each such pseudotrajectory, we estimate the measure of parameters for which it fails Pn . This bounds the total parameter measure for which Pn fails by a finite sum over the periodic pseudotrajectories of length n. Application of Newton interpolation polynomials to estimate the measure of parameters that fail Pn for a given periodic pseudotrajectory of length n is the second idea. We outline application of these ideas to two quite different problems: • Growth of number of periodic points for prevalent diffeomorphisms (Kaloshin–Hunt). • Palis’ conjecture on finititude of number of “localized” sinks for prevalent surface diffeomorphisms (Gorodetski–Kaloshin). Mathematics Subject Classification (2000). 37C05, 37C50, 37D25, 37C29. Keywords. Discretization method, Newton interpolation polynomials, prevalence, pseudotrajectory, growth of number of periodic points, Newhouse phenomenon.

1. Introduction A classical problem in dynamics, geometry, and topology is the description of generic behavior. Given a set of objects what are the properties of a generic element of the set? This question applies to diffeomorphisms, Riemannian metrics, linear operators, and vector fields, just to give several examples. The traditional approach is based on the category theorem of Baire. A countable intersection of open, dense sets is called a residual, or topologically generic, set. The Baire category theorem says that topologically generic sets of a complete metric space (or, more generally, Baire space) are dense. The book of Oxtoby [O] provides a rich variety of topologically generic mathematical objects. However, in many different areas of mathematics examples ∗ Supported by NSF grant DMS0104087. † Supported by an Alfred Sloan Research Fellowship, American Institute of Mathematics Fellowship, and NSF Grant No. DMS-0300229


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Anton Gorodetski, Brian Hunt, and Vadim Kaloshin

of “wild behavior” of topologically generic objects have been detected (see [HSY], [Ka2], [OY], [Si] and references there). In this paper we are concerned with generic properties in dynamics, particularly those that are not generic topologically but are generic in a measure-theoretic sense. In the 1960s two main theories in dynamical systems were developed, one of which was designed for conservative systems and called KAM for Kolmogorov–Arnold– Moser and the other was constructed for general dynamical systems (nonconservative, dissipative) and called hyperbolic. Kolmogorov [Ko], in his plenary talk of ICM 1954, pointed out that a different notion of genericity may be appropriate: “In order to obtain negative results concerning insignificant or exceptional character of some phenomenon we shall apply the following, somewhat haphazard, technique: if in a class K of functions f (x) one can introduce a finite number of functionals F1 (f ), F2 (f ), . . . , Fr (f ), which in some sense can naturally be considered as taking “arbitrary” values in general F1 (f ) = C1 , F2 (f ) = C2 , . . . , Fr (f ) = Cr from some domain of the r-dimensional space of points C = (C1 , . . . , Cr ), then any phenomenon that takes place only if C belongs to a set of zero r-dimensional measure will be regarded exceptional and subject to “neglect”.” A somewhat similar way to define a measure-theoretic genericity, often called prevalence, is the following: We call a property P prevalent if for a generic1 finiteparameter family {fε }ε∈B for Lebesgue almost every parameter ε the corresponding fε satisfies P . If complement of a property is prevalent such a property is called shy. We shall discuss prevalence further in Section 9. There are many examples when topological genericity and measure-theoretic genericity do not coincide. We just mention a few of them (see [HSY], [Ka2], [OY] for many more). • Diophantine numbers form a set of full measure on the line R, but are topologically negligible (that is the complement of the set is topologically generic). • For a topologically generic, even open dense, set of circle maps preserving orientation there is a finite number of attracting and repelling periodic orbits. All other orbits accumulate to these orbits both forward or backward in time. However, as the famous example of Arnold, called Arnold tongues, shows in the family fα,ε : θ → θ + α + ε sin θ that the smaller ε is, the smaller is the measure of α values such that fα,ε has this property. Moreover, the main result of KAM theory says that for consevative systems close to integrable most, in a measure-theoretic sense, motions are quasiperiodic. • In general dynamical systems a dream of the 1960s was to prove that a generic dynamical system is structurally stable. However, this dream evaporated by the end of 1 We give a rigorous definition in Section 9.

Newton interpolation polynomials and discretization method

29

that decade. One of the beautiful counterexamples is due to Newhouse [N1], [N2]. He shows that there is an open set in the space of diffemorphisms of a compact manifold such that a generic diffeomorphism in this open set has infinitely many coexisting sinks (attracting periodic orbits). Below we show in some weak sense this phenomenon is shy (see Section 7). This phenomenon is closely related to Palis’ program [Pa] which is discussed next. Let Diffr (M) be the space of C r diffeomorphisms of a smooth compact manifold M with the uniform C r -topology, where dim M ≥ 2, and let f ∈ Diffr (M). The main focus of the present paper is the space of general (nonconservative) diffeomorphisms Diffr (M). The authors believe that the method presented here also applies to conservative systems. While examples such as Newhouse’s show that on open subsets of Diffr (M), “wild” phenomena that are not structurally stable can be topologically generic, a measure-theoretic point of view may be more appropriate to describe the dynamical behavior that would typically be observed by a scientist. In the influential paper J. Palis [Pa] proposed a new global view of generic dynamics based on measure theory. He stated the following conjectures on finititude of attractors and their metric stability: (I) Denseness of finititude of attractors – there is C r (r ≥ 1) dense set D of diffeomorphisms in Diffr (M) such that each element of D has finitely many attractors, the union of whose basins of attraction has full measure; (II) Existence of physical (SRB) mesure – each attractor of an element of D supports a physical measure that represents the limiting distribution for Lebesgue almost every initial condition in its basin; (III) Metric stability of basins of attraction – for each element in D and each of its attractors, for almost all small C r perturbations in generic k-parameter families of diffeomorphisms in Diffr (M), k ∈ N, there are finitely many attractors whose union of basins is nearly equal in the sense of Lebesgue measure to the basin of the initial attractor; such perturbed attractors support a physical measure. Such results have been established for certain examples of dynamical systems. Lyubich [Ly] for the quadratic family of 1-dimensional maps and Avila–Lyubich–de Melo [ALM] for a generic family of analytic unimodal 1-dimensional maps showed that for almost all parameters the attractors are either periodic sinks or carry an absolutely continuous invariant measure. For the 1-dimensional Schrödinger cocycles Avila–Krikorian [AK] showed that for all analytic or C ∞ potentials and almost all rotation numbers the corresponding cocycle is either non-uniformly hyperbolic or reducible. In this paper we discuss two important topologically negligible dynamical properties that are in fact prevalent. One property is (stretched) exponential growth of the number of periodic points and the other is finiteness of number of coexisting “localized” sinks for surface diffeomorphisms. We hope that the method, outlined in this article, brings a better understanding of prevalent properties of Diffr (M) in the direction of Palis’ conjectures and other important dynamical properties.

30


2. Elementary events and a sample result Here we expose ideas in a general setting. Consider a family of diffeomorphisms {fε }ε∈B ⊂ Diffr (M) of a compact manifold with a probability measure μ supported on the set of parameters B. To avoid distracting details we postpone specification of μ and B. Let us fix a certain property P of periodic points of period n. In both cases that we will consider, P is some form of quantitative hyperbolicity. We split the problem into two parts. • Estimate the measure of the set μ(Bn ) ≤ μn , Bn = {ε ∈ B : fε has a periodic orbit that does not satisfy P } ⊂ B. • Derive some dynamically interesting properties from this estimate. The second part essentially depends on the problem. As for the first part, application of the discretization method and Newton interpolation polynomials give a uniform approach to get a required estimate. First, we discuss the problem of growth of the number of periodic points (see Theorem 2.1 below). For γ > 0 we say that x = f n (x) is (n, γ )-hyperbolic if all eigenvalues of the linearization df n (x) are at least γ -away from the unit circle2 . For γ > 0 this is a weak analog of Kupka–Smale property. Fix some c > 0 and a decaying to zero sequence of positive numbers c = {cγn }n∈Z+ . We say that the map fε satisfies the inductive hypothesis of order n with constants c, denoted fε ∈ I H (n, c), if for all k ≤ n all periodic orbits of period k are (k, cγk )-hyperbolic. Consider a sequence of “bad” sets in the parameter space / I H (n, c)}. Bn (c) = {ε ∈ B : fε ∈ I H (n − 1, c), but fε ∈

(1)

In other words, Bn (c) is the set of “bad” parameter values ε ∈ B for which all periodic points with period strictly less than n are sufficiently hyperbolic, but there is a periodic point of period n that is not (n, cγn )-hyperbolic. Our goal is to find an upper bound μ{Bn (c)} ≤ μn (c)

(2)

for the measure of the set values. Then the sum over n of (2) of “bad” parameter gives an upper bound μ n Bn (c) ≤ n≥1 μn (c) on the set of all parameters ε for which fε has a periodic point of some period n that is not (n, cγn )-hyperbolic. If the sum converges and n≥1 μn (c) = μ(c) → 0 as c → 0, then for μ-almost every ε there is c > 0 such that for every n every periodic point of period n is (n, cγn )-hyperbolic. This statement (almost) implies that all periodic points of period n are at least ≈ cγn -apart and, therefore, the number of periodic points is bounded by ≈ (cγn )− dim M 2 In [KH1] we use a stronger property of hyperbolicity of periodic points (see Section 2 of that paper).


31

(see [KH1], Proposition 1.1.6). Thus, the key to prove a statement that a certain property is prevalent, i.e. holds for almost every parameter value, is an estimate of the probability (2) of a “bad” event. One could replace the property of hyperbolicity of periodic points by another property and still the key is to get an estimate of the probability to fail a certain dynamical property. Our goal is to outline the proof of the following result: Theorem 2.1 ([KH1], [Ka3], [Ka4]). For a prevalent set of diffeomorphisms f ∈ Diff r (M), with 1 < r < ∞, and for all δ > 0 there exists C = C(δ) such that Pn (f ) := #{ isolated x ∈ M : f n (x) = x} ≤ exp(Cn1+δ ). Density of diffeomorphisms with this property is the classical result of Artin– Mazur [AM] (see also [Ka2] for a simple proof). In [Ka1], using [GST], it is shown that diffeomorphisms having an arbitrary ahead given growth along a subsequence are topologically generic. In Section 7 we briefly describe application of the method of the paper to Newhouse phenomenon from [GK].

3. Strategy to estimate probability of a “bad” event: discretization method The goal of this section is to outline how one can get estimate (2). Usually we do not know where is a “bad” trajectory, which fails P , and what are the dynamics in its neighborhood. So our analysis will be implicit. More exactly, we shall consider all possible trajectories in the family {fε }ε∈B and the worst case scenario for each of them. In order to fail the inductive hypothesis of order n with constants c, a diffeomorphism fε should have a periodic, but not (n, cγn )-hyperbolic point x = fεn (x). There is a continuum of possible n-tuples {xk }0≤k≤n such that for some ε ∈ B we have f (xk ) = xk+1 (mod n) and x0 is not (n, cγn )-hyperbolic. Instead of looking at the continuum of n-tuples, we discretize this space and consider only those n-tuples {xk }0≤k≤n that lie on a particular grid, denoted Iγñ , and replace trajectories by γñ pseudotrajectories. If we choose the grid spacing γñ small enough, then every (almost) periodic point of period n that is not sufficiently hyperbolic will have a corresponding γñ -pseudotrajectory of length-n on the grid that also has small hyperbolicity. In this way we reduce the problem of bounding the measure of a set of “bad” parameters corresponding to a particular length-n γñ -pseudotrajectory on the chosen grid. Thus, the basic requirement for the grid size γñ is that every real periodic trajectory {xk = fεk (x0 )}0≤k≤n of length n can be approximated by a γñ -pseudotrajectory {x˜k }0≤k≤n so that if x0 is periodic but not (n, cγn )-hyperbolic, then the n-tuple {x˜k }0≤k≤n is not (n, cγn /2)-hyperbolic (see [KH1], sect. 3.2 and [GK], sect. 8 for various definitions).

32


n We call an n-tuple {xk }n−1 k=0 ⊂ Iγñ a γñ -pseudotrajectory associated to some ε (or to the map fε ) if for each k = 0, . . . , n − 1 we have dist (fε (xk−1 ), xk ) ≤ γñ and we call it a γñ -pseudotrajectory associated to B (or the family {fε }ε∈B ) if it is associated to some ε ∈ B. The naive idea of estimate (2) consists of two steps: Step 1. Estimate the number of different γñ -pseudotrajectories #n (γñ ) associated to B; Step 2. For an n-tuple {xk }0≤k≤n−1 ⊂ Iγnñ estimate the measure

μ{ε ∈ B : {xk }0≤k≤n−1 is a γñ -pseudotrajectory associated to ε which is γñ -periodic but not (n, cγn /2)-hyperbolic} ≤ μn (cγn , γñ ).

(3)

Then the product of two numbers #n (γñ ) and μn (cγn , γñ ) that are obtained in Steps 1 and 2 gives the required estimate. In fact, this simpleminded scheme requires modifications discussed at the end of the next section (see (10–13)). We start with the second step. For simplicity we shall discuss 1-dimensional maps (see [KH1], sect. 3). In [KH1], sect. 4.2 we discuss difficulties arising to extend this method to multidimensional maps. See also [GK], sect. 10 (resp. [Ka4], sect. 7–8), where 2-dimensional (resp. N -dimensional) case is considered. To treat the multidimensional case one use very similar ideas, however, technical difficulties arising due to multidimensionality are fairly involved. Now we show how to estimate probability (3) within a particular polynomial family and then show how to do Step 1 and incorporate the method into the global framework.

4. Newton interpolation polynomials and an estimate of probability of a γñ -periodic but not (n, cγn /2)-hyperbolic γñ -pseudotrajectory of length n Let M be an interval [−1, 1] and Iγñ ⊂ [−1, 1] be a γñ -grid. Fix an n-tuple of points {xk }n−1 k=0 ⊂ Iγñ . Consider the following 2n-parameter family of maps: fu (x) = f (x) +

2n−1 k=0

uk

k−1

(x − xj

(mod n) ).

j =0

This family is nothing but the Newton 2n−1interpolation k−1 polynomials associated to the n-tuple {xk }n−1 . Denote φ (x) = u u k k=0 j =0 (x − xj (mod n) ). Notice that k=0 φu (x0 ) = u0 , φu (x1 ) = u0 + u1 (x1 − x0 ), φu (x2 ) = u0 + u1 (x2 − x0 ) + u2 (x2 − x0 )(x2 − x1 ), .. .

33


.. . φu (xn−1 ) = u0 + u1 (xn−1 − x0 ) + . . . + un−1 (xn−1 − x0 ) . . . (xn−1 − xn−2 ), φu (x0 ) =

∂ ∂x

2n−1 k=0

uk

k j =0

(4)

(x − xj (mod n) ) x=x , 0

.. . φu (xn−1 ) =

2n−1 k

∂ uk (x − xj (mod n) ) x=x . n−1 ∂x k=0

j =0

These formulas are very useful for dynamics. For a given map f and an initial point x0 , the image fu (x0 ) = f (x0 ) + φu (x0 ) of x0 depends only on u0 . Furthermore the image can be set to any desired point by choosing u0 appropriately – we say then that it depends only and nontrivially on u0 . If x0 , x1 , and u0 are fixed, the image fu (x1 ) of x1 depends only on u1 , and as long as x0 = x1 it depends nontrivially on u1 . More generally for 0 ≤ k ≤ n − 1, if distinct points {xj }jk=0 and coefficients {uj }jk−1 =0 are fixed, then the image fu (xk ) of xk depends only and nontrivially on uk . Suppose now that an n-tuple of pairwise distinct points {xj }jn−1 =0 and Newton coef-

ficients {uj }jn−1 =0 are fixed. Then derivative fu (x0 ) at x0 depends only and nontrivially

on un . Likewise for 0 ≤ k ≤ n − 1, if distinct points {xj }jn−1 =0 and Newton coefficients

{uj }jn+k−1 are fixed, then the derivative fu (xk ) at xk depends only and nontrivially on =0 un+k . As Figure 1 illustrates, these considerations show that for any map f and any desired trajectory of distinct points with any given derivatives along it, one can choose Newton coefficients {uk }2n−1 k=0 and explicitly construct a map fu = f + φu with such a trajectory. While the parametrization depends on the n-tuple, the family is equivalent by a change of parameter coordinates (see Section 5) to the family {fε }ε of perturbations by degree 2n − 1 polynomials, given by (14). Using these properties of Newton interpolation polynomials we can easily estimate probability (3). Let us split this compound dynamic event into simple ones and use the above properties: 1. 2.

n. n + 1.

|fε (x0 ) − x1 | ≤ γñ ; |fε (x1 ) − x2 | ≤ γñ ; .. . |fε (xn−1 ) − x0 | ≤ γñ ; n−1 |fε (xj )| − 1 ≤ cγn /2. j =0

(5)

34

Anton Gorodetski, Brian Hunt, and Vadim Kaloshin x0

xk

x1

xk

···

fu (x0 )

fu (xk )

uk

u0

···

fu (x0 )

fu (xk )

un

un+k

Figure 1. Newton coefficients and their action.

First, we find probabilities of these events with respect to u-parameters (see [KH1], sect. 3.3 for more details). It turns out that the map relating ε-parameters and uparameters is one-to-one, linear, and volume-preserving (see Section 5). Notice that in (4) and Figure 1, the image fu (x0 ) of x0 is independent of uk for all k > 0. Therefore, the position of fu (x0 ) depends only on u0 . For the 1-dimensional Lebesgue measure of the u0 ’s we have Leb {u0 : |fu (x0 ) − x1 | = |f (x0 ) + u0 − x1 | ≤ γñ } ≤ 2γñ . Fix u0 . Similarly, the position of fu (x1 ) depends only on u1 (see (4) and Figure 1). Thus, we have Leb u1 : |fu (x1 ) − x2 | = |f (x1 ) + u0 + u1 (x1 − x0 ) − x2 | ≤ γñ } ≤

2γñ . |x1 − x0 |

Inductively for k = 2, . . . , n − 1, fix u0 , . . . , uk−1 . Then the position of fu (xk ) depends only on uk . Moreover, for k = 2, . . . , n − 2 we have k m−1

um (xk − xj ) − x0 ≤ γñ Leb uk : |fu (xk ) − xk+1 | = f (xk ) + j =0

m=0

≤ k−1 j =0

2γñ |xk − xj |

,

35


and for k = n − 1 we have 2γñ Leb un−1 : |fu (xn−1 ) − x0 | ≤ γñ ≤ n−2 . xn−1 − xj j =0

(6)

In particular, the parameter un−1 is responsible for (n, γn )-periodicity of the n-tuple {xk }0≤k≤n . This formula estimates the “measure of periodicity”. Choose u0 , . . . , un−1 so that the n-tuple {xk }n−1 k=0 is a (n, γñ )-periodic γñ -pseudotrajectory. Notice that parameters un , un+1 , . . . , u2n−1 do not change the γñ -pseudotrajectory {xk }n−1 k=0 . Fix now parameters u0 , . . . , u2n−2 and vary only u2n−1 . Then for any C 1 -smooth map g : I → I , consider the 1-parameter family gu2n−1 (x) = g(x) + (x − xn−1 )

n−2

(x − xj )2 .

j =0

2 Since the corresponding monomial (x −xn−1 ) jn−2 =0 (x −xj ) has zeroes of the second order at all points xk , except the last one xn−1 , we have n−1

(gu2n−1 ) (xj ) = g (xn−1 ) + u2n−1

j =0

n−2

|xn−1 − xj |

2

n−2

j =0

g (xj ).

(7)

j =0

To get the final estimate, we use the fact that we are interested only in maps from the family {fu }u . Suppose |f u (xn−1 )| is uniformly bounded by some M1 . For condition (x ) must lie in [1 − cγ /2, 1 + cγ /2]. If this f (n + 1) of (5) to hold, jn−1 n n =0 u j (x ) ≥ (1 − cγ /2)/M for all u occurs for any u2n−1 , then jn−2 f n 1 2n−1 , because =0 u j this product does not depend on u2n−1 . Using (7) and the fact that 1 − cγn /2 ≥ 1/2, we get Leb u2n−1

n−1 cγ 2cγn n : |fu (xj )| − 1 ≤ . ≤ M1 n−2 2 2 j =0 |xn−1 − xj |

(8)

j =0

This formula estimates the “measure of hyperbolicity”. We can combine all these estimates and get Lebn+1 {(u0 , . . . , un−1 , u2n−1 ) : fu satisfies conditions (5) and fu C 1 ≤ M1 } n−1 2γñ 2M1 cγn ≤ . m−1 2 n−2 j =0 xm − xj m=1 j =0 xn−1 − xj

(9)

This completes Step 2, but leaves many open questions which we shall discuss while treating Step 1. The estimate of Step 1 then breaks down as follows: #n (γñ ) ≈

# of initial points in Iγñ

×

# of γñ -pseudotrajectories per initial point

(10)

36


And up to an exponential function of n, the estimate of Step 2 breaks down like:

μn (cγn , γñ ) ≈

Measure of periodicity (6)

×

Measure of hyperbolicity (8)

# of γñ -pseudotrajectories per initial point

(Roughly speaking, the terms in the numerator represent respectively the measure of parameters for which a given initial point will be (n, γñ )-periodic and the measure of parameters for which a given n-tuple is (n, cγn )-hyperbolic; they correspond to estimates (6) and (8) in the next section.) Thus after cancellation, the estimate of the measure of “bad” set Bn (c) associated to almost periodic, not sufficiently hyperbolic trajectories becomes: Measure of bad parameters

≤

# of initial points of Iγñ

×

Measure of periodicity (6)

×

Measure of hyperbolicity (8)

(11)

1/4 and supConsider only pseudotrajectories having jn−2 =0 |xn−1 − xj | ≥ (cγn ) −n pose γñ = M1 cγn . Then up to exponential function of n the first term on the right hand side of (11) is of order (cγn )−1 . The second term has an upper bound of order (cγn )3/4 , and the third term is at most of order (cγn )1/2 , so that the product on the right-hand side of (11) is of order at most (cγn )1/4 (up to an exponential function in n). If cγn is exponentially small with a large exponent in n, then μn (cγn , γñ ) is at most exponentially small. This discussion motivates the following Definition 4.1. A trajectory x0 , . . . , xn−1 of length n of a diffeomorphism f ∈ Diff r (B N ), where xk = f k (x0 ), is called (n, γ )-simple if n−2

|xn−1 − xk | ≥ γ 1/4 .

k=0

A point x0 is called (n, γ )-simple if its trajectory {xk = f k (x0 )}n−1 k=0 of length n is (n, γ )-simple. Otherwise a point (resp. a trajectory) is called non-(n, γ )-simple. If a trajectory is simple, then perturbation of this trajectory by Newton interpolation polynomials is effective. The product of distances is a quantitative characteristic of recurrent properties of a trajectory. If it is small enough, then there are close returns of it to x0 before time n. Even though most of properties of periodic orbits do not depend on a starting point, it turns out that for the above product, even asymptotically, it does matter where to choose the starting point. A good example to look at is periodic trajectories in a neighborhood of a planar homoclinic tangency (see [KH1], sect. 2.4 for more). It motivates the following

37


Definition 4.2. A point x is called essentially (n, γ )-simple if for some nonnegative j < n, the point f j (x) is (n, γ )-simple. Otherwise a point is called essentially non-(n, γ )-simple. In (11) we consider only (n, cγn )-simple pseudotrajectories. To study nonsimple pseudotrajectories we look for their simple almost periodic parts. More exactly, for each non-(n, cγn )-simple pseudotrajectory we find such a close return, say xk , that k−1 k−1 {xj }jn−1 =0 is almost equal to n/k copies of {xj }j =0 and {xj }j =0 is (k, cγk )-simple. Due to n−1 closeness, sufficient hyperbolicity of {xj }jk−1 =0 implies sufficient hyperbolicity {xj }j =0 . Then investigation of the measure of nonhyperbolicity of nonsimple pseudotrajectory reduces to the measure of nonhyperbolicity of its simple almost periodic parts. Thus to obtain μn (cγn , γñ ) from (3) we arrive at the following scheme:

Measure of bad parameters associated to periodic nonhyperbolic orbits Measure of bad parameters associated to simple periodic (I) + nonhyperbolic orbits

&

(12)

Measure of bad parameters associated to nonsimple periodic (II) nonhyperbolic orbits

Measure of bad parameters associated to nonsimple periodic nonhyperbolic orbits Partition of nonsimple periodic orbits into simple almost periodic parts (II.A)

=

≤

(13)

Measure of bad parameters associated to short non-simple almost periodic nonhyperbolic orbits (II.B)

As a matter of fact (13) requires additional comments, since the left hand side is a number, while the right hand side is not. To estimate the number from the left hand side we do two step procedure described in the right hand side. First, we do a certain partition (II.A) and then estimate a different number (II.B), which turn out to be an upper bound for the left hand side. This diagram summarizes the problems we face in the proof. • Part (I): how to estimate the measure of parameter values (11) associated with simple periodic nonhyperbolic orbits; • Part (II.A): how to partition a nonsimple periodic orbit into almost periodic parts so that hyperbolicity of an almost periodic part implies hyperbolicity of the whole orbit; The part (II.B) (how to estimate the measure associated with (11) simple periodic nonhyperbolic shorter orbits) can be treated in the same way as part (I), even though the actual details are usually quite involved (see [KH1], sect. 3.5–3.6).

38


5. How to collect all simple (almost) periodic pseudotrajectories: the Distortion and Collection Lemmas In this section for the model family we show how one can justify heuristic estimates (10 – 11). The model family is the family of perturbations of a C 2 map f : I → I, I = [−1, 1] such that f (I ) strictly belongs to I fε (x) = f (x) +

2n−1

ε = (ε0 , . . . , ε2n−1 ).

εk x k ,

(14)

k=0

This is a 2n-parameter family. Assume that parameters belong to a brick, called the brick of standard thickness with width τ (see [KH1], sect. 3.1 in the 1-dimensional case, [KH1], sect. 4.3, [Ka4], sect. 8.3 in the N-dimensional case, [GK], sect. 2.3 and 11.2 for modified definitions in the 2-dimensional case applicable to the problem of finiteness of localized sinks) τ st . (τ ) = {εk }2n−1 : for all 0 ≤ k < 2n, |ε | < H B 0.

660

Yves Le Jan

b) The mapping defined in a) is a bijection between solutions of (4.1) and probability measures on [0, 1] with mean 1/2. The Feller convolution semigroup associated with a measure m is denoted {νtm , t ≥ 0} or ν m . c) K W is associated with δ1/2 and ϕ c with 21 (δ0 + δ1 ). Let us now describe the domination relations. Definition 4.1.2. Let m1 and m2 be probability measures on [0, 1]. a) m1 is swept by m2 if and only if for all positive convex function f , f dm1 .

f dm2 ≤

b) m2 is a barycenter of m1 if and only if there exists a measurable map ψ : [0, 1] → [0, 1] such that ψ ∗ m1 = m2 and ψ ∗ (I · m1 ) = I · m2 (where I denotes the identity function). It can easily be seen that a) and b) define partial order relations. The order defined in a) is the balayage order. The fact that m2 is a barycenter of m1 is equivalent to saying that if U1 is a random variable of law m1 , then there exists a σ (U1 )-measurable random variable U2 of law m2 such that E[U1 |U2 ] = U2 . In [16], a domination and a weak domination relation between (laws of) stochastic flow of kernels is defined: Let ν 1 and ν 2 be two Feller convolution semigroups. We recall that Definition 3.3 in [16] essentially says that ν 1 dominates ν 2 if and only if there is a joint realisation (K 1 , K 2 ) such that K 1 (resp. K 2 ) is a stochastic flow of kernels associated to ν 1 (resp. to ν 2 ) satisfying E[K 1 |K 2 ] = K 2 and σ (K 2 ) ⊂ σ (K 1 ). One says that ν 1 weakly dominates ν 2 when only the conditional expectation assumption is verified (σ (K 2 ) needs not be a sub-σ -field of σ (K 1 )). A full understanding of the solutions of a general SDE should involve a classification of the solutions according to these domination relations. As we will see in the following section, this is not achieved yet even in relatively simple cases. Theorem 4.1.3. Let m1 and m2 be two probability measures on [0, 1] with mean 1/2. a) ν m1 dominates ν m2 if and only if m2 is a barycenter of m1 . b) ν m1 weakly dominates ν m2 if and only if m1 is swept by m2 .

5. Stochastic flows of kernels and SDEs: an example on the circle Notation. In all the following we will denote by S the unit circle R/2π Z, by m the Lebesgue measure on S and by P (S) the set of Borel probability measures on S. Let (W , F W , PW ) be the canonical probability space of a sequence of independent W denote the σ -field Wiener processes (Wtk , k ≥ 0, t ≥ 0). For all s < t let Fs,t generated by the random variables Wvk − Wuk , s ≤ u < v ≤ t and k ≥ 0. Being

661

New developments in stochastic dynamics

given (ak )k≥0 a sequence of nonnegative numbers such that k≥0 ak2 < ∞, we set C(z) = k≥0 ak2 cos(kz). Note that all real positive definite functions on S can be written in this form and that C(0) = k≥0 ak2 . 5.1. Flows of diffeomorphisms. Assume that k≥1 k 2 ak2 < ∞. Then by a stochastic version of Gronwall’s lemma it can be shown that for each x0 ∈ S the stochastic differential equation (SDE) xt = x0 + a0 Wt0 t t 2k−1 2k + ak sin(kxs )dWs + cos(kxs )dWs 0

k≥1

(5.1)

0

has a unique strong solution. These solutions can be considered jointly to form a stochastic flow of diffeomorphisms (ϕs,t )s 2 the condition k≥1 k 2 ak2 < ∞ is satisfied. When α is close to the boundary case, α = 2, the Lyapounov exponent gets close to −∞. In the following we will define stochastic flows corresponding to the case α ≤ 2 which are not flows of diffeomorphisms. 5.2. Wiener chaos expansion: Lipschitz case. Suppose that k k 2 ak2 < ∞ and let ϕs,t be the flow defined in the previous section. For any function f ∈ C(S), x ∈ S and s ≤ t, f ϕs,t (x) belongs to the Wiener space L2 (PW ). Following the original idea of [26], its Wiener chaos expansion can be explicitly computed as follows.

662

Yves Le Jan

Proposition 5.2.1. For all s ≤ t and f ∈ C(S), n Js,t f (x) in L2 (PW ), f ϕs,t (x) = Pt−s f (x) +

(5.2)

n≥1

where J n is defined recursively as follows (denoting ck the function x → cos(kx) and sk the function x → sin(kx)): t n+1 n Js,u ((Pt−u f ) )(x)dWu0 Js,t f (x) = a0 s t n + ak Js,u (sk (Pt−u f ) )(x)dWu2k−1 k≥1

s

+

t

s

n Js,u (ck (Pt−u f ) )(x)dWu2k

0 =P for n ≥ 0 with Js,t t−s .

Remark 5.2.2. The chaos expansion (5.2) can be extended to all f ∈ L2 (m), the two terms being equal in L2 (m ⊗ PW ). 5.3. Non-Lipschitz case. From now on we assume k∈N k 2 ak2 = ∞. In this case, using Gronwall’s inequality, the existence of a strong solution to the SDE (5.1) cannot be proven. But the series giving the Wiener chaos expansion of f ϕs,t in the Lipschitz case for f ∈ L2 (m) also converges in L2 (m ⊗ PW ) in the non-Lipschitz case. n of random operators acting of L2 (m) recursively: We can construct a family Ss,t 0 =P 2 let Ss,t t−s and for f ∈ L (m) and n ≥ 0 set t n+1 n f = Pt−s f + a0 Ss,u Ss,t ((Pt−u f ) )dWu0 0 t t n 2k−1 n 2k + ak Ss,u (sk (Pt−u f ) )dWu + Ss,u (ck (Pt−u f ) )dWu . k≥1

s

s

n f )2 ] ≤ P 2 It can be seen that for all n, E[(Ss,t t−s f and n Ss,t f =

n

k Js,t f,

(5.3)

k=0 k f belongs to the k-th Wiener chaos. Thus all these terms are orthogonal where Js,t n and Ss,t f converges in L2 (m ⊗ PW ) towards a limit we denote by Ss,t f . The family S = (Ss,t ) of random operators acting on L2 (m) satisfies the following.

(i) Cocycle property: Ss,u = Ss,t St,u for all s < t < u.

663


(ii) Stationary increments: for all s ≤ t, Ss,t and S0,t−s have the same law. (iii) Independent increments: for t0 ≤ · · · ≤ tn , St0 ,t1 , . . . , Stn−1 ,tn are independent. (iv) Solution of the SDE t Ss,t f = f + a0 Ss,u (f )dWu0 s t t 2k−1 2k + ak Ss,u (sk f )dWu + Ss,u (ck f )dWu (5.4) k≥1

+ for all f ∈

C(0) 2

H 2 (S)

s

t

s

Ss,u f du,

s

and all s < t.

Moreover, S is the unique family of random operators acting on L2 (m) verifyWing E[(Ss,t f )2 ] ≤ Pt−s f 2 , satisfying (i), (ii), (iii), (iv) and such that Ss,t is Fs,t measurable. Obviously Ss,t 1 = 1, and it can be proved that Ss,t is nonnegative as follows. Consider an independent stationary Brownian motion Bt with diffusion coefficient C(0) on S. Set, for k ≥ 1, t t t2k−1 = Wt2k−1 + ak W sk (Bs )dBs − ak a0 sk (Bs )dWs0 0 0 t t 2l−1 2l − ak al sk sl (Bs )dWs + sk cl (Bs )dWs 0

l≥1

and, for k ≥ 0,

0

t ck (Bs )dBs − ak a0 ck (Bs )dWs0 0 0 t t 2l−1 2l − ak al ck sl (Bs )dWs + ck cl (Bs )dWs .

t2k = Wt2k + ak W

l≥1

t

0

0

These formulas are obtained by conditioning the “velocity differential” at time t and forms a family of independent Wiener processes. Set site Bt to be dBt . Then W , Bs = x]. S˜s,t f (x) = E[f (Bt )|W It is clear that S˜ is nonnegative and that S˜ verifies the properties listed above ((i), (ii), . This implies S˜ = S and proves that S is nonnegative. (iii) and (iv)) with respect to W Two cases may occur: (a) Ss,t f 2 = (Ss,t f )2 for all f ∈ L∞ (m). (b) Ss,t f 2 > (Ss,t f )2 for some f ∈ L∞ (m), and in fact for all non constant f ∈ L∞ (m).

664

Yves Le Jan

5.4. n-point motions. Let Pt be the family of random operators acting on L∞ (m⊗n ) defined by (n)

(n)

Pt f1 ⊗ · · · ⊗ fn = E[S0,t f1 ⊗ · · · ⊗ S0,t fn ]. (n)

Properties (i), (ii) and (iii) imply that Pt is a Markovian semigroup. As in the case (n) of Rd or Sd studied in [16], one can show that the isotropy implies that Pt is a Feller semigroup acting on C(Sn ). (n) The n-point motion of (Ss,t ) is the diffusion on Sn associated with Pt . The generator A(n) of this diffusion is given by A(n) =

1 C(xi − xj )∂xi ∂xj . 2

(5.5)

1≤i,j ≤n

The case (a) appears when the diagonal is absorbing for the two-point motion. If this is not the case we are in case (b). 5.5. Diffusive or coalescing? In case (a) it can be shown (using the Feller property) that there exists a flow of random mappings ϕ = (ϕs,t ) such that for all s ≤ t and all f ∈ L2 (m), we have Ss,t f = f ϕs,t in L2 (m ⊗ PW ). Furthermore, ϕs,t : (S × W , B(S) ⊗ F W ) → (S, B(S)) is measurable and solves the SDE (5.1). W) In case (b) it can be shown that there exists a flow of random kernels K = (Ks,t W 2 2 such that Ss,t f = Ks,t f in L (m ⊗ PW ) for all s ≤ t and all f ∈ L (m). The stochastic flow of kernels will be called diffusive when the kernels are not induced by maps, which clearly happens in case (b). This flow solves the SDE in the sense that for all f ∈ C 2 (S), s ≤ t and x ∈ S, t t W W W Ks,t f =f + ak Ks,u (sk f )dWu2k−1 + Ks,u (ck f )dWu2k s

k≥1

t

+ a0 s

W Ks,u (f )dWu0

+

C(0) 2

s

t s

(5.6) W Ks,u f du.

In the following the flow ϕ (in case (a)) or the flow K W (in case (b)) will be called the Wiener solution of the SDE (5.1). Since (Ss,t ) is the unique solution of (5.4) which is W -measurable, the Wiener solution ϕ (or K W ) is the unique solution of SDE (5.1) Fs,t W -measurable. (or of (5.6)) which is Fs,t A diffusive flow is called diffusive with hitting if the two-point motion hits the diagonal = {(x, x), x ∈ S} . The diffusion zt ∈ [0, 2π) such that zt = Xt − Yt modulo 2π , where (Xt , Yt ) is the two point motion, has a natural scale. The speed measure m of this diffusion is z z−x given by m(dz) = (C(0) − C(z))−1 dz. Let κ be defined by κ(z) = π C(0)−C(x) dx. Note that κ(0+) = ∞ implies that m((0, 2π )) = ∞.


665

Theorem 5.5.1. 1) If κ(0+ ) = ∞ then the Wiener solution is a stochastic flow of maps, which is not a coalescing flow. 2) If m((0, 2π)) = ∞ and κ(0+ ) < ∞ then the Wiener solution is a coalescing flow. 3) If m((0, 2π)) < ∞ then the Wiener solution is a diffusive flow with hitting. Corollary 5.5.2. Let ak2 = k −(1+α) with α > 0. 1) If α > 2, then the Wiener solution is a stochastic flow of C 1 -diffeomorphisms. 2) If α = 2 then the Wiener solution is a stochastic flow of maps, which is not a coalescing flow. 3) If α ∈ [1, 2) then the Wiener solution is a coalescing flow. 4) If α ∈ (0, 1) then the Wiener solution is a diffusive flow with hitting. Remark 5.5.3. The case α = 2 has been studied in [1], [8], [22]. It is shown in particular that the maps of the flow are homeomorphisms. 5.6. Extension of the noise and weak solution. Let (, A, P) be an extension of the probability space (W , F W , PW ). We say that a measurable flow of maps ϕ = (ϕs,t ) is W -measurable. Similarly, a weak solution of (5.1) if it satisfies (5.1) without being Fs,t a measurable flow of kernels K = (Ks,t ) will be called weak (generalized) solution W -measurable. of the SDE (5.1) if it satisfies (5.6) without being Fs,t We have seen that uniqueness is verified if one assumes in addition Wiener meaW -measurable for all s ≤ t. surability: Ks,t is Fs,t (n),c can be conIn case (b) a different consistent system of Feller semigroups Pt (n),c (n) associated with Xt , structed by considering the coalescing n-point motion Xt c the n-point motion of the Wiener solution. A measurable flow of coalescing maps ϕs,t (n),c whose n-point motion is Xt can be defined on an extension (, A, P) of the probability space (W , F W , PW ). This coalescing flow also solves the SDE (5.1). It is a weak solution. c = σ (ϕ c , s ≤ u ≤ v ≤ t). Then (F c ) For s ≤ t set Fs,t u,v s,t s≤t defines a noise. W c It can be seen (for details see [16]) that Fs,t ⊂ Fs,t (this property also holds for any flow solution of SDE). This solution being different from the Wiener solution implies W = F c . The noise (F c ) cannot be generated by Brownian motions. It is a Fs,t s,t s,t non-classical noise (see also [27], [28]). The Wiener solution K W can be recovered by filtering: W c W Ks,t f = E[f ϕs,t |Fs,t ], for all f ∈ C(S). It can be shown that in case (a) there is no weak solution different in law from the c is the only solution which is a flow of maps. There Wiener solution. In case (b), ϕs,t are certainly other “intermediate” kernel solutions similar to the sticky flows, but they have not been constructed yet.

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Yves Le Jan

Final remarks. Similar results hold in a more general context, especially in the case of Sd and Rd (including d = 1). In fact, for isotropic flows in dimension d ≥ 2, a different phase appears, in which the Wiener solution is a diffusive flow without hitting. This solution cannot be represented by filtering a coalescing solution defined on an extended probability space and there are no weak (generalized) solutions. In dimension 2 and 3 the coalescing phase (where the Wiener solution is a coalescing flow) and the phase of non uniqueness (where the Wiener solution is diffusive with hitting) still occurs. Many important questions remain open: for example, the nature of the noises when they are not classical, the possible relations with rough paths ([20]), and the classifications of all solutions, starting with the isotropic case.

References [1] Airault, Hélène, and Ren, Jiagang, Modulus of continuity of the canonic Brownian motion “on the group of diffeomorphisms of the circle”. J. Funct. Anal. 196 (2) (2002), 395–426. [2] Arratia, R.A., Brownian motion on the line. Ph D Thesis. University of Wisconsin, Madison, 1979. [3] Baxendale, Peter, Brownian motions in the diffeomorphism group. I. Compositio Math. 53 (1) (1984), 19–50. [4] Bernard, Denis, Gaw¸edzki, Krzysztof, and Kupiainen, Antti, Slow modes in passive advection. J. Statist. Phys. 90 (3–4) (1998), 519–569. [5] Darling, R. W. R., Constructing nonhomeomorphic stochastic flows. Mem. Amer. Math. Soc. 70 (376) (1987), vi + 97pp. [6] E, Weinan, and Vanden Eijnden, Eric, Generalized flows, intrinsic stochasticity, and turbulent transport. Proc. Natl. Acad. Sci. USA 97 (15) (2000), 8200–8205 (electronic). [7] Elworthy, K. D., Stochastic differential equations on manifolds. In Probability towards 2000 (New York, 1995), Lecture Notes in Statist. 128, Springer-Verlag, New York 1998, 165–178. [8] Fang, Shizan, Canonical Brownian motion on the diffeomorphism group of the circle. J. Funct. Anal. 196 (1) (2002), 162–179. [9] Gawedzki, K., and Kupiainen, A., Universality in turbulence: an exactly solvable model. In Low-dimensional models in statistical physics and quantum field theory (Schladming, 1995), Lecture Notes in Phys. 469, Springer-Verlag, Berlin 1996, 71–105. [10] Gaw¸edzki, Krzysztof, and Vergassola, Massimo, Phase transition in the passive scalar advection. Phys. D 138 (1-2) (2000), 63–90. [11] Harris, Theodore E., Coalescing and noncoalescing stochastic flows in R1 . Stochastic Process. Appl. 17 (2) (1984), 187–210. [12] Kunita, Hiroshi, Stochastic flows and stochastic differential equations. Cambridge Stud. Adv. Math. 24, Cambridge University Press, Cambridge 1990. [13] Le Jan, Y., and Lemaire, S., Products of Beta matrices and sticky flows. Probab. Theory Related Fields 130 (1) (2004), 109–134.


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[14] Le Jan, Yves, On isotropic Brownian motions. Z. Wahrsch. Verw. Gebiete 70 (4) (1985), 609–620. [15] Le Jan, Yves, and Raimond, Olivier, Integration of Brownian vector fields. Ann. Probab. 30 (2) (2002), 826–873. [16] Le Jan, Yves, and Raimond, Olivier, Flows, coalescence and noise. Ann. Probab. 32 (2) (2004), 1247–1315. [17] Le Jan, Yves, and Raimond, Olivier, Sticky flows on the circle and their noises. Probab. Theory Related Fields 129 (1) (2004), 63–82. [18] Le Jan, Yves, and Raimond, Olivier, Flows associated to Tanaka’s SDE. Alea 1 (2005), 21–34 (electronic). [19] Le Jan, Yves, and Watanabe, Shinzo, Stochastic flows of diffeomorphisms. In Stochastic analysis (Katata/Kyoto, 1982), North-Holland Math. Library 32, North-Holland, Amsterdam 1984, 307–332. [20] Lyons, Terry, and Qian, Zhongmin, System control and rough paths. Oxford Mathematical Monographs, Oxford University Press, Oxford 2002. [21] Ma, Zhi-Ming, and Xiang, Kai-Nan, Superprocesses of stochastic flows. Ann. Probab. 29 (1) (2001), 317–343. [22] Malliavin, Paul, The canonic diffusion above the diffeomorphism group of the circle. C. R. Acad. Sci. Paris Sér. I Math. 329 (4) (1999), 325–329. [23] Raimond, Olivier, Flots browniens isotropes sur la sphère. Ann. Inst. H. Poincaré Probab. Statist. 35 (3) (1999), 313–354. [24] Tsirelson, Boris, Nonclassical stochastic flows and continuous products. Probab. Surv. 1 (2004), 173–298 (electronic). [25] Tsirelson, Boris, Scaling limit, noise, stability. In Lectures on probability theory and statistics, Lecture Notes in Math. 1840, Springer-Verlag, Berlin 2004, 1–106. [26] Veretennikov, A. Ju., and Krylov, N. V., Explicit formulae for the solutions of stochastic equations. Mat. Sb. (N.S.) 100 (142) (2) (1976), 266–284, 336. [27] Warren, J., Splitting: Tanaka’s sde revisited. arXiv:math.PR/9911115. [28] Watanabe, S., The stochastic flow and the noise associated to Tanaka’s stochastic differential equation. Ukraïn. Mat. Zh. 52 (9) (2000), 1176–1193; English translation Ukrainian Math. J. 52 (2) (2000), 1346–1365. Département Mathématique, Université Paris-Sud, Bâtiment 425, 91405 Orsay Cedex, France E-mail: [email protected]

Stochastic classification models Peter McCullagh and Jie Yang∗

Abstract. Two families of stochastic processes are constructed that are intended for use in classification problems where the aim is to classify units or specimens or species on the basis of measured features. The first model is an exchangeable cluster process generated by a standard Dirichlet allocation scheme. The set of classes is not pre-specified, so a new unit may be assigned to a previously unobserved class. The second model, which is more flexible, uses a marked point process as the mechanism generating the units or events, each with its associated class and feature. The conditional distribution given the superposition process is obtained in closed form for one particular marked point process. This distribution determines the conditional class probabilities, and thus the prediction rule for subsequent units. Mathematics Subject Classification (2000). Primary 62H30; Secondary 68T10. Keywords. Cluster process, Cox process, Dirichlet process, Gauss–Ewens process, lack of interference, marked point process, permanent polynomial, Random subset, supervised learning.

1. Introduction 1.1. Classification. The problem of numerical taxonomy is to classify individual specimens or units u on the basis of measured variables or features x(u) ∈ X. The units may be anything from tropical insects to bitmap images of handwritten digits or vocalizations of English words. The feature variables may be length or width or weight measurements in the case of insects, or the Fourier transformation at certain frequencies in the case of spoken words. The choice of feature variables is an important problem in its own right, but this matter is of little concern in the present paper. A deterministic classification model is a rule or algorithm that associates with each feature value x ∈ X a class y(x) ∈ C. Ordinarily the model must be primed or trained on a sample of units with measured features and known classes. In the dialect of artificial intelligence and computer science, the classifier learns the characteristics peculiar to each class and classifies subsequent units accordingly. When the training is over, each subsequent input is a feature value x(u ) for a new unit, and the output is the assigned class. The error rate is the fraction of wrong calls. A stochastic classification model is a process determining a rule that associates with each feature value x a probability distribution p( · ; x) on the set of classes. ∗ We are grateful to Jim Pitman for helpful comments. Support for this research was provided by NSF Grant DMS-0305009.


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Once again, the classification model must be primed or trained on a sample of units with measured features and known classes. In statistical language, the classifier is a statistical model with unknown parameters to be estimated from the training data. Subsequent units are classified in the usual stochastic sense by computing the conditional distribution given the training data and the feature value for the new unit. Three stochastic models are described in the sections that follow. The first of these is a regression model with independent components in which the feature values are treated as covariates. The second is an exchangeable cluster process closely related to Fisher’s discriminant model, but different in several fundamental ways. The third model is also an exchangeable cluster process, called a permanent cluster process because the conditional distributions are expressed in terms of permanent polynomials. The distinction between a closed classification model with a pre-determined set of labelled classes, and an open model with unlabelled classes is emphasized. A model of the latter type has a mathematical framework that permits a new unit to be assigned to a class that has not previously been observed and therefore does not have a name. The goal is to construct a classification model with no more than 4–5 parameters to be estimated regardless of the number of classes or the dimension of the feature space. In this way, the technically difficult problems associated with consistency and parameter estimation in high dimensional models are evaded. Ideally, the model should be capable of adapting to classification problems in which one or more classes occupies a non-convex region, or even several disconnected regions, in the feature space. 1.2. Remarks on the literature. The literature on stochastic classification is very extensive, the modern theory beginning with Fisher’s discriminant model ([12]). Logistic regression models emerged in the 1960s, and with the advent of faster computing, smoothed versions using penalized likelihood became more popular. Stochastic models used in the statistical literature are sometimes complicated, but they are frequently of the most elementary form with independent components such that log(pr(Y (u) = r | X)) = fr (X(u)). The goal is to estimate the functions fr under certain smoothness conditions, which are enforced through penalty functions added to the log likelihood. For a good overview see [29], [15], [27] or [16]. At the more mathematical end of the statistical spectrum, the same model with independent components is frequently used, with f belonging to a suitable space of functions, usually a Besov space. The stated mathematical goal is to obtain the best estimate of f under the most adverse conditions in very large samples ([9]). Smoothing is usually achieved by shrinkage or thresholding of coefficients in a wavelet expansion. The past decade has seen an upsurge of work in the computer science community under the headings of artificial intelligence, data mining and supervised learning.

Stochastic classfication models

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Methods used include neural nets, support vector machines and tree classifiers. The emphasis is primarily on algorithms, regularization, efficiency of computation, how best to combine weak classifiers ([13]), and so on. Few algorithms and methods of this type have an overt connection with a generative stochastic process beyond the simple additive form with independent components. In the Bayesian literature, more complicated processes are constructed using mixture models with Dirichlet priors for the class frequencies ([11], [2], [24], [14]). The cluster process in Section 3 is in fact a simple special case of a more general classification model ([4], [8]). It is used here mainly for illustrative purposes because the distributions can be studied analytically, which is rare for processes generated by Dirichlet allocation schemes. The semi-parametric models described in Section 4 are of a different type. They are based on Cox processes ([5]) with a baseline intensity measure μ treated as an unknown parameter. One major attraction for practical work is that the conditional distribution of the class labels given the observed features does not depend on the baseline measure. The unknown nuisance parameter is eliminated by conditioning rather than by integration, and this conditional distribution is the basis for inference and classification.

2. Logistic discrimination 2.1. Non-interference and regression models. Let U be the set of units, the infinite set of objects such as plots or subjects or specimens, on which the process Y is defined. A covariate x : U → X is a function on the units, the values of which are thought to have an effect on the distribution. In a logistic regression model it is the class Y (u) ∈ C that is regarded as the response, and the measured feature x(u) is the covariate. In practical work, it is often helpful to distinguish between covariates such as sex, age and geographical position that are intrinsic to the unit, and treatment variables such as medication or variety that can in principle be controlled by the experimenter. For mathematical purposes it is more useful to distinguish between a covariate as a function on the units, and a relationship as a function on pairs of units. Examples of the latter include distance if the units are arrayed in space, temporal ordering for time points, genetic or familial relationships if the units are individual organisms, or a block factor as an equivalence relation on units. The statistical distinction, roughly speaking, is that a covariate affects one-dimensional marginal distributions, while a relationship affects bivariate distributions. For present purposes, however, distinctions of this sort are unnecessary. A regression model is a process in which the joint distribution of the response (Y (u1 ), . . . , Y (un )) on n units is determined by the covariate values x = (x(u1 ), . . . , x(un )) on those units. We write Pn ( · ; x) for the joint distribution on an ordered set of n distinct units, implying that two sets of units having the

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same ordered list of covariate values, also have the same distribution. In other words, if (x(u1 ), . . . , x(un )) = (x(u1 ), . . . , x(un )) then (Y (u1 ), . . . , Y (un )) and (Y (u1 ), . . . , Y (un )) are both distributed as Pn ( · ; x). In general, the probability assigned to an event A ⊂ C n depends on the covariate vector (x1 , . . . , xn ). However, the lack of interference condition Pn (A; (x1 , . . . , xn )) = Pn+1 (A × C; (x1 , . . . , xn , xn+1 ))

(2.1)

implies that the probability assigned by Pn+1 to the event A × C does not depend on the final component xn+1 of x. The failure of this condition means that the probability assigned by P2 to an event of the form Y (u1 ) = 0 depends on the value of x(u2 ). Since the value assigned by P1 to the same event depends only on x(u1 ), the two probability distributions are mutually inconsistent. At the very least, interference of this sort may lead to ambiguities in the calculation of probabilities. Consider two disjoint sets of units with associated vectors X(1) , Y (1) , X(2) , Y (2) , all regarded as random variables. Lack of interference is equivalent to the condition that the response Y (1) be conditionally independent of X(2) given X(1) . The condition is asymmetric in X and Y . As a consequence, the covariate value on unit u has no effect on the joint distribution for other units. The same term is used in the applied statistical literature ([6], section 2.4; [26]) with a similar meaning, though usually interpreted as a physical or biological property of the system rather than a mathematical property of the model. Without this property, it is difficult to give the model a causal interpretation, so lack of interference is often taken for granted as a logical necessity in applications involving deliberate intervention or assignment of treatment to units. For applications in which the x-values are generated by a process, the preceding argument is not compelling, and the non-interference condition is in fact unduly restrictive. The classification model in Section 3 is derived from an exchangeable bivariate process (Y (u), X(u))u∈U with finite-dimensional distributions Qn . The conditional distributions Qn ( · | X = x) determine the joint classification probabilities for n units having the given covariate values as generated by the process. This is not a regression model because the non-interference condition (2.1) is not satisfied by the conditional distributions. As a result, the response distribution for a set of units selected on the basis of their covariate values is not easily determined and is not equal to Qn ( · | X = x). We argue that condition (2.1) is unnecessarily strong for certain applications, and that a weaker condition is sufficient for applications in which intervention does not arise. Consider a family of distributions Pn ( · ; x), one such distribution for each covariate configuration. It may happen that there exists a bivariate process with distributions Qn such that, for each covariate configuration x and each event A ⊂ C n , the conditional distributions satisfy Pn (A; x) = Qn (A | X = x). The distributions {Pn ( · ; x)} are then said to be weakly compatible with one another. If such a bivariate process exists, it is not unique because the marginal distribution of the X-process is arbitrary. Since the units in the bivariate process have no covariates to distinguish one

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from another, the bivariate process is ordinarily exchangeable. Lack of interference implies weak compatibility, but the converse is false. 2.2. Logistic regression. In a logistic regression model, the components Y (u1 ), . . . are independent, so the joint distributions are determined by the one-dimensional marginal distributions. The dependence on x is determined by a suitable collection of discriminant functions, fj : X → R, which could be the coordinate projections if X = Rq , but might include quadratic or other non-linear functions. For a unit u whose feature value is x = x(u) the class probabilities are βrj fj (x), log pr(Y (u) = r) = j

where the coefficients βrj are parameters to be estimated from the training data. In particular, if there are only two classes, the log odds for class 0 are log(pr(Y (u) = 0)/ pr(Y (u) = 1)) = (β0j − β1j )fj (x). (2.2) j

For a model with k classes and q linearly independent discriminant functions, the number of parameters is q(k − 1), which can be large. The lack of interference condition is automatically satisfied by the logistic regression model, and in fact by any similar model with independent components.

3. An exchangeable cluster process 3.1. Random permutations and random partitions. A partition B of the set [n] = {1, . . . , n} is a set of disjoint non-empty subsets called blocks whose union is the whole set. The symbol #B denotes the number of blocks, and for each block b ∈ B, #b is the number of elements. The partition is also an equivalence relation on [n], i.e. a function B : [n] × [n] → {0, 1} that is reflexive, symmetric and transitive. Finally, B is also a symmetric binary matrix with components B(i, j ). No distinction is made in the notation between B as a set of subsets, B as a matrix, and B as an equivalence relation. If the partition is regarded as a matrix, #B is its rank. Denote by Bn the set of partitions of [n]. Thus, B2 = {12, 1|2} has two elements, and B3 has five elements 123,

12|3,

13|2,

23|1,

1|2|3,

where 13|2 is an abbreviation for {{1, 3}, {2}}, containing two blocks. The 15 elements of B4 can be grouped by block sizes as follows 1234,

123|4 [4],

12|34 [3],

12|3|4 [6],

1|2|3|4

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where 12|34 [3] is an abbreviation for the three distinct partitions 12|34, 13|24, 14|23, each having two blocks of size two. The number of elements in Bn is the nth Bell number, the coefficient of t n /n! in the generating function exp(et − 1). The first few values are 1, 2, 5, 15, 52, 203, 877,…, increasing rapidly with n. Consider a probability distribution on the symmetric group Sn in which the probability assigned to the permutation σ depends on the number of cycles as follows: pn (σ ) = λ#σ (λ)/ (n + λ),

(3.1)

where λ > 0, and the ratio of gamma functions is the required normalizing constant. This is the exponential family generated from the uniform distribution with weight function λ#σ , canonical parameter log λ and canonical statistic #σ the number of cycles. It is evident that the distribution is invariant under the action of the group on itself by conjugation, so pn is finitely exchangeable. Less obvious but easily verified is the fact that pn is the marginal distribution of pn+1 under the natural deletion operation σ → σ from Sn+1 into Sn , which operates as follows. Write σ in cycle form, for example σ = (1, 3)(5)(2, 6, 4) for n = 5, and delete element n + 1 = 6 giving σ = (1, 3)(5)(2, 4). This construction, together with the associated Chinese restaurant process, is described by Pitman ([24], section 4). The projection Sn+1 → Sn is not a group homomorphism, but successive deletions are commutative. For each λ > 0, these distributions determine an exchangeable permutation process closely related to the Ewens process on partitions. The cycles of the permutation σ ∈ Sn determine a partition of the set [n], and thus a map Sn → Bn . The inverse image of B ∈ Bn contains b∈B (#b) permutations all having the same probability. Thus, the marginal distribution on partitions induced by (3.1) is (λ)λ#B pn (B; λ) = (#b) (3.2) (n + λ) b∈B

for B ∈ Bn and λ > 0 ([10], [1]). This distribution is symmetric in the sense that for each permutation σ : [n] → [n], the permuted matrix (B σ )ij = Bσ (i),σ (j ) has the same distribution as B. The partition B σ has the same block sizes as B, which are maximal invariant, and the probability pn (B; λ) depends only on the block sizes. In addition if B ∼ pn+1 ( · ; λ) is a random partition of [n + 1], the leading n × n submatrix B is a random partition of [n] whose distribution is pn ( · ; λ) ([19]). For each λ > 0, the sequence of distributions {pn } determines an exchangeable process called the Ewens partition process. For further details, see Pitman ([25]). The Ewens process is by no means the only example of an exchangeable partition process, but it is one of the simplest and most natural, and it is sufficient to illustrate the ideas in the sections that follow. Some simple extensions are described by Pitman ([24]). 3.2. Gauss–Ewens cluster process. A cluster process with state space X is an infinite sequence of X-valued random variables X(u) for u ∈ U, together with a random

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partition B : U × U → {0, 1}, which determines the clusters. An observation on a finite set of units {u1 , . . . , un } consists of the values X(u1 ), . . . , X(un ) together with the components of the matrix Bij = B(ui , uj ). The finite-dimensional distributions on Bn × Xn with densities pn satisfy the obvious Kolmogorov consistency condition: pn+1 (B , x1 , . . . , xn+1 ) dxn+1 pn (B, x1 , . . . , xn ) = B :φB =B X

where φ : Bn+1 → Bn is the deletion operator that removes the last row and column. In the Gauss–Ewens process, X = Rq is a vector space. The observation (B, (X1 , . . . , Xn )) on a finite set of n units has a joint density in which B is a partition with distribution (3.2). The conditional distribution given B is Gaussian with constant mean vector, here taken to be zero, and covariance matrix B = In ⊗ + B ⊗ 1 , where , 1 are q × q covariance matrices. In component form cov(Xir , Xj s | B) = δij rs + Bij 1 rs . This construction implies that X is a sum of two independent processes, one i.i.d. on the units, and one with i.i.d. components for each block. If X = R, the coefficient matrices are scalars and the joint density is pn (B, x) =

(λ)λ#B (#b) × (2π )−n/2 |B |−1/2 exp(−x B−1 x/2). (n + λ) b∈B

2 2 It is helpful here to re-parameterize by writing xb for the mean in block 1 /σ b, θ = σ for the ratio of variance components, wb = #b/(1 + θ #b) and x = wb xb / wb , in which case we have |B |−1/2 = σ −n (1 + θ #b)−1/2 ,

x

B−1 x

=

b∈B

(S 2 (b) + wb xb2 )/σ 2 ,

b∈B

where S 2 (b) is the sum of squares for block b. A permutation of the units sends X1 , . . . , Xn to Xσ (1) , . . . , Xσ (n) and also transforms the components of B in such a way that the i, j component of B σ is Bσ (i)σ (j ) . Evidently, the distribution pn is unaffected by such permutations, so the Gauss–Ewens process is infinitely exchangeable. As it stands, the Gauss–Ewens process is not a mixture of independent and identically distributed processes because the observation space Bn × Xn for a finite set of n units is not an n-fold product space. However, if the blocks are labelled at random, the new process is equivalent in every way to the original, and the new process does follow the de Finetti characterization ([25], p. 44).

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3.3. Conditional distributions. Given the observed list of feature values x = (x1 , . . . , xn ), the conditional distribution on partitions induced by the one-dimensional Gauss–Ewens process is pn (B | x) ∝ λ(#b)(1 + θ #b)−1/2 exp((−S 2 (b) − wb xb2 )/(2σ 2 )). b∈B

This is a distribution of the product-partition type pn (B) ∝ cohesion function

b∈B

C(b; x) ([17]) with

C(b; x) = λ(#b)(1 + θ #b)−1/2 exp((−S 2 (b) − wb xb2 )/(2σ 2 )) depending on the feature values of the units in block b only. In particular, C(b; x) does not depend on #B or on n. Evidently, two sets of units having the same ordered list of feature values are assigned the same conditional distribution. The marginal distribution on Bn induced from pn+1 ( · | (x, xn+1 )) by deleting the last component, depends on the value of xn+1 , so these conditional distributions do not determine a process. However, there is no contradiction here because these are conditional distributions, and the two conditioning events are different. Since they are derived from a bivariate process, the distributions are weakly compatible with one another in the sense of Section 2.1. For the multivariate Gauss–Ewens process, the conditional distributions are not of the product-partition type unless the coefficient matrices are proportional, i.e. 1 = θ . When this condition is satisfied, the cohesion function is an obvious multivariate analogue of the univariate version. Product partition distributions are certainly convenient for use in applied work, but the great majority of product partition models are incompatible with any process. Consider for example, the product partition model with cohesion function C(b, x) = λ, independent of the covariate values. For λ = 1, the distributions are uniform on each Bn . But the distribution on Bn induced from the uniform distribution on Bn+1 is not uniform. The Ewens distributions with cohesion function λ(#b) are the only product partition models that are compatible with an exchangeable process. 3.4. Stochastic classification. Given the observation (B, x(u1 ), . . . , x(un )) on n units, plus the feature value x(u ) on a subsequent unit, we aim to calculate the conditional distribution pn+1 ( · | data) on Bn+1 given the observed values generated by the process. The only missing piece of information is the block to which unit u is assigned, so the conditional distribution is determined by the probabilities assigned to the events u → b for those blocks b ∈ B or b = ∅. A straightforward calculation for a product partition model shows that C(b ∪ {u }, (x, x ))/C(b, x) b ∈ B, pr(u → b | data) ∝ C({u }, x ) b = ∅,

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where (x, x ) is the complete list of n + 1 observed feature values. For b ∈ B, the cohesion ratio for the univariate Gauss–Ewens process is #bγ 1/2 exp(−γ (x − θ#bxb /(1 + θ #b))2 /(2σ 2 )) where γ = (1 + θ#b)/(1 + θ(#b + 1)). If θ#b is large, blocks whose sample means are close to x have relatively high probability, which is to be expected. The predictive distribution for the general multivariate Gauss–Ewens process involves a ratio of multivariate normal densities. Although preference is given to larger blocks, the predictive distribution also puts more weight on those classes whose block means are close to x . If x is sufficiently far removed from all observed block means, the empty set (new class) is given relatively greater weight. When the empty set is excluded from consideration the parameter λ has no effect, and the predictive distribution is roughly the same as that obtained from the Fisher discriminant model with prior probabilities proportional to class sizes.

4. Point process models 4.1. Permanent polynomial. To each square matrix K of order n there corresponds a polynomial of degree n, t #σ K1σ (1) · · · Knσ (n) per t (K) = σ

where the sum runs over permutations of {1, . . . , n}, and #σ is the number of cycles. The conventional permanent is the value at t = 1, and the determinant is det(K) = per −1 (−K). The coefficient of t is the sum of cyclic products cyp(K) = lim t −1 per t (K) = K1σ (1) · · · Knσ (n) . t→0

σ :#σ =1

For certain types of patterned matrices, the permanent polynomial can be evaluated in closed form or by recursion. Consider, for example, the matrix J of order n such that Jii = ζ and Jij = 1 otherwise. The permanent polynomial is the value fn (t) obtained by recursion fn (t) fn+1 (t) ζt n = t n hn+1 (t) hn (t) starting with f0 (t) = h0 (t) = 1. In particular, for ζ = 1 and t = λ we obtain the value fn (λ) = (n + λ)/ (λ), which is the normalizing constant in the distribution (3.1). 4.2. Gaussian moments. The permanent polynomial arises naturally in statistical work associated with factorial moment measures of Cox processes as follows. Let Z be

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a zero-mean real Gaussian process on X with covariance function cov(Z(x), Z(x )) = K(x, x )/2. The joint cumulant and the joint moment of the squared variables |Z(x1 )|2 , . . . , |Z(xn )|2 are

cumn |Z(x1 )|2 , . . . , |Z(xn )|2 = cyp[K](x1 , . . . , xn )/2,

E |Z(x1 )|2 · · · |Z(xn )|2 = per 1/2 [K](x1 , . . . , xn ), where [K](x1 , . . . , xn ) is the symmetric matrix of order n whose entries are K(xi , xj ). More generally, if (x) = |Z1 (x)|2 + · · · + |Zk (x)|2 is the sum of squares of k independent and identically distributed Gaussian processes, we have cumn ( (x1 ), . . . , (xn )) = α cyp[K](x1 , . . . , xn ), E( (x1 ) · · · (xn )) = per α [K](x1 , . . . , xn )

(4.1)

with α = k/2 ([22]). Thus, if is the intensity function for a doubly stochastic Poisson process, the nth order product density at x = (x1 , . . . , xn ) is per α [K](x). In other words, the expected number of ordered n-tuples of distinct events occurring in an infinitesimal ball of volume dx centered at x ∈ Xn is per α [K](x) dx. The analogous result for zero-mean complex-valued processes with covariance function cov(Z(x), Z(x )) = K(x, x ) and as defined above is the same except that α = k rather than k/2. A proof for α = 1 can be found in Macchi ([21]), and for general k in McCullagh and Møller ([22]). Although K is Hermitian, the polynomial is real because inverse permutations have conjugate coefficients. 4.3. Convolution semi-group properties. Permanent polynomials also have a semigroup convolution property that is relevant for probability calculations connected with the superposition of independent processes. In describing this property, it is helpful to regard the points x = {x1 , . . . , xn } as distinct and unordered, so x is a finite subset of X. Since per α [K](x1 , . . . , xn ) is a symmetric function of x, we may write per α [K](x) without ambiguity for non-empty sets. For the empty subset, per α [K](∅) = 1. It is shown in McCullagh and Møller ([22]) that per α [K](w) per α [K](w) = per α+α [K](x) (4.2) w⊂x

where the sum is over all 2n subsets, and w is the complement of w in x. Suppose that per α [K](x) is the density at x, with respect to some product measure μ(dx1 ) · · · μ(dxn ), of a finite point process in X. The convolution property implies that the superposition of two independent processes having the same covariance function K has a distribution in the same family with parameter α + α . Furthermore, the ratio per α [K](w) per α [K](w) q(w; x) = (4.3) per α+α [K](x)

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determines a probability distribution on the subsets of x. If in fact some components of x are duplicated, these duplicates must be regarded as distinct units that happen to have the same x-value, and q is then regarded as a distribution on subsets of the n units. In the extreme case where all components are identical, all components of the matrix [K](x) are equal, and the distribution reduces to q(w; x) =

(#w + α) (#w + α ) (α + α ) . (n + α + α ) (α) (α )

In other words, #w has the beta-binomial distribution. The statistical construction ensures that the polynomial per α (K) is positive at all positive half-integer values of α provided only that K is real symmetric and positive semi-definite. In view of the convolution property, it is natural to ask whether the permanent polynomial of a real symmetric positive semi-definite matrix is positive for all α ≥ 1/2. The numerical evidence on this point is compelling, but so far there is no proof. On the one hand, there exist positive semi-definite symmetric matrices such that per α (K) < 0 for values in the interval 0 < α < 1/2. On the other hand, extensive numerical work has failed to produce a positive semi-definite matrix such that the permanent polynomial has a root whose real part exceeds one half. Although no proof is offered, it seems safe to proceed as if per α (K) ≥ 0 for all α ≥ 1/2 and positive semi-definite symmetric K. In applications where the covariance function is non-negative, the permanent polynomial is clearly positive for all α > 0. 4.4. A marked point process. Consider a Poisson process X in X with intensity measure μ. In the first instance, X is a counting measure in X such that the number of events X(A) has the Poisson distribution with mean μ(A). In addition, for nonoverlapping sets A, A , the event counts X(A) and X(A ) are independent. The process is said to be regular if it has no multiple events at the same point and is finite on compact sets. In that case X is a random subset of X such that X ∩ A is finite for compact sets A. For linguistic convenience, we use the terminology associated with random sets rather than the terminology associated with random measures or multisets. All processes are assumed to be regular. A Poisson process driven by a random intensity measure (x)μ(dx) is called a doubly stochastic Poisson process, or a Cox process. Details of such processes can be found in the books by Kingman ([20]) and Daley and Vere-Jones ([7]). Let μ be a non-random measure in X serving as a baseline for the construction of subsequent point processes. For probabilistic purposes, μ is a fixed measure defined on a suitable algebra of subsets of X that includes all singletons. For statistical purposes, μ is a parameter to be estimated, if necessary, from the data. Given a random non-negative intensity function (x), the associated Cox process is such that the expected number of events occurring in an infinitesimal ball dx centered at x is E( (x)) μ(dx). Likewise, the expected number of ordered pairs of distinct events in the infinitesimal product set dx dx at (x, x ) is E( (x) (x )) μ(dx) μ(dx ), and

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so on. In general, for x = (x1 , . . . , xn ), m(n) (x) = E( (x1 ) · · · (xn )) is called the nth order product density at x ∈ Xn . These expectations are the densities of the factorial moment measures of the process with respect to the product measure μn . The order is implicit from the argument x ∈ Xn , so we usually write m(x) rather than m(n) (x). Ordinarily, in typical ecological applications or studies of the spatial interactions of particles, an observation on a point process consists of a census X ∩ S of all events occurring in the bounded set S. The observation tells us not only that an event occurred at certain points in S, but also that no events occurred elsewhere in S. For the sorts of applications with which we are concerned, however, the training sample is not exhaustive, so the observation is regarded as a sample of the events in X. Such an observation tells us only that an event occurred at certain points in X, and says nothing about the occurrence or non-occurrence of events elsewhere. Suppose now that X(1) , . . . , X(k) are k independent Cox process on X driven by independent random intensity functions 1 (x), . . . , k (x), all relative to the same measure μ. The marked process can be represented by the pair (X, y) in which X = ∪X(r) is the superposition process, and y : X → C is the list of labels. Then the rth component process X(r) = y −1 (r) is the inverse image of label r. Let x ⊂ X be a given finite point configuration consisting of n points. Given that x ⊂ X, i.e. that the superposition process contains x, each event x ∈ x has a label y(x) in the marked process so there are k n possible values for the labels of the events in x. Denote by x (r) the subset x ∩ y −1 (r), possibly empty, consisting of those events in x having label r. The conditional distribution of the class labels given x ⊂ X is proportional to the product of the product densities of the component processes mr (x (r) ) . (4.4) pn (y | x) = r∈C m. (x) In this expression, mr (x (r) ) is the product density of order #x (r) at x (r) for the process labelled r, and m. (x) is the nth order product density for the superposition process at x. For the empty set, mr (∅) = 1. A key point to note is that the conditional distribution of the class labels depends only on the product densities, and not on the baseline measure μ. The conditional distribution of the unlabelled partition B is obtained by ignoring labels, in effect by multiplying by the combinatorial coefficient k!/(k − #B)!. Since the combinatorial coefficient depends on the number of blocks, the conditional distribution of the unlabelled partition is not a product partition model, but it is a distribution of Gibbs type ([25], p. 26) These conditional distributions do not determine a regression model because they fail to satisfy the lack of interference condition (2.1). However, they are derived from a bona fide bivariate process, so they are mutually compatible in the weak sense.

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In this context of prediction, it may be helpful to think of each event as a unit or specimen, in such a way that x(u) is the position or feature value of the event, and y(u) is the label. To classify a new unit or event u such that x(u ) = x , it is sufficient to calculate the conditional distribution as determined by pn+1 given the extended configuration x = x ∪ {x } plus the labels of those points in x. The conditional probabilities are proportional to the ratio of product densities pn+1 (y(u ) = r | data) ∝ mr (x (r) ∪ {x })/mr (x (r) )

(4.5)

for r ∈ C. 4.5. Specific examples. We consider two examples, one in which the intensity is the square of a Gaussian process with product density (4.1), and one in which the intensity is log normal. Permanent process. Suppose that each component process is a permanent process and that the product density for process r is mr (x) = per αr [K](x). Then the product density for the superposition process is per α. [K](x) and the conditional distribution of the labels given x is pn (y | x) =

per α1 [K](x (1) ) · · · per αk [K](x (k) ) per α. [K](x)

.

(4.6)

This distribution determines a random labelled partition of the given events into k classes, some of which may be empty. It is the ‘multinomial’ generalization of (4.3), and is closed under aggregation of classes. For a new unit u such that x(u ) = x , the conditional probability of class r is proportional to the permanent ratio pn+1 (y(u ) = r | data) ∝ per αr [K](x (r) , x )/ per αr [K](x (r) ). This expression is restricted to the set of k classes in C, but it may include classes for which x (r) is empty, i.e. named classes that do not occur in the training sample. In the extreme case where x is empty, the probability of class r is αr /α. regardless of x . The derivation of the conditional distribution from the marked point process requires each α to be a half-integer, and K to be positive semi-definite. Alternatively, K could be Hermitian and αr a whole integer. However, if K is non-negative on X, the distribution (4.6) exists for arbitrary αr > 0, even if K is not positive semi-definite. We shall therefore consider the limit in which αr = α and k → ∞ such that α. = kα = λ > 0 is held fixed. The limit distribution for the unlabelled partition is λ#B b∈B cyp[K](x (b) ) pn (B | x; λ) = , (4.7) per λ [K](x) which is a product partition model, and reduces to the Ewens distribution if K is constant on X. For a new unit u such that x(u ) = x , the conditional probability of

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assignment to block b is pn+1 (u → b | data) ∝

cyp[K](x (b) , x )/ cyp[K](x (b) ) λK(x , x )

b ∈ B, b = ∅.

Our experience with these classification rules is restricted to the simplest versions of the model in which X is Euclidean space and K(x, x ) = exp(−|x − x |2 /ρ 2 ) or similar versions such as exp(−|x − x |/ρ). On the whole, the smoother version is better, and the value of α in (4.6) has only minor effects. It is necessary to select a suitable value of the range parameter ρ, but the qualitative conclusions are the same for all ρ. The region in the X-space for which the predictive probability of class r is high need not be convex or simply connected. In that sense, both of these classification rules are qualitatively different from the one derived from the Gauss–Ewens process. Log Gaussian Cox processes. Suppose that each component process is log Gaussian, i.e. log r is a Gaussian process with mean and variance E log r (x) = θr (x),

cov(log r (x), log r (x )) = Kr (x, x ).

Then the nth order product density at x = (x1 , . . . , xn ) is

θr (xj ) + 21 Kr (xi , xj ) . mr (x) = exp j

ij

Given that x occurs in the superposition process, the conditional distribution of the labels satisfies log pn (y | x) = θy(x) (x) + 21 Ky(x) (x, x ) + const. x∈x

x,x ∈x y(x)=y(x )

Finally, a new unit with x(u ) = x generated from the process is assigned to class r with probability log pn+1 (y(u ) = r | data) = θr (x ) + 21 Kr (x , x ) +

Kr (x , x) + const.

x∈x (r)

Thus, if θr (x) = j βrj fj (x) as in Section 2.2, and there are only two classes with K0 = K1 = K, the conditional log odds that the new unit is assigned to class 0 are

(β0j − β1j )fj (x ) +

j

coinciding with (2.4) when K = 0.

x∈x (0)

K(x , x) −

x∈x (1)

K(x , x),

(4.8)

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4.6. Numerical illustration. A simple artificial example suffices to illustrate the qualitative difference between classification models based on Cox processes, and classification models of the type described in Section 3. We use the two-class permanent model (4.6) with α1 = α2 = 1. The feature space is a 3 × 3 square in the plane, the covariance function is K(x, x ) = exp(−x −x 2 /ρ 2 ) with ρ = 0.5, and the true class is determined by a 3 × 3 chequerboard pattern with white in the center square. The training data consists of 90 units, with 10 feature values uniformly distributed in each small square as shown in the first panel of Figure 1. The second panel is a density plot, and the third panel a contour plot, of the conditional probability that a new unit at that point is assigned to class ‘white’. These probabilities were computed by an approximation using a cycle expansion for the permanent ratio. x2 0.0

0.5

1.0

1.5

pts 2.0

2.5

3.0

0.0

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1.5

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0.5

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Figure 1. Predictive probability of class I using a permanent model.

For the parameter values chosen, the range of predictive probabilities depends to a moderate extent on the configuration of x-values in the training sample, but the extremes are seldom below 0.1 or above 0.9 for a configuration of 90 points with 10 in each small square. The range of predictive probabilities decreases as ρ increases, but the 50% contour is little affected, so the classification is fairly stable. Given that the correct classification is determined by the chequerboard rule, the error rate for the permanent model using this particular training configuration can be computed exactly: it is around 13% for a point chosen uniformly at random from the large square. This error rate is a little misleading because most of those errors occur near an internal boundary where the predictive probability is close to 0.5. Gross errors are rare.

5. Parameter estimation Let (y, x) be the training data, and let x be the feature value for a subsequent unit. In principle, the likelihood function should be computed for the full data including the

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value for the subsequent unit. In practice, it is more convenient to base the likelihood on the training data alone, i.e. pn (y, x; θ) at the parameter point θ . Ordinarily, the information sacrificed by ignoring the additional factor is negligible for large n, and the gain in simplicity may be substantial. Likelihood computations are straightforward for logistic regression models, and the same is true for the Gauss–Ewens process, but the state of affairs is more complicated for point process models. Consider a marked permanent process model with αr = α, in which X is a Euclidean space and K(x, x ) = exp(−x − x 2 /ρ 2 ). The parameters of the process are the scalars α, ρ plus the baseline measure μ. However, the conditional likelihood given the observation x from the training sample depends only on α, ρ, and the predictive distribution also depends only on (α, ρ). In this setting, the distribution of x is governed largely by the baseline measure μ, so the information for (α, ρ) in the superposition process must be negligible. Accordingly, we use the conditional likelihood instead of the full likelihood, for parameter estimation. Even though the most troublesome component of the parameter has been eliminated, computation of the likelihood for the remaining parameters does present difficulties. In the case of the log Gaussian model, the normalizing constant is not available in closed form. In the case of the permanent models (4.6) or (4.7), for which the normalizing constants are available, the only remaining obstacle is the calculation of cyclic products and permanent polynomials. The permanent of a large matrix is notoriously difficult to compute exactly ([28]), and the permanent polynomial appears to be even more challenging. For α = 1, polynomial-time algorithms are available for fixed-rank matrices ([3]). In addition, the existence of polynomial-time Monte Carlo algorithms for non-negative matrices, has been demonstrated but not implemented ([18]). Our experience for positive definite matrices is less pessimistic than the preceding remarks suggest. Reasonably accurate polynomial-time continued-fraction approximations for the ratio of permanent polynomials can be developed without resorting to Monte Carlo approximation. We use a cycle expansion whose accuracy improves as α increases. Here, reasonably accurate means within 2–3% for typical covariance matrices of order n = 100, and for α ≥ 1/2. These expansions, which were used in the construction of Figure 1, will be described elsewhere.

References [1] Aldous, D., Probability distributions on cladograms. In Random Discrete Structures (ed. by D. Aldous and R. Pemantle), IMA Vol. Math. Appl. 76, Springer-Verlag, New York 1995, 1–18. [2] Antoniak, C. E., Mixtures of Dirichlet processes with applications to Bayesian nonparametric problems. Ann. Statist. 2 (1974), 1152–1174. [3] Barvinok, A. I., Two algorithmic results for the traveling salesman problem. Math. Oper. Res. 21 (1996), 65–84.


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[4] Blei, D., Ng, A., Jordan, M., Latent Dirichlet allocation. J. Mach. Learn. Res. 3 (2003), 993–1022. [5] Cox, D. R., Some statistical methods connected with series of events. J. Roy. Statist. Soc. Ser. B 17 (1955), 129–164. [6] Cox, D. R., Planning of Experiments. Wiley Publ. Appl. Statist., Wiley, New York 1958. [7] Daley D., Vere-Jones, D., An Introduction to the Theory of Point Processes. 2nd edition, Probab. Appl. (N. Y.), Springer-Verlag, New York 2003. [8] Daumé, H., Marcu, D., A Bayesian model for supervised clustering with the Dirichlet process prior. J. Mach. Learn. Res. 6 (2005), 1551–1577. [9] Donoho, D., Johnstone, I., Kerkyacharian, G., Picard, D., Wavelet shrinkage: asymptopia? J. Roy. Statist. Soc. Ser. B 57 (1995), 301–369. [10] Ewens, W. J., The sampling theory of selectively neutral alleles. Theoret. Population Biology 3 (1972), 87–112. [11] Ferguson, T., A Bayesian analysis of some nonparametric problems. Ann. Statist. 1 (1973), 209–230. [12] Fisher, R. A., The use of multiple measurements in taxonomic problems. Annals of Eugenics 7 (1936), 179–188. [13] Freund,Y., Schapire, R., Experiments with a new boosting algorithm. In Machine Learning: Proceedings of the Thirteenth International Conference, Morgan Kauffman, San Francisco 1996, 148–156. [14] Gopalan, R., Berry, D., Bayesian multiple comparisons using Dirichlet process priors. J. Amer. Statist. Assoc. 93 (1998), 1130–1139. [15] Green, P., Silverman, B., Nonparametric Regression and Generalized Linear Models. Monogr. Statist. Appl. Probab. 58, Chapman and Hall, London, 1994. [16] Hastie, T., Tibshirani, R., Friedman, J., The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer Ser. Statist., Springer-Verlag, New York 2001. [17] Hartigan, J. A., Partition models. Comm. Statist. Theory Methods 19 (1990), 2745–2756. [18] Jerrum, M., Sinclair, A., Vigoda, E., A polynomial-time approximation algorithm for approximating the permanent of a matrix with non-negative entries. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, ACM Press, New York 2001, 712–721. [19] Kingman, J. F. C., Mathematics of Genetic Diversity. CBMS-NSF Regional Conf. Ser. in Appl. Math. 34, SIAM, Philadelphia, PA, 1980. [20] Kingman, J. F. C., Poisson Processes. Oxford Stud. Probab. 3, Clarendon Press, Oxford University Press, Oxford 1993. [21] Macchi, O., The coincidence approach to stochastic point processes. Adv. in Appl. Probab. 7 (1975), 83–122. [22] McCullagh, P., Møller, J., The permanent process. 2005; available via http://www.stat. uchicago.edu/~pmcc/permanent.pdf. [23] Minc, H., Permanents. Encyclopedia Math. Appl. 6, Addison-Wesley, Reading, MA, 1978. [24] Pitman, J., Some developments of the Blackwell-MacQueen urn scheme. In Statistics, Probability and Game Theory: Papers in Honor of David Blackwell (ed. by T. S. Ferguson et al.), IMS Lecture Notes Monogr. Ser. 30, Hayward, CA, 1996, 245–267.

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[25] Pitman, J., Combinatorial Stochastic Processes (Ecole d’Eté de Probabilités de Saint-Flour XXXII, 2002). Lecture Notes in Math. 1875, Springer-Verlag, Berlin 2006. [26] Rubin, D., Which Ifs have causal answers? J. Amer. Statist. Assoc. 81 (1986), 961–962. [27] Ripley, B., Pattern recognition and Neural Networks. Cambridge University Press, Cambridge 1996. [28] Valiant, L. G., The complexity of computing the permanent. Theoret. Comput. Sci. 8 (1979), 189–201. [29] Wahba, G., Spline Models for Observational Data. CBMS-NSF Regional Conf. Ser. in Appl. Math. 59. SIAM, Philadelphia, PA, 1990. Department of Statistics, University of Chicago, 5734 S. University Ave, Chicago, IL 60637, U.S.A. E-mail: [email protected] Department of Statistics, University of Chicago, 5734 S. University Avenue, Eckhart 108, Chicago, IL 60637 U.S.A. E-mail: [email protected]

Random partitions and instanton counting Andrei Okounkov∗

Abstract. We summarize the connection between random partitions and N = 2 supersymmetric gauge theories in 4 dimensions and indicate how this relation extends to higher dimensions. Mathematics Subject Classification (2000). Primary 81T13; Secondary 14J60.

1. Introduction 1.1. Random partitions. A partition of n is a monotone sequence λ = (λ1 ≥ λ2 ≥ · · · ≥ 0) of nonnegative integers with sum n. The number n is denoted |λ| and called the size of λ. A geometric object associated to a partition is its diagram; it contains λ1 squares in the first row, λ2 squares in the second row and so on. An example, flipped and rotated by 135 can be seen in Figure 1. Partitions naturally label many basic objects

Figure 1. The diagram of λ = (10, 8, 7, 4, 4, 3, 2, 2, 1, 1), flipped and rotated by 135 . Bullets indicate the points of S(λ). The profile of λ is plotted in bold.

in mathematics and physics, such as e.g. conjugacy classes and representations of the symmetric group S(n), and very often appear as modest summation ranges or indices. A simple but fruitful change of perspective, which I wish to stress here, is to treat sums over partitions probabilistically, that is, treat them as expectations of some functions of a random partition. ∗ The author thanks Packard Foundation for partial financial support.


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Andrei Okounkov

A survey of the theory of random partitions may be found in [33]. Of the several natural measures on partitions, the Plancherel measure MPlanch (λ) =

(dim λ)2 , n!

|λ| = n,

(1)

stands out as the one with deepest properties and widest applications. Here dim λ is the dimension of the corresponding representation of S(n). This is a probability measure on partitions of n. It can be viewed as a distinguished discretization of the GUE ensemble of the random matrix theory. Namely, a measure on partitions can be made a point process on a lattice by placing particles in positions S(λ) = λi − i + 21 ⊂ Z + 21 . Figure 1 illustrates the geometric meaning of this transformation. An important theme of recent research was to understand how and why for a Plancherel random partition of n → ∞ the particles S(λ) behave like the eigenvalues of a random Hermitian matrix. See [3], [4], [14], [31] and e.g. [15], [32], [33] for a survey. In these notes, we consider a different problem, namely, the behavior of (1) in a periodic potential, that is, additionally weighted by a multiplicative periodic function of the particles’ positions. This leads to new phenomena and new applications. As we will see, the partition function of MPlanch in a periodic potential is closely related to Nekrasov partition function from supersymmetric gauge theory. This relationship will be reviewed in detail in Section 2 and its consequences may be summarized as follows. 1.2. Instanton counting. In 1994, Seiberg and Witten proposed an exact description of the low-energy behavior of certain supersymmetric gauge theories [38], [39]. In spite of the enormous body of research that this insight has generated, only a modest progress was made towards its gauge-theoretic derivation. This changed in 2002, when Nekrasov proposed in [28] a physically meaningful and mathematically rigorous definition of the regularized partition function Z for supersymmetric gauge theories in question. Supersymmetry makes the gauge theory partition function the partition function of a gas of instantons. Nekrasov’s idea was to use equivariant integration with respect to the natural symmetry group in lieu of a long-distance cut-off for the instanton gas. He conjectured that as the regularization parameter ε → 0 ln Z ∼ −

1 F ε2

where the free energy F expressed by the Seiberg–Witten formula in terms of periods of a certain differential dS on a certain algebraic curve C. This conjecture was proven in 2003 by Nekrasov and the author for a list of gauge theories with gauge group U (r), namely, pure gauge theory, theories with matter

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Random partitions and instanton counting

fields in fundamental and adjoint representations of the gauge group, as well as 5dimensional theory compactified on a circle [29]. Simultaneously, independently, and using completely different ideas, the formal power series version of Nekrasov’s conjecture was proven for the pure U (r)-theory by Nakajima and Yoshioka [25]. The methods of [29] were applied to classical gauge groups in [30] and to the 6-dimensional gauge theory compactified on a torus in [11]. Another algebraic approach, which works for pure gauge theory with any gauge group, was developed by Braverman [5] and Braverman and Etingof [6]. In these notes, we outline the results of [29] in the simplest, yet fundamental, case of pure gauge theory. As should be obvious from the title, the main idea is to treat the gauge theory partition function Z as the partition function of an ensemble of random partitions. The ε → 0 limit turns out to be the thermodynamic limit in this ensemble. What emerges in this limit is a nonrandom limit shape, an example of which may be seen in Figure 6. This is a form of the law of large numbers, analogous, for example, to Wigner’s semicircle law for the spectrum of a large random matrix. The limit shape is characterized as the unique minimizer ψ of a certain convex functional S(ψ), leading to F = min S. We solve the variational problem explicitly and the limit shape turns out to be an algebraic curve C is disguise. Namely, the limit shape is essentially the graph of the function

x

dS, x0

where dS is the Seiberg–Witten differential. Thus all ingredients of the answer appear very naturally in the proof. Random matrix theory and philosophy had many successes in mathematics and physics. Here we have an example when random partitions, while structurally resembling random matrices, offer several advantages. First, the transformation into a random partition problem is geometrically natural and exact. Second, the discretization inherent in partitions regularizes several analytic issues. For further examples along these lines the reader may consult [33]. 1.3. Higher dimensions. The translation of the gauge theory problem into a random partition problem is explained in Section 2. In Section 3, we analyze the latter problem, in particular, derive and solve the variational problem for the limit shape. Section 4 summarizes parallel results for 3-dimensional partitions, where similar algebraic properties of limit shapes are now proven in great generality. The surprising fact that free energy F is given in terms of periods of a hidden algebraic curve C is an example of mirror symmetry. A general program of interpreting mirror partners as limit shapes was initiated in [34]. Known results about the limit shapes of periodically weighted 3-dimensional partitions, together with the

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conjectural equality of Gromov–Witten and Donaldson–Thomas theories of projective algebraic 3-folds [22] can be interpreted as a verification of this program for toric Calabi–Yau 3-folds. See [35] for an introduction to these ideas. Note that something completely different is expected to happen in dimensions > 3, where the behavior of both random interfaces and Gromov–Witten invariants changes qualitatively.

2. The gauge theory problem 2.1. Instantons. We begin by recalling some basic facts, see [9] for an excellent mathematical treatment and [8], [10], [44] for a physical one. This will serve as motivation for the introduction of Nekrasov’s partitions function in (8) below. In gauge theories, interactions are transmitted by gauge fields, that is, unitary connections on appropriate vector bundles. In coordinates, these are matrix-valued functions Ai (x) that define covariant derivatives ∇i =

∂ + Ai (x), ∂xi

A∗i = −Ai .

We consider the most basic case of the trivial bundle R4 × Cr over the flat Euclidean space-time R4 , where such coordinate description is global. The natural (Yang–Mills) energy functional for gauge fields is L2 -norm squared F 2 of the curvature F = ∇i , ∇j dxi ∧ dxj . The path integral in quantum gauge theory then takes the form DA exp −β F 2 × · · · ,

(2)

connections/G

where dots stand for terms involving other fields of the theory and G is the group of gauge transformations g : R4 → U (r) acting by ∇ → g ∇ g −1 . In these notes, we will restrict ourselves to pure gauge theory, which is already quite challenging due to the complicated form of the energy. A parallel treatment of certain matter fields can be found in [29]. Our goal is to study (2) as function of the parameter β (and boundary conditions at infinity, see below). A head-on probabilistic approach to this problem would be to make it a theory of many interacting random matrices through a discretization of space-time. This is a fascinating topic about which I have nothing to say. In a different direction, when β 0, the minima of F 2 should dominate the integral.


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In supersymmetric gauge theory, there is a way to make such approximation exact, thereby reducing the path integral to the following finite-dimensional integrals. Local minima of F 2 are classified by a topological invariant c2 ∈ Z, 1 c2 = tr F 2 , 8π 2 R4 called charge, and satisfy a system of first order PDEs F ± F = 0,

(3)

where is the Hodge star operator on 2-forms on R4 . With the plus sign, (3) corresponds to c2 > 0 and is called the anti-self-duality equation. Its solutions are called instantons. Minima with c2 < 0 are obtained by reversing the orientation of R4 . The ASD equations (3) are conformally invariant and can be transported to a punctured 4-sphere S 4 = R4 ∪{∞} via stereographic projection. From the removable singularities theorem of Uhlenbeck it follows that any instanton on R4 extends, after a gauge transformation, to an instanton on S 4 . Thus we can talk about the value of an instanton at infinity. Let G0 be the group of maps g : S 4 → U (r) such that g(∞) = 1. Modulo G0 , instantons on S 4 with c2 = n are parametrized by a smooth manifold M(r, n) of real dimension 4rn. Naively, one would like the contribution from charge n instantons to (2) to be the volume of M(r, n) in a natural symplectic structure. However, M(r, n) is noncompact (and its volume is infinite) for two following reasons. Approximately, an element of M(r, n) can be imagined as a nonlinear superposition of n instantons of charge 1. Some of those may become point-like, i.e. their curvature may concentrate in a δ-function spike, while others may wander off to infinity. A partial compactification of M(r, n), constructed by Uhlenbeck, which replaces point-like instanton by just points of R4 , takes care of the first problem but not the second. Nekrasov’s idea was to use equivariant integration to regularize the instanton contributions. 2.2. Equivariant regularization. The group K = SU(2) × SU(r) acts on M(r, n) by rotations of R4 = C2 and constant gauge transformation, respectively. Our plan is to use this action for regularization. Let us start with the following simplest example: suppose we want to regularize the volume of R2 . A gentle way to do it is to introduce a Gaussian well 1 2 2 e−tπ(x +y ) dx dy = , t ≥ 0 (4) t R2 and thus an effective cut-off at the |t|−1/2 scale. Note that the Hamiltonian flow on R2 generated by H = 21 (x 2 + y 2 ) with respect to the standard symplectic form

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ω = dx ∧ dy is rotation about the origin with angular velocity one. This makes (4) a simplest instance of the Atiyah–Bott–Duistermaat–Heckman equivariant localization formula [2]. We will use localization in the following complex form. Let T = C∗ act on a complex manifold X with isolated fixed points XT . Suppose that the action of U (1) ⊂ T is generated by a Hamiltonian H with respect to a symplectic form ω. Then e−2π tH (x) e ω−2π tH = , (5) det t|Tx X X T x∈X

where t should be viewed as an element of Lie(T ) ∼ = C, an so it acts in the complex tangent space Tx X to a fixed point x ∈ X. While (5) is normally stated for compact manifolds X, example (4) shows that with care it can work for noncompact ones, too. Scaling both ω and H to zero, we get from (5) a formal expression 1 def 1 = , (6) det t|Tx X X T x∈X

which does not depends on the symplectic form and vanishes if X is compact. A theorem of Donaldson identifies instantons with holomorphic bundles on C2 = R4 and thus gives a complex description of M(r, n). Concretely, M(r, n) is the moduli space of rank r holomorphic bundles E → CP2 with given 2nd Chern class c2 (E ) = n and a given trivialization along the line L∞ = CP2 \ C2 at infinity. Note that existence of such trivialization implies that c1 (E ) = 0. A similar but larger moduli space M(r, n) of torsion-free sheaves, see e.g. [12], [24], is a smooth partial compactification of M(r, n). The complexification of K KC = SL(2) × SL(r) acts on M(r, n) by operating on C2 and changing the trivialization at infinity. Equivariant localization with respect to a general t ∈ Lie(K) t = (diag(−iε, iε), diag(ia1 , . . . , iar ))

(7)

combines the two following effects. First, it introduces a spatial cut-off parameter ε as in (4). Second, it introduces dependence on the instanton’s behavior at infinity through the parameters ai . While the first factor in K works to shepherd run-away instantons back to the origin, the second works to break the gauge invariance at infinity. In supersymmetric gauge theories, the parameters ai correspond to the vacuum expectation of the Higgs field and thus are responsible for masses of gauge bosons. In short, they are live physical parameters.

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2.3. Nekrasov partition function. We are now ready to introduce our main object of study, the partition function of the pure (N = 2 supersymmetric) U (r) gauge theory: Z(ε; a1 , . . . , ar ; ) = Zpert 2rn 1, (8) n≥0

M(r,n)

where the integral is defined by (6) applied to (7), = exp(−4π 2 β/r), and Zpert is a certain perturbative factor to be discussed below. The series in (8) is denoted Zinst . Because of factorials in denominators, see (24), Zinst converges whenever we avoid zero denominators, that is, on the complement of ai − aj ≡ 0

mod ε.

(9)

In essence, these factorials are there because the instantons are unordered. Also note that Z is an even function of ε and a symmetric function of the ai ’s. Since by our regularization rule 1 4 1 = 2, vol R = ε R4 we may expect that as ε → 0 ln Z(ε; a; ) ∼ −

1 F (a; ), ε2

where F is the free energy. At first, the poles (9) of Zinst , which are getting denser and denser, may look like a problem. Indeed, poles of multiplicity O(ε−1 ) may affect the free energy, but it is a question of competition with the other terms in Zinst , which the pole-free terms win if |ai − aj | 0. As a result, either by passing to a subsequence of ε, or by restricting summation in Zinst to the relevant pole-free terms, we obtain a limit Finst = − lim ε2 Zinst , which is analytic and monotone far enough from the walls of the Weyl chambers. Recall that Weyl chambers for SU(r) are the r! cones obtained from

ai = 0 C+ = a1 > a2 > · · · > ar , by permuting the coordinates. As |ai − aj | get small, poles do complicate the asymptotics. This is the origin of cuts in the analytic function F (a), ai ∈ C. Nekrasov conjectured in [28] that the free energy F is the Seiberg–Witten prepotential, first obtained in [38], [39] through entirely different considerations. It is defined in terms of a certain family of algebraic curves.

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2.4. Seiberg–Witten geometry. In the affine space of complex polynomials of the form P (z) = zr + O(zr−2 ) consider the open set U of polynomials such that P (z) = ±2r

(10)

has 2r distinct roots. Over U , we have a g-dimensional family of complex algebraic curves C of genus g = r − 1 defined by 1 r = P (z), P ∈ U . (11) w+ w The curve (11) is compactified by adding two points ∂C = {w = 0, ∞}. Let M ⊂ U be the set of P (z) for which all roots of (10) are real. The corresponding curves C are called maximal and play a special role, see e.g. [40]. They arise, for example, as spectral curves of a periodic Toda chain [41]. A maximal curve C has r real ovals, as illustrated in Figure 2. Note that for z ∈ R, w is either real or lies on the unit circle |w| = 1. 2

1

–3

–2

–1

0

1

2

3

z –1

–2

Figure 2. w (bold) and w for w + 1/w = z3 − 3.5 z and z ∈ R.

The intervals P −1 ([−2r , 2r ]) ⊂ R on which |w| = 1 are called bands. The intervals between the bands are called gaps. The smaller (in absolute value) root w of the equation (11) can be unambiguously defined for z ∈ C \ {bands}. On the corresponding sheet of the Riemann surface of w, we define cycles αi ∈ H1 (C − ∂C),

βi ∈ H1 (C, ∂C),

i = 1, . . . , r

(12)

as illustrated in Figure 3, where dotted line means that βi continues on the other sheet. Note that αi ∩ βj = δij and that αi = −αi ,

βi = βi ,

(13)

where bar stands for complex conjugation. The ovals in Figure 2 represent the cycles αi and βi − βi+1 .

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β1

β2

β3

β1 − β2

α1

α2

α3

Figure 3. Cycles βi go from w = ∞ to w = 0. Bold segments indicate bands.

The Seiberg–Witten differential dS =

1 dw r z =± 1 + O z−2 dz 2πi w 2π i

is holomorphic except for a second order pole (without residue) at ∂C. Its derivatives with respect to P ∈ U are, therefore, holomorphic differentials on C. In fact, this gives TP U ∼ = holomorphic diff. on C. Nondegeneracy of periods implies the functions def ai = dS, ai = 0,

(14)

αi

which are real on M by (13), are local coordinates on U , as are ∨ def ∨ ai − ai+1 = 2πi dS, ai∨ = 0. βi −βi+1

(15)

Further, there exists a function F (a; ), which is real and convex on M, such that ∂ ∂ ∨ F = − ai∨ − ai+1 . (16) − ∂ai ∂ai+1 Indeed, the Hessian of F equals (−2πi) times the period matrix of C, hence symmetric (and positive definite on M). The function F is called the Seiberg–Witten prepotential. Note that F is multivalued on U and, in fact, its monodromy played a key role in the argument of Seiberg and Witten. By contrast, M is simply-connected, indeed a ∨ : M → C+ is a diffeomorphism, see e.g. [18] for a more general result. Note that the periods (15) are the areas enclosed by the images of real ovals of C under (z, w) → (z, ln |w|). A similar geometric interpretation of the ai ’s will be given in (40) below. In particular, the range A = a(M)

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of the coordinates (14) is a proper subset of C− = −C+ . At infinity of M, we have ai ∼ {roots of P },

a1 a2 · · · ar .

2.5. Main result. We have now defined all necessary ingredients to confirm Nekrasov’s conjecture in the following strong form: Theorem 1 ([29]). For a ∈ A, − lim ε2 ln Z(ε; a; ) = F (a; ), ε→0

(17)

where F is the Seiberg–Witten prepotential (16). At the boundary of A, free energy has a singularity of the form 2 F = − ai∨ − aj∨ ln ai∨ − aj∨ + · · · where dots denote analytic terms. This singularity is one of the main physical features of the Seiberg–Witten theory. In broad strokes, the logic of the proof was explained in the Introduction. We now proceed with the details.

3. The random partition problem 3.1. Fixed points contributions. A rank 1 torsion-free sheaf on C2 is a fancy name to call an ideal I of C[x, y]. Any partition λ defines one by Iλ = (x λ1 , x λ2 y, x λ3 y 2 , . . . ) ⊂ C[x, y]. It is easy to see that all torus-fixed points of M(r, n) have the form F=

r

Iλ(k) ,

λ(k) = n,

(18)

k=1

where λ(k) is an r-tuple of partitions. Our goal now is to compute the character of the torus action in the tangent space to the fixed point (18) and thus the contribution of F to the sum in (6). By construction of M(r, n), its tangent space at F equals Ext1P2 (F, F(−L∞ )). From the vanishing of the other Ext-groups we conclude tr et |Ext1

P2

(F,F(−L∞ ))

= XO ⊕r (t) − X F (t),

where X F (t) is the character X F (t) = tr et |χC2 (F,F)

(19)

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of the infinite-dimensional virtual representation χC2 (F, F) = Ext0C2 (F, F) − Ext1C2 (F, F) + Ext2C2 (F, F). Any graded free resolution of F gives X F (t) = |G F (t)|2 ,

t ∈ Lie(K),

where G F (t) is, up to a factor, the character of F itself Gλ(1) ,...,λ(r) (t) = (e−iε/2 − eiε/2 ) tr et |F

=

r

e

iak

k=1

∞ j =1

(k) exp iε λj − j + 21 .

(20)

It is also a natural generating function of the r-tuple λ(k) . Note that the weight of any F is real and positive, being a product of purely imaginary numbers in conjugate pairs. 3.2. Perturbative factor. In the spirit of the original uncompactified gauge theory problem on R4 , we would like to drop the first term in (19) and declare its contribution canceled by Zpert . In view of (20), this requires a regularization of the following product ∞ r i(ak − ak + ε(j − j )). Zpert “=” k,k =1 j,j =1

A natural regularization is provided by Barnes’ double -function (21), see e.g. [37]. For c1 , c2 ∈ R and w 0, define ∞ 1 dt s e−wt ζ2 (s; w | c1 , c2 ) = t . (s) 0 t (1 − e−ci t ) This has a meromorphic continuation in s with poles at s = 1, 2. Define 2 (w | c1 , c2 ) = exp

d ζ (s; w | c1 , c2 )s=0 . ds

(21)

Through the difference equation w 2 (w) 2 (w + c1 + c2 ) = 2 (w + c1 ) 2 (w + c2 ) it extends to a meromorphic function of w. We define i(ak − ak ) iε −iε −1 , Zpert = 2 . k,k

(22)

(23)

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where 2 is analytically continued to imaginary arguments using w2

2 (Mw | Mc, −Mc ) = M 2c2

1 − 12

2 (w | c, −c ),

M∈ / (−∞, 0].

The scaling by is introduced in (23) to make (8) homogeneous of degree 0 in a, ε, and . Note also 2 (0| 1, −1) = e−ζ (−1) . Our renormalization rule (23) fits nicely with the following transformation of the partition function Z. 3.3. Dual partition function. For r = 1, the weight of Iλ in (8) equals n 1 2 2n −1 MPlanch (λ), det t T M(1,n) = Iλ n! ε2

(24)

where MPlanch is the Plancherel measure (1) and the prefactor is the Poisson weight with parameter 2 /ε2 . For r > 1, we will transform Z into the partition function (29) of the Plancherel measure in a periodic potential with period r. Let a function ξ : Z + 21 → R be periodic with period r and mean 0. The energy (λ) of the configuration S(λ) in the potential ξ is defined by Abel’s rule def ξ(x) = lim ξ(x) ezx . (λ) = z→+0

x∈S(λ)

x∈S(λ)

Grouping the points of S(λ) modulo r uniquely determines an r-tuple of partitions λ(k) , known as r-quotients of λ, and shifts sk ∈ Q such that r S(λ) = r S λ(k) + sk

(25)

k=1

and rs ≡ ρ

mod r Zr0 ,

ρ=

r−1 2

, . . . , 1−r , 2

where Zr0 denotes vectors with zero sum. It follows from (25) that Gλ (ε/r) = Gλ(1) ,...,λ(r) (ε; εs).

Letting ε → 2πik, k = 1, . . . , r − 1, in (26) gives (λ) = (s, ξ ) = si ξi , ξi = ξ 21 − i , while the ε → 0 limit in (26) yields (k) sk2 1 − r2 λ + |λ| = r + . 2 24

(26)

(27)

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Using these formulas and the difference equation (22), we compute (ξ, a) def exp Z(rε; a; ) Z ∨ (ε; ξ1 , . . . , ξr ; ) = rε2 r

(28)

a∈ε(ρ+r Z0 )

=e

ζ (−1)+ πi 24

1 2|λ|− 12 dim λ 2 (λ) exp . ε |λ|! ε

(29)

λ

We call (28) the dual partition function. By (29), it equals the partition function of a periodically weighted Plancherel measure on partitions. While it will play no role in what follows, it may be mentioned here that Z ∨ is a very interesting object to study not asymptotically but exactly. For example, Toda equation for ln Z ∨ may be found in Section 5 of [29]. 3.4. Dual free energy. Define the dual free energy by F ∨ (ξ ; ) = − lim ε2 ln Z ∨ . ε→0

(30)

Since (28) is a Riemann sum for Laplace transform, we may expect that F ∨ (ξ ; ) = minr

a∈R0

1 1 F (a; ) − (ξ, a) 2 r r

(31)

that is, up to normalization, F ∨ is the Legendre transform of F . This is because the asymptotics of Laplace transform is determined by one point – the maximum. Our plan is apply to same logic to the infinite-dimensional sum (29), namely, to show that its ε → 0 asymptotics is determined by a single term, the limit shape. The law of large numbers, a basic principle of probability, implies that on a large scale most random system are deterministic: solids have definite shape, fluids obey the laws of hydrodynamics, etc. Only magnification reveals the full randomness of nature. In the case at hand, the weight of a partition λ in (29), normalized by the whole sum, defines a probability measure on the set of partitions. This measure depends on a parameter ε and as ε → 0 it clearly favors partitions of larger and larger size. In fact, the expected size of λ grows as ε −2 . We thus expect the diagram of λ, scaled by ε in both directions, to satisfy a law of large numbers, namely, to have a nonrandom limit shape. By definition, this limit shape will dominate the leading ε → 0 asymptotics of Z ∨ . In absence of the periodic potential , such analysis is a classical result of Logan–Shepp and Vershik–Kerov [21], [42], [43]. Note that the maximum in (31) is over all of a, including the problematic region where |ai − aj | get small. However, this region does not contribute to F ∨ as the convexity of free energy is lost there. We will see this reflected in the following properties of F ∨ : it is strictly concave, analytic in the interior of the Weyl chambers, and singular along the chambers’ walls.

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3.5. Variational problem for the limit shape. The profile of a partition λ is, by definition, the piecewise linear function plotted in bold in Figure 1. Let ψλ be the profile of λ scaled by ε in both directions. The map λ → ψλ embeds partitions into the convex set of functions ψ on R with Lipschitz constant 1 and |ψ| = |ψ(x) − |x|| dx < ∞. The Lipschitz condition implies 1/2

ψ1 − ψ2 C ≤ ψ1 − ψ2 L1 ,

ψ1 , ψ2 ∈ ,

and so is complete and separable in the L1 -metric. Some function of a partition have a natural continuous extension to , for example 1 1 ix Gλ (ε) = iε/2 e (ψλ (x) − |x|) dx , 1− e − e−iε/2 2 while others, specifically the ones appearing in (29), do not. An adequate language for dealing with this is the following. Let f (λ) ≥ 0 be a function on partitions depending on the parameter ε. We say that it satisfies a large deviation principle with action (rate) functional Sf (ψ) if for any set A ⊂ f (λ) ⊂ inf Sf , inf Sf ⊂ R ∪ {+∞}, (32) − lim ε 2 ln ψλ ∈A

A

A

where lim denotes all limit points, A and A stand for the interior and closure of A, respectively. For the Plancherel weight (24), Logan–Shepp and Vershik–Kerov proved a large deviation principle with action 1 |x − y| Spl (ψ) = (1 + ψ (x))(1 − ψ (y)) ln dx dy. (33) 2 x 0, | z| < 1 } as in Figure 5. The slits in go along z = −1 + 2i/r,

i = 1, . . . , r − 1,

and their lengths are, essentially, the critical values of the polynomial P (z). The bands and gaps are preimages of the horizontal and vertical segments of ∂, respectively.

Figure 5. Conformal map defined by a maximal curve.

We claim that

ψ = R ,

(37)

where the polynomial P (z) is determined by the relation (39) below. The equations (35) are verified for (37) as follows. Since (z) = O(z−1 ), z → ∞, we have the Hilbert transform relation 1 P.V. ∗ R = π R . x Integrating it once and using (36) to fix the integration constant, we get (L ∗ ) = π. Therefore, the function L ∗ is constant on the bands and strictly increasing on the gaps, hence (37) satisfies (35) with ξi+1 − ξi = π (x) dx (38) ith gap

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Integrating (38) by parts and using definitions from Section 2.4 gives ξ =−

a∨ , r

(39)

thus every limit shape ψ comes from a maximal curve. For example, the limit shape corresponding to the curve from Figure 2 is plotted in Figure 6. Note also that for C ∈ M, we have r (40) ai = (Ii−1 − Ii ) , 2 where Ii is the intercept of the ith facet of the limit shape. In particular, A ⊂ C− .

2

1

0 –2

–1

0

1

2

Figure 6. Limit shape corresponding to the curve from Figure 2. Thin segments are facets.

For given ξ ∈ C− , consider the distribution of the r-quotients λ(i) of the partition λ, as defined in Section 3.3. For the shifts sk in (25) we have using (27) εs → −

∂F ∨ , ∂ξ

ε → 0,

in probability. Observe that ∂ ∂ ∨ F (ξ ) = S (ψ ) ∂ξ ∂ξ ∂ since the other term, containing ∂ξ ψ , vanishes by the definition of a maximum. Definitions and integration by parts yield

∂ ∂ − − ∂ξi ∂ξi+1

F

∨

=

ai − ai+1 . r

By (26), this means that the resulting sum over the r-quotients λ(i) is the original partition function Z with parameters a ∈ A. This concludes the proof.

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4. The next dimension 4.1. Stepped surfaces An obvious 3-dimensional generalization of a partition, also known as a plane partition can be seen on the right. More generally, we consider stepped surfaces, that is, continuous surfaces glued out of sides of a unit cube, spanning a given polygonal contour in R3 , and projecting 1-to-1 in the (1, 1, 1) direction, see Figure 7. Note that stepped surfaces minimize the surface area for given boundary conditions, hence can be viewed as zero temperature limit of the interface in the 3D Ising model. The most natural measure on stepped surfaces is the uniform one with given boundary conditions, possibly conditioned on the volume enclosed. It induces Plancherellike measures on 2-dimensional slices. Stepped surfaces are in a natural bijection with fully packed dimers on the hexagonal lattice and Kasteleyn theory of planar dimers [16] forms the basis of most subsequent developments. The following law of large numbers for stepped surfaces was proven in [7]. Let Cn be a sequence of boundary contours such that each Cn can be spanned by at least one stepped surface. Suppose that n−1 Cn converge to a given curve C ⊂ R3 . Then, scaled by n−1 , uniform measures on stepped surfaces spanning Cn converge to the δ-measure on a single Lipschitz surface spanning C – the limit shape. This limit shape formation is clearly visible in Figure 7.

Figure 7. A limit shape simulation. The frozen boundary is the inscribed cardioid.

The limit shape is the unique minimizer of the following functional. Let the surface be parameterized by x3 = h(x3 − x1 , x3 − x2 ), where h is a Lipschitz function

705


with gradient in the triangle with vertices (0, 0), (0, 1), (1, 0). Let be the planar region enclosed by the projection of C in the (1, 1, 1) direction. We will use (x, y) = (x3 − x1 , x2 − x1 ) as coordinates on . The limit shape is the unique minimizer of Sstep (h) = σstep (∇h) dx dy, (41)

where, in the language of [20], the surface tension σstep is the Legendre dual of the Ronkin function of the straight line z + w = 1.

(42)

We recall that for a plane curve P (z, w) = 0, its Ronkin function [23] is defined by dz dw 1 . (43) R(x, y) = x log |P (z, w)| 2 |z|=e (2πi) z w y |w|=e

The gradient ∇R always takes values in the Newton polygon (P ) of the polynomial P , so (P ) is naturally the domain of the Legendre transform R ∨ . For the straight line as above, the Newton polygon is evidently the triangle . The surface tension σstep is singular and not strictly convex at the boundary of , which leads to formation of facets and edges in the limit shape (which can be clearly seen in Figure 7). This models facet formation in natural interfaces, e.g. crystalline surfaces, and is the most interesting aspect of the model. Note that facets are completely ordered (or frozen). The boundary between the ordered and disordered (or liquid) regions is known as the frozen boundary. The following transformation of the Euler-Lagrange equation for (41) found in [19] greatly facilitates the study of the facet formation. Namely, in the liquid region we have 1 ∇h = (arg w, − arg z), (44) π where the functions z and w solve the differential equation wy zx + =c z w

(45)

and the algebraic equation (42). Here c is the Lagrange multiplier for the volume constraint h = const, the unconstrained case is c = 0. At the boundary of the liquid region, z and w become real and the ∇h starts to point in one of the coordinate directions. The first-order quasilinear equation (45) is, essentially, the complex Burgers equation zx = zzy and, in particular, it can be solved by complex characteristics as follows. There exists an analytic function Q(z, w) such that Q(e−cx z, e−cy w) = 0.

(46)

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In other words, z(x, y) can be found by solving (42) and (46). In spirit, this is very close to Weierstraß parametrization of minimal surfaces in terms of analytic data. Frozen boundary can only develop if Q is real, in which case the roots (z, w) and (¯z, w) ¯ of (46) coincide at the frozen boundary. At a smooth point of the frozen boundary, the multiplicity of this root will be exactly two, hence ∇h has a square-root singularity there. As a result, the limit shape has an x 3/2 singularity at the generic point of the frozen boundary, thus recovering the well-known Pokrovsky–Talapov law [36] in this situation. At special points of the frozen boundary, triple solutions of (46) occur, leading to a cusp singularity. One such point can be seen in Figure 7. Remarkably, for a dense set of boundary condition the function Q is, in fact, a polynomial. Consequently, the frozen boundary takes the form R(ecx , ecy ) = 0, where R is the polynomial defining the planar dual of the curve Q = 0. This allows to use powerful tools of algebraic geometry to study the singularities of the solutions, see [19]. The precise result proven there is Theorem 2 ([19]). Suppose the boundary contour C is a connected polygon with 3k sides in coordinate directions (cyclically repeated) which can be spanned by a Lipschitz function with gradient in . Then Q = 0 is an algebraic curve of degree k and genus zero. For example, for the boundary contour in Figure 7 we have k = 3 (one of the boundary edges there has zero length) and hence R is the dual of a degree 3 genus 0 curve – a cardioid. The procedure of determining Q from the boundary conditions is effective and can be turned into a practical numeric homotopy procedure, see [19]. Higher genus frozen boundaries occur for multiply-connected domains, in fact, the genus of Q equals the genus of the liquid region. Of course, for a probabilist, the law of large numbers is only the beginning and the questions about CLT corrections to the limit shape and local statistics of the surface in various regions of the limit shape follow immediately. Conjecturally, the limit shape controls the answers to all these questions. For example, the function e−cx z defines a complex structure on the liquid region and, conjecturally, the Gaussian correction to the limit shape is given by the massless free field in the corresponding conformal structure. In the absence of frozen boundaries and without the volume constraint, this is proven in [17]. See e.g. [15], [17], [20], [32] for an introduction to the local statistics questions. 4.2. Periodic weights. Having discussed periodically weighted Plancherel measure and a 3-dimensional analog of the Plancherel measure, we now turn to periodically weighted stepped surfaces. This is very natural if stepped surfaces are interpreted as crystalline interfaces. Periodic weights are introduced as follows: we weight each square by a periodic function of x3 − x1 and x2 − x1 (with some integer period M). The role previously played by the straight line (42) is now played by a certain higher degree curve P (z, w) = 0, the spectral curve of the corresponding periodic


707

Kasteleyn operator. In particular, the surface tension σstep is now replaced by the Legendre dual of the Ronkin function of P , see [20]. We have deg P = M and the coefficients of P depend polynomially on the weights. The main result of [20], known as maximality, says that for real and positive weights the curve P is always a real algebraic curve of a very special kind, namely, a Harnack curve, see [23]. Conversely, as shown in [18], all Harnack curves arise in this way. Harnack curves are, in some sense, the best possible real curves; their many remarkable properties are discussed in [23]. One of several equivalent definitions of a Harnack curve is that the map (z, w) → (log |z|, log |w|)

(47)

from P (z, w) = 0 to R2 is 1-to-1 on the real locus of P and 2-to-1 over the rest. The image of P = 0 under (47) is known as the amoeba of P . Note from (43) that the gradient ∇R of the Ronkin function of P is nonconstant precisely for (x, y) in the amoeba of P . In other words, the Ronkin function has a facet (that is, a linear piece) over every component of the amoeba complement. The 2-to-1 property implies that the number of compact facets of Ronkin function equals the (geometric) genus of the curve P . Each of these facets translates into the singularity of the surface tension and, hence, into facets with the same slope in limit shapes. By Wulff’s theorem, the Ronkin function itself is a minimizer, corresponding to its own (“crystal corner”) boundary conditions. An example of the Ronkin function of a genus 1 Harnack curve can be seen in Figure 8.

Figure 8. The (curved part of the) Ronkin function of a genus 1 curve. Its projection to the plane is the amoeba.

Maximality implies persistence of facets, namely, for fixed period M, there will be M−1 M−1 compact facets of the Ronkin function and corresponding singularities 2 2

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of the surface tension, except on a codimension 2 subvariety of the space of weights. It also implies e.g. the following universality of height fluctuations in the liquid region Var(h(a) − h(b)) ∼

1 ln a − b, π

a − b → ∞.

Remarkably, formulas (44), (45), and (46) need no modifications for periodic weights. Replacing (42) by P (z, w) = 0 is the only change required, see [19]. From our experience with periodically weighted Plancherel measure, it is natural to expect that, for some special boundary conditions, the partition function of periodically weighted stepped surfaces will encode valuable physical information. A natural choice of “special boundary conditions” are the those of a crystal corner, when we require the surface to be asymptotic to given planes at infinity, as in Figure 8. For convergence of the partition function, one introduces a fugacity factor q vol , where the missing volume is measured with respect to the “full corner”. I hope that further study will reveal many special properties of such crystal corner partition functions. Their extremely degenerate limits have been identified with allgenera, all-degree generating functions for Donaldson–Thomas invariants of toric Calabi–Yau threefolds. Namely, as the periodic weights become extreme, all limit shapes, and the Ronkin function in particular, degenerate to piecewise linear functions. This is known as the tropical limit. The only remaining features of limit shapes are the edges and the triple points, where 2 and 3 facets meet, respectively. In this tropical limit, the partition function becomes the partition function of ordinary, unweighted, 3D partitions located at triple points. These 3D partitions may have infinite legs along the edges, as in Figure 9 and through these legs they interact with their neighbors. This description precisely matches the localization formula for Donaldson–Thomas invariants of the toric threefold whose toric polyhedron is given by the piecewise linear limit shape, see [22].

Figure 9. Two 3D partitions connected at an angle through an infinite leg.

Donaldson–Thomas theory of any 3-fold has been conjectured to be equivalent, in a nontrivial way, to the Gromov–Witten theory of the same 3-fold in [22]. For the toric Calabi–Yau 3-folds, this specializes to the earlier topological vertex conjecture of [1]. It is impossible to adequately review this subject here, see [35] for an introduction.


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This is also related to the supersymmetric gauge theories considered in Section 2, or rather their 5-dimensional generalizations, via a procedure called geometric engineering of gauge theories. See for example [13] and references therein. I find such close and unexpected interaction between rather basic statistical models and instantons in supersymmetric gauge and string theories very exciting and promising. The field is still full of wide open questions and, in my opinion, it is also full of new phenomena waiting to be discovered.

References [1] Aganagic, M., Klemm, A., Marino, M., Vafa, C., The Topological Vertex. Comm. Math. Phys. 254 (2005), 425–478. [2] Atyiah, M., Bott, R., The moment map and equivariant cohomology. Topology 23 (1) (1984), 1–28. [3] Baik, J., Deift, P., Johansson, K., On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999), 1119–1178. [4] Borodin, A., Okounkov, A., Olshanski, G., Asymptotics of the Plancherel measures for symmetric groups. J. Amer. Math. Soc. 13 (3) (2000), 481–515. [5] Braverman, A., Instanton counting via affine Lie algebras. I. Equivariant J -functions of (affine) flag manifolds and Whittaker vectors. In Algebraic structures and moduli spaces, CRM Proc. Lecture Notes 38, Amer. Math. Soc., Providence, RI, 2004, 113–132. [6] Braverman, A., Etingof, P., Instanton counting via affine Lie algebras II: from Whittaker vectors to the Seiberg-Witten prepotential. math.AG/0409441. [7] Cohn, H., Kenyon, R., Propp, J., A variational principle for domino tilings. J. Amer. Math. Soc. 14 (2) (2001), 297–346. [8] D’Hoker, E., Phong, D., Lectures on supersymmetric Yang-Mills theory and integrable systems. In Theoretical physics at the end of the twentieth century, CRM Ser. Math. Phys., Springer-Verlag, New York 2002, 1–125. [9] Donaldson, S, Kronheimer, P., The geometry of four-manifolds. Oxford Math. Monogr., The Clarendon Press, New York 1990. [10] Dorey, N., Hollowood, T., Khoze, V., Mattis, M., The calculus of many instantons. Phys. Rep. 371 (4–5) (2002), 231–459. [11] Hollowood, T., Iqbal, A., Vafa, C., Matrix models, geometric engineering, and elliptic genera. hep-th/0310272. [12] Huybrechts, D., Lehn, M., The geometry of moduli spaces of sheaves. Aspects Math. E31, Vieweg, Braunschweig 1997. [13] Iqbal,A., Kashani-Poor, A.-K., The vertex on a strip. hep-th/0410174. [14] Johansson, K., Discrete orthogonal polynomial ensembles and the Plancherel measure. Ann. of Math. 153 (1) (2001), 259–296. [15] Johansson, K., Random matrices and determinantal processes. math-ph/0510038. [16] Kasteleyn, P., Graph theory and crystal physics. In Graph Theory and Theoretical Physics, Academic Press, London 1967, 43–110.

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[17] Kenyon, R., Height fluctuations in honeycomb dimers. math-ph/0405052. [18] Kenyon, R., Okounkov, A., Planar dimers and Harnack curves. math.AG/0311062. [19] Kenyon, R., Okounkov, A., Limit shapes and complex Burgers equation. math-ph/0507007. [20] Kenyon, R., Okounkov, A., Sheffield, S., Dimers and amoebae. math-ph/0311005. [21] Logan, B., Shepp, L., A variational problem for random Young tableaux. Adv. Math. 26 (1977), 206–222. [22] Maulik, D., Nekrasov, N., Okounkov, A., Pandharipande, R., Gromov-Witten theory and Donaldson-Thomas theory, I. & II. math.AG/0312059, math.AG/0406092. [23] Mikhalkin, G., Amoebas of algebraic varieties and tropical geometry. In Different faces of geometry, Int. Math. Ser. (N. Y.), Kluwer/Plenum, New York 2004, 257–300. [24] Nakajima, H., Lectures on Hilbert schemes of points on surfaces. Univ. Lecture Ser. 18, Amer. Math. Soc., Providence, RI, 1999. [25] Nakajima, H., Yoshioka, K., Instanton counting on blowup. I. 4-dimensional pure gauge theory. Invent. Math. 162 (2005), 313–355. [26] Nakajima, H., Yoshioka, K., Lectures on instanton counting. Algebraic structures and moduli spaces, CRM Proc. Lecture Notes, 38, Amer. Math. Soc., Providence, RI, 2004, 31–101. [27] Nakajima, H., Yoshioka, K., Instanton counting on blowup. II. K-theoretic partition function. math.AG/0505553. [28] Nekrasov, N., Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7 (5) (2003), 831–864. [29] Nekrasov, N., Okounkov, A., Seiberg-Witten Theory and Random Partitions. In The Unity of Mathematics (ed. by P. Etingof, V. Retakh, I. M. Singer), Progr. Math. 244, Birkhäuser, Boston, MA, 2006, 525–596. [30] Nekrasov, N., Shadchin, S., ABCD of instantons. Comm. Math. Phys. 252 (2004), 359–391. [31] Okounkov, A., Random matrices and random permutations. Internat. Math. Res. Notices 2000 (20) (2000), 1043–1095. [32] Okounkov, A., Symmetric function and random partitions. In Symmetric functions 2001: surveys of developments and perspectives (ed. by S. Fomin), Kluwer Acad. Publ., Dordrecht 2002, 223–252. [33] Okounkov, A., The uses of random partitions. math-ph/0309015. [34] Okounkov, A., Reshetikhin, N., Vafa, C., Quantum Calabi-Yau and Classical Crystals. In The Unity of Mathematics (ed. by P. Etingof, V. Retakh, I. M. Singer), Progr. Math. 244, Birkhäuser, Boston, MA, 2006, 597–618. [35] Okounkov, A., Random surfaces enumerating algebraic curves. In Proceedings of Fourth European Congress of Mathematics, EMS, Zürich 2005, 751–768. [36] Pokrovsky, V., Talapov, A., Theory of two-dimensional incommensurate crystals. Soviet Phys. JETP 78 (1) (1980), 269–295. [37] Ruijsenaars, S., On Barnes’ multiple zeta and gamma functions. Adv. Math. 156 (1) (2000), 107–132. [38] Seiberg, N., Witten, E., Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nuclear Phys. B 426 (1994), 19–52; Erratum ibid. 430 (1994), 485–486.


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[39] Seiberg, N., Witten, E., Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD. Nuclear Phys. B 431 (1994), 484–550. [40] Sodin, M., Yuditskii, P., Functions that deviate least from zero on closed subsets of the real axis. St. Petersburg Math. J. 4 (2) (1993), 201–249. [41] Toda, M., Theory of nonlinear lattices. Springer Ser. Solid-State Sci. 20, Springer-Verlag, Berlin 1981. [42] Vershik, A., Kerov, S., Asymptotics of the Plancherel measure of the symmetric group and the limit form of Young tableaux. Soviet Math. Dokl. 18 (1977), 527–531. [43] Vershik, A., Kerov, S., Asymptotics of maximal and typical dimensions of irreducible representations of a symmetric group. Funct. Anal. Appl. 19 (1) (1985), 21–31 . [44] Witten, E., Dynamics of quantum field theory. In Quantum fields and strings: a course for mathematicians (ed. by P. Deligne, P. Etingof, D. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D. Morrison and E. Witten), Vol. 2, Amer. Math. Soc., Providence, RI, IAS, Princeton, NJ, 1999, 1119–1424. Department of Mathematics, Princeton University, Fine Hall, Washington Road, Princeton, New Jersey, 08544, U.S.A. E-mail: [email protected]

Estimation in inverse problems and second-generation wavelets Dominique Picard and Gérard Kerkyacharian

Abstract. We consider the problem of recovering a function f when we receive a blurred (by a linear operator) and noisy version: Yε = Kf + εW˙ . We will have as guides 2 famous examples of such inverse problems: the deconvolution and the Wicksell problem. The direct problem (K is the identity) isolates the denoising operation. It cannot be solved unless accepting to estimate a smoothed version of f : for instance, if f has an expansion on a basis, this smoothing might correspond to stopping the expansion at some stage m. Then a crucial problem lies in finding an equilibrium for m, considering the fact that for m large, the difference between f and its smoothed version is small, whereas the random effect introduces an error which is increasing with m. In the true inverse problem, in addition to denoising, we have to ‘inverse the operator’ K, an operation which not only creates the usual difficulties, but also introduces the necessity to control the additional instability due to the inversion of the random noise. Our purpose here is to emphasize the fact that in such a problem there generally exists a basis which is fully adapted to the problem, where for instance the inversion remains very stable: this is the singular value decomposition basis. On the other hand, the SVD basis might be difficult to determine and to numerically manipulate. It also might not be appropriate for the accurate description of the solution with a small number of parameters. Moreover, in many practical situations the signal provides inhomogeneous regularity, and its local features are especially interesting to recover. In such cases, other bases (in particular, localised bases such as wavelet bases) may be much more appropriate to give a good representation of the object at hand. Our approach here will be to produce estimation procedures keeping the advantages of a localisation properly without loosing the stability and computability of SVD decompositions. We will especially consider two cases. In the first one (which is the case of the deconvolution example) we show that a fairly simple algorithm (WAVE-VD), using an appropriate thresholding technique performed on a standard wavelet system, enables us to estimate the object with rates which are almost optimal up to logarithmic factors for any Lp loss function and on the whole range of Besov spaces. In the second case (which is the case of the Wicksell example where the SVD basis lies in the range of Jacobi polynomials) we prove that a similar algorithm (NEED-VD) can be performed provided one replaces the standard wavelet system by a second generation wavelet-type basis: the needlets. We use here the construction (essentially following the work of Petrushev and co-authors) of a localised frame linked with a prescribed basis (here Jacobi polynomials) using a Littlewood–Paley decomposition combined with a cubature formula. Section 5 describes the direct case (K = I ). It has its own interest and will act as a guide for understanding the ‘true’ inverse models for a reader who is not familiar with nonparametric statistical estimation. It can be read first. Section 1 introduces the general inverse problem and describes the examples of deconvolution and Wicksell’s problem. A review of standard methods is given with a special focus on SVD methods. Section 2 describes the WAVE-VD procedure. Section 3 and 4 give a description of the needlets constructions and the performances of the NEED-VD procedure. Mathematics Subject Classification (2000). 62G07, 62G20, 62C20. Keywords. Nonparametric estimation, denoising, inverse models, thresholding, Meyer wavelet, singular value decomposition, Littlewood–Paley decomposition. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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1. Inverse models Let H and K be two Hilbert spaces. K is a linear operator: f ∈ H → Kf ∈ K. The standard linear ill-posed inverse problem consists of recovering a good approximation fε of f , solution of g = Kf, (1) when only a perturbation gε of g is observed. In this paper we will consider the case where this perturbation is an additive stochastic white noise. Namely, we observe Yε defined by the following equation: ˙ H, K, Yε = Kf + ε W,

(2)

where ε is the amplitude of the noise. It is supposed to be a small parameter which will tend to 0. Our error will be measured in terms of this small parameter. ˙ g)K , ξ(h) := (W, ˙ h)K W˙ is a K-white noise: i.e. for any g, h in K, ξ(g) := (W, form a random gaussian vector, centered, with marginal variance g2K , h2K , and covariance (g, h)K (with the obvious extension when one considers k functions instead of 2). Equation (2) means that for any g in K, we observe Yε (g) := (Yε , g)K = (Kf, g)K + εξ(g) where ξ(g) ∼ N(0, g2 ), and Yε (g), Yε (h) are independent random variables for orthogonal functions g and h. The case where K is the identity is called the ‘direct model’ and is summarized as a memento in Section 5. The reader who is unfamiliar with nonparametric statistical estimation is invited to consult this section, which will act as a guide for understanding the more general inverse models. In particular it is recalled therein that the model (2) is in fact an approximation of models appearing in real practical situations, for instance the case where (2) is replaced by a discretisation. 1.1. Two examples: the problem of deconvolution and Wicksell’s problem 1.1.1. Deconvolution. The following problem is probably one of the most famous among inverse problems in signal processing. In the deconvolution problem we consider the following operator. In this case let H = K be the set of square integrable periodic functions with the standard L2 ([0, 1]) norm and consider 1 γ (u − t)f (t) dt ∈ H, (3) f ∈ H → Kf = 0

where γ is a known function of H, which is generally assumed to be a regular function (often in the sense that its Fourier coefficients γˆk behave like k −ν ). A very common 1 example is also the box-car function: γ (t) = 2a I{[−a, a]}(k). The following figures show first four original signals to recover, which are wellknown test-signals of the statistical literature. They provide typical features which are difficult to restore: bumps, blocks and Doppler effects. The second and third series of

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Estimation in inverse problems and second-generation wavelets

pictures show their deformation after blurring (i.e. convolution with a regular function) and addition of a noise. These figures show how the convolution regularizes the signal, making it very difficult to recover, especially the high frequency features. A statistical investigation of these signals can be found in [22]. A variant of this problem consists in observing Y1 , . . . , Yn , n independent and identically distributed random variables where each Yi may be written as Yi = Xi +Ui , where Xi and Ui again are independent, the distribution of Ui is known and of density γ and we want to recover the common density of the Xi ’s. The direct problem is the case where Ui = 0, for all i, and is corresponding to a standard density estimation problem (see Section 5.1) . Hence the variables Ui are acting as perturbations of the Xi ’s, whose density is to be recovered. (a) Lidar

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1.1.2. Wicksell’s problem. Another typical example is the following classical Wicksell problem [42]. Suppose a population of spheres is embedded in a medium. The spheres have radii that may be assumed to be drawn independently from a density f . A random plane slice is taken through the medium and those spheres that are intersected by the plane furnish circles the radii of which are the points of observation Y1 , . . . , Yn . The unfolding problem is then to infer the density of the sphere radii from the observed circle radii. This unfolding problem also arises in medicine, where

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the spheres might be tumors in an animal’s liver [36], as well as in numerous other contexts (biological, engineering,…), see for instance [9]. Following [42] and [23], Wicksell’s problem corresponds to the following operator: H = L2 ([0, 1], dμ) dμ(x) = (4x)−1 dx, K = L2 ([0, 1], dλ) dλ(x) = 4π −1 (1 − y 2 )1/2 dy 1 π 2 −1/2 (x 2 − y 2 )−1/2 f (x)dμ. Kf (y) = y(1 − y ) 4 y

Notice, however, that in this presentation, again in order to avoid additional technicalities, we handle this problem in the white noise framework, which is simpler than the original problem expressed above in density terms. 1.2. Singular value decomposition and projection methods. Let us begin with a quick description of well-known methods in inverse problems with random noise. Under the assumption that K is compact, there exist 2 orthonormal bases (SVD bases) (ek ) of H and (gk ) of K, respectively, and a sequence (bk ), tending to 0 when k goes to infinity, such that Kek = bk gk ,

K ∗ gk = bk ek

if K ∗ is the adjoint operator. For the sake of simplicity we suppose in the sequel that K and K ∗ are into. Otherwise we have to take care of the kernels of these operators. The bases (ek ) and (gk ) are called singular value bases, whereas the bk ’s are simply called singular values. Deconvolution. In this standard case simple calculations prove that the SVD bases (ek ) and (gk ) both coincide with the Fourier basis. The singular values are corresponding to the Fourier coefficients of the function γ : bk = γˆk .

(4)

Wicksell. In this case, following [23], we have the following SVD: ek (x) = 4(k + 1)1/2 x 2 Pk0,1 (2x 2 − 1), gk (y) = U2k+1 (y). Pk0,1 is the Jacobi polynomial of type (0, 1) with degree k, and Uk is the second type Chebyshev polynomial with degree k. The singular values are bk =

π (1 + k)−1/2 . 16

(5)


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1.2.1. SVD method. The singular value decomposition (SVD) of K, bk f, ek gk , Kf = k

gives rise to approximations of the type fε =

N

bk−1 yε , gk ek ,

k=0

where N = N(ε) has to be chosen properly. This SVD method is very attractive theoretically and can be shown to be asymptotically optimal in many situations (see Mathé and Pereverzev [31], Cavalier and Tsybakov [6], Mair and Ruymgaart [29]). It also has the big advantage of performing a quick and stable inversion of the operator. However, it suffers from different types of limitations. The SVD bases might be difficult to determine as well as to numerically manipulate. Secondly, while these bases are fully adapted to describe the operator K, they might not be appropriate for the accurate description of the solution with a small number of parameters. Also in many practical situations the signal provides inhomogeneous regularity, and its local features are especially interesting to recover. In such cases other bases (in particular localised bases such as wavelet bases) may be much more appropriate to give a good representation of the object at hand. 1.2.2. Projection methods. Projection methods which are defined as solutions of (1) restricted to finite dimensional subspaces HN and KN (of dimension N) also give rise to attractive approximations of f , by properly choosing the subspaces and the tuning parameter N (Dicken and Maass [10], Mathé and Pereverzev [31] together with their non linear counterparts Cavalier and Tsybakov [6], Cavalier et al. [7], Tsybakov [41], Goldenshluger and Pereverzev [19], Efromovich and Koltchinskii [16]). In the case where H = K and K is a self-adjoint operator, the system is particularly simple to solve since the restricted operator KN is symmetric positive definite. This is the socalled Galerkin method. Obviously, restricting to finite subspaces has similar effects and can also be seen as a Tychonov regularisation, i.e. minimizing the least square functional penalised by a regularisation term. The advantage of the Galerkin method is to allow the choice of the basis. However the Galerkin method suffers from the drawback of being unstable in many cases. Comparing the SVD and Galerkin methods exactly states one main difficulty of the problem. The possible antagonism between the SVD basis where the inversion of the system is easy, and a ‘localised’ basis where the signal is sparsely represented, will be the issue we are trying to address here. 1.3. Cut-off, linear methods, thresholding. The reader may profitably look at Subsections 5.3 and 5.4, where the linear methods and thresholding techniques are presented in detail in the direct case.

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SVD as well as Galerkin methods are very sensitive with respect to the choice of the tuning parameter N(ε). This problem can be solved theoretically. However the solution heavily depends on prior assumptions of regularity on the solution, which have to be known in advance. In the last ten years, many nonlinear methods have been developed especially in the direct case with the objective of automatically adapting to the unknown smoothness and local singular behavior of the solution. In the direct case, one of the most attractive methods is probably wavelet thresholding, since it allies numerical simplicity to asymptotic optimality on a large variety of functional classes such as Besov or Sobolev classes. To adapt this approach in inverse problems, Donoho [11] introduced a wavelet-like decomposition, specifically adapted to the operator K (wavelet–vaguelette-decomposition) and provided a thresholding algorithm on this decomposition. In Abramovitch and Silverman [1], this method was compared with the similar vaguelette– wavelet-decomposition. Other wavelet approaches, might be mentioned such as Antoniadis and Bigot [2], Antoniadis et al. [3] and, especially for the deconvolution problem, Penski and Vidakovic [37], Fan and Koo [17], Kalifa and Mallat [24], Neelamani et al. [34]. Later, Cohen et al. [8] introduced an algorithm combining a Galerkin inversion with a thresholding algorithm. The approach developed in the sequel is greatly influenced by these previous works. The accent we put here is on constructing (when necessary) new generation wavelet-type bases well adapted to the operator K, instead of sticking to the standard wavelet bases and reducing the range of potential operators covered by the method.

2. Wave-VD-type estimation We explain here the basic idea of the method, which is very simple. Let us expand f using a well-suited basis (‘the wavelet-type’ basis’, to be defined later): f = (f, ψλ )H ψλ . Using Parseval’s identity we have βλ = (f, ψλ )H = fi ψλi for fi = (f, ei )H and i ψλ = (ψλ , ei )H . Let us put Yi = (Yε , gi )K . We then have Yi = (Kf, gi )K + εξi = (f, K ∗ gi )K + εξi = fj ej , K ∗ gi + εξi = bi fi + εξi , j

H

where the ξi ’s are forming a sequence of independent centered gaussian variables with variance 1. Furthermore, Yi βˆλ = ψi bi λ i

719


is such that E(βˆλ ) = βλ (i.e. its average value is βλ ). It is a plausible estimate of βλ . Let us now put ourselves in a multiresolution setting, taking λ = (j, k) for j ≥ 0, k belonging to a set χj , and consider fˆ =

J

t (βˆj k )ψj k ,

j =−1 k∈χj

where t is a thresholding operator. (The reader who is unfamiliar with thresholding techniques is referred to Section 5.4.) (6) t (βˆj k ) = βˆj k I {|βˆj k | ≥ κtε σj }, tε = ε log 1/ε, where I {A} denotes the indicator function of the set A∗ . Here κ is a tuning parameter of the method which will be properly chosen later. A main difference here with the direct case is the fact that the thresholding is depending on the resolution level through the constant σj which also will be stated more precisely later. Our main discussion will concern the choice of the basis (ψj k ). In particular, we shall see that coherence properties with the SVD basis are of special interest. We will particularly focus on two situations (corresponding to the two examples discussed in the introduction). In the first type of cases, the operator has as SVD bases the Fourier basis. In this case, this ‘coherence’ is easily obtained with ‘standard’ wavelets (still, not any kind of standard wavelet as will be seen). However, more difficult problems (and typically Wicksell’s problem) require, when we need to mix these coherence conditions with the desired property of localisation of the basis, the construction of new objects: second generation-type wavelets. 2.1. WAVE-VD in a wavelet scenario. In this section we take {ψj k , j ≥ −1, k ∈ χj } to be a standard wavelet basis. More precisely, we suppose as usual that ψ−1 stands for the scaling function and, for any j ≥ −1, χj is a set of order 2j contained in N. Moreover, we assume that the following properties are true. There exist constants cp , Cp , dp such that p

p

cp 2j ( 2 −1) ≤ ψj k p ≤ Cp 2j ( 2 −1) , p p p u ψ |uk |ψj k p for any sequence uk . k j k ≤ Dp k∈χj

p

p

(7) (8)

k∈χj

It is well known (see for instance Meyer [32]) that wavelet bases provide characterisations of smoothness spaces such as Hölder spaces Lip(s), Sobolev spaces Wps s for a range of indices s depending on the wavelet ψ. as well as Besov spaces Bpq s For the scale of Besov spaces which includes as particular cases Lip(s) = B∞∞ (if s s s ∈ N) and Wp = Bpp (if p = 2), the characterisation has the following form: j [s+ 1 − 1 ]

2 p β s ∼ 2 If f = βj k ψj k , then f Bpq (9) j · lp j ≥−1 l . j ≥−1 k∈Z

q

720


As in Section 5, we consider the loss of a decision fˆ if the truth is f as the Lp norm fˆ − f p , and its associated risk Efˆ − f p . p

Here E denotes the expectation with respect to the random part of the observation yε . The following theorem is going to evaluate this risk, when the strategy is the one introduced in the previous section, and when the true function belongs to a Besov ball s (M) ⇐⇒ f s ≤ M). One nice property of this estimation procedure (f ∈ Bπ,r Bpq is that it does not need the a priori knowledge of this regularity to get a good rate of convergence. If (ek ) is the SVD basis introduced in Section 1.2, bk are the singular values and ψjik = ei , ψj k , we consider the estimator fˆ defined in the beginning of Section 2. Theorem 2.1. Assume that 1 < p < ∞, 2ν + 1 > 0 and σj2 :=

2 ψjik i

bi

≤ C22j ν for all j ≥ 0.

(10)

−2

s (M) with π ≥ 1, s ≥ 1/π, r ≥ 1 Put κ 2 ≥ 16p, 2J = [tε ] 2ν+1 . If f belongs to Bπ,r

p (with the restriction r ≤ π if s = (2ν + 1) 2π − 21 ), then we have p Efˆ − f p ≤ C log(1/ε)p−1 [ε 2 log(1/ε)]αp ,

(11)

with s 1 + 2(ν + s) s − 1/π + 1/p α= 1 + 2(ν + s − 1/π)

α=

if s ≥ (2ν + 1) if

1 π

p 2π

−

≤ s < (2ν + 1)

1 2

p 2π

, −

1 2

.

Remarks. 1. Condition (10) is essential here. As will be shown later, this condition is linking the wavelet system with the singular value decomposition of the kernel K. If we set ourselves in the deconvolution case, the SVD basis is the Fourier basis in such a way that ψjik is simply the Fourier coefficient of ψj k . If we choose as wavelet basis the periodized Meyer wavelet basis (see Meyer [32] and Mallat [30]), conditions (7) and (8) are satisfied. In addition, as the Meyer wavelet has the remarkable property of being compactly supported in the Fourier domain, simple calculations prove that, for any j ≥ 0, k, the number of i’s such that ψjik = 0 is finite and equal to 2j . Then if we assume to be in the so-called ‘regular’ case (bk ∼ k −ν , for all k), it is easy to establish that (10) is true. This condition is also true for more general cases in the deconvolution setting such as the box-car deconvolution, see [22], [27]. 2. These results are minimax (see [43]) up to logarithmic factors. This means that if we consider the best estimator in its worst performance over a given Besov


721

ball, this estimator attains a rate of convergence which is the one given in (11) up to logarithmic factors. 3. If we compare these results with the rates of convergence obtained in the direct model (see Subsections 5.3 and 5.4), we see that the difference (up to logarithmic terms) essentially lies in the parameter ν which acts as a reducing factor of the rate of convergence. This parameter quantifies the extra difficulty offered by the inverse problem. It is often called coefficient of illposedness. If we recall that in the deconvolution case, the coefficients bk are the Fourier coefficients of the function γ , the illposedness coefficient then clearly appears to be closely related to the regularity of the blurring function. This result has been proved in the deconvolution case in [22]. The proof of the theorem is given in Appendix I. 2.2. WAVE-VD in Jacobi scenario: NEED-VD. We have seen that the results given above are true under the condition (10) on the wavelet basis. Let us first appreciate how the condition (10) links the ‘wavelet-type’ basis to the SVD basis (ek ). To see this let us put ourselves in the regular case: bi ∼ i −ν . (By this we mean more precisely that there exist two positive constants c and c such that c i −ν ≤ bi ≤ ci −ν .) If (10) is true, we have

i 2 ψj k 2j ν C2 ≥ . bi m+1 m m 2 ≤i −1/2, β > −1/2.

For the sake of simplicity, let us suppose α ≥ β. (Otherwise we can exchange the parameters.) Let Pk be the normalized Jacobi polynomial for this weight. We suppose that these polynomials appear as SVD basis of the operator K, as it is the case for the Wicksell problem with β = 0, α = 1, bk ∼ k −1/2 . 4.1. Needlets and condition (10). Let us define the ‘needlets’ as constructed above: j −1 ψj,ηk (x) = a(l/2 ˆ )Pl (x)Pl (ηk ) bj,ηk . (17) l

The following proposition asserts that such a construction always implies the condition (10) in the regular case. Proposition 4.1. Assume that ψj,ηk is a frame. If bi ∼ i −ν then σj2

:=

ψjik 2 i

bi

≤ C22j ν .

727


Proof. Suppose the family ψj,ηk is a frame (not necessarily tight). As the elements of a frame are bounded and the set {i, ψjik = 0} is included in the set {C1 2j , . . . , C2 2j }, we have

ψjik 2 ≤ C2j ν ψj,ηk 2 ≤ C 2j ν . 2 bi i

4.2. Convergence results in the Jacobi case. The following theorem is the analogous of Theorem 2.1 in this case. As can be seen, the results there are at the same time more difficult to obtain (the following theorem does not cover the same range as the previous one) and richer since they furnish new rates of convergence. Theorem 4.2. Suppose that we are in the Jacobi case as stated above (α ≥ β > − 21 ). We put tε = ε log 1/ε, 2

−

2J = tε 1+2ν ,

choose κ ≥ 16p 1 + α2 − α+1 p + , and suppose that we are in the regular case, i.e. 1 ν>− . 2 is such that

bi ∼ i −ν , Then, if f =

it follows that

j

k βj,ηk ψj,ηk

p 1/p

|βj,ηk |p ψj,ηk p

≤ ρj 2−j s ,

(ρj ) ∈ lr ,

μp p Efˆ − f p ≤ C[log(1/ε)]p−1 ε log(1/ε)

with 1. if p < 2 +

1 α+1/2 ,

then μ=

2. if p > 2 +

1 α+1/2 ,

s s+ν+

1 2

;

then μ=

s s+ν+α+1−

2(1+α) p

.

This theorem is proved in Kerkyacharian et al. [26]. Simulation results on these methods are given there, showing that their performances are far above the usual SVD methods in several cases. It is interesting to notice that the rates of convergence which are obtained here agree with the minimax rates evaluated in Johnstone and Silverman 1 [23] where the case p = 2 is considered. But the second case (p > 2 + α+1/2 ) shows a rate of convergence which is new in the literature. In [26], where the whole range of Besov bodies is considered, more atypical rates are given.

728


5. Direct models (K = I ): a memento 5.1. The density model. The most famous nonparametric model consists in observing n i.i.d. random variables having a common density f on the interval [0, 1], and in trying to give an estimation of f . A standard route to perform this estimation consists in expanding the density f in an orthonormal basis {ek , k ∈ N} of a Hilbert space H – assuming implicitly that f belongs to H: f = θl el . l∈N

If H happens to be the space L2 = {g : [0, 1] → R, g22 := observe that 1

θl =

1 0

g 2 < ∞}, we

el (x)f (x)dx = Eel (Xi ).

0

Replacing the expectation by the empirical one leads to a standard estimate for θl : θˆl =

1 el (Xi ). n n

i=1

At this step, the simplest choice of estimate for f is obviously: fˆm =

m

θˆl el .

(18)

i=1

5.2. From the density to the white noise model. Before analysing the properties of the estimator defined above, let us observe that the previous approach (representing f by its coefficients {θk , k ≥ 0}, leads to summarize the information in the following sequence model: {θˆk , k ≥ 0}. (19) We can write θˆk =: θk + uk , with 1 [ek (Xi ) − θk ], n n

uk =

i=1

The central limit theorem is a relatively convincing argument that the model (19) may be approximated by the following one: ηk (20) θˆk = θk + √ , k ≥ 0 , n where the ηk ’s are forming a sequence of i.i.d. gaussian, centered variables with fixed variance σ 2 , say. Such an approximation requires more delicate calculations than these quick arguments and is rigourously proved in Nussbaum [35], see also Brown

729


and Low [5]. This model is the sequence space model associated to the following global observation, the so-called white noise model (with ε = n−1/2 ): dYt = f (t)dt + εdWt , t ∈ [0, 1], where for any ϕ ∈ L2 ([0, 1], dt), [0,1] ϕ(t)dYt = [0,1] f (t)ϕ(t)dt +ε [0,1] ϕ(t)dWt is observable. (20) formally consists in considering all the observables obtained for ϕ = ek for all k in N. Among nonparametric situations, the white noise model considered above is one of the simplest, at least technically. Mostly for this reason, this model has been given a central place in statistics, particularly by the Russian school, following Ibraguimov and Has’minskii (see for instance their book [20]). However it arises as an appropriate large sample limit to more general nonparametric models, such as regression with random design, or non independent spectrum estimation, diffusion models – see for instance [21], [4],. . . . 5.3. The linear estimation: how to choose the tuning parameter m? In (18), the choice of m is crucial. To better understand the situation let us have a look at the risk of the strategy fˆm . If we consider that, when deciding fˆm when f is the truth, we have a loss of order fˆm − f 22 , then our risk will be the following mathematical expectation: Efˆm − f 22 .

Of course this way of measuring our risk is arguable since there is no particular reason for the L2 norm to reflect well the features we want to recover in the signal. For instance, an L∞ -norm could be preferred because it is easier to visualize. In general, several Lp norms are considered (as it is the case in Sections 2.1 and 4.2). Here we restrict to the L2 case for sake of simplicity. To avoid technical difficulties, we set ourselves in the case of a white noise model, considering that we observe the sequence defined in (20). Hence, E(θˆl − θl )2 =

1 n

1

el (x)2 dx =

0

1 := ε2 . n

We are now able to obtain Efˆm − f 22 =

l≤m

(θˆl − θl )2 +

l>m

θl2 ≤ mε 2 +

θl2 .

l>m

Now assume that f belongs to the following specified compact set of l2 : θl2 ≤ Mk −2s for all k ∈ N∗ , l>k

(21)

730


for some s > 0 which is here an index of regularity directly connected to the size of the compact set in l2 containing the function f . Then we obtain Efˆm − f 22 ≤ mε 2 + Mm−2s .

We observe that the RHS is the sum of two factors: one (called the stochastic term) is increasing in m and reflects the fact that because of the noise, the more coefficients we have to estimate, the larger the global error will be. The second one (called the bias term or approximation term) does not depend on the noise and is decreasing in m. −2 The RHS is optimised by choosing m = m∗ (s) =: c(s, M)ε 1+2s . Then −4s

Efˆm∗ (s) − f 2 ≤ c (s, M)ε 1+2s .

Let us observe that the more f is supposed to be regular (in the sense the larger s is), the less coefficients we need to estimate: a very irregular function (s close to 0) requires almost as much as ε−2 = n coefficients, which corresponds to estimate as many coefficients as the number of available observations – in the density model for instance. The rate obtained in (5.3) can be proved to be optimal in the following sense (minimax): if we consider the best estimator in its worst performance over the class of functions verifying (21), this estimator attains a rate of convergence which is (up to a constant) the one given in (5.3). See Tsybakov [40] for a detailed review of the minimax point of view. 5.4. The thresholding estimation. Let us now suppose that the constant s, which plays an essential role in the construction of the previous estimator is not known. This is realistic, since it is extremely rare to know in advance that the function we are seeking has a specified regularity. Also, the previous approach takes very seriously into account the order in which the basis is taken. Let us now present a very elegant way of addressing at the same time both of these issues. The thresholding techniques which have been known for long by engineers in electronic and telecommunications, was introduced in statistics in Donoho and Johnstone [14] and later in a series of papers on wavelet thresholding [12], [13]. It allies numerical simplicity to asymptotic optimality. It starts from a different kind of observation. Let us introduce the following estimate: B f = θˆk I{|θˆk | ≥ κtε }ek . (22) k=0

Here the point of view is the following. We choose B very large (i.e. almost corresponding to s = 0): B = ε −2 log 1/ε. But instead of keeping all the coefficients θk such that k is between 0 and B, we decide to kill those which are not above the threshold tε . The intuitive justification of

731


this choice is as follows. Assuming that f has some kind of regularity condition like (21) (unknown, but real...), essentially means that the coefficients θk of f are of small magnitude except perhaps a small number of them. Obviously, in the reconstruction of f , only the large coefficients will be significant. tε is chosen in such a way that the noise θˆk − θk due to the randomness of the observation might be neglected: tε = ε[log 1/ε]−1/2 . Now let us assume another type of condition on f – easily interpreted by the fact that f is sparsely represented in the basis (ek ) – namely: there exists a positive constant 0 < q < 2 such that sup λq #{k, |θk | ≥ λ} ≤ M λ>0

Ef− f 22 =

for all k ∈ N∗ ,

(θˆl I{|θˆl | ≥ κtε } − θl )2 +

l≤B

≤

(23)

θl2

l>B

(θˆl − θl ) I{|θl | ≥ κtε /2} + 2

l

+

θl2 I{|θl | ≤ 2κtε }

l

[(θˆl − θl )2 + θl2 ]I{|θˆl − θl | ≥ κtε /2} +

l≤B

θl2 .

l>B

Now, using the probabilistic bounds P(|θˆl − θl | ≥ λ) ≤ 2 exp −

E(θˆl − θl )2 = ε 2 ,

λ2 2ε2

for all λ > 0,

and the fact that condition (23) implies θl2 I{|θl | ≤ 2κtε } ≤ Ctε2−q , l

we get Ef− f 22 ≤ Mε 2 tε−q + C tε2−q + ε κ

2 /8

B+

θl2 .

l>B

It remains now to choose κ 2 ≥ 32 in order to get Ef− f 22 ≤ C tε2−q +

θl2 ,

l>B

and if we assume in addition to (23) that 2−q θl2 ≤ Mk − 2 l>k

for all k ∈ N∗ ,

(24)

732


then we get Ef− f 22 ≤ C"tε2−q

Note that the interesting point in this construction is that the regularity conditions imposed on the function f are not known by the statistician, since they do not enter into the construction of the procedure. This property is called adaptation. 2 . It is not difficult Now, to compare with the previous section, let us take q = 1+2s to prove that as soon as f verifies (21), it automatically verifies (23) and (24). Hence fand fˆm∗(s) have the same rate of convergence up to a logarithmic term. If we neglect this logarithmic loss, we substantially gain here the fact that we need not know the apriori regularity conditions on the aim function. It can also be proved that in fact conditions (23) and (24) are defining a set which is substantially larger than the set defined by condition (21): for instance its entropy is strictly larger (see [28]).

6. Appendix: Proof of Theorem 2.1 In this proof, C will denote an absolute constant which may change from one line to the other. We can always suppose p ≥ π. Indeed, if π ≥ p it is very simple to see j [s+ 21 − p1 ]

s (M) is included into B s (M): as 2 that Bπ,r p,r (since χj is of cardinality 2j ). First we have the following decomposition:

1

1

βj · lp ≤ 2j [s+ 2 − π ] βj · lπ

J p p p Efˆ − f p ≤ 2p−1 E (t (βˆj k ) − βj k )ψj k + βj k ψj k p

j =−1 k∈χj

j >J k∈χj

p

=: I + II. s (M), using standard embedding The term II is easy to analyse: since f belongs to Bπ,r results (which in this case simply follows from direct comparisons between lq norms) s−( π1 − p1 )+

we have that f also belong to Bp,r

(M ), for some constant M . Hence

−J [s−( 1 − 1 ) ] βj k ψj k ≤ C2 ¯ π p + . j >J k∈χj

p

s−( 1 − 1 )+

π p is always larger that α, which is not Then we only need to verify that 1+2ν difficult. Bounding the term I is more involved. Using the triangular inequality together

733


with Hölder’s inequality and property (8) for the second line, we get I ≤ 2p−1 J p−1

J

p E (t (βˆj k ) − βj k )ψj k

j =−1

≤2

p−1 p−1

J

Dp

p

k∈χj

J

p E|t (βˆj k ) − βj k |p ψj k p .

j =−1 k∈χj

Now, we separate four cases: J

p E|t (βˆj k ) − βj k |p ψj k p

j =−1 k∈χj

=

J

p E|t (βˆj k ) − βj k |p ψj k p I {|βˆj k | ≥ κtε σj } + I {|βˆj k | < κtε σj }

j =−1 k∈χj

≤

J p E|βˆj k − βj k |p ψj k p I {|βˆj k | ≥ κtε σj } j =−1 k∈χj

κ κ I |βj k | ≥ tε σj + I |βj k | < tε σj 2 2 p p ˆ + |βj k | ψj k p I {|βj k | ≤ κtε σj } I {|βj k | ≥ 2κtε σj } + I {|βj k | < 2κtε σj }

≤: Bb + Bs + Sb + Ss.

ψjik Yi −bi fi i ψ = ε i i ξi bi is a centered gaussian random jk bi i ψi ψ 2 variable with variance ε2 i bjik . Also recall that we set σj2 =: i [ bjik ]2 ≤ C22j ν and denote by sq the qth absolute moment of the gaussian distribution when centered Notice that βˆj k − βj k =

and with variance 1. Then, using standard properties of the gaussian distribution, for any q ≥ 1 we have q E|βˆj k − βj k |q ≤ sq σj ε q ,

P{|βˆj k − βj k | ≥

κ 2 tε σj } ≤ 2εκ /8 . 2

Hence Bb ≤

J

p

p

sp σj εp ψj k p I {|βj k | ≥

j =−1 k∈χj

Ss ≤

J j =−1 k∈χj

p

κ tε σj }, 2

|βj k |p ψj k p I {|βj k | < 2κtε σj }

734


and 1/2 κ [E|βˆj k − βj k |2p ]1/2 P |βˆj k − βj k | ≥ tε σj 2 j =−1 k∈χj κ p ψj k p I |βj k | < tε σj 2 J 1/2 p κ 2 p s2p σj εp 21/2 εκ /16 ψj k p I |βj k | < tε σj ≤ 2

Bs ≤

J

j =−1 k∈χj

≤C

J

1

2jp(ν+ 2 ) εp εκ

2 /16

≤ Cεκ

2 /16

.

j =−1

Now, if we remark that the βj k ’s are necessarily all bounded by some constant (des (M), and using (7), pending on M) since f belongs to Bπ,r

Sb ≤

J

|βj k |p ψj k p P{|βˆj k − βj k | ≥ 2κtε σj }I {|βj k | ≥ 2κtε σj } p

j =−1 k∈χj

≤

J

p

|βj k |p ψj k p 2εκ

2 /8

I {|βj k | ≥ 2κtε σj }

j =−1 k∈χj

≤C

J

p

2j 2 ε κ

2 /8

≤ Cε

p κ2 8 − 2(2ν+1)

.

j =−1

It is easy to check that in all cases, if κ 2 ≥ 16p the terms Bs and Sb are smaller than the rates given in the theorem. Using (7) and condition (10), for any z ≥ 0 we have

Bb ≤ Cε p

J

p

2j (νp+ 2 −1)

j =−1

≤ Cεp

J

k∈χj p

2j (νp+ 2 −1)

j =−1

≤ Ctε p−z

κ I |βj k | ≥ tε σj 2

J j =−1

|βj k |z [tε σj ]−z

k∈χj p

2j [ν(p−z)+ 2 −1]

k∈χj

|βj k |z .

735


Also, for any p ≥ z ≥ 0, Ss ≤ C

J

p

2j ( 2 −1)

j =−1

p−z

|βj k |z σj

[tε ]p−z

k∈χj

≤ C[tε ]p−z

J

p

2j (ν(p−z)+ 2 −1)

j =−1

|βj k |z .

k∈χj

So in both cases we have the same bound to investigate. We will write this bound in the following form (forgetting the constant): I + II = tε

p−z1

j0 j =−1

+ tε p−z2

p

2j [ν(p−z1 )+ 2 −1]

|βj k |z1

k∈χj

J

p

2j [ν(p−z2 )+ 2 −1]

j =j0 +1

|βj k |z2 .

k∈χj

on The constants zi and j0 will be chosen depending

the cases.

1 p Let us first consider the case where s ≥ ν + 2 π − 1 . Put q=

p(2ν + 1) 2(s + ν) + 1

and observe that, on the considered domain, q ≤ π and p > q. In the sequel it will be used that we automatically have s = ν + 21 pq − 1 . Taking z2 = π we get II ≤ tε

p−π

J

p

2j [ν(p−π )+ 2 −1]

j =j0 +1

Now, as p 1 − +ν 2q π and

|βj k |π .

k∈χj

p 1 1 −1 =s+ − q 2 π 1

1

|βj k |π = 2−j (s+ 2 − π ) τj

k∈χj s (M) and (6)), we can with (τj )j ∈ lr (this is a consequence of the fact that f ∈ Bπ,r write jp(1− π )(ν+ 1 ) q 2 τπ II ≤ tε p−π 2 j j =j0 +1

≤ Ctε

π 1 p−π j0 p(1− q )(ν+ 2 )

2

.

736


The last inequality is true for any r ≥ 1 if π > q and for r ≤ π if π = q. Notice that π = q is equivalent to s = (ν + 21 )( πp − 1). Now if we choose j0 such that 2

j0 pq (ν+ 21 )

∼ tε −1 we get the bound tε p−q

which exactly gives the rate asserted in the theorem for this case. As for the first part of the sum (before j0 ), we have, taking now z1 = q , with 1 1 1 1 π q q π q ≤ π, so that 2j k∈χj |βj k | ≤ 2j k∈χj |βj k | , and using again (6), I ≤ tε

j0

p− q

p

q )+ 2 −1] 2j [ν(p−

−1

q |βj k |

k∈χj

j0 q q q q )+ p2 − π ] ≤ tε p− 2j [ν(p− |βj k |π ] π −1

≤ tε

p− q

j0

2

k∈χj q q j [(ν+ 21 )p(1− q )] τ j

−1 q ≤ Ctε p− 2

q j0 [(ν+ 21 )p(1− q )]

≤ Ctε p−q . The last two lines are valid if q is chosen strictly smaller than q (this is possible since π ≥ q).

Let us now consider the case where s < ν + 21 pq − 1 , and choose p q=

2 s + ν − π1 + 1 s−1/π+1/p )(1+2ν)

> in such a way that we easily verify that p−q = 2 1+2(ν+s−1/π) , q−π = (p−π 1 2 s+ν− π +1 have s + 21 − π1 =

1 π.

Furthermore we also 0, because s is supposed to be larger than p

p 1 − + ν − 1 . 2q q q s (M), Hence taking z1 = π and using again the fact that f belongs to Bπ,r I ≤ tε p−π

j0

p

2j [ν(p−π)+ 2 −1]

−1

≤ tε p−π

j0

2

k∈χj

j [(ν+ 21 − p1 ) pq (q−π )] π τj

−1

≤ Ctε

1 1 p p−π j0 [(ν+ 2 − p ) q (q−π )]

2

.

|βj k |π

737


This is true since ν +

−

now take 2

∼

1 2 j0 pq (ν+ 21 − p1 )

1 p is also strictly positive tε −1 we get the bound

because of our constraints. If we

tε p−q which is the rate stated in the theorem for this case. Again, for II, we have, taking now z2 = q > q (> π ), J q q q )+ p2 −1] 2j [ν(p− |βj k | II ≤ tε p− j =j0 +1

q ≤ Ctε p−

k∈χj

2

j [(ν+ 21 − p1 ) pq (q− q )]

q

zjπ

j =j0 +1 q ≤ Ctε p− 2

j0 [(ν+ 21 − p1 ) pq (q− q )]

≤ Ctε p−q .

References [1] Abramovich, F., and Silverman, B. W., Wavelet decomposition approaches to statistical inverse problems. Biometrika 85 (1) (1998), 115–129 . [2] Antoniadis, A., and Bigot, J., Poisson inverse models. Preprint, Grenoble 2004. [3] Antoniadis, A., Fan,J., and Gijbels, I., A wavelet method for unfolding sphere size distributions. Canad. J. Statist. 29 (2001), 251–268. [4] Brown, Lawrence D., Cai, T. Tony, Low, Mark G., and Zhang, Cun-Hui Asymptotic equivalence theory for nonparametric regression with random design. Ann. Statist. 30 (3) (2002), 688–707. [5] Brown, Lawrence D., and Low, Mark G., Asymptotic equivalence of nonparametric regression and white noise. Ann. Statist. 24 (6) (1996), 2384–2398. [6] Cavalier, Laurent, and Tsybakov, Alexandre, Sharp adaptation for inverse problems with random noise. Probab. Theory Related Fields 123 (3) (2002), 323–354. [7] Cavalier, L., Golubev, G. K., Picard, D., and Tsybakov., A. B., Oracle inequalities for inverse problems. Ann. Statist. 30 (3) (2002), 843–874. [8] Cohen, Albert, Hoffmann, Marc, and Reiß, Markus, Adaptive wavelet Galerkin methods for linear inverse problems. SIAM J. Numer. Anal. 42 (4) (2004), 1479–1501 (electronic). [9] Cruz-Orive, L. M., Distribution-free estimation of sphere size distributions from slabs showing overprojections and truncations, with a review of previous methods. J. Microscopy 131 (1983), 265–290. [10] Dicken, V., and Maass, P., Wavelet-Galerkin methods for ill-posed problems. J. Inverse Ill-Posed Probl. 4 (3) (1996), 203–221. [11] Donoho, David L., Nonlinear solution of linear inverse problems by wavelet-vaguelette decomposition. Appl. Comput. Harmon. Anal. 2 (2) (1995), 101–126.

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[12] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Picard, D., Wavelet shrinkage: Asymptopia? J. Royal Statist. Soc. Ser. B 57 (1995), 301–369. [13] Donoho, D. L., Johnstone, I. M., Kerkyacharian, G., and Picard, D., Density estimation by wavelet thresholding. Ann. Statist. 24 (1996), 508–539. [14] Donoho, D. L., Johnstone, I. M., Minimax risk over p -balls for q -error. Probab. Theory Related Fields 99 (1994), 277–303. [15] Duffin, R. J., and Schaeffer, A. C., A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72 (1952), 341–366. [16] Efromovich, Sam, and Koltchinskii, Vladimir, On inverse problems with unknown operators. IEEE Trans. Inform. Theory 47 (7) (2001), 2876–2894. [17] Fan, J., and Koo, J. K., Wavelet deconvolution. IEEE Trans. Inform. Theory 48 (3) (2002), 734–747. [18] Frazier, M., Jawerth, B., and Weiss, G., Littlewood-Paley theory and the study of function spaces. CBMS Reg. Conf. Ser. Math. 79, Amer. Math. Soc., Providence, RI, 1991. [19] Goldenshluger, Alexander, and Pereverzev, Sergei V., On adaptive inverse estimation of linear functionals in Hilbert scales. Bernoulli 9 (5) (2003), 783–807. [20] Ibragimov, I. A., and Hasminskii, R. Z., Statistical estimation. Appl. Math. 16, SpringerVerlag, New York 1981. [21] Jähnisch, Michael, and Nussbaum, Michael, Asymptotic equivalence for a model of independent non identically distributed observations. Statist. Decisions 21 (3) (2003), 197–218. [22] Johnstone, Iain M., Kerkyacharian, Gérard, Picard, Dominique, and Raimondo, Marc, Wavelet deconvolution in a periodic setting. J. R. Stat. Soc. Ser. B Stat. Methodol. 66 (3) (2004), 547–573. [23] Johnstone, Iain M., and Silverman, Bernard W., Discretization effects in statistical inverse problems. J. Complexity 7 (1) (1991), 1–34. [24] Kalifa, Jérôme, and Mallat, Stéphane, Thresholding estimators for linear inverse problems and deconvolutions. Ann. Statist. 31 (1) (2003), 58–109. [25] Kerkyacharian, G., Petrushev, P., Picard, D., and Xu, Y., Localized polynomials and frames induced by laguerre functions. Preprint, 2005. [26] Kerkyacharian, G., Picard, D., Petrushev, P., and Willer, T., Needvd: second generation wavelets for estimation in inverse problems. Preprint, LPMA 2006. [27] Kerkyacharian, G., Picard, D., and Raimondo, M., Adaptive boxcar deconvolution on full lebesgue measure sets. Preprint, LPMA 2005. [28] Kerkyacharian, G., and Picard, D., Thresholding algorithms and well-concentrated bases. Test 9 (2) (2000). [29] Mair, Bernard A., and Ruymgaart, Frits H., Statistical inverse estimation in Hilbert scales. SIAM J. Appl. Math. 56 (5) (1996), 1424–1444. [30] Mallat, Stéphane, A wavelet tour of signal processing. Academic Press Inc., San Diego, CA, 1998. [31] Mathé, Peter, and Pereverzev, Sergei V., Geometry of linear ill-posed problems in variable Hilbert scales. Inverse Problems 19 (3) (2003), 789–803. [32] Meyer, Yves, Ondelettes et opérateurs. I. Actualités Mathématiques. Hermann, Paris 1990.


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[33] Narcowich, F. J., Petrushev, P., and Ward, J. M., Localized tight frames on spheres. SIAM J. Math. Anal. 38 (2) (2006), 574–594. [34] Neelamani, R., Choi, H., and Baranuik, R., Wavelet-based deconvolution for ill-conditioned systems. Preprint, 2000; http://www-dsp.rice.edu/publications/pub/neelsh98icassp. pdf. [35] Nussbaum, Michael, Asymptotic equivalence of density estimation and Gaussian white noise. Ann. Statist. 24 (6) (1996), 2399–2430. [36] Nyshka, D., Wahba, G., Goldfarb, S., and Pugh, T., Cross validated spline methods for the estimation of three-dimensional tumor size distributions from observations on twodimensional cross sections. J. Amer. Statist. Assoc. 79 (1984), 832–8464. [37] Pensky, M., and Vidakovic, B., Adaptive wavelet estimator for nonparametric density deconvolution. Ann. Statist. 27 (1999), 2033–2053. [38] Petrushev, P., and Xu,Y., Localized polynomials frames on the interval with jacobi weights. J. Fourier Anal. Appl. 11 (5) (2005), 557–575. [39] Petrushev, P., and Xu, Y., Localized polynomials kernels and frames (needlets) on the ball. 2005. IMI 2005. [40] Tsybakov, Alexandre B., Introduction à l’estimation non-paramétrique. Math. Appl. (Berlin) 41, Springer-Verlag, Berlin 2004. [41] Tsybakov, Alexandre, On the best rate of adaptive estimation in some inverse problems. C. R. Acad. Sci. Paris Sér. I Math. 330 (9) (2000), 835–840. [42] Wicksell, S. D., The corpuscle problem: a mathematical study of a biometric problem. Biometrika 17 (1925), 84–99. [43] Willer, T., Deconvolution in white noise with a random blurring effect. Preprint, LPMA 2005.

LPMA, Universités Paris VII et Paris X, CNRS, 175 rue du Chevaleret, 75013 Paris, France

Conformal restriction properties Wendelin Werner∗

Abstract. We give an introduction to some aspects of recent results concerning conformally invariant measures. We focus in this note on the conformal restriction properties of some measures on curves and loops in the plane, and see that these properties in fact almost characterize the measures and allow to classify them. For example, there basically exists a unique measure μ on the set of self-avoiding loops in the plane, such that for any two conformally equivalent domains D and D , the restrictions of μ to the set of loops remaining in D and in D are conformally equivalent. This enables to show that a priori different discrete models define the same curves in the scaling limit and exhibit some surprising symmetries. It gives also a way to tie links between these concrete measures on curves and conformal field theory. Important roles in this theory are played by Brownian loops and by the Schramm–Loewner Evolutions (SLE). Most of the results described in this paper were derived in joint work with Greg Lawler, and Oded Schramm. Mathematics Subject Classification (2000). Primary 60K35; Secondary 82B27, 60J65, 30Cxx.

Keywords. Conformal invariance, random curves, random loops, Brownian motion, percolation.

1. A very brief introduction The last years have seen progress in the mathematical understanding of random twodimensional structures arising as scaling limits of two-dimensional systems from statistical physics. These probabilistic questions are related to complex analysis considerations (because conformal invariance plays an important role in the description of these objects) and to conformal field theory (that had been developed by theoretical physicists precisely to understand these questions). Mathematically speaking, one can broadly distinguish two types of questions: Firstly, proving the convergence of the natural discrete lattice-based models from statistical physics to conformally invariant scaling limits. This aspect based on specific lattice models will be discussed in Schramm’s and Smirnov’s papers in the present proceedings, and will not be the main focus of the present paper. The second type of questions is to define directly the possible continuous limiting objects and to study their ∗ The author acknowledges the support of the Institut Universitaire de France.


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properties. Two ideas have emerged and can be fruitfully combined to study these continuous objects: The Schramm–Loewner Evolutions (SLE) are random planar curves that are explicitly defined via iterations of random conformal maps, and they appear to be the only ones that combine conformal invariance with a certain Markov property. This shows that they are the only possible conformally invariant scaling limits of interfaces of the critical lattice models. Another instrumental idea is to study how close or how different the random objects defined in different but close domains are, and to see what the conformally invariant possibilities are. This very last approach will be the main focus of the present survey. We warn the reader that we will here remain on a rather general introductory level.

2. Conformal invariance of planar Brownian paths In this section, we first recall Paul Lévy’s result on conformal invariance of planar Brownian paths. We then describe some conformally invariant measures on Brownian loops and Brownian excursions. 2.1. Paul Lévy’s theorem. Consider a simple random walk (Sn , n ≥ 0) on the square lattice Z2 (but in fact any planar lattice with some rotational symmetry would do) started from the origin (i.e. S0 = 0). At each integer time, this random walk moves independently to one of its four neighbors with probability 1/4. In other words, the probability that the first n steps of S are exactly a given nearest-neighbor path on the consequence of the central limit theorem that when lattice is equal to 4−n . It is a simple √ N → ∞, the law of (S[2N t] / N, t ≥ 0) converges in some suitable topology to that of a continuous random two-dimensional path (Bt , t ≥ 0) with Gaussian independent increments called planar Brownian motion. It should be noted that planar Brownian paths have a rather complicated geometry. Even if their Lebesgue measure in the plane is almost surely equal to zero,√the Hausdorff dimension of a Brownian path is equal to 2 (this can be related to the N normalization of the simple random walk). Also, there almost surely exists exceptional points of any (including infinite) multiplicity on planar Brownian paths (see [24] and the references therein). Elementary properties of Gaussian random variables show that the law of the process B is invariant under rotations in the plane, and that it is also scale-invariant (this is also quite clear from the normalization of the random walk) in the following sense: For each given λ > 0, the laws of (Bλ2 t , t ≥ 0) and of (λBt , t ≥ 0) are identical. In other words, if one looks at the path of a Brownian motion with a magnifying glass, one sees exactly a Brownian motion, but running at a faster “speed”. Paul Lévy (see e.g. [25]) has observed more than fifty years ago that planar Brownian paths exhibit conformal invariance properties that generalize scale-invariance and rotation-invariance, and that we are now describing:

Conformal restriction

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Consider two given conformally equivalent planar domains D and D : These are two open subsets of C such that there exists an angle-preserving (and orientationpreserving) bijection (i.e. a conformal map) from D onto D . Recall that when D and D are two simply connected proper open subsets of the plane, then by Riemann’s mapping Theorem, there exists a three-dimensional family of such conformal maps from D onto D . Consider a point z in D and define its image z = (z). Then, define a planar Brownian motion (Bt , t ≥ 0) that is started from B0 = z, and denote by T its exit time from the domain D (i.e. T = inf{t ≥ 0 : Bt ∈ / D}). For each t < T , one can therefore define (Bt ), and when t → T , (Bt ) hits the boundary of D = (D). Then: Theorem 2.1 (Paul Lévy). The path ((Bt ), t ≤ T ) is a time-changed planar Brownian motion in D , started at z and stopped at its first exit time of D . The time-change means that there exists a (random continuous increasing) timereparametrization t = t (s) such that ((Bt (s) ), s ≥ 0) is exactly a Brownian motion in D . In order to state exact conformal invariance properties, we will from now on consider paths defined “modulo increasing time reparametrization”. Lévy’s Theorem is nowadays usually viewed as a standard application of stochastic calculus (Itô’s formula). It has led to probabilistic approaches to aspects of potential theory and complex analysis. 2.2. Brownian excursions, Brownian loops. It might be desirable to define conformally invariant random objects in a domain D ⊂ C, but where no marked point in D is given. In Lévy’s Theorem, the starting point of the Brownian path is such a special prescribed point. There are (at least) two natural ways to get rid of it without loosing conformal invariance, that both give rise to infinite measures (i.e. measures with an infinite total mass) on Brownian curves. • A first possibility, described in [22], is to consider Brownian paths that start and end at the boundary of D: Call an excursion in D a continuous path (e(t), 0 ≤ t ≤ T ) such that e(0, T ) ⊂ D and e(0) ∈ ∂D, e(T ) ∈ ∂D. Then, for each D, one can define an infinite measure excD on the set of “Brownian” excursions in D (with unprescribed time-length) in such a way that the image under a conformal map from D onto D of the measure excD is identical to excD modulo time-change. One way to describe the measure in the case where D is equal to the unit disc U (and therefore in the case of all other simply connected domains D via conformal invariance) is to take the limit when ε goes to zero of ε−1 times the law of a Brownian motion started uniformly on the circle (1 − ε)∂ U and stopped at its first hitting time of the unit circle ∂ U. One can also view these measures excD as the scaling limits (when δ → 0) of the measures on discrete excursions on approximations of D by a subset of δ Z2 that assign a mass 4−n to each discrete excursion with n steps (see e.g. [12] for precise estimates).

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Figure 1. A Brownian excursion in the upper half-plane.

Figure 2. Its conformal image in the unit disc.

• A second possibility, described in [23], that will be important in the present paper, is to consider loops instead of open-ended paths. We say that a continuous planar path (t , 0 ≤ t ≤ T ) is a rooted loop if 0 = T . The term rooted is used to emphasize that with this definition, there is a marked point on the loop, namely the starting point 0 . Note that it is possible to re-root a given loop by defining (t = t+t0 , 0 ≤ t ≤ T ) for a given fixed t0 (where is extended into a T -periodic function). We may want to say however that and define in fact the same unrooted loop. Hence, we call an unrooted loop the equivalence class of a rooted loop modulo the equivalence defined by this re-rooting procedure. In order to simplify the conformal invariance statements, we will also say that an unrooted loop is defined modulo increasing continuous time-reparametrizations. Then [23], there exists a measure M on the set of unrooted (Brownian) loops in the plane with strong conformal invariance properties: For any two conformally ) denotes the equivalent open domains D and D = (D), if MD (resp. MD measure M restricted to the set of loops that stay in D (resp. D ), then the image measure of MD under the conformal map from D onto D is exactly the measure MD . One can view this measure M as the limit when δ goes to zero of the measures on discrete unrooted loops in δ Z2 that assign a mass 4−n to each loop with n

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Figure 3. A Brownian loop.

steps (see e.g. [20] for precise estimates). A direct construction of M goes as follows ([23]): It is easy to define the law Pz,T of a Brownian loop with a given time-length T that starts and ends at a given point z. This can be viewed as the conditioning of a Brownian path (Bt , t ≤ T ) started from B0 = z by the event BT = z (this event has zero probability but it is no big deal to make sense of this). Then, one can define a measure M˜ on rooted Brownian loops by integrating the starting point z with respect to the Lebesgue measure in the plane, and the time-length by the measure dT /T 2 . Then, M is just the measure ˜ on the set of unrooted loops induced by M. Note that excD and M are infinite measures (this follows readily from the scale invariance of M and from the scale-invariance of the excursion measure excH in the upper half-plane), so that we can in both cases choose a normalization constant as we wish (i.e. multiply the measures by a well-chosen constant). In fact, the different descriptions of the measures that we did (and will) give differ by a multiplicative constant, and we will not really care here about the exact choice of the constant in the definition of M. Since these are measures on Brownian paths, they are supported on the set of paths with Hausdorff dimension equal to two, but that the mass of the set of paths that go through any given prescribed point z is equal to zero. In a way, both these measures are invariant under a larger class of conformal transformations than the killed Brownian motions defined in the previous subsection because no marked starting point is prescribed. Just as killed Brownian motions describe conformally invariant quantities associated to a given marked point such as the harmonic measure, these two measures define also natural conformally invariant quantities that can be related to extremal distances or Schwarzian derivatives for

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instance. Let us finally define a further useful Brownian measure, the Brownian excursion measure with prescribed endpoints: The excursion measure excU can be decomposed according to the starting and endpoints of the Brownian excursions. This gives rise for each A = B on ∂ U to a probability measure eU,A,B on the set of Brownian excursions from A to B in U. This defines (not surprisingly) again a conformally invariant family of probability measures (eD,A,B ) where (D, A, B) spans the set T the set of triplets (D, A, B) such that D is a simply connected proper subset of C and A and B denote two distinct prime ends of D. When the boundary of D is a smooth self-avoiding loop, this means that A and B are two distinct boundary points. When is a conformal map from D onto D , then “(A)” and “(B)” are then by definition distinct prime ends of D = (D).

3. Conformal restriction We have so far defined some measures on Brownian paths with conformal invariance properties. This means that for each (simply connected) domain, we had a measure mD on paths in D, and that the family (mD ) is conformally invariant (i.e. mD = m(D) ). But when D ⊂ D, it is also natural to compare mD with the measure mD restricted to those paths that stay in D . The conformal restriction property basically requires that these two measures coincide (and that conformal invariance also holds). 3.1. Loops. Suppose that ν is a measure on loops in the plane. As in the rest of the paper, the loops are unrooted and defined modulo increasing time-reparametrizations. For each open domain D, we define νD to be the measure ν restricted to the set of loops that stay in D. Definition 3.1. We say that ν satisfies conformal restriction (resp. conformal restriction for simply connected domains) if for any open domain (resp. open simply connected domain) D and any conformal map : D → (D), one has νD = ν(D) . We have already seen one measure satisfying conformal restriction in the previous section: The measure M on Brownian loops in the plane. Let us now describe a simple argument that shows that all measures that satisfy conformal restriction are closely related. Before that, let us introduce the notion of the filling of a loop. If γ is a loop in the plane, we define its filling K(γ ) to be the complement of the unbounded connected component of C \ γ . In other words, K(γ ) is obtained by filling in all the bounded connected components of the complement of γ . Clearly, any measure on loops defines a measure on their fillings, and we can also define the conformal restriction property for measures on fillings.

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Proposition 3.2 ([42]). Up to multiplication by a positive constant, there exists a unique measure on fillings that satisfies conformal restriction for simply connected domains. It can be defined as the measure on filling of Brownian loops.

Figure 4. A self-avoiding loop.

Proof (sketch). In this proof, we will always discuss conformal restriction for simply connected domains. The existence part of the proposition follows from the fact that the measure M on Brownian loops exists and satisfies conformal restriction (so that the fillings of Brownian loops satisfy conformal restriction as well). It remains to prove the uniqueness statement. Consider the family U of conformal maps ϕ from some (unprescribed) simply connected subset U of the unit disc U containing the origin onto the unit disc U, such that ϕ(0) = 0 and ϕ (0) is a positive real number. Riemann’s mapping theorem shows that for any simply connected domain U ⊂ U with 0 ∈ U , there exists a unique ϕ = ϕU ∈ U from U onto U. Note that U is closed under composition: If ϕU and ϕV are in U, then so is ψ = ϕU ϕV (it is a conformal map from ϕV−1 ϕU−1 (U) = ϕV−1 (U ) onto U with the right properties at the origin). Note that of course, log ψ (0) = log ϕU (0) + log ϕV (0). It is also straightforward to check that ϕV (0) ≥ 1 because V ⊂ U.

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Suppose now that a measure ν on fillings satisfies conformal restriction. Let us define for each ϕU ∈ U, A(ϕU ) = ν({K : 0 ∈ K, K ⊂ U, K ⊂ U }). This is the mass of fillings containing the origin, that stay in U but not in U . Then, it is easy to see that A(ϕU ϕV ) = A(ϕU ) + A(ϕV ). Indeed, there are two types of fillings that contain the origin, stay in U but not in ϕV−1 ϕU−1 (U): • Those that do not stay in V = ϕV−1 (U) and the set of these fillings has a ν-mass equal to A(ϕV ) by definition. • Those that stay in V = ϕV−1 (U) but not in ϕV−1 (ϕU−1 )(U) = ϕV−1 (U ). But by conformal invariance (via the mapping ϕV ), this set is conformally equivalent to the set of loops that stay in U and not in U . So, its ν-mass is A(ϕU ). Rather soft considerations (for instance involving Loewner’s approximation of any mapping in U by iterations of slit mappings) then imply that the functional A is necessarily of the form A(ϕU ) = c log ϕU (0) for a positive constant c. Hence, it follows by conformal invariance that for each z ∈ D ⊂ D, the ν-mass of the set of fillings that contain z, stay in the simply connected domain D but not in the simply connected domain D is equal to c times the logarithm of the derivative at z of the conformal map from D onto D that fixes z and has positive derivative at z. Soft arguments (of the type “a finite measure is characterized by its values on a intersection-stable set that generates the σ -field”) then show that (for each choice of c) this characterizes the measure ν uniquely. This implies the uniqueness part of 2 the proposition. It is possible to show that the boundary of a Brownian loop is almost surely a selfavoiding loop (the fact that it is a continuous loop is straightforward, but the fact that it has no double point requires some estimates, see e.g. [4]). Hence, the proposition shows that modulo multiplication by a positive constant, there is a unique measure μ on self-avoiding loops that satisfies conformal restriction for simply connected domains. As we shall see later, it turns out that it satisfies also the general conformal restriction property. In [15], [16] (see also Schramm’s contribution in these proceedings), it is proved that the Hausdorff dimension of the outer boundary of a Brownian path is almost surely 4/3 (the proof uses SLE considerations and we shall explain why later in this paper). Hence: Corollary 3.3. For the (up-to-constants) unique measure on fillings that satisfies conformal restriction, the boundary of the filling is almost surely a self-avoiding loop with dimension 4/3.

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3.2. The chordal case. Suppose that for each (D, A, B) ∈ T , we have the law PD,A,B of a random excursion from A to B in D. We say that the family (PD,A,B ) is conformally invariant if for any D, A, B and any conformal map from D onto some domain D = (D), the image measure of PD,A,B under is the measure P(D),(A),(B) . This implies in particular that PD,A,B is invariant under any conformal map from D onto itself that preserves the boundary points A and B. For instance, for D = H, A = 0 and B = ∞, this means that PH,0,∞ is scale-invariant (i.e. for each λ > 0, γ and λγ have the same law modulo time-reparametrization). We then say that the probability measure PD,A,B is conformally invariant. Conversely, if one has a probability measure P on excursions from A0 to B0 in D0 for some given triplet (D0 , A0 , B0 ) that is conformally invariant, one can simply define for each D, A, B in T the measure PD,A,B to be the conformal image of P under a conformal map from (D0 , A0 , B0 ) onto (D, A, B). The obtained family (PD,A,B ) is then conformally invariant. We say that the family (PD,A,B ) is restriction-invariant if for any D, A, B, and any simply connected subset D of D such that the distance between {A, B} and D \ D is positive (this implies in particular that A and B are on ∂D ), one has PD,A,B ( · | γ ⊂ D ) = PD ,A,B ( · ). In other words, if γ is defined under PD,A,B , the conditional law of γ given γ ⊂ D is exactly PD ,A,B . Definition 3.4. We say that the probability measure PD,A,B for some (D, A, B) ∈ T satisfies conformal restriction if: • It is conformally invariant. • The conformally invariant family that it defines is restriction-invariant Note that an excursion γ from A to B in D defines also a filling K(γ ), and that one can generalize the conformal restriction property to fillings also. For a fixed triplet D, A, B, we call DD,A,B the set of all simply connected domains D ⊂ D such that the distance between D \ D and {A, B} is strictly positive. For each such D , we define a conformal map from D back onto D with (A) = A and (B) = B. In the case where ∂D is smooth in the neighborhood of A and B, one can define (A) and (B) (which are real numbers) and note that the product of these two derivatives does not depend on which (in the possible one-dimensional family of maps) one did choose. When ∂D is not smooth in the neighborhood of A and B, it is still possible to make sense of the quantity “ (A) (B)” by conformal invariance (map D onto the unit disc, and look at the corresponding quantity for the image of A, B and D ). In short, the quantity (A) (B) is a conformally invariant quantity that measures how smaller D is compared to D, seen from the two points/prime ends A and B.

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Theorem 3.5 ([17]). For each triple (D, A, B) ∈ T , there exists exactly (and in particular: no more than) a one-parameter family of measures on fillings that satisfy conformal restriction. It is parametrized by a number α ∈ [5/8, ∞) and for each α, α the corresponding measure PD,A,B is characterized by the property that for each D ∈ DD,A,B , α PD,A,B (K ⊂ D ) = ( (A) (B))α . Proof (sketch). The uniqueness part is analogous to the loop case: By conformal invariance, we may choose D, A, B to be U, −1, 1. Then, the set D := DU,−1,1 is the family of simply connected subsets U of U such that U \ U is at positive distance from 1 and −1. For each such U , we define ψ = ψU to be the unique conformal map from U onto U such that ψ(−1) = −1, ψ(1) = 1 and ψ (−1) = 1. The family of these conformal maps is closed under composition, and for two such maps ψ1 and ψ2 , (ψ1 ψ2 ) (1) = ψ1 (1)ψ2 (1). Suppose that the measure P on fillings of excursions from −1 to 1 in U satisfies conformal restriction. We then define for each such U ∈ D, A(ψU ) = P (K ⊂ U ). Conformal restriction implies readily that A(ψU ψV ) = A(ψU ) × A(ψV ) for all U and V in D, and this leads to the fact that there exists a positive constant α such that P (K ⊂ U ) = A(ψU ) = ψU (1)α . But the probability measure P is fully characterized by the knowledge of all the probabilities P (K ⊂ U ) for U ∈ D. It then remains to see that for each α ≥ 5/8, these identities indeed describe a probability measure on fillings, and that when α < 5/8, no such measure exists. The way we prove this in [17] is that we explicitly construct the measure when α ≥ 5/8 using the Schramm–Loewner Evolution (SLE) process. For α < 5/8, we also construct what would be the unique possible candidate (that satisfies a weaker condition – called the one-sided conformal restriction property – than the conformal restriction property that we described) for P (via SLE or Brownian means), and we 2 show that this candidate fails to satisfy the actual conformal restriction property. It is easy to check that the Brownian excursions from A to B in D (and their fillings therefore also) defined by eD,A,B do satisfy conformal restriction for α = 1, so that 1 for PD,A,B the boundary of the filling is almost surely supported on sets of Hausdorff dimension 4/3. Let us give a partial description of the boundary of these fillings for general α in terms of Brownian excursions. Let us stick to case of the triplet U, −1, 1. Suppose that K is a filling satisfying conformal restriction. Then it turns out that K ∩ ∂ U = {−1, 1} and that the complement of K in U consists of two connected components: The upper one O + such that ∂O + contains the upper half-circle ∂+ := {eiθ , θ ∈ (0, π)} and the lower one O − , such that ∂O − contains the lower semi-circle ∂− . The boundary

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Figure 5. A Brownian excursion from −1 to 1 in the unit disc (sketch).

of O + (resp. O − ) then consists of the upper (resp. lower) semi-circle and a continuous curve γ + (resp. γ − ) joining −1 to +1 in U. It is then not difficult to see that the law of γ + is characterized by the fact that for any U ∈ D, such that U \ U is at positive distance of the lower semi-circle (i.e. U \ U is attached to the upper semi-circle) P α (γ + ⊂ U ) = ϕU (1)α (we will call D + this subset of D). One way to construct such a random curve uses a Poissonization argument and the Brownian excursion measure that we described earlier. Since a similar Poissonization argument will be useful in another setup a little bit later, let us briefly describe this classical idea in abstract terms: Suppose that N is a σ -finite measure without atoms on some space X. We can define the law of a random countable family X = {Xj , j ∈ J } of elements of X in such a way that: • For each A1 , A2 ⊂ X in the σ -field on which N is defined, such that A1 ∩A2 = ∅, the random families X ∩ A1 and X ∩ A2 are independent. • For each A1 as above, the probability that X∩A1 is empty equals exp(−N(A1 )). The law of X is in fact characterized by these two properties. It is easy to see that for each A, the cardinality of X ∩ A is a Poisson random variable with mean N(A) (so that it is a.s. infinite if and only if N(A) = ∞). X is called a Poisson point process with intensity N. Note that if X1 and X2 are two independent Poisson point processes on the same space X with respective intensity N1 and N2 , then X1 ∪ X2 is a Poisson point process with intensity N1 + N2 . Using this idea, one can define on the same probability space a collection (Xc , c ≥ 0) of Poisson point processes in such a way that Xc ⊂ Xc for all c ≤ c , and such that the intensity of Xc is cN. One intuitive way to view this is to say that with time, elements of X appear independently. During a time-interval dt, an element of

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a set A ⊂ X will appear with probability dt × N(A). Then, Xc denotes the family of elements that did appear before time c. Let us now use this construction for a measure N on the space of excursions in U. More precisely, we define X the set of excursions in U that start and end on the lower semi-circle ∂− , and we define N to be excU restricted to this set of excursions. Hence, for each c, the previous procedure defines a random countable collection of Brownian excursions Ec = (ej , j ∈ Jc ) starting and ending on the negative halfcircle. Despite the fact that this collection is almost surely infinite (because the total mass of N is infinite), the total number of excursions of diameter greater than ε is almost surely finite for all positive ε (because the N-mass of this set of excursions is finite). In particular, this implies that the “upper boundary” γ + of the union of all excursions in Ec does not intersect the upper semi-circle ∂+ . It does not exit a given U ∈ D + if and only if no excursion in Ec does exit U , and by definition, this happens with probability exp(−cN({γ : γ ⊂ U })). The conformal restriction property of the excursion measure shows that for each U ∈ D + , the image under ϕU of the measure N restricted to the set of excursions that stay in U is exactly equal to N. It follows readily from this fact that exp(−N({γ : γ ⊂ U })) = ϕU (1)α1 for some α1 . Hence, for each α > 0, if one chooses c = α/α1 , the curve γ + does indeed satisfy P (γ + ⊂ U ) = ϕU (1)cα1 = ϕU (1)α . The fact that α < 5/8 is not possible corresponds to the fact that the probability that γ + goes “below” the origin becomes larger than 1/2, which is not possible for symmetry reasons if it is equal to the upper boundary of a filling satisfying conformal restriction. For more precise statements and also other possible descriptions of the joint law of (γ + , γ − ), see [17], [39], [40].

4. Related models So far, we have defined only measures on Brownian curves, and we have basically shown that any measure satisfying conformal restriction defines the same outer boundary as that of these Brownian measures. The theory becomes interesting when we note that some a priori different measures do also satisfy conformal restriction. 4.1. Percolation. We now very briefly describe the percolation model that has been proved by Smirnov [35] to be conformally invariant in the scaling limit. Consider the honeycomb lattice (the regular tiling of the plane by hexagons) with mesh size δ. Each hexagon is colored independently in black or in white with probability 1/2. Then, we are interested in the connectivity properties of the set of white (resp. black) cells. We call white (resp. black) cluster a connected component of the union of the white (resp. black) cells. This model is sometimes called “critical site-percolation on the triangular lattice”.

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Figure 6. A rescaled large percolation cluster.

By now classical arguments due to Russo, Seymour and Welsh show that the number of clusters that are of diameter ε > 0 in the unit disc remains tight when δ → 0. This suggests the existence of a scaling limit for the joint law on all clusters when δ (in an appropriately chosen topology). Smirnov [35] proved the existence of the limit of certain observables (the crossing probabilities) and their conformal invariance. A consequence of this result is [36] that it is possible to use SLE computations from [14], [15] and earlier results from Kesten [11] to deduce the existence and the value of the critical exponents for critical percolation as predicted by theoretical physicists such as Cardy, Duplantier, Saleur (see e.g. the references in [36]). But, we would here like to focus on the conformal restriction aspect of the scaling limit of percolation and its consequences. We will remain on a heuristic level, but what follows can be made rigorous: A percolation configuration is described by its white (say) clusters (Cj , j ∈ J ). Smirnov’s result can be shown to imply (see [5]) the convergence in law of this family when δ → 0 to the joint law of a collection of “clusters” (Cj , j ∈ J ) in the plane. A slightly weaker statement is that the measure on clusters π δ that assigns to each possible cluster the probability that this cluster indeed occurs converges when δ → 0 towards a measure on “clusters” π. The measure π satisfies conformal restriction. This is due to the combination of conformal invariance (due to Smirnov’s result) and of the independence properties of percolation from which restriction immediately follows in the scaling limit. Hence: Proposition 4.1. The measure π on scaling limits of critical percolation clusters satisfies conformal restriction. So, π defines exactly the same fillings as (a multiple) of the Brownian loop measure M, and it defines a measure on outer boundaries that is exactly a multiple of μ. In other words, the shape of the outer perimeter of a very large percolation cluster has (in the scaling limit) the same law than the outer boundary of a Brownian loop.

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4.2. The self-avoiding walk conjectures. A classical open problem is to understand the behavior of very long self-avoiding paths, sampled uniformly among all such long self-avoiding paths on some planar lattice with a given starting point and a given length N, in the limit when N → ∞. It is believed that in the scaling limit (for regular periodic lattices with some rotational symmetry) these paths exhibit conformal invariance properties. This led to various striking predictions by theoretical physicists concerning this model and its critical exponents. For instance, it is believed that the diameter of a typical self-avoiding path with N steps is of the order of N 3/4 . This can be loosely phrased in terms of “fractal dimension” since it means that one requires N steps of size N −3/4 to cover a long self-avoiding walk of macroscopic size on the lattice N −3/4 Z2 . More precisely, this could mean that in the scaling limit, self-avoiding walks converge in law to some continuous measure on paths supported on the set of paths with dimension equal to 4/3. Note that the number of self-avoiding walks of length N on Z2 that start at the origin can easily (via a sub-multiplicativity argument) be shown to behave like λN+o(N) when N → ∞, where λ is a positive real number called the connectivity constant of Z2 . One of the striking conjectures in this field is the more precise prediction λN N 11/32+o(1) by Nienhuis [28]. Here are two possible ways to state this existence of scaling-limit conjecture (in the case of the square lattice): • Self-avoiding loops: The measure on self-avoiding loops on δ Z2 that assigns a mass λ−N to each loop with N steps has a (non-trivial) limit when δ → 0. • Excursions: The probability measure on self-avoiding excursions from −1 to 1 in an approximation of U by a sublattice of δ Z2 that assigns a probability proportional to λ−N to each excursion with N steps converges (when δ → 0) to a (non-trivial) scaling limit. In the first case, the scaling limit is then a measure S supported on the set of loops in the plane. In the second one, it is then a probability measure P S on the set of excursions from −1 to 1 in U. If one assumes furthermore that these measures exhibit conformal invariance properties, then S should be a measure on self-avoiding loops satisfying conformal restriction: By the previously described results, it is therefore a multiple of the measure μ on outer boundaries of Brownian loops and of the measure on outer boundaries of percolation clusters. Similarly, we get that P S should satisfy chordal conformal restriction. Hence, it should be a measure on excursions without double points that coincides with one of the P α ’s. This gives an explanation (but not a proof) of the 4/3-dimension conjecture for self-avoiding walks. Let us note that in his book [26], Mandelbrot had already proposed the name “self-avoiding Brownian motion” for the outer boundary of a planar Brownian loop.


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The above results show that this would be indeed an appropriate name.

5. Related SLEs The (chordal) Schramm–Loewner Evolutions (SLE) first introduced in [31] are conformally invariant random planar excursions in a domain with prescribed endpoints. They are defined via iterations of random conformal maps and they are the only ones satisfying a certain Markov property. Since the discrete analogue of this Markovian property is obviously satisfied by the interfaces of many discrete lattice-models from statistical physics (including for instance percolation), this shows that if these discrete interfaces converge to conformally invariant scaling limits, then they have to be one of the SLE curves. For details on the definition and properties of SLEs, their relations (conjectured and proved) to lattice-models, there are now many surveys, lecture notes, a book (e.g. [13], [37] and the references therein); see also Schramm’s contribution to the present ICM proceedings. There exists a one-parameter family of SLE’s: For each κ > 0, the SLE with parameter κ (in short: SLEκ ) is a mathematically well-defined random planar excursion joining prescribed boundary points in a simply connected domain [30], [18]. One can then see if these random excursions satisfy conformal restriction (in the chordal case). It turns out that: Proposition 5.1 ([17]). SLE 8/3 is a random excursion without double points that satisfies chordal restriction. Its law is exactly P 5/8 . No other SLE satisfies chordal conformal restriction. In fact, one can prove that it is the only measure supported on excursions without double points that satisfies chordal conformal restriction (i.e. that for all α > 5/8, the measure P α is not supported on self-avoiding curves). Hence, the SLE8/3 is the conjectural scaling limit of self-avoiding excursions, i.e. P S = P 5/8 . Not surprisingly given all what we have said so far, it can be proved directly that it is supported on the set of excursions with Hausdorff dimension 4/3 [17], [2]. The computation of the critical exponents for SLE (e.g. [14], [15]) allow also to recover the physicists’ predictions on critical exponents such as the 11/32 mentioned above (see e.g. [19]). Also, there is a rather direct relation between discrete self-avoiding loops and selfavoiding excursions (the self-avoiding excursion tells how to finish a loop if we know part of it). This suggests a direct relation between the outer boundaries of planar Brownian loops and the SLE8/3 processes. Indeed (see e.g. [23]), it is possible to define a measure on SLE8/3 loops and to see that it is a measure on self-avoiding loops in the plane that satisfies conformal restriction: Proposition 5.2. The measure μ can be viewed as a measure on SLE 8/3 loops. In fact, this has a deeper consequence, which is not really surprising if one thinks

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of μ in terms of the conjectural scaling limit S of the measure on discrete self-avoiding loops: Theorem 5.3 ([42]). The measure μ on self-avoiding loops satisfies conformal restriction also for non-simply connected domains D. A particular instance of the theorem is that the measure μ is invariant under the inversion z → 1/z. This implies [42] that the inner boundaries of Brownian loops (and those of the scaling limits of critical percolation clusters) have exactly the same distribution than the outer boundaries. More precisely, if one looks at the boundary of the connected component that contains the origin of the complement of a Brownian loop (defined under M) then it is defined under exactly the same measure as the outer boundary. This is by no means an obvious fact. Another consequence is the following: Corollary 5.4 ([42]). It is possible to extend the definition of the planar measure μ on self-avoiding loops to any Riemann surface (possibly with boundaries) in such a way that conformal restriction still holds. This gives a direct description of various conformally invariant quantities in the framework of Riemann surfaces. The SLE6 process can be shown (see e.g. [14], [15]) to be the only SLE satisfying a so-called locality property that makes it the only possible candidate for the (conformally invariant) scaling limit of percolation interfaces. In fact, using Smirnov’s result [35] and ideas, it is possible to deduce [5] that SLE6 is indeed this scaling limit for critical percolation on the triangular lattice. Hence, it should not be surprising that it is possible to define directly (from the definition of SLE6 ) conformally invariant measures on loops and excursions that satisfy conformal restriction (see e.g. [17]). This is one of the ways to see that chordal restriction for α = 2 is very closely related to the loop measure μ.

6. Restriction defect Most models arising from statistical physics should however not satisfy conformal restriction in the scaling limit. Self-avoiding walks and percolation are in this respect rather exceptional cases. We now describe how one can extend the conformal restriction property to cover the more generic cases. It is useful to start with a specific model to illustrate the basic ideas and to show why the Brownian loop-soup can be useful. 6.1. Loop-erased random walks. Suppose that S = (Sn , n ≤ N) is a discrete nearest neighbor-walk of length N on a finite connected graph G. It is as a path joining the two points o = S0 and e = SN that can have double points. One can however associate to S a path from o to e without double-points by following S and


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erasing the loops as they appear. This gives rise to the loop-erasure L = L(S) of S. It is the only simple path from o = L0 to e = Lp (the length p of L is not greater than N but it can be smaller and it depends on the length of the loops erased during this procedure) with the property that for each i ≤ p − 1, Li+1 = Sni +1 , where ni = sup{n ≤ N : Sn = Li }. If we are given the two points o and e, we can choose S randomly to be a simple random walk on the graph, started at o and stopped at its first hitting of e. Its looperasure L = L(S) is then the so-called loop-erased random walk from o to e. It has many nice combinatorial features, that are not obvious at first sight. For instance, the law of the loop-erased random walk from o to e and of the loop-erased random walk from e to o are the same (modulo time-reversal of course). It can also be interpreted as the law of the unique (simple) path joining o to e in a spanning tree chosen uniformly among all spanning trees of the graph G (i.e. choose uniformly a subgraph of G with just one connected component but no cycle, and look at the unique path joining o to e in this subgraph). This result by Pemantle [29] has been extended by Wilson into a complete construction of a uniformly chosen spanning tree of G using loop-erased random walks [43]. It shows that loop-erased random walks belong to a wider general class of models from statistical physics (the random-cluster models) that includes also the Ising models. A fine-grid approximation of the Brownian excursion measure eU,−1,1 goes as follows: Consider a fine-mesh approximation of the unit disc with two boundary points o and e close to −1 and 1, and consider a simple random walk started from o, stopped at e, and conditioned to exit U through e. Theorem 6.1 ([18]). The loop-erasure of this discrete excursion converges when the mesh-size converges to zero to a conformally invariant scaling limit, the SLE 2 from −1 to 1 in U. Similarly, for any triplet (D, A, B) ∈ T , the loop-erasure of a fine-grid approximation of an excursion defined under eD,A,B converges to the SLE 2 from A to B in D. For a given U ⊂ U that still has −1 and 1 on its boundary, it happens with positive probability that the loop-erasure L of the discrete excursion S stays in U , but that the path S does exit U (i.e. one of the erased loops went out of U ). This feature pertains in the scaling limit and shows that conformal restriction is not satisfied by SLE2 . The lack of restriction can be quantified in terms of the erased random walk loops (i.e. in the scaling limit in terms of a quantity involving Brownian loops). More precisely, for a given simple nearest neighbor path from o ∼ −1 to e ∼ 1 on the δ Z2 -approximation of U ⊂ U, the ratio between the probability that L = l for the LERW from o to e in U and the probability that L = l for the LERW in U is given by Fδ (l) = cst(U )PU (none of the erased loops did exit U |L = l). This function Fδ converges to a non-trivial function F when δ → 0 that measures the restriction-defect of SLE2 and that can be expressed in terms of Brownian loops.

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6.2. The Brownian loop soup. Consider the (properly normalized) Brownian loopmeasure M. Recall that it is a measure on the set of unrooted Brownian loops in the entire plane. For each c > 0, we define a Poisson point process with intensity cM. This is a random countable collection {bj , j ∈ J } of Brownian loops in the plane. For each domain D, we define J (D) = {j ∈ J : bj ⊂ D}. It is clear from the definition that this corresponds to a Poisson point process with intensity cMD . In [23], we show that: Proposition 6.2 ([23]). The function F (l) is equal to the probability that no loop in the loop-soup with intensity 2MU intersects both the excursion l and the complement of U . This indicates that the loops that have been erased correspond to the loops in the loop-soup that the path l intersects. This is not so surprising if one thinks of Wilson’s algorithm (that in some sense shows that the law of the constructed uniform spanning tree is independent of the erased loops). It shows [23] that if one adds to an SLE2 the loops that it intersects in a Brownian loop-soup, one recovers exactly a path satisfying conformal restriction (in fact with parameter α = 1, the one of the Brownian excursion excU,−1,1 ). A similar coupling of the SLEκ ’s for κ < 8/3 with a Brownian loop-soup of parameter c = c(κ) = (8 − 3κ)(6 − κ)/2κ. By adding the loops of this loop-soup to the SLE curve, one compensates its lack of restriction and constructs a filling that satisfies conformal restriction with parameter α = (6 − κ)/2κ. These relations correspond to the relation between the central charge (−c), the highest weight (α) and the degeneracy factor (κ/4) of degenerate highest-weight representations of the Virasoro Algebra, as predicted by conformal field theory (see e.g. [9], [1], [3]). 6.3. Loop-soup clusters, CLEs. This does not describe the type of restrictiondefects of the SLE’s with parameter κ > 8/3 that should arise as scaling limits of various lattice models, corresponding in the physics language to models with positive central charge. Loosely speaking, these are the curves that are attracted by the boundaries of a domain (as opposed for instance to the SLE2 that was “repelled” from the boundary). The previous case κ < 8/3 corresponded to a negative central charge. For this, it is useful to consider the geometry of the union of all loops in a loopsoup of intensity cμU (recall that the measure μU corresponds to the outer boundaries of the Brownian loops defined by MU ). This loop-soup is a countable collection Cc = {j , j ∈ Jc } of self-avoiding loops in the unit disc that can overlap with each other. Recall that can couple all Cc ’s in such a way that c → Cc is increasing. When c is large and fixed, it is not difficult to see that almost surely every point in U is surrounded by a loop in Cc , so that all the loops hook up into one single connected component i.e. the set j ∈Jc j has just one connected component. On the other hand, when c is small, it is also easy for instance by coupling this problem with the so-called fractal percolation (sometimes also called Mandelbrot percolation) studied in [7], [27] to see that this phenomenon does not pertain: The

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Figure 7. A CLE (very very sketchy).

set j ∈Jc j has countably many connected components. The outermost boundaries of these clusters of loops define a family of non-overlapping and non-nested loops uc = {uci , i ∈ Ic } in U. This leads to the following definition [34]: Definition 6.3. Suppose that u = {ui , i ∈ I } is a random collection of non-intersecting and non-nested self-avoiding loops in U. We say that it is a simple conformal loopensemble (CLE) if the following properties hold: • It is invariant under the conformal transformation from U onto itself. This allows to define the law PU of the collection of loops in any simply connected domain U by taking the conformal image of u. • Let U be any simply connected subset of U with d(U \ U, 1) > 0. Consider I = {i ∈ I, ui ⊂ U } and let U˜ denote the connected component of U \ i∈I ui that has 1 on its boundary. Then, conditionally on {ui , i ∈ I }, the law of {ui , i ∈ I and ui ⊂ U˜ } is PU˜ . Loosely speaking, this means that each loop (once it is discovered) plays the role of the boundary of the domain in which the others are yet to be discovered. Note that a CLE almost surely is an infinite collection of loops (because the number of loops contained in U and in U˜ ⊂ U have the same law). The previous considerations show that the outermost boundaries of cluster of loops for sub-critical (i.e. for small c) loop-soups, are conformal loop ensembles (so that CLEs exist). This gives rise to measures on loops that do not satisfy conformal restriction, but have the same type of restriction defect as that of SLEs for κ ∈ (8/3, 4]. The intensity c of the loop-soup corresponds to the central charge of the model. Conformal loop-ensembles (and SLEs) arise also in the context of level-lines (or flow-lines) of the Gaussian Free Field [33] in the ongoing work of Oded Schramm and

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Figure 8. The CLE in U˜ when U is the semi-disc.

Scott Sheffield [32]. Combining all these arguments should [34] describe all CLEs as loop-cluster boundaries and their boundaries as SLE loops for κ ≤ 4. Acknowledgments. I would like to express many thanks to Greg Lawler and Oded Schramm for the opportunity to interact and work with them during these last years. I also thank Pierre Nolin for Figures 4 and 6.

References [1] Bauer, M., Bernard, D., Conformal transformations and the SLE partition function martingale. Ann. Henri Poincaré 5 (2004), 289–326. [2] Beffara, V., The dimensions of SLE curves. math.PR/0211322, 2002. [3] Belavin, A. A., Polyakov, A. M., Zamolodchikov, A. B., Infinite conformal symmetry in two-dimensional quantum field theory. Nuclear Phys. B 241 (1984), 333–380. [4] Burdzy, K., Lawler, G. F., Non-intersection exponents for random walk and Brownian motion II. Estimates and application to a random fractal. Ann. Probab. 18 (1990), 981–1009. [5] Camia, F., Newman, C., The Full Scaling Limit of Two-Dimensional Critical Percolation. Preprint, 2005; arXiv:math.PR/0504036. [6] Cardy, J. L., Conformal invariance and surface critical behavior. Nuclear Phys. B 240 (1984), 514–532. [7] Chayes, J. T., Chayes, L., Durrett, R., Connectivity properties of Mandelbrot’s percolation process. Probab. Theory Related Fields 77 (1988), 307–324. [8] Duplantier, B., Conformal fractal geometry and boundary quantum gravity. In Fractal Geometry and applications, a jubilee of Benoît Mandelbrot. Proc. Symp. Pure Math. 72, Part II, Amer. Math. Soc., Providence, RI, 2004, 365–482.


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[9] Friedrich, R., Werner, W., Conformal restriction, highest-weight representations and SLE. Comm. Math. Phys. 243 (2003), 105–122. [10] Garban, C., Trujillo-Ferreras, J. A., The expected area of the Brownian loop is π/5. Comm. Math. Phys. 264 (3) (2006), 797–810. [11] Kesten, H., Scaling relations for 2D-percolation. Comm. Math. Phys. 109 (1987), 109–156. [12] Kozdron, M., On the scaling limit of simple random walk excursion measure in the plane. Preprint, 2005; arXiv:math.PR/0506337. [13] Lawler, G. F., Conformally invariant processes in the plane. Math. Surveys Monogr. 144, Amer. Math. Soc., Providence, RI, 2005. [14] Lawler, G. F., Schramm, O., Werner, W., Values of Brownian intersection exponents I: Half-plane exponents. Acta Math. 187 (2001), 236–273. [15] Lawler, G. F., Schramm, O., Werner, W., Values of Brownian intersection exponents II: Plane exponents. Acta Math. 187 (2001), 275–308. [16] Lawler, G. F., Schramm, O., Werner, W., The dimension of the Brownian frontier is 4/3. Math. Res. Lett. 8 (2001), 401–411. [17] Lawler, G. F., Schramm, O., Werner, W., Conformal restriction properties. The chordal case. J. Amer. Math. Soc. 16 (2003), 917–955. [18] Lawler, G. F., Schramm, O., Werner, W., Conformal invariance of planar loop-erased random walks and uniform spanning trees. Ann. Probab. 32 (2004), 939–996. [19] Lawler, G. F., Schramm, O., Werner, W., On the scaling limit of planar self-avoiding walks. In Fractal Geometry and applications, a jubilee of Benoît Mandelbrot, Proc. Symp. Pure Math. 72, Part II, Amer. Math. Soc., Providence, RI, 2004, 339–364. [20] Lawler, G. F., Trujillo-Ferreras, J. A., Random walk loop-soup. Trans. Amer. Math. Soc., to appear. [21] Lawler, G. F., Werner, W., Intersection exponents for planar Brownian motion. Ann. Probab. 27 (1999), 1601–1642. [22] Lawler, G. F., Werner, W., Universality for conformally invariant intersection exponents. J. Europ. Math. Soc. 2 (2000), 291–328. [23] Lawler, G. F., Werner, W., The Brownian loop-soup. Probab. Theory Related Fields 128 (2004), 565–588. [24] Le Gall, J. F., Some properties of planar Brownian motion. In École d’Été de Probabilités de Saint-Flour XX—1990 (ed. by P. L. Hennequin), Lecture Notes in Math. 1527, SpringerVerlag, Berlin 1992, 111–235. [25] Lévy, P., Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris 1948. [26] Mandelbrot, B. B., The Fractal Geometry of Nature. W. H. Freeman and Co., San Francisco, CA, 1982. [27] Meester, R., Roy, R., Continuum Percolation. Cambridge Tracts in Math. 119, Cambridge University Press, Cambridge 1996. [28] Nienhuis, B., Exact critical exponents for the O(n) models in two dimensions. Phys. Rev. Lett. 49 (1982), 1062–1065. [29] Pemantle, R., Choosing a spanning tree for the integer lattice uniformly. Ann. Probab. 19 (1991), 1559–1574. [30] Rohde, S., Schramm, O., Basic properties of SLE. Ann. of Math. (2) 161 (2005), 879–920.

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[31] Schramm, O., Scaling limits of loop-erased random walks and uniform spanning trees. Israel J. Math. 118 (2000), 221–288. [32] Schramm, O., Sheffield, S., in preparation. [33] Sheffield, S., Gaussian Free Fields for mathematicians. Preprint, 2003; arXiv:math.PR/ 0312099. [34] Sheffield, S., Werner, W., in preparation. [35] Smirnov, S., Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333 (2001), 239–244. [36] Smirnov, S., Werner, W., Critical exponents for two-dimensional percolation. Math. Res. Lett. 8 (2001), 729–744. [37] Werner, W., Random planar curves and Schramm-Loewner Evolutions. In Lectures on probability theory and statistics, Lecture Notes in Math. 1840, Springer-Verlag, Berlin 2004, 107–195. [38] Werner, W., SLEs as boundaries of clusters of Brownian loops. C. R. Acad. Sci. Paris Sér. I Math. 337 (2003), 481–486. [39] Werner, W., Girsanov’s Theorem for SLE(κ, ρ) processes, intersection exponents and hiding exponents. Ann. Fac. Sci. Toulouse 13 (2004), 121–147. [40] Werner, W., Conformal restriction and related questions. Probab. Surv. 2 (2005), 145–190. [41] Werner, W., Some recent aspects of conformally invariant systems. Lecture Notes from Les Houches summer school, Preprint, 2005; arXiv:math.PR/0511268. [42] Werner, W., The conformal invariant measure on self-avoiding loops. Preprint, 2005; arXiv:math.PR/0511605. [43] Wilson, D. B., Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing, ACM, New York 1996, 296–303. Université Paris-Sud, Laboratoire de mathématiques, Bât. 425, 91405 Orsay, France and Ecole Normale Supérieure, Département de mathématiques et applications, 45, rue d’Ulm, 75230 Paris cedex 05, France E-mail: [email protected]

The complexity of generating functions for integer points in polyhedra and beyond Alexander Barvinok∗

Abstract. Motivated by the formula for the sum of the geometric series, we consider various classes of sets S ⊂ Zd of integer points for which an a priori “long” Laurent series or polynomial m m∈S x can be written as a “short” rational function f (S; x). Examples include the sets of integer points in rational polyhedra, integer semigroups, and Hilbert bases of rational cones, among others. We discuss applications to efficient counting and optimization and open questions. Mathematics Subject Classification (2000). Primary 05A15; Secondary 68W30, 11P21, 52C07, 11H06. Keywords. Lattice point, rational polytope, generating function, rational function, Laurent polynomial, integer semigroup, Hilbert basis, efficient counting, computational complexity.

1. Introduction Our inspiration comes from a formula for the sum of a finite geometric series: n m=0

xm =

1 − x n+1 . 1−x

(1.1)

We look at the formula from several points of view. Geometrically, the left hand side of (1.1) represents the sum over all integer points in a one-dimensional polytope. Namely, with every integer point m we associate a monomial x m and then consider the sum over all integer points in the interval [0, n]. From the computational complexity point of view, the left hand side of (1.1) is a “long” polynomial whereas the right hand side of (1.1) is a “short” rational function. More precisely, to write an integer m we need about log m digits or bits. Consequently, to write the left hand side of (1.1), we need about n log n bits. On the other hand, to write the right hand side of (1.1) we need only about log n bits. Thus the left hand side is exponentially longer than the right hand side. Finally, let us read (1.1) from right to left. We can ask how to extract various facts about the set S of integer points in the interval [0, n] from the rational function ∗ The author is grateful to Microsoft (Redmond) for hospitality during his work on this paper. This work was partially supported by NSF Grant DMS 0400617


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encoding. For example, to compute the number |S| of points we substitute x = 1 into the right hand side of (1.1). Although x = 1 is a pole of the rational function, we can compute the desired value by applying l’Hospital’s rule. Let Rd be Euclidean space with the standard basis e1 , . . . , ed , so a point x ∈ Rd is identified with the d-tuple x = (ξ1 , . . . , ξd ) of its coordinates, and let Zd ⊂ Rd be the standard integer lattice, that is the set of points with integer coordinates. With every integer point m = (μ1 , . . . , μd ) we associate the Laurent monomial μ

μ

x m = x1 1 . . . xd d in d complex variables x = (x1 , . . . , xd ). We agree that xi0 = 1. Let S ⊂ Zd be a finite set and let us consider the sum xm. f (S; x) = m∈S

Thus f (S; x) is a Laurent polynomial that is the generating function of the set S. We are interested in the following general questions: • For which sets S ⊂ Zd a potentially long Laurent polynomial f (S; x) can be written as a short rational function? • What information about the set S can be extracted from f (S; x) given as a short rational function? The paper is organized as follows. In Section 2, we discuss necessary preliminaries from the theory of computational complexity, define what “long” and “short” means and show that if S is the set of integer points in a rational polyhedron P ⊂ Rd then the generating function f (S; x) can be computed in polynomial time as a short rational function, provided the dimension d of the ambient space is fixed in advance. We discuss applications to efficient counting and optimization and practical implementations of the algorithms. In Section 3, we discuss what information can we extract from a set S ⊂ Zd defined by its generating function f (S; x) written as a rational function. In particular, we show that if S1 , S2 ⊂ Zd are two finite sets defined by their rational generating functions f (S1 ; x) and f (S2 ; x), then the generating function f (S; x) of their intersection S = S1 ∩ S2 can be computed in polynomial time as a rational function. In Section 4, we show that if S ⊂ Z+ is an integer semigroup with a fixed number d of generators, then f (S; x) can be computed in polynomial time as a short rational function. This result is obtained as a corollary of a more general result that the projection of the set of integer points in a rational polytope admits a polynomial time computable rational generating function. We mention some other examples such as Hilbert bases of rational cones. In Section 5, we consider the results of Sections 2 and 4 in the general context of Presburger arithmetic. We argue that the “natural” class of sets S ⊂ Zd with short rational generating functions f (S; x) would have been the class of sets defined by

The complexity of generating functions for integer points in polyhedra and beyond

765

formulas of Presburger arithmetic where all combinatorial parameters (the number of variables and Boolean operations) are fixed and only numerical constants are allowed to vary. As the paper is being written, this is still a conjecture. In Section 6, we try to identify the natural boundaries of the developed theory. We also discuss the emerging picture of what happens if the dimension d of the ambient space is allowed to grow.

2. Rational polyhedra Formula (1.1) admits an extension to general rational polyhedra. Definition 2.1. The set P ⊂ Rd of solutions to a system of finitely many linear inequalities is called a polyhedron: d αij ξj ≤ βi , i = 1, . . . , n . P = (ξ1 , . . . , ξd ) :

(2.1)

j =1

Here αij and βi are real numbers. A bounded polyhedron is called a polytope. A polyhedron P is called rational if in (2.1) one can choose all αij and βi integer. To state an analogue of formula (1.1) we need to discuss the notion of the input size. As we remarked earlier, to write an integer a we need roughly log2 (|a|+1)+1 bits. Consequently, to define a rational polyhedron P ⊂ Rd by the inequalities (2.1) we need about L = n(d + 1) + log2 (|αij | + 1) + log2 (|βi | + 1) (2.2) i,j

i

bits. The number L is called the input size of representation (2.1) of P . We are interested in the computational complexity of formulas and algorithms. In particular, we are interested in polynomial time algorithms, that is, in the algorithms whose running time is at most LO(1) , where L is the input size. In what follows, often the dimension d of the ambient space will be fixed in advance and the algorithms will run in polynomial time for any fixed dimension d. In other words, the running time of such an algorithm is at most Lφ(d) for some function φ. We use [28] as a general reference in the area of computational complexity and algorithms. Let P ⊂ Rd be a rational polyhedron with a vertex (equivalently, a non-empty polyhedron without lines), possibly unbounded, and let S = P ∩ Zd be the set of integer points in P . To simplify notation, we denote the generating function f (S; x) = xm, m∈S

where S = P ∩ Zd , just by f (P , x).

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Alexander Barvinok

It is not hard to show that there exists a non-empty open set U ⊂ Cd such that for all x ∈ U the series f (P , x) = xm m∈P ∩Zd

converges absolutely and uniformly on compact subsets of U to a rational function in x. It turns out that this rational function can be efficiently computed as long as the dimension d of the ambient space is fixed in advance. The following result was proved, essentially, in [3] although the formal statement and better complexity bounds did not appear until [4]. Theorem 2.2. Let us fix d. Then there exists a polynomial time algorithm, which, for a rational polyhedron P ⊂ Rd without lines defined by inequalities (2.1) computes the generating function f (P , x) = xm m∈P ∩Zd

in the form f (P , x) =

εi

i∈I

where εi ∈ {−1, 1}, vi , uij ∈

Zd ,

x vi (1 − x ui1 ) . . . (1 − x uid )

,

(2.3)

and uij = 0 for all i, j .

The complexity of the algorithm is LO(d) , where L is the input size of P defined by (2.2). In particular, the number |I | of terms in (2.3) is LO(d) , which is why we call (2.3) a short rational function. Rational cones play the crucial role in the proof of Theorem 2.2. 2.1. Rational cones. A non-empty rational polyhedron K is called a rational cone if for every x ∈ K and λ ≥ 0 we have λx ∈ K. We are interested in pointed rational cones, that is, cones not containing lines (equivalently, cones for which 0 is the vertex). A basic example of a pointed rational cone is provided by the non-negative orthant Rd+ consisting of the points with non-negative coordinates. The generating function for the set of integer points in Rd+ is a multiple geometric series f (Rd+ , x) =

xm =

d i=1

m∈Zd+

1 . 1 − xi

A unimodular cone K is the set of non-negative linear combinations of a given basis u1 , . . . , ud of the lattice Zd . Up to an integral change of coordinates, a unimodular cone K looks like the non-negative orthant Rd+ . Consequently, the generating function for the set of integer points in K is a multiple geometric series f (K, x) =

m∈K∩Zd

x = m

d i=1

1 . 1 − x ui

767


It is well known that any rational cone K can be subdivided into unimodular cones, cf., for example, Section 2.6 of [16]. However, even for d = 2, the number of the unimodular cones may have to be exponentially large in the input size: consider the cone K ⊂ R2 spanned by (1, 0) and (1, n) for a positive integer n. Nevertheless, where exists a computationally efficient procedure for constructing a more general decomposition of a rational cone into unimodular cones. Definition 2.3. For a set A ⊂ Rd , let [A] : Rd −→ R be the indicator of A defined by 1 if x ∈ A [A](x) = 0 if x ∈ / A. Let P (Qd ) be the vector space (over C) spanned by the indicators [P ] of rational polyhedra P ⊂ Rd . We call P (Qd ) the algebra of rational polyhedra. The vector space P (Qd ) possesses an interesting and useful algebra structure, cf. [26], which we do not discuss here. The idea is to write the indicator [K] of a given rational cone K ⊂ Rd as a linear combination of indicators of unimodular cones. For d = 2 such an efficient procedure has long been known via the continued fractions method, cf., for example, [22]. We give a simple example below. Suppose that K ⊂ R2 is the cone spanned by vectors (1, 0) and (31, 164). Writing the continued fraction expansion, we obtain 164 =5+ 31

1 3+

1

,

1 4 so we write 164/31 = [5; 3, 2, 4]. Next, we compute the convergents 2+

[5; 3, 2] = 5 +

1 3+

1 2

=

37 , 7

[5; 3] = 5 +

1 16 = , 3 3

and

[5] =

5 1

and notice that [K] = [K0 ] − [K1 ] + [K2 ] − [K3 ] + [K4 ], where K0 is spanned by (1, 0) and (0, 1), K1 is spanned by (0, 1) and (1, 5), K2 is spanned by (1, 5) and (3, 16), K3 is spanned by (3, 16) and (7, 37), and K4 is spanned by (7, 37) and (31, 164). Since Ki turn out to be unimodular for i = 0, 1, 2, 3, 4, we get the short rational function expression f (K, x) =

1 1 1 + − 5 5 (1 − x1 )(1 − x2 ) (1 − x2 )(1 − x1 x2 ) (1 − x1 x2 )(1 − x13 x216 ) 1 1 + . − 3 7 16 7 37 37 (1 − x1 x2 )(1 − x1 x2 ) (1 − x1 x2 )(1 − x131 x2164 )

768

Alexander Barvinok

A polynomial time algorithm for computing a unimodular cone decomposition in any (fixed in advance) dimension d was suggested in [3]. Using triangulations, it is not hard to reduce the case of an arbitrary rational cone to that of a simple rational cone K ⊂ Rd d K= λi ui : λi ≥ 0 i=1

spanned by linearly independent vectors u1 , . . . , ud ∈ Zd , which may not, however, constitute a basis of the lattice Zd . As a measure of how far is K from being unimodular, we introduce the index ind(K) of K as the index of the sublattice generated by u1 , . . . , ud in the ambient lattice Zd . Thus ind(K) is a positive integer and ind(K) = 1 if and only if K is a unimodular cone. Let us consider the parallelepiped =

d

λi ui : |λi | ≤ ind−1/d (K) for i = 1, . . . , d .

i=1

Then is a convex body symmetric about the origin and vol = 2d . Therefore, by the Minkowski Theorem there is a non-zero point w ∈ ∩ Zd , cf., for example, Section VII.3 of [5]. Moreover, such a point w can be constructed in polynomial time as long as the dimension d is fixed, cf. Section 6.7 of [17]. Replacing w by −w if needed, we can also ensure that w lies in the same halfspace as u1 , . . . , ud . Let Ki be the cone spanned by u1 , . . . , ud with the vector ui replaced by w and let εi = 1 or εi = −1 depending on whether this replacement preserves or reverses the orientation of the set u1 , . . . , ud (we choose εi = 0 if we obtain a linearly dependent set). Then we observe that [K] =

d

εi [Ki ] ± indicators of lower-dimensional cones, and

i=1

ind(Ki ) ≤ ind

(d−1)/d

(K)

(2.4)

if dim Ki = d.

As we iterate the above procedure, on the nth step, we obtain a decomposition of the cone K as a linear combination of at most d n cones Ki (not counting smallerdimensional cones) with d−1 n ind(Ki ) ≤ (ind(K)) d . To ensure that all Ki are unimodular, we can choose n = O(d log log ind(K)), which results in a polynomial time algorithm for a fixed d. To prove a weaker version of Theorem 2.2 (with d replaced by d + 2 in (2.3) and 2 LO(d ) complexity) one can note that a rational polyhedron P ⊂ Rd without lines


769

can be represented as the section of a pointed rational cone K ⊂ Rd+1 by the affine hyperplane ξd+1 = 1. Consequently, we have f (P , x) =

∂ f (K, (x, xd+1 )) x =0 . d+1 ∂xd+1

(2.5)

2.2. Using identities in the algebra of polyhedra. The following remarkable result was proved by A. G. Khovanskii and A. V. Pukhlikov [23], and, independently, by J. Lawrence [25]. Theorem 2.4. Let P (Qd ) be the vector space spanned by the indicators of rational polyhedra and let C(x) be the vector space of rational functions in d complex variables x = (x1 , . . . , xd ). There exists a linear transformation F : P (Qd ) −→ C(x) such that the following holds. (1) If P ⊂ Rd is a rational polyhedron with a vertex then F ([P ]) = f (P , x), where f (P , x) is the rational function defined as the sum of the series xm m∈P ∩Zd

when the series converges absolutely. (2) If P ⊂ Rd is a rational polyhedron without vertices then F ([P ]) = 0. Proof. Let us fix a decomposition Rd =

αi [Qi ]

(2.6)

i∈I

for some rational polyhedra Qi with vertices and some numbers αi . Multiplying (2.6) by [P ], we get [P ] = αi [P ∩ Qi ], (2.7) i∈I

from which we deduce that P (Qd ) is spanned by indicators of rational polyhedra with vertices. Suppose that we have a linear relation βj [Pj ] = 0 (2.8) j ∈J

for some polyhedra Pj with vertices. Multiplying (2.8) by [Qi ], we get j ∈J

βj [Pj ∩ Qi ] = 0.

770

Alexander Barvinok

Since Qi has a vertex and Pj ∩ Qi ⊂ Qi , there exists a non-empty open set Ui ⊂ Cd such that for all x ∈ Ui all the series defining f (Pj ∩ Qi , x) converge absolutely and uniformly on compact subsets of Ui . Therefore, we must have βj f (Pj ∩ Qi , x) = 0 for all i ∈ I. j ∈J

Similarly, from (2.7) we get f (Pj , x) =

αi f (Pj ∩ Qi , x)

for all j ∈ J.

i∈I

Combining the last two equations, we conclude that βj f (Pj , x) = αi βj f (Pj ∩ Qi , x) = 0. j ∈j

(2.9)

i∈I,j ∈J

Thus a linear dependence (2.8) among indicators of rational polyhedra Pj with vertices implies the corresponding linear dependence (2.9) among the generating functions f (Pj , x). Therefore, the correspondence [P ] −→ f (P , x) extends to a linear transformation F : P (Qd ) −→ C(x). It remains to show that F ([P ]) = 0 if P is a rational polyhedron with a line. We observe that if P = P + u is a translation of P by a lattice vector u, we must have f (P , x) = x u f (P , x) for all rational polyhedra P with vertices. By linearity, we must have F ([P + u]) = x u F ([P ]) for all rational polyhedra P . However, if P contains a line then there is a vector u ∈ Zd \ {0} such that P + u = P . Therefore, we must have F ([P ]) = 0 for P with a line. 2 Theorem 2.4 provides a powerful tool for computing the generating function of the set of integer points in a rational polyhedron. The following “duality trick” going back to the seminal paper of M. Brion [11] turns out to be particularly useful. Let ·, · be the standard scalar product in Rd and let K ⊂ Rd be a cone. The cone K ∗ = {x ∈ Rd : x, y ≥ 0 for all y ∈ K} is called the dual to K. It is easy to see that if K is rational (resp. unimodular) cone then K ∗ is a rational (resp. unimodular) cone, and that if K contains a line (resp. lies in a proper subspace of Rd ) then K ∗ lies in a proper subspace of Rd (resp. contains a line). A standard duality argument implies that (K ∗ )∗ = K for closed convex cones K. A less obvious observation is that duality preserves linear relations among indicators of closed convex cones: αi [Ki ] = 0 implies αi [Ki∗ ] = 0, i∈I

i∈I

771


see, for example, Section IV.1 of [5] for a proof. Now, to compute the generating function f (K, x) one can do the following. First, we compute the dual cone K ∗ , and, iterating (2.4), we compute unimodular cones Ki and numbers εi ∈ {−1, 1} such that εi [Ki ] modulo indicators of lower-dimensional cones. [K ∗ ] ≡ i∈I

Then, dualizing again, we get εi [Ki∗ ] [K] ≡

modulo indicators of cones with lines.

(2.10)

i∈I

In view of Theorem 2.4, cones with lines can be ignored as far as generating functions are concerned. This gives us εi f (Ki∗ , x). f (K, x) = i∈I

Ki∗

are unimodular cones, this completes computation of f (K, x). This trick Since 2 allows us to reduce the complexity of the algorithm in Theorem 2.2 from LO(d ) to LO(d) , where L is the size of the input. Another important identity is Brion’s Theorem [11], which expresses the generating function of the set of integer points in P as the sum of generating functions for the sets of integer points in the tangent (supporting) cones at the vertices of P . Namely, for a vertex v of a polyhedron P let us define the tangent cone Kv as Kv = {x : εx + (1 − ε)v ∈ P for all sufficiently small ε > 0}. We note that Kv is not a cone per se but rather a translation of the cone Kv − v. Theorem 2.5 (Brion’s Theorem). For a rational polyhedron P we have f (P , x) = f (Kv , x), v

where the sum is taken over all vertices of P and the identity is understood as the identity among rational functions. Discovered by M. Brion [11], Theorem 2.5 started an avalanche of research. The original proof of Theorem 2.5 was based on algebro-geometric methods. Later, elementary proofs were discovered in [23] and [25]. One can deduce Theorem 2.5 from Theorem 2.4 and an elementary identity [Kv ] modulo indicators of polyhedra with lines, [P ] ≡ v

cf. Section VIII.4 of [5].

772

Alexander Barvinok

Theorem 2.5 together with the unimodular decomposition of Section 2.1 and the duality trick provide the proof of Theorem 2.2 as stated. Another advantage of using Theorem 2.5 is that it allows us to understand how the generating function f (P , x) changes as the facets of P move parallel to themselves so that the combinatorial structure of P does not change. In this case, the tangent cones Kv get translated by vectors linearly depending on the displacements of the facets of P . Writing Kv as combinations of translated unimodular cones Ki + v as in (2.10), we notice that as far as lattice points are concerned, a rational translation Ki + v of a unimodular cone Ki is equivalent to a certain integer translation Ki + u: (Ki + v) ∩ Zd = (Ki + u) ∩ Zd

for some u ∈ Zd

and hence we have f (Ki + v, x) = f (Ki + u, x) = x u f (Ki , x). If K = Rd+ then u is obtained from v by rounding up the coordinates to the nearest integer. The case of a general unimodular cone differs by a unimodular linear transformation, see [4] for details. 2.3. Implementation. The algorithm of Theorem 2.2 appears to be practical. First, it was implemented by J. De Loera et al. [12], who wrote the LattE (Lattice point enumeration) software package. The authors of LattE discovered that often the most practically efficient way to handle computations is to represent a polyhedron P as a hyperplane section of a higher-dimensional cone as in (2.5) and then use the “dualized” decomposition (2.10). The package allows one to compute the number of integer points in a given rational polytope. Formally speaking, to compute the number |P ∩ Zd | of integer points in a given rational polytope P , we should substitute x = (1, . . . , 1) into the rational function f (P , x). However, we need to be careful since this particular value is a pole of every fraction in (2.3). Nevertheless, the substitution can be done efficiently, see Section 3.1 and [3], [4], [7], and [12] for details. In addition, LattE allows one to compute the Ehrhart (quasi)-polynomial of a given rational polytope P , that is, to find a formula for the number of integer points in the dilated polytope nP , where n is a positive integer, see also Section 6.1. Testing whether a given rational polyhedron P contains an integer point, or, equivalently, whether f (P , x) ≡ 0 is a non-trivial problem related to the general integer programming problem of optimizing a given linear function on the set P ∩ Zd . LattE package contains also an implementation of an integer programming algorithm based on rational functions f (P , x). Another implementation, called barvinok , was written by S. Verdoolaege, see [36]. Among other features, the implementation allows one to obtain closed explicit formulas for the number of integer points in a parametric polytope as a function of displacement parameters when the facets of the polytope move parallel to themselves, see Theorem 2.5 and the subsequent discussion.


773

There is an extensive literature devoted to the lattice point enumeration in polytopes, whether from algorithmic, structural, or application points of view. For the classical Ehrhart theory in the context of enumerative combinatorics, see [34] and [9] for a clever simplification of the proofs of the main results of the theory. For an approach featuring Dedekind sums and other analytic tools, see [8]. It does not seem to be possible to survey all the literature in the paper. In addition to already mentioned papers, we provide only a few references among many good papers which appeared after the survey [4]. Efficient counting in special situations with applications to computational questions in representation theory and network flows is discussed in [2]. For a recent advance connecting lattice point counting with algebraic geometry, see [29]. For a computationally efficient version of the Euler–Maclaurin formula, satisfying, in addition, some natural “local” conditions, see [10].

3. Operations on sets and generating functions Motivated in part by Theorem 2.2, let us consider sets S ⊂ Zd defined by their generating functions xm f (S; x) = m∈S

written as rational functions in the form x ai εi . f (S; x) = (1 − x bi1 ) . . . (1 − x bik )

(3.1)

i∈I

Here I is a finite set of indices, εi ∈ Q, ai , bij ∈ Zd , and bij = 0 for all i, j . To avoid ambiguity, we assume that either S is finite, or, if S is infinite, then there is a nonempty open set U ⊂ Cd such that the series defining f (S; x) converges absolutely and uniformly on compact subsets of U and for every fraction in (3.1) there is the Laurent series (multiple geometric series) expansion x ai x ai +μ1 bi1 +···+μk bik = (1 − x bi1 ) . . . (1 − x bik ) k (μ1 ,...,μk )∈Z+

in U . To indicate the computational complexity level of our set S, we consider the two parameters fixed in formula (3.1): the number d of variables and the number k of binomials in the denominator of each fraction. Note that if we happen to have a smaller number of binomials in some fraction, we can formally “pad” it to k by multiplying both the numerator and denominator of the fraction by some artificial binomials. Since k is fixed, that would increase the length of the formula by a constant factor. Next, we discuss what information about the set S can be extracted from f (S; x) given in the form of (3.1).

774

Alexander Barvinok

3.1. Monomial substitutions and differentiation. One piece of information we can get is the cardinality |S| of a finite set S. To compute |S|, we would like to substitute x = (1, . . . , 1) in (3.1), but this should be done carefully since this particular value of x is the pole of every single fraction in (3.1). The procedure is introduced in [3]. We choose a sufficiently generic vector c ∈ Zd , c = (γ1 , . . . , γd ), so that c, bij = 0 for all i, j . For a τ ∈ C, let x(τ ) = (eτ γ1 , . . . , eτ γd ). Thus we want to compute lim f (S; x(τ )).

τ −→0

Let us compute αi = c, ai and Then f (S; x(τ )) =

i∈I

εi

βij = c, bij .

eαi τ . (1 − eβi1 τ ) . . . (1 − eβik τ )

(3.2)

Next, we note that f (S; x(τ )) is a meromorphic function in τ and that we want to compute the constant term of its Laurent expansion in the neighborhood of τ = 0. To do that, we deal with every fraction separately. We write each fraction of (3.2) as k eαi τ −k αi τ gij (τ ), =τ e (1 − eβi1 τ ) . . . (1 − eβik τ )

where gij (τ ) =

j =1

τ . 1 − eβij τ

Now, each gij (τ ) is an analytic function of τ and we compute is Taylor series expansion pij (τ ) up to the τ k+1 term: τ ≡ pij (τ ) mod τ k+1 . 1 − eβij τ Similarly, we compute a polynomial qi (τ ) such that eαi τ ≡ qi (τ ) mod τ k+1 . Finally, successively multiplying polynomials mod τ k+1 we compute the polynomial hi (τ ) with deg hi ≤ k such that qi pi1 . . . pik ≡ hi Letting h(τ ) =

mod τ k+1 .

hi (τ ),

i∈I

we conclude that the coefficient of τ k in h(τ ) is the desired value of (3.2) at τ = 0 and hence is the value f (S; x) at x = (1, . . . , 1). We note that the procedure has a


775

polynomial time complexity even if both k and d are allowed to vary and if we allow different numbers ki ≤ k of binomials in different fractions of (3.1). A more general operation which can be computed in polynomial time is that of a monomial substitution. Let f (x) be an expression of the type (3.1). Let z = (z1 , . . . , zn ) be a new set of variables, let l1 , . . . , ld ∈ Zn be vectors, and let φ : Cn −→ Cd be the transformation defined by (z1 , . . . , zn ) −→ (x1 , . . . , xd )

where xi = zli .

If the image φ(Cn ) does not lie in the set of poles of f , one can define a rational function g(z) = f (φ(z)). Function g can be computed in polynomial time in the form zqi g(z) = δi , (1 − zbi1 ) . . . (1 − zbiki ) i∈I where δi ∈ Q, qi , bij ∈ Zn , bij = 0 for all i, j and ki ≤ k for all i ∈ I . The case of l1 = · · · = ld = 0 corresponds to the case of x = (1, . . . , 1) considered above. As above, the general case of a monomial substitution is handled by a one-parametric perturbation and computation with univariate polynomials. Details can be found in [7] (the assumption that k is fixed in advance is not needed there). The operation of monomial substitution has the following geometric interpretation. Let T : Rd −→ Rn be the linear transformation whose matrix in the standard bases consists of the integer column vectors l1 , . . . , ld . Let S ⊂ Zd be a set and suppose that for all m ∈ T (S) the set T −1 (m) ∩ S is finite. The monomial substitution xi = zli into the generating function f (S; x) produces the weighted generating function g(z) of the image T (S) ⊂ Zn , where each monomial zm for m ∈ T (S) is counted with multiplicity |T −1 (m) ∩ S|. Another useful operation is that of differentiation. Let p be a d-variate polynomial. We can write ∂ ∂ m p(m)x = p x1 , . . . , xd f (S; x). ∂x1 ∂xd m∈S

As long as k is fixed in advance, the result can be computed in polynomial time in the form x qi δi , (1 − x bi1 )γi1 . . . (1 − x bik )γik

i∈I

where δi ∈ Q, ai , bij ∈ Zd , bij = 0, and γij are non-negative integers such that γi1 + · · · + γik ≤ k + deg p for all i, see [6]. This observation is used in [6], see also [10] and [13]. One corollary of Theorem 2.2 is that we can efficiently perform set-theoretic operations (intersection, union, difference) of finite sets defined by (3.1). The following result is proved in [7].

776

Alexander Barvinok

Theorem 3.1. Let us fix positive integers d and k. Then there exists a polynomial time algorithm, which, for any two finite sets S1 , S2 ⊂ Zd given by their rational generating functions f (S1 ; x) =

αi

x pi (1 − x ai1 ) . . . (1 − x aik )

(3.3)

βi

x qi (1 − x bi1 ) . . . (1 − x bik )

(3.4)

i∈I1

and f (S2 ; x) =

i∈I2

computes the generating function f (S; x) of their intersection S = S1 ∩ S2 in the form x ui f (S; x) = γi , (1 − x vi1 ) . . . (1 − x vis ) i∈I

where s ≤ 2k. Proof. The idea of the proof is to linearize the operation of intersection of sets. Suppose we have two Laurent series g1 (x) = ρ1m x m and g2 (x) = ρ2m x m . m∈Zd

m∈Zd

Let us define their Hadamard product g1 (x) g2 (x) as g(x) = ρm x m where ρm = ρ1m ρ2m . m∈Zd

Then, clearly, f (S1 ∩ S2 ; x) = f (S1 ; x) f (S2 ; x). Without loss of generality, we assume that there is a non-empty open set U ⊂ Cd such that for all x ∈ U and every fraction of (3.3) and (3.4) we have the multiple geometric series expansions: x pi = (1 − x ai1 ) . . . (1 − x aik ) and

x qi = (1 − x bi1 ) . . . (1 − x bik )

x pi +μ1 ai1 +···+μk aik

(3.5)

x qi +ν1 bi1 +···+νk bik .

(3.6)

(μ1 ,...,μk )∈Zk+

(ν1 ,...,νk )∈Zk+

As usual, we assume that for all x ∈ U the convergence in (3.5) and (3.6) is absolute and uniform on all compact subsets of U . To ensure that such a set U indeed exists, we choose a sufficiently generic linear function : Rd −→ R and make sure that


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(aij ), (bij ) > 0 for all i, j by reversing, if necessary, the direction of aij and bij via the identity 1 x −a = − . 1 − xa 1 − x −a Here we use that S1 and S2 are finite so that f (S1 ; x) and f (S2 ; x) are, in fact, Laurent polynomials. Since the Hadamard product is a bilinear operation on series, in order to compute f (S1 ; x) f (S2 ; x) it suffices to compute the Hadamard product of every pair of series (3.5) and (3.6). In the space R2k of 2k-tuples (x, y), where x = (ξ1 , . . . , ξk ) and y = (η1 , . . . , ηk ), let us introduce the polyhedron

ξ 1 , . . . , ξ k ; η1 , . . . , η k ≥ 0 Qi = (x, y) : (3.7) pi + ξ1 ai1 + · · · + ξk aik = qi + η1 bi1 + · · · + ηk bik and let Z2k ⊂ R2k be the standard integer lattice. Since the Hadamard product is bilinear and for monomials we have x m if m1 = m2 = m m1 m2 x x = 0 if m1 = m2 , the Hadamard product of the series (3.5) and (3.6) can be expressed as the sum x pi +μ1 ai1 +···+μk aik , (3.8) (m,n)∈Qi ∩Z2k

where m = (μ1 , . . . , μk ) and n = (ν1 , . . . , νk ). On the other hand, (3.8) is obtained from the generating function f (Qi , z) with z = (z1 , . . . , z2k ) by the monomial substitution zi = x ai for i = 1, . . . , k

and zi = 1 for i = k + 1, . . . , 2k

(3.9)

and multiplication by x pi . We use Theorem 2.2 to compute f (Qi , z). The monomial substitution (3.9) can also be computed in polynomial time, cf. Section 3.1. 2 Therefore, one can compute the generating functions of the union and difference: f (S1 ∪ S2 ; x) = f (S1 ; x) + f (S2 ; x) − f (S1 ∩ S2 ; x) and f (S1 \ S2 ; x) = f (S1 ; x) − f (S1 ∩ S2 ; x).

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Theorem 3.1 allows us to work with generating functions (3.1) directly as with data structures bypassing any more explicit descriptions of sets S in question. Of course, there is a price to pay: with every set-theoretic operation, the complexity level of the set, the number k of binomials in the denominator of each fraction in (3.1), doubles. From the definition (3.7) of Qi we can notice that in a sufficiently general position we will have dim Qi = 2k −d, so we would be able to choose s = 2k −d in Theorem 3.1. Theorem 3.1 admits an extension to infinite sets S1 and S2 provided there is a nonempty open set U ⊂ Cd such that the multiple geometric series expansions (3.5) and (3.6) hold for all fractions in (3.3) and (3.4). K. Woods [38] used the construction of the Hadamard product to show that in any fixed dimension there is a polynomial time algorithm to check if a given integer is a period of the Ehrhart quasi-polynomial of a given rational polytope.

4. Beyond polyhedra: projections There are other interesting sets admitting short rational generating functions (3.1). We start with examples. 4.1. Integer semigroups. Let S be the semigroup generated by positive coprime integers a1 and a2 , that is, the set of all non-negative integer combinations of a1 and a2 : S = {μ1 a1 + μ2 a2 : μ1 , μ2 ∈ Z+ }. It is not hard to show that f (S; x) =

1 − x a1 a2 (1 − x a1 )(1 − x a2 )

(the series defining f (S; x) converges for all |x| < 1). Let S be the semigroup generated by positive coprime integers a1 , a2 , and a3 , S = {μ1 a1 + μ2 a2 + μ3 a3 : μ1 , μ2 , μ3 ∈ Z+ }. Then there exist positive integers p1 , p2 , p3 , p4 , and p5 , not necessarily distinct, such that 1 − x p1 − x p2 − x p3 + x p4 + x p5 f (S; x) = . (1 − x a1 )(1 − x a2 )(1 − x a3 ) This interesting result was rediscovered a number of times. It was explicitly stated by M. Morales [27]; the proof was not published though. Independently, the proof was rediscovered by G. Denham [14]. Both proofs are algebraic and based on the interpretation of f (S; x) as the Hilbert series of a graded ring C[t a1 , t a2 , t a3 ]. In this special case (a Cohen–Macaulay ring of codimension 2), the Hilbert series can


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be computed via the Hilbert–Burch Theorem, cf. also [18]. Meanwhile, a combinatorial proof of a somewhat weaker result (up to 12 monomials in the numerator) independently appeared in [35]. The pattern breaks down for semigroups with d ≥ 4 generators, meaning that if we choose the denominator of f (S; x) in the form (1 − x a1 ) . . . (1 − x ad ), the number of monomials in the numerator does not remain constant for a particular value of d, and, moreover, grows exponentially with the input size of a1 , . . . , ad . As shown in [35], for d = 4 the number of the monomials in the numerator can grow as fast as min1/2 {a1 , a2 , a3 , a4 }, whereas the input size is only about log(a1 a2 a3 a4 ). Nevertheless, the generating function f (S; x) admits a short rational function representation for any number d of generators fixed in advance. The following result was proved in [7]. Theorem 4.1. Let us fix d. Then there exists a positive integer s = s(d) and a polynomial time algorithm, which, given positive integers a1 , . . . , ad , computes the generating function f (S; x) of the semigroup S=

d

μi ai : μ1 , . . . , μd ∈ Z+

i=1

generated by a1 , . . . , ad in the form f (S; x) = αi i∈I

x pi , (1 − x bi1 ) . . . (1 − x bis )

(4.1)

where αi ∈ Q, pi , bij ∈ Z and bij = 0 for all i, j . In particular, for any fixed d, the number |I | of fractions in (4.1) is bounded by a polynomial in the input size, that is, in log(a1 . . . ad ). Theorem 4.1 is obtained as a corollary of a more general result that the projection of the set of integer points in a rational polytope of a fixed dimension admits a short rational generating function [7]. Theorem 4.2. Let us fix d. Then there exists a number s = s(d) and a polynomial time algorithm, which, given a rational polytope P and a linear transformation T : Rd −→ Rk such that T (Zd ) ⊂ Zk , computes the generating function f (S; x) for S = T (P ∩ Zd ), S ⊂ Zk , in the form x pi f (S; x) = , (4.2) (1 − x bi1 ) . . . (1 − x bis ) i∈I

where αi ∈ Q, pi , bij ∈ Zk and bij = 0 for all i, j . One can observe that Theorem 4.1 is a corollary of Theorem 4.2. Indeed, let T : Rd −→ R be the linear transformation defined by T (ξ1 , . . . , ξd ) = a1 ξ1 + · · · + ad ξd .

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Then the semigroup S generated by a1 , . . . , ad is the image S = T (Zd+ ) of the set Zd+ of integer points in the rational polyhedron Rd+ ⊂ Rd . The polyhedron Rd+ is unbounded, so Theorem 4.2 cannot be applied immediately. However, it is not hard to show that S ⊂ Z+ stabilizes after a while (if a1 , . . . , ad are coprime then S includes all sufficiently large positive integers). Thus only the initial interval of S is of interest, to get which we replace Rd+ by a sufficiently large simplex d P = (ξ1 , . . . , ξd ) : ξi ≤ t and ξi ≥ 0 for i = 1, . . . , d , i=1

see [7] for details. We sketch the proof of Theorem 4.2 below. Without loss of generality we assume that dim ker T = d − k. The proof then proceeds by induction on d − k. If d = k we are in the situation of Theorem 2.2. We note that for any k and d, if the restriction T : P ∩ Zd −→ S is one-to-one, we can compute the generating function f (S; x) from that of the set P ∩ Zd using an appropriate monomial substitution, cf. Section 3.1. Otherwise, the monomial substitution will account for each point m ∈ S with the multiplicity equal to the number of the points in P ∩ Zd mapped onto m. Thus our goal is to eliminate multiplicities. The case of d = k + 1 illuminates some of the ideas used in the proof for an arbitrary d − k. Suppose that T : Rk+1 −→ Rk ,

(ξ1 , . . . , ξk+1 ) −→ (ξ1 , . . . , ξk )

is the projection (this is a sufficiently general case). Let Sˆ = P ∩ Zk+1 and let us consider the restriction T : Sˆ −→ S. Then, for every point m ∈ S, the preimage T −1 (m) ⊂ Sˆ is the set of integer points in the interval T −1 (m) ∩ P which all agree in their first k coordinates and disagree in the last coordinate. Let ek+1 be the last basis vector and let us consider Y = Sˆ \ ( Sˆ + ek+1 ). In words: we subtract from Sˆ its translation by 1 in the last coordinate. Then the restriction T : Y −→ S is one-to-one since the preimage T −1 (m) ⊂ Y consists of the single point in T −1 (m) ⊂ Sˆ with the smallest last coordinate. Now, Sˆ is the set of integer points in a rational polytope and we compute its generating function using Theorem 2.2. Then we compute the generating function of Y using Theorem 3.1. Finally, we obtain f (S; x) by substituting xk+1 = 1 in the generating function f (Y ; (x, xk+1 )), cf. Section 3.1. Let us consider the case of general k and d. Let pr : Zk+1 −→ Zk be the natural projection, pr(μ1 , . . . , μk+1 ) = (μ1 , . . . , μk ). Let Tˆ : Zd −→ Zk+1 be a linear transformation which is a lifting of T so that pr(Tˆ (m)) = T (m) for all m ∈ Zd . We define Sˆ = Tˆ (S), Sˆ ⊂ Zk+1 , and consider the restriction pr : Sˆ −→ S.


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For every m ∈ S the preimage pr −1 (m) ⊂ Sˆ consists of the points which differ in their last coordinate only. Suppose that we managed to construct Tˆ in such a way that the set pr −1 (m) ⊂ Sˆ has small gaps, meaning that there exists a constant l = l(d) such that if there are two points in pr −1 (m) whose (k + 1)st coordinates differ by more than l, there must be a point in pr −1 (m) lying strictly between them. In this case, we compute f (S; x) as follows. Let us define Y = Sˆ \

l

(Sˆ + j ek+1 ).

j =1

In words: we subtract from Sˆ its l translates by 1, . . . , l in the last coordinate. Because of the small gap property, the restriction pr : Y −→ S is one-to-one: now, the preimage pr −1 (m) ⊂ Y consists of the single point in pr −1 (m) ⊂ Sˆ with the smallest last coordinate. Using the induction hypothesis, we compute the generating function ˆ Then, applying Theorem 3.1 l times, we compute the generating function of Y . of S. Finally, f (S; x) is obtained from f (Y ; (x, xk+1 )) by the substitution xk+1 = 1, see Section 3.1. In general, we cannot construct a lifting Tˆ with the small gap property but the next best thing is possible. Namely, we can construct in polynomial time a decomposition Rk = i Qi of Rk into a union of non-overlapping rational polyhedra Qi such that for each piece Si = S ∩ Qi a lifting Tî with the small gap property indeed exists. The generating functions f (Si ; x) are computed as above and then patched together into a single generating function f (S; x). The construction of such polyhedra Qi and liftings Tî is based on the results of [21] and [20]. The main tool is the following Flatness Theorem, see, for example, Section 6.7 of [17] or Section VII.8 of [5]. Theorem 4.3 (Flatness Theorem). For each dimension d there exists a constant ω(d) with the following property: if V is a d-dimensional real vector space, ⊂ V is a lattice of rank d, ∗ ⊂ V ∗ is the reciprocal lattice, and K ⊂ V is a convex compact set with non-empty interior such that K ∩ = ∅ then there is an ∈ ∗ \ {0} such that (4.3) max (x) − min (x) ≤ ω(d). x∈K

x∈K

In words: a lattice-free convex body is flat in some lattice direction. The number in the left hand side of (4.3) is called the width of K with respect to and denoted width(K, ). The infimum of width(K, ) over all ∈ ∗ is called the lattice width of K and denoted width(K). A simple and crucial observation relating the lattice width and the small gap property is that if for ∈ ∗ we have width(K, ) ≤ γ width(K) then the gaps between the consecutive integers in the set (K ∩ ) do not exceed γ ω(d). We go back to finish the sketch of the proof of Theorem 4.2. Let = Zk ∩ ker(T ) be the lattice in ker(T ). For y ∈ Rd , let Py = P ∩T −1 (x) be the fiber of the polytope P over x. We will measure the lattice width of Py with respect to . The results of [21]

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Alexander Barvinok

and [20] allow us to construct a polyhedral decomposition Rk = i Qi and vectors i ∈ ∗ such that for all y ∈ Qi we have either width(Py , i ) ≤ 2 width(Py ) or width(Py , i ) ≤ 1. We then define Tî (x) = (T (x), i (x)) if T (x) ∈ Qi . This completes the sketch of proof of Theorem 4.2. 4.2. Applications. Theorem 4.1 implies polynomial time solvability of a variety of problems about integer semigroups. Suppose that the generators a1 , . . . , ad are coprime. As is known, all sufficiently large integers lie in the semigroup S generated by a1 , . . . , ad . In the situation when the number d of generators is fixed, R. Kannan [20] constructed a polynomial time algorithm to compute the largest integer not in S. Theorem 4.1 implies that one can compute in polynomial time the number of positive integers not in S, the number of integers in S belonging to a particular interval, etc. Unlike the algorithm of Theorem 2.2, the algorithms of Theorems 4.1 and 4.2 seem to be unimplementable at the moment. Indeed, the way Theorem 4.2 is proved gives s = d (d) at best and, similarly, in Theorem 4.1. It is not clear at the moment whether a smaller value of s is possible. In Theorem 4.1, apart from d = 1, 2, 3, the value of d = 4 seems to indicate a possibility of a “special treatment”. The approach of [33] combined with the continued fraction method, see Section 2.1, may lead to a practically efficient algorithm to compute f (S; x). Theorem 4.2 implies that some other interesting sets admit short rational generating functions. One class of such sets consists of the Hilbert bases of rational cones. / S, is called the Let K ⊂ Rd be a pointed rational cone. The set S ⊂ K ∩ Zd , 0 ∈ (minimal) Hilbert basis of the semigroup K ∩ Zd if every point in K ∩ Zd can be represented as a sum of some points in S and if no point in S is a sum of other points in S. In other words, S consists of the points in K ∩ Zd that cannot be written as a sum of non-zero points in K ∩ Zd . Theorem 4.2 implies that as long as the dimension d remains fixed, given a rational cone K, the generating function f (S; x) can be computed in polynomial time as a short rational function of the type (3.1). Consequently, the number |S| of points in the Hilbert basis of K ∩ Zd can be computed in polynomial time. To deduce this result from Theorem 4.2, let Q ⊂ K be a rational polyhedron containing all integer points in K except 0 (to get Q from K, we cut the vertex of K by a hyperplane), let P = Q × Q ⊂ Rd ⊕ Rd = R2d and let T be the projection P −→ K, T (x, y) = x + y. Then the Hilbert basis S is the complement in Q ∩ Zd of the image T (P ∩ Z2d ). The obstacle that the polyhedron Q is not bounded, so Theorem 4.2 cannot be applied immediately, can be easily fixed since only the “initial part” of the semigroup K ∩ Zd is of interest, see [7]. Another class of sets allowing short rational generating functions via Theorem 4.2 are the test sets in integer programming, see [30].


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It should be noted that the short rational function description provides only very general characterization of the set. For example, many of the fine properties of test sets [30] do not seem to be picked up by rational generating functions and some empirically observed phenomena are still waiting for their explanation. For structural results (without complexity estimates) regarding f (S; x), where S is the projection of the set of integer points in a rational polyhedron, see [24].

5. Beyond projections: Presburger arithmetic Let us consider formulas we can construct by using integer variables, operations of addition, subtraction, and multiplication by an integer constant (but not multiplication of two integer variables), comparison (, =), Boolean operations (“and”, “or”, “not”), and quantifiers (∀, ∃). The realm of such formulas is Presburger arithmetic. Thus the set P ∩ Zd of integer points in a rational polyhedron can be described by a quantifier-free formula of Presburger arithmetic: the set P ∩ Zd consists of the d-tuples of integer variables that satisfy a number of linear constraints with constant integer coefficients. Similarly, the projection T (P ∩ Zd ) of the set of integer points in a polyhedron is described by a formula of Presburger arithmetic with existential quantifiers only (no quantifier alternations). With a little work, Theorem 2.2 can be extended as follows. Let us fix the number d of variables. Then there exists a polynomial time algorithm, which, given a quantifierfree formula F of Presburger arithmetic, computes the generating function f (S; x) of the set S ⊂ Zd defined by F as a rational function (2.3). Some routine precautions regarding convergence of the series defining f (S; x), if S is infinite, should be taken. The general case of a set defined by a quantifier-free formula F reduces to that of the set integer points in a rational polyhedron by some more or less straightforward “cutting and pasting” of polyhedra. Since the dimension d of the ambient space is fixed, this cutting and pasting can be performed in polynomial time. Theorem 4.2 can be extended as follows. Let us fix the number of variables and the number of Boolean operations used. Then there exists a polynomial time algorithm, which, given a formula F of Presburger arithmetic without quantifier alternations, computes the generating function f (S; x) of the finite set S ⊂ Zk defined by F as a rational function (4.2). Note that here we have to fix not only the number of variables but also the number of Boolean operations. For example, unless P = NP one cannot hope to compute the generating function of the projection of the set of integer points in a union of rational polytopes if the number of polytopes is allowed to vary, cf. Section 5.3 of [37] and [31]. One can ask whether the results can be extended even further. Let us fix the number of variables and the number of Boolean operations, making numerical constants essentially the only parameters of the formula. Is there a polynomial time algorithm which computes the generating function (3.1) of the (finite) set S of points described by such a formula? This indeed seems very plausible, see the discussion in Chapter V

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of [37]. Intuitively, such sets should have some “hidden periodicity” and short rational generating functions should reveal that periodicity. Besides, it seems hard to prove that a particular finite, but large, set S ⊂ Zd does not admit a short rational generating function: if a particular candidate expression for f (S; x) is not short, one can argue that we have not searched hard enough and that there is another, better candidate. We mention that the result of R. Kannan [19] establishes polynomial time solvability of decision problems for formulas with not more than one quantifier alternation. If the number of variables is not fixed, the complexity of decision problems in Presburger arithmetic is double exponential by the result of M. Fischer and M. Rabin [15].

6. Concluding remarks One can ask whether some of the technique discussed in this paper can be extended to lattice points satisfying some non-linear constraints. The answer seems to be “no”. For example, lattice points in the standard Euclidean ball exhibit phenomena explained not by rational but rather by theta functions. Let Bn = {(ξ1 , ξ2 , ξ3 , ξ4 ) : ξ12 + ξ22 + ξ32 + ξ42 ≤ n} √ be the Euclidean ball of radius equal to n. Jacobi’s formula asserts √ that the number |Bn ∩ Z4 | − |Bn−1 ∩ Z4 | of integer points on the sphere of radius n is equal to 8 r 4 r|n

(in words: eight times the sum of divisors of n that are not divisible by four). One can then show ([1]) that if one can count points in a 4-dimensional ball efficiently (in polynomial time), one can factor integers efficiently (in randomized polynomial time). We note also that lattice points in irrational polyhedra exhibit a very interesting behavior, see [32]. 6.1. Large dimensions. Almost everywhere in this paper we assumed that the dimension d of the ambient space is fixed in advance. But what if the dimension is allowed to grow? Given a rational polyhedron P ⊂ Rd , it is an NP-hard problem to determine whether P ∩ Zd = ∅ (even when P is a rational simplex). Thus there is little hope to compute the generating function f (P , x) in polynomial time. However, it appears that some interesting “residues” or “shadows” of f (P , x) can be efficiently computed even when the dimension d is allowed to grow, cf. [10] and [6]. The number e(P ) = |P ∩ Zd | of integer points in a rational polyhedron is an example of a lattice invariant valuation, see [26]. That is, the map P −→ e(P ) extends to a linear functional on the space spanned by the indicators [P ] of rational polyhedra, cf. Definition 2.3, and the linear functional is invariant under lattice shifts:


785

e(P ) = e(P + u), u ∈ Zd . One can ask if there is another lattice invariant valuation ν on rational polytopes which is efficiently computable in interesting cases and which, in some sense, approximates the counting valuation e(P ). For example, the volume vol P may serve as the “0th” approximation to e(P ). With every lattice invariant valuation ν one can associate the expression ν(nP ) =

d

νi (P ; n)ni ,

(6.1)

i=0

where nP is a dilation of P by an integer factor n and the coefficients νi (P ; n) are quasi-periodic: νi (P ; n + t) = νi (P ; n) provided tP is a polytope with integer vertices, cf. [26]. In the case of the counting valuation e, the expression (6.1) is called the Ehrhart quasi-polynomial of P and ed (P ; n) = vol P . As the kth approximation to the counting valuation e we consider a lattice invariant valuation ν which agrees with e in the k + 1 highest terms: νi (P ; n) = ei (P ; n) for

i = d, d − 1, . . . , d − k.

A natural goal is to construct such a valuation ν, which is computable in polynomial time (at least, in some interesting cases) for any k fixed in advance. Abstractly speaking, to define the counting valuation e, we have to choose a finitedimensional real vector space V and a lattice ⊂ V . Then we define e(P ) = |P ∩| for every polytope P ⊂ V such that the vertices of tP belong to for some integer t. Apparently, to make a canonical choice of ν, we have to fix some additional structure in V . In [6] a canonical valuation ν is constructed for rational polytopes whose facets are parallel to hyperplanes from a given finite collection of hyperplanes. Valuation ν agrees with e in the k +1 highest terms and for any fixed k valuation ν is polynomially computable on polytopes with the number facets exceeding the dimension d by not more than a constant fixed in advance (in particular, on rational simplices). In [10] a different canonical valuation μ is constructed provided a scalar product on V is chosen. Valuation μ also agrees with e on the k + 1 highest terms and polynomially computable on the same class of polytopes.

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Barvinok, A., A Course in Convexity. Graduate Studies in Mathematics 54, American Mathematical Society, Providence, RI, 2002.

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Barvinok, A., Computing the Ehrhart quasi-polynomial of a rational simplex. Math. Comput. 75 (2006), 1449–1466.

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[10] Berline, N., Vergne, M., Local Euler-Maclaurin formula for polytopes. Preprint, arXiv math.CO/0507256, 2005. [11] Brion, M. Points entiers dans les polyèdres convexes. Ann. Sci. École Norm. Sup. (4) 21 (1988), 653–663. [12] De Loera, J. A., Hemmecke, R., Tauzer, J., Yoshida, R., Effective lattice point counting in rational convex polytopes. J. Symbolic Comput. 38 (2004), 1273–1302; see also http://www.math.ucdavis.edu/~latte/ [13] De Loera, J. A., Hemmecke, R., Köppe, M., Weismantel, R., Integer polynomial optimization in fixed dimension. Math. Oper. Res., to appear. [14] Denham, G., Short generating functions for some semigroup algebras. Electron. J. Combin. 10 (2003), Research Paper 36, 7 pp. (electronic). [15] Fischer, M. J., Rabin, M. O., Super-exponential complexity of Presburger arithmetic. In Complexity of Computation (Proc. SIAM-AMS Sympos., New York, 1973), SIAM-AMS Proc. VII, Amer. Math. Soc., Providence, R.I., 1974, 27–41. [16] Fulton, W., Introduction to Toric Varieties. Annals of Mathematics Studies 131, Princeton University Press, Princeton 1993. [17] Grötschel, M., Lovász, L., Schrijver, A., Geometric Algorithms and Combinatorial Optimization. Second edition, Algorithms and Combinatorics 2, Springer-Verlag, Berlin 1993. [18] Herzog, J., Generators and relations of abelian semigroups and semigroup rings. Manuscripta Math. 3 (1970), 175–193. [19] Kannan, R., Test sets for integer programs, ∀∃ sentences. In Polyhedral Combinatorics (Morristown, NJ, 1989), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 1, Amer. Math. Soc., Providence, RI, 1990, 39–47. [20] Kannan, R., Lattice translates of a polytope and the Frobenius problem. Combinatorica 12 (1992), 161–177. [21] Kannan, R., Lovász, L., Scarf, H. E., The shapes of polyhedra. Math. Oper. Res. 15 (1990), 364–380. [22] Khinchin, A. Ya., Continued Fractions. The University of Chicago Press, Chicago, IL, London 1964. [23] Khovanskii, A. G., Pukhlikov, A. V., The Riemann-Roch theorem for integrals and sums of quasipolynomials on virtual polytopes. Algebra i Analiz 4 (4) (1992), 188–216; English translation in St. Petersburg Math. J. 4 (4) (1993), 789–812.


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[24] Khovanskii, A. G., Sums of finite sets, orbits of commutative semigroups and Hilbert functions. Funktsional. Anal. i Prilozhen. 29 (2) (1995), 36–50, 95; English translation in Funct. Anal. Appl. 29 (2) (1995), 102–112. [25] Lawrence, J., Rational-function-valued valuations on polyhedra. In Discrete and Computational Geometry (New Brunswick, NJ, 1989/1990), DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 6, Amer. Math. Soc., Providence, RI, 1991, 199–208. [26] McMullen, P., Valuations and dissections. In Handbook of Convex Geometry, Vol. B, NorthHolland, Amsterdam 1993, 933–988. [27] M. Morales, Syzygies of monomial curves and a linear diophantine problem of Frobenius. Preprint, Max-Planck-Institut für Mathematik, Bonn 1986. [28] Papadimitriou, C. H., Computational Complexity. Addison-Wesley Publishing Company, Reading, MA, 1994. [29] Pommersheim, J., Thomas, H., Cycles representing the Todd class of a toric variety. J. Amer. Math. Soc. 17 (2004), 983–994. [30] Scarf, H. E., Test sets for integer programs. In Lectures on Mathematical Programming (ISMP97, Lausanne, 1997), Math. Programming 79 (1997), no. 1-3, Ser. B, 355–368. [31] Schöning, U., Complexity of Presburger arithmetic with fixed quantifier dimension. Theory Comput. Syst. 30 (1997), 423–428. [32] Skriganov, M. M., Ergodic theory on SL(n), Diophantine approximations and anomalies in the lattice point problem. Invent. Math. 132 (1998), 1–72. [33] Shallcross, D., Neighbors of the origin for four by three matrices. Math. Oper. Res. 17 (1992), 608–614. [34] Stanley, R. P., Enumerative Combinatorics. Vol. 1, corrected reprint of the 1986 original, Cambridge Studies in Advanced Mathematics 49, Cambridge University Press, Cambridge 1997. [35] Székely L. A., Wormald, N. C., Generating functions for the Frobenius problem with 2 and 3 generators. Math. Chronicle 15 (1986), 49–57. [36] Verdoolaege, S., Woods, K., Bruynooghe M., Cools R., Computation and manipulation of enumerators of integer projections of parametric polytopes. Preprint Katholieke Universiteit Leuven, Dept. of Computer Science, Report CW 392, 2005; see also http://www.kotnet.org/~skimo/barvinok/ [37] Woods, K. M., Rational generating functions and lattice point sets. Diss. University of Michigan, 2004. [38] Woods, K., Computing the period of an Ehrhart quasipolynomial. Electron J. Combin. 12 (2005), Research paper 34, 12 pp. (electronic). Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1043, U.S.A. E-mail: [email protected]

Rational and algebraic series in combinatorial enumeration Mireille Bousquet-Mélou

Abstract. Let A be a class of objects, equipped with an integer size such that for all n the number a n of objects of size n is finite. We are interested in the case where the generating function n an t n is rational, or more generally algebraic. This property has a practical interest, since one can usually say a lot on the numbers an , but also a combinatorial one: the rational or algebraic nature of the generating function suggests that the objects have a (possibly hidden) structure, similar to the linear structure of words in the rational case, and to the branching structure of trees in the algebraic case. We describe and illustrate this combinatorial intuition, and discuss its validity. While it seems to be satisfactory in the rational case, it is probably incomplete in the algebraic one. We conclude with open questions. Mathematics Subject Classification (2000). Primary 05A15; Secondary 68Q45. Keywords. Enumerative combinatorics, generating functions, rational and algebraic power series, formal languages.

1. Introduction The general topic of this paper is the enumeration of discrete objects (words, trees, graphs,…) and more specifically the rational or algebraic nature of the associated generating functions. Let A be a class of discrete objects equipped with a size: size : A → N A → |A|. Assume that for all n, the number an of objects of size n is finite. The generating function of the objects of A, counted by their size, is the following formal power series in the indeterminate t: A(t) := an t n = t |A| . (1) n≥0

A∈A

To take a very simple example, if A is the set of words on the alphabet {a, b} and the size of a word is its number of letters, then the generating function is n≥0 2n t n = 1/(1 − 2t). Generating functions provide both a tool for solving counting problems, and a concise way to encode their solution. Ideally, one would probably dream of finding a closed formula for the numbers an . But the world of mathematical objects would be extremely poor if this was always possible. In practise, one is usually happy with an Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Mireille Bousquet-Mélou

expression of the generating function A(t), or even with a recurrence relation defining the sequence an , or a functional equation defining A(t). Enumerative problems arise spontaneously in various fields of mathematics, computer science, and physics. Among the most generous suppliers of such problems, let us cite discrete probability theory, the analysis of the complexity of algorithms [56], [44], and the discrete models of statistical physics, like the famous Ising model [5]. More generally, counting the objects that occur in one’s work seems to answer a natural curiosity. It helps to understand the objects, for instance to appreciate how restrictive are the conditions that define them. It also forces us to get some understanding of the structure of the objects: an enumerative result never comes for free, but only after one has elucidated, at least partly, what the objects really are. We focus in this survey on objects having a rational, or, more generally, algebraic generating function. Rational and algebraic formal power series are well-behaved objects with many interesting properties. This is one of the reasons why several classical textbooks on enumeration devote one or several chapters to these series [43], [74], [75]. These chapters give typical examples of objects with a rational [resp. algebraic] generating function (GF). After a while, the collection of these examples builds up a general picture: one starts thinking that yes, all these objects have something in common in their structure. At the same time arises the following question: do all objects with a rational [algebraic] GF look like that? In other words, what does it mean, what does it suggest about the objects when they are counted by a rational [algebraic] GF? This question is at the heart of this survey. For each of the two classes of series under consideration, we first present a general family of enumerative problems whose solution falls invariably in this class. These problems are simple to describe: the first one deals with walks in a directed graph, the other with plane trees. Interestingly, these families of objects admit alternative descriptions in language theoretic terms: they correspond to regular languages, and to unambiguous context-free languages, respectively. The words of these languages have a clear recursive structure, which explains directly the rationality [algebraicity] of their GF. The series counting words of a regular [unambiguous context-free] language are called N-rational [N-algebraic]. It is worth noting that a rational [algebraic] series with non-negative coefficients is not necessarily N-rational [N-algebraic]. Since we want to appreciate whether our two generic classes of objects are good representatives of objects with a rational [algebraic] GF, the first question to address is the following: do we always fall in the class of N-rational [N-algebraic] series when we count objects with a rational [algebraic] GF? More informally, do these objects exhibit a structure similar to the structure of regular [context-free] languages? Is such a structure usually clearly visible? That is to say, is it easy to feel, to predict rationality [algebraicity]? We shall see that the answer to all these questions tends to be yes in the rational case (with a few warnings...) but is probably no in the algebraic case. In particular, the rich world of planar maps (planar graphs embedded in the sphere) abounds in candidates for non-N-algebraicity. The algebraicity of the associated GFs has been

Rational and algebraic series in combinatorial enumeration

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known for more than 40 years (at least for some families of maps), but it is only in the past 10 years that a general combinatorial explanation of this algebraicity has emerged. Moreover, the underlying constructions are more general that those allowed in context-free descriptions, as they involve taking complements. Each of the main two sections ends with a list of questions. In particular, we present at the end of Section 3 several counting problems that are simple to state and have an algebraic GF, but for reasons that remain mysterious. The paper is sometimes written in an informal style. We hope that this will not stop the reader. We have tried to give precise references where he/she will find more details and more material on the topics we discuss. In particular, this survey borrows a lot to two books that we warmly recommend: Stanley’s Enumerative Combinatorics [74], [75], and Flajolet & Sedgewick’s Analytic Combinatorics [43]. Notation and definitions. Given a (commutative) ring R, we denote by R[t] the ring of polynomials in t having coefficients in R. A Laurent series in t is a series of the form A(t) = n≥n0 an t n , with n0 ∈ Z and an ∈ R for all n. If n0 ≥ 0, we say that A(t) is a formal power series. The coefficient of t n is denoted an := [t n ]A(t). The set of Laurent series forms a ring, and even a field if R is a field. The quasi-inverse of A(t) is the series A∗ (t) := 1/(1 − A(t)). If A(t) is a formal power series with constant term 0, then A∗ (t) is a formal power series too. In most occasions, the series we consider are GFs of the form (1) and thus have rational coefficients. However, we sometimes consider refined enumeration problems, in which every object A is weighted, usually by a monomial w(A) in some additional indeterminates x1 , . . . , xm . The weighted GF is then A∈A w(A)t |A| , so that the coefficient ring is Q[x1 , . . . , xm ] rather than Q. We denote [[k]] = {1, 2, . . . , k}. We use the standard notation N, Z, Q, and P := {1, 2, 3, . . . }.

2. Rational generating functions 2.1. Definitions and properties. The Laurent series A(t) with coefficients in the field R is said to be rational if it can be written in the form A(t) =

P (t) Q(t)

where P (t) and Q(t) belong to R[t]. There is probably no need to spend a lot of time explaining why such series are simple and well-behaved. We refer to [74, Ch. 4] and [43, Ch. IV] for a survey of their properties. Let us review briefly some of them, in the case where R = Q. The set of (Laurent) rational series is closed under sum, product, derivation, reciprocals – but not under integration as shown by A(t) = 1/(1 − t). The coefficients an of a rational series A(t) satisfy a linear recurrence relation with constant coefficients:

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for n large enough, an = c1 an−1 + c2 an−2 + · · · + ck an−k . The partial fraction expansion of A(t) provides a closed form expression of these coefficients of the form: k an = Pi (n)μni (2) i=0

where the μi are the reciprocals of the roots of the denominator Q(t), and the Pi are polynomials. In particular, if A(t) has non-negative integer coefficients, its radius of convergence ρ is one its the poles (Pringsheim) and the “typical” asymptotic behaviour of an is (3) an ∼ κρ −n nd where d ∈ N and κ is an algebraic number. The above statement has to be taken with a grain of salt: all poles of minimal modulus may actually contribute to the dominant term in the asymptotic expansion of an , as indicated by (2). Let us add that Padé approximants allow us to guess whether a generating function whose first coefficients are known is likely to be rational. For instance, given the 10 first coefficients of the series A(t) = t + 2 t 2 + 6 t 3 + 19 t 4 + 61 t 5 + 196 t 6 + 629 t 7 + 2017 t 8 + 6466 t 9 + 20727 t 10 + O(t 11 ), it is easy to conjecture that actually A(t) =

t (1 − t)3 . 1 − 5 t + 7 t2 − 4 t3

Padé approximants are implemented in most computer algebra packages. For instance, the relevant Maple command is convert/ratpoly. 2.2. Walks on a digraph. We now introduce our typical “rational” objects. Let G = (V , E) be a directed graph with (finite) vertex set V = [[p]] and (directed) edge set E ⊂ V ×V . A walk of length n on G is a sequence of vertices w = (v0 , v1 , . . . , vn ) such that for all i, the pair (vi , vi+1 ) is an edge. Such a walk goes from v0 to vn . We denote |w| = n. Now assign to each directed edge e a weight (an indeterminate) xe . Define the weight xw of the walk w as the product of the weights of the edges it visits: more precisely, n−1 xw = x(vi ,vi+1 ) . i=0

See Figure 1 (a) for an example. Let X denote the (weighted) adjacency matrix of G: for i and j in [[p]], the entry Xi,j is xe if (i, j ) = e is an edge of G and 0 otherwise.

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a 1

1

4

2

c

c y

3

5

c

x

(a)

(b)

a

a

2

4

c

a

a

b

b

3

b

5

c

b c

Figure 1. (a) A weighted digraph. The default value of the weight is 1. (b) A deterministic automaton on the alphabet {a, b, c, a, b, c}. The initial state is 1 and the final states are 2 and 3.

Let Wi,j (t) be the weighted generating function of walks going from i to j : Wi,j (t) = xw t |w| . w : i ;j

It is well-known, and easy to prove, that Wi,j is a rational function in t with coefficicients in Q[xe , e ∈ E] (see [74, Thm. 4.7.1]). Theorem 2.1. The series Wi,j (t) is the (i, j )-entry in the matrix (1 − tX)−1 . This theorem reduces the enumeration of walks on a digraph to the calculation of the inverse of a matrix with polynomial coefficients. It seems to be little known in the combinatorics community that this inverse matrix can be computed by studying the elementary cycles of the digraph G. This practical tool relies on Viennot’s theory of heaps of pieces [81]. Since it is little known, and often convenient, let us advertise it here. It will be illustrated further down. An elementary cycle of G is a closed walk w = (v0 , v1 , . . . , vn−1 , v0 ) such that v0 , . . . , vn−1 are distinct. It is defined up to a cyclic permutation of the vi . That is, (v1 , v2 , . . . , vn−1 , v0 , v1 ) is the same cycle as w. A collection γ = {γ1 , . . . , γr } of (elementary) cycles is non-intersecting if the γi are pairwise disjoint. The weight xγ of γ is the product of the weights of the γi . We denote |γ | = |γi |. Proposition 2.2 ([81]). The generating function of walks going from i to j reads Wi,j (t) = where D=

γ ={γ1 ,...,γr }

(−1)r xγ t |γ |

and

Ni,j , D

Ni,j =

(−1)r xw xγ t |w|+|γ | .

w;γ ={γ1 ,...,γr }

The polynomial D is the alternating generating function of non-intersecting collections of cycles. In the expression of N, γ a non-intersecting collection of cycles and w a self-avoiding walk going from i to j , disjoint from the cycles of γ .

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To illustrate this result, let us determine the generating function of walks going from 1 to 2 and from 1 to 3 on the digraph of Figure 1 (a). This graph contains 4 cycles of length 1, 2 cycles of length 2, 2 cycles of length 3 and 1 cycle of length 4. By forming all non-intersecting collections of cycles, one finds: D(t) = 1−(3+x)t +(3+3x −2)t 2 +(−1−3x +3+x −2)t 3 +(x −1−x +1+x −y)t 4 = 1 − (3 + x)t + (1 + 3x)t 2 − 2xt 3 + (x − y)t 4 . There is only one self-avoiding walk (SAW) going from 1 to 2, and one SAW going from 1 to 3 (via the vertex 2). The collections of cycles that do not intersect these walks are formed of loops, which gives N1,2 = t (1 − t)2 (1 − xt) and

N1,3 = t 2 (1 − t)2 .

Hence the generating function of walks that start from 1 and end at 2 or 3 is: W1,2 + W1,3 =

N1,2 + N1,3 t (1 − t)2 (1 + t − xt) . = D 1 − (3 + x)t + (1 + 3x)t 2 − 2xt 3 + (x − y)t 4

(4)

2.3. Regular languages and automata. There is a very close connection between the collection of walks on a digraph and the words of regular languages. Let A be an alphabet, that is, a finite set of symbols (called letters). A word on A is a sequence u = u1 u2 . . . un of letters. The number of occurrences of the letter a in the word u is denoted |u|a . The product of two words u1 u2 . . . un and v1 v2 . . . vm is the concatenation u1 u2 . . . un v1 v2 . . . vm . The empty word is denoted ε. A language on A is a set of words. We define two operations on languages: – the product LK of two languages L and K is the set of words uv, with u ∈ L and v ∈ K; this product is easily seen to be associative, – the star L∗ of the language L is the union of all languages Lk , for k ≥ 0. By convention, L0 is reduced to the empty word ε. A finite state automaton on A is a digraph (V , E) with possibly multiple edges, together with: – a labelling of the edges by letters of A, that is to say, a function L : E → A, – an initial vertex i, – a set Vf ⊂ V of final vertices. The vertices are usually called the states of the automaton. The automaton is deterministic if for every state v and every letter a, there is at most one edge labelled a starting from v. To every walk on the underlying multigraph, one associates a word on the alphabet A by reading the letters met along the walk. The language L recognized by the automaton is the set of words associated with walks going from the initial state i to


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one of the states of Vf . For j ∈ V , let Lj denote the set of words associated with walks going from i to j . These sets admit a recursive description. For the automaton of Figure 1 (b), one has L = L2 ∪ L3 with L1 = {ε}, L2 = L1 c ∪ L2 a ∪ L3 a ∪ L4 c,

L4 = L2 a ∪ L3 a ∪ L4 a ∪ L5 b,

L3 = L2 c ∪ L3 b ∪ L3 c,

L5 = L2 c ∪ L3 b ∪ L3 c ∪ L5 b.

Remarkably, there also exists a non-recursive combinatorial description of the languages that are recognized by an automaton [52, Thms. 3.3 and 3.10]. Theorem 2.3. Let L be a language on the alphabet A. There exists a finite state automaton that recognizes L if and only if L can be expressed in terms of finite languages on A, using a finite number of unions, products and stars of languages. If these conditions hold, L is said to be regular. Moreover, there exists a deterministic automaton that recognizes L. Regular languages and walks on digraphs. Take a deterministic automaton, and associate with it a weighted digraph as follows: the vertices are those of the automaton, and for all vertices j and k, if m edges go from j to k in the automaton, they are replaced by a single edge labelled m in the digraph. For instance, the automaton of Figure 1 (b) gives the digraph to its left, with x = y = 2. Clearly, the length GF of words of L is the GF of (weighted) walks of this digraph going from the initial vertex i to one of the final vertices of Vf . For instance, according to (4), the length GF of the language recognized by the automaton of Figure 1 (b) is A(t) =

t (1 − t)3 . 1 − 5 t + 7 t2 − 4 t3

(5)

Take a regular language L recognized by a deterministic automaton A. There exists another deterministic automaton that recognizes L and does not contain multiple edges. The key is to create a state (j, a) for every edge labelled a ending at j in the automaton A. The digraph associated with this new automaton has all its edges labelled 1, so that there exists a length preserving bijection between the words of L and the walks on the digraph going from a specified initial vertex v0 to one of the vertices of a given subset Vf of vertices. Conversely, starting from a digraph with all edges labelled 1, together with a specified vertex v0 and a set Vf of final vertices, it is easy to construct a regular language that is in bijection with the walks of the graph going from v0 to Vf (consider the automaton obtained by labelling all edges with distinct letters). This shows that counting words of regular languages is completely equivalent to counting walks in digraphs. In particular, the set of rational series obtained in both types of problems coincide, and have even been given a name: n Definition 2.4. A series A(t) = n≥0 an t with coefficients in N is said to be N-rational if there exists a regular language having generating function A(t) − a0 .

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The description of regular languages given by Theorem 2.3 implies that the set of N-rational series contains the smallest set of series containing N[t] and closed under sum, product and quasi-inverse. The converse is true [71, Thm. II.5.1]. There exists a simple way to decide whether a given rational series with coefficients in N is N-rational [71, Thms. II.10.2 and II.10.5]. Theorem 2.5. A series A(t) = n≥0 an t n with coefficients in N is N-rational if and only if there exists a positive integer p such that for all r ∈ {0, . . . , p}, the series anp+r t n Ar,p (t) := n≥0

has a unique singularity of minimal modulus (called dominant). There exist rational series with non-negative integer coefficients that are not Nrational. For instance, let α be such that cos α = 3/5 and sin α = 4/5, and define an = 25n cos(nα)2 . It is not hard to see that an is a non-negative integer. The associated series A(t) reads A(t) =

1 − 2t + 225t 2 . (1 − 25t)(625t 2 + 14t + 1)

It has 3 distinct dominant poles. As α is not a rational multiple of π, the same holds for all series A0,p (t), for all values of p. Thus A(t) is not N-rational. 2.4. The combinatorial intuition of rational generating functions. We have described two families of combinatorial objects that naturally yield rational generating functions: walks in a digraph and words of regular languages. We have, moreover, shown that the enumeration of these objects are equivalent problems. It seems that these families convey the “right” intuition about objects with a rational GF. By this, we mean informally that: (i) “every” family of objects with a rational GF has actually an N-rational GF, (ii) for almost all families of combinatorial objects with a rational GF, it is easy to foresee that there will be a bijection between these objects and words of a regular language. Point (ii) means that most of these families F have a clear automatic structure, similar to the automatic structure of regular languages: roughly speaking, the objects of F can be constructed recursively using unions of sets and concatenation of cells (replacing letters). A more formal definition would simply paraphrase the definition of automata. Point (i) means simply that I have never met a counting problem that would yield a rational, but not N-rational GF. This includes problems coming from algebra, like growth functions of groups. On the contrary, Point (ii) only concerns purely combinatorial problems (but I do not want to be asked about the border between

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combinatorics and algebra). It admits very few counter-examples. Some will be discussed in Section 2.5. For the moment, let us illustrate the two above statements by describing the automatic structure of certain classes of objects (some being rather general), borrowed from [74, Ch. 4]. 2.4.1. Column-convex polyominoes. A polyomino is a finite union of cells of the square lattice, whose interior is connected. Polyominoes are considered up to a translation. A polyomino is column-convex (cc) if its intersection with every vertical line is connected. Let an be the number of cc-polyominoes having n cells, and let A(t) be the associated generating function. We claim that these polyominoes have an automatic structure. Consider a cc-polyomino P having n cells. Let us number these cells from 1 to n as illustrated in Figure 2. The columns are visited from left to right. In the first column, cells are numbered from bottom to top. In each of the other columns, the lowest cell that has a left neighbour gets the smallest number; then the cells lying 8 3 7 13 2 6 9 10 15 1 4 11 14 5 12

a a a a a a c c a c c b c b b

Figure 2. A column-convex polyomino, with the numbering and encoding of the cells.

below it are numbered from top to bottom, and finally the cells lying above it are numbered from bottom to top. Note that for all i, the cells labelled 1, 2, . . . , i form a cc-polyomino. This will be essential in our description of the automatic structure of these objects. Associate with P the word u = u1 . . . un on the alphabet {a, b, c} defined by – ui = c (like Column) if the ith cell is the first to be visited in its column, – ui = b (like Below) if the ith cell lies below the first visited cell of its column, – ui = a (like Above) if the ith cell lies above the first visited cell of its column. Then, add a bar on the letter ui if the ith cell of P has a South neighbour, an East neighbour, but no South-East neighbour. (In other words, the barred letters indicate where to start a new column, when the bottommost cell of this new column lies above the bottommost cell of the previous column.) This gives a word v on the alphabet {a, b, c, a, b, c}. It is not hard to see that the map that sends P on the word v is a size-preserving bijection between cc-polyominoes and words recognized by the automaton of Figure 1 (b). Hence by (5), the generating function of column-convex polyominoes is [76]: t (1 − t)3 A(t) = . 1 − 5 t + 7 t2 − 4 t3

798


2.4.2. P -partitions. A partition of the integer n into at most k parts is a nondecreasing k-tuple λ = (λ1 , . . . , λk ) of nonnegative integers that sum to n. This classical number-theoretic notion is generalized by the notion of P-partitions. Let P be a natural partial order on [[k]] (by natural we mean that if i < j in P , then i < j in N). A P -partition of n is a k-tuple λ = (λ1 , . . . , λk ) of nonnegative integers that sum to n and satisfy λi ≤ λj if i ≤ j in P . Thus when P is the natural total order on [[k]], a P -partition is simply a partition1 . We are interested in the following series: t |λ| , FP (t) = λ

where the sum runs over all P -partitions and |λ| = λ1 + · · · + λk is the weight of λ. The case of ordinary partitions is easy to analyze: every partition can be written in a unique way as a linear combination c1 λ(1) + · · · + ck λ(k)

(6)

where λ(i) = (0, 0, . . . , 0, 1, 1, . . . , 1) has exactly i parts equal to 1 and ci ∈ N. The weight of λ(i) is i, and one obtains: FP (t) =

1 . (1 − t)(1 − t 2 ) . . . (1 − t k )

(7)

The automatic structure of (ordinary) partitions is transparent: since they are constructed by adding a number of copies of λ(1) , then a number of copies of λ(2) , and so on, there is a size preserving bijection between these partitions and walks starting from 1 and ending anywhere in the following digraph: [1] 1

[2] [2]

2 [3]

[3] [3]

3

[4] [4]

4

[4] [4]

Note that this graph corresponds to k = 4, and that an edge labelled [ ] must be understood as a sequence of edges. These labels do not correspond to multiplicities. Observe that the only cycles in this digraph are loops. This, combined with Proposition 2.2, explains the factored form of the denominator of (7). Consider now the partial order on [[4]] defined by 1 < 3, 2 < 3 and 2 < 4. The partitions of weight at most 2 are (0, 0, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), (1, 0, 1, 0), (0, 0, 1, 1), (0, 0, 2, 0), (0, 0, 0, 2), 1A P -partition is usually defined as an order-reversing map from [[k]] to N [74, Section 4.5]. Both notions are of course completely equivalent.

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so that FP (t) = 1 + 2t + 4t 2 + O(t 3 ). If one is brave enough to list P -partitions of weight at most 20, the Padé approximant of the truncated series thus obtained is remarkably simple: FP (t) =

1 + t + t2 + t3 + t4 + O(t 21 ), (1 − t)(1 − t 2 )(1 − t 3 )(1 − t 4 )

and allows one to make a (correct) conjecture. It turns out that the generating function of P -partitions is always a rational series of denominator (1 − t)(1 − t 2 ) . . . (1 − t k ). Moreover, P -partitions obey our general intuition about objects with a rational GF. The following proposition, illustrated below by an example, describes their automatic structure: the set of P -partitions can be partitioned into a finite number of subsets; in each of these subsets, partitions have a structure similar to (6). Recall that a linear extension of P is a bijection σ on [[k]] such that σ (i) < σ (j ) if i < j in P . Proposition 2.6 ([74], Section 4.5). Let P be a natural order on [[k]]. For every P -partition λ, there exists a unique linear extension σ of P such that for all i, λσ (i) ≤ λσ (i+1) , the inequality being strict if σ (i) > σ (i + 1). We say that λ is compatible with σ . Given a linear extension σ , the P -partitions that are compatible with σ can be written in a unique way as a linear combination with coefficients in N: λ(σ,0) + c1 λ(σ,1) + · · · + ck λ(σ,k)

(8)

where λ(σ,0) is the smallest P -partition compatible with σ : (σ,0)

λσ (j ) = |{i < j : σ (i) > σ (i + 1)}|

for 1 ≤ j ≤ k,

and for 1 ≤ i ≤ k, (σ,i)

(σ,i)

(λσ (1) , . . . , λσ (k) ) = (0, 0, . . . , 0, 1, 1, . . . , 1) has exactly i parts equal to 1. Thus the GF of these P -partitions is FP ,σ (t) =

t e(σ ) (1 − t)(1 − t 2 ) . . . (1 − t k )

where e(σ ) is a variant of the Major index of σ : e(σ ) = (k − i). i:σ (i)>σ (i+1)

Example. Let us return to the order 1 < 3, 2 < 3 and 2 < 4. The 5 linear extensions are 1234, 2134, 1243, 2143 and 2413. Take σ = 2143. The P -partitions λ that are compatible with σ are those that satisfy λ2 < λ1 ≤ λ4 < λ3 . The smallest of them is thus λ(σ,0) = (1, 0, 2, 1). Then λ(σ,1) = (0, 0, 1, 0), λ(σ,2) = (0, 0, 1, 1), λ(σ,3) = (1, 0, 1, 1) and λ(σ,4) = (1, 1, 1, 1).

800


2.4.3. Integer points in a convex polyhedral cone ([74], Sec. 4.6). Let H be a finite collection of linear half-spaces of Rm of the form c1 α1 +· · ·+cm αm ≥ 0, with ci ∈ Z. We are interested in the set E of non-negative integer points α = (α1 , . . . , αm ) lying in the intersection of those half-spaces. For instance, we could have the following set E , illustrated in Figure 3 (a): E = {(α1 , α2 ) ∈ N2 : 2α1 ≥ α2 and 2α2 ≥ α1 }.

(9)

α3 α2

(a)

(b)

α1

α2

α1

Figure 3. Integer points in a polyhedral cone.

Numerous enumerative problems (including P -partitions) can be formulated in terms of linear inequalities as above. The generating function of E is E(t) = t |α| , α∈E

where |α| = α1 + · · · + αm . In the above example, E(t) = 1 + t 2 + 2t 3 + t 4 + 2t 5 + 3t 6 + 2t 7 + O(t 8 ). The set E is a monoid (it is closed under summation). In general, it is not a free monoid. Geometrically, the set C of non-negative real points in the intersection of the half-spaces of H forms a pointed convex polyhedral cone (the term pointed means that it does not contain a line), and E is the set of integer points in C. The simplicial case. In the simplest case, the cone C is simplicial. This implies that the monoid E is simplicial, meaning that there exists linearly independent vectors α (1) , . . . , α (k) such that E = {α ∈ Nm : α = q1 α (1) + · · · + qk α (k) with qi ∈ Q, qi ≥ 0}. This is the case in Example (9), with α (1) = (1, 2) and α (2) = (2, 1). The interior of E (the set of points of E that are not on the boundary of C) is then E = {α ∈ Nm : α = q1 α (1) + · · · + qk α (k) with qi ∈ Q, qi > 0}.

(10)

801


Then there exists a finite subset D of E [resp. D of E ] such that every element of E [resp. E ] can be written uniquely in the form α = β + c1 α (1) + · · · + ck α (k) ,

(11)

with β ∈ D [resp. β ∈ D] and ci ∈ N [74, Lemma 4.6.7]. In our running example (9), taken with α (1) = (1, 2) and α (2) = (2, 1), one has D = {(0, 0), (1, 1), (2, 2)} while D = {(1, 1), (2, 2), (3, 3)}. Compare (11) with the structure found for P partitions (8). Thus E and E have an automatic structure and their GFs read |β| t |β| β∈D t β∈D E(t) = k resp. E(t) = k . |α (i) | ) |α (i) | ) i=1 (1 − t i=1 (1 − t In Example (9), one thus obtains E(t) =

1 − t + t2 1 + t2 + t4 = (1 − t 3 )2 (1 − t)(1 − t 3 )

and

E(t) = t 2 E(t).

The general case. The set E can always be partitioned into a finite number of sets F of the form (10), where F is a simplicial monoid [74, Ch. 4, Eq. (24)]. Thus E , as a finite union of sets with an automatic structure, has an automatic structure as well. The associated generating function E(t) is N-rational, with a denominator which is a product of cyclotomic polynomials. Consider, for example, the set E = {(α1 , α2 , α3 ) ∈ N3 : α3 ≤ α1 + α2 }. The cone C of non-negative real points α satisfying α3 ≤ α1 +α2 is not simplicial, as it has 4 faces of dimension 2, lying respectively in the hyperplanes αi = 0 for i = 1, 2, 3 and α3 = α1 +α2 (Figure 3 (b)). But it is the union of two simplicial cones C1 and C2 , obtained by intersecting C with the half-spaces α1 ≥ α3 and α1 ≤ α3 , respectively. Let E1 [resp. E2 ] denote the set of integer points of C1 [resp. C2 ]. The fastest way to obtain the generating function E(t) is to write E(t) = E1 (t) + E2 (t) − E12 (t)

(12)

where E12 (t) counts integer points in the intersection of C1 and C2 (that is, in the plane α1 = α3 ). Since E1 , E2 and E1 ∩ E2 are simplicial cones (of dimension 3, 3 and 2 respectively), the method presented above for simplicial cones applies. Indeed, E1 [resp. E2 ; E12 ] is the set of linear combinations (with coefficients in N) of (1, 0, 1), (0, 1, 0) and (1, 0, 0) [resp. (1, 0, 1), (0, 1, 0) and (0, 1, 1); (1, 0, 1) and (0, 1, 0)]. This implies: E(t) =

1 (1 − t)2 (1 − t 2 )

+

1 1 1 + t + t2 − = . 2 2 2 (1 − t)(1 − t ) (1 − t)(1 − t ) (1 − t)(1 − t 2 )2

802


However, the “minus” sign in (12) prevents us from seeing directly the automatic nature of E (the difference of N-rational series is not always N-rational). This structure only becomes clear when we write E as the disjoint union of the interiors of all simplicial monoids induced by the triangulation of C into C1 and C2 . These monoids are the integer points of the faces (of all possible dimensions) of C1 and C2 . As there are 12 such faces (more precisely, 1 [resp. 4, 5, 2] faces of dimension 0 [resp. 1, 2, 3]), this gives E as the disjoint union of 12 sets having an automatic structure of the form (10). 2.5. Rational generating functions: more difficult questions. 2.5.1. Predicting rationality. We wrote in Section 2.4 that it is usually easy to foresee, to predict when a class of combinatorial objects has a rational GF. There are a few exceptions. Here is one of the most remarkable ones. Example 2.7 (Directed animals). A directed animal with a compact source of size k is a finite set of points A on the square lattice Z2 such that: – the points (−i, i) for 0 ≤ i < k belong to A; they are called the source points, – all the other points in A can be reached from one of the source points by a path made of North and East steps, having all its vertices in A. See Figure 4 for an illustration. A similar notion exists for the triangular lattice. It turns out that these animals have extremely simple generating functions [50], [10].

Figure 4. Compact-source directed animals on the square and triangular lattices.

Theorem 2.8. The number of compact-source directed animals of cardinality n is 3n−1 on the square lattice, and 4n−1 on the triangular lattice. The corresponding GFs are respectively t/(1 − 3t) and t/(1 − 4t), and are as rational as a series can be. There is at the moment no simple combinatorial intuition as to why these animals have rational GFs. A bijection between square lattice animals and words on a 3-letter alphabet was described in [50], but it does not shed a clear light on the structure of these objects. Still, there is now a convincing explanation of the algebraicity of these series (see Section 3.4.2). Example 2.9 (The area under Dyck paths). Another family of (slightly less natural) examples is provided by the enumeration of points lying below certain lattice paths.


803

For instance, let us call Dyck path of length 2n any path P on Z2 formed of steps (1, 1) and (1, −1), that starts from (0, 0) and ends at (2n, 0) without ever hitting a point with a negative ordinate. The area below P is the number of non-negative integer points (i, j ), with i ≤ 2n, lying weakly below P (Figure 5). It turns out that the sum of the areas of Dyck paths of length 2n is simply a(P ) = 4n . P :|P |=2n

Again, the rationality of the associated generating function does not seem easy to predict, but there are good combinatorial reasons explaining why it is algebraic. See [33], [65] for a direct explanation of this result, references, and a few variations on this phenomenon, first spotted by Kreweras [58].

Figure 5. The 5 Dyck paths of length 6 and the 43 = 64 points lying below.

Finally, let us mention that our optimistic statement about how easy it is to predict the rationality of a generating function becomes less and less true as we move from purely combinatorial problems to more algebraic ones. For instance, it is not especially easy to foresee that a group has an automatic structure [39]. Let us give also an example coming from number theory. Let P (x) ≡ P (x1 , . . . , xr ) be a polynomial with integer coefficients, and take p a prime. For n ≥ 0, let an be the number of x ∈ (Z/pn Z)r such that P (x) = 0 mod pn . Then the generating function n an t n is rational. A related result holds with p-adic solutions [37], [53]. 2.5.2. Computing a rational generating function. Let us start with an elementary, but important observation. Many enumerative problems, including some very hard, can be approximated by problems having a rational GF. To take one example, consider the notoriously difficult problem of counting self-avoiding polygons (elementary cycles) on the square lattice. It is easy to convince oneself that the generating function of SAP lying in a horizontal strip of height k is rational for all k. This does not mean that it will be easy (or even possible, in the current state of affairs) to compute the corresponding generating function when k = 100. Needless to say, there is at the moment no hope to express this GF for a generic value of k. The generating function of SAP having 2k horizontal steps can also be seen to be rational. Moreover, these SAP can be described in terms of linear inequalities (as in Section 2.4.3), which implies that the denominator of the corresponding series Gk is a product of cyclotomic polynomials. But again, no one knows what this series is for a generic value of k, or even for k = 100. Still, some progress have been made recently, since it has been

804


proved that the series Gk have more and more poles as k increases, which means that their denominators involve infinitely many cyclotomic polynomials [68]. This may be considered as a proof of the difficulty of this enumerative problem [51]. In general, computing the (rational) generating function of a family of objects depending on a parameter k may be non-obvious, if not difficult, even if the objects are clearly regular, and even if the final result turns out to be nice. A classical example is provided by the growth functions of Coxeter groups [61]. Here is a more combinatorial example. A partition λ = (λ1 , . . . , λk ) is said to be a k-Lecture Hall partition (k-LHP) if λ2 λk λ1 ≤ ≤ ··· ≤ . 0≤ 1 2 k Since these partitions are defined by linear inequalities, it follows from Section 2.4.3 that their weight generating function is rational, with a denominator formed of cyclotomic polynomials. Still, there is no clear reason to expect that [15]: λ k−LHP

t |λ| =

1 (1 − t)(1 − t 3 ) . . . (1 − t 2k−1 )

.

Several proofs have been given for this result and variations on it. See for instance [16], [35] and references in the latter paper. Some of these proofs are based on a bijection between lecture hall partitions and partitions into parts taken in {1, 3, . . . , 2k − 1}, but these bijections are never really simple [82], [40]. 2.5.3. N-rationality. As we wrote in Section 2.4, we do not know of a counting problem that would yield a rational, but not N-rational series. It would certainly be interesting to find one (even if it ruins some parts of this paper). Let us return to Soittola’s criterion for N-rationality (Theorem 2.5). It is not always easy to prove that a rational series has non-negative coefficients. For instance, it was conjectured in [46] that for any odd k, the number of partitions of n into parts taken in {k, k + 1, . . . , 2k − 1} is a non-decreasing function of n, for n ≥ 1. In terms of generating functions, this means that the series q+

1−q (1 − q k )(1 − q k+1 ) . . . (1 − q 2k−1 )

has non-negative coefficients. This was only proved recently [67]. When k is even, a similar result holds for the series q+

1−q (1 − q k )(1 − q k+1 ) . . . (1 − q 2k )(1 − q 2k+1 )

.

Once the non-negativity of the coefficients has been established, it is not hard to prove that these series are N-rational. This raises the question of finding a family of combinatorial objects that they count.


805

3. Algebraic generating functions 3.1. Definitions and properties. The Laurent series A(t) with coefficients in the field R is said to be algebraic (over R(t)) if it satisfies a non-trivial algebraic equation: P (t, A(t)) = 0 where P is a bivariate polynomial with coefficients in R. We assume below R = Q. Again, the set of algebraic Laurent series possesses numerous interesting properties [75, Ch. 6], [43, Ch. VII]. It is closed under sum, product, derivation, reciprocals, but not under integration. These closure properties become effective using either the theory of elimination or Gröbner bases, which are implemented in most computer algebra packages. The coefficients an of an algebraic series A(t) satisfy a linear recurrence relation with polynomial coefficients: for n large enough, p0 (n)an + p1 (n)an−1 + p2 (n)an−2 + · · · + pk (n)an−k = 0. Thus the first n coefficients can be computed using a linear number of operations. There is no systematic way to express the coefficients of an algebraic series in closed form. Still, one can sometimes apply the Lagrange inversion formula: Proposition 3.1. Let and be two formal power series and let U ≡ U (t) be the unique formal power series with no constant term satisfying U = t (U ). Then for n > 0, the coefficient of t n in (U ) is: [t n ] (U ) =

1 n−1 [t ] (t) (t)n . n

Given an algebraic equation P (t, A(t)) = 0, one can decide whether there exists a series U (t) and two rational series and satisfying U = t (U )

and

A = (U ).

(13)

Indeed, such series exist if and only if the genus of the curve P (t, a) is zero [1, Ch. 15]. Moreover, both the genus and a parametrization of the curve in the form (13) can be determined algorithmically. Example 3.2 (Finding a rational parametrization). The following algebraic equation was recently obtained [22], after a highly non-combinatorial derivation, for the GF of certain planar graphs carrying a hard-particle configuration: 0 = 23328 t 6 A4 + 27 t 4 (91 − 2088 t)A3 + t 2 (86 − 3951 t + 46710 t 2 + 3456 t 3 )A2 + (1 − 69 t + 1598 t 2 − 11743 t 3 − 14544 t 4 )A − 1 + 66 t − 1495 t 2 + 11485 t 3 + 128 t 4 .

(14)

806


The package algcurves of Maple, and more precisely the commands genus and parametrization, reveal that a rational parametrization is obtained by setting (3 U + 7) 9 U 2 + 33 U + 37 t = −3 . (3 U + 1)4 Of course, this is just the net result of Maple, which is not necessarily very meaningful for combinatorics. Still, starting from this parametrization, one obtains after a few attempts an alternative parametrizing series V with positive coefficients: V =

t . (1 − 2V )(1 − 3V + 3V 2 )

(15)

The main interest of such a parametrization for this problem does not lie in the possibility of applying the Lagrange inversion formula. Rather, it suggests that a more combinatorial approach exists, based on the enumeration of certain trees, in the vein of [19], [27]. It also gives a hint of what these trees may look like. Another convenient tool borrowed from the theory of algebraic curves is the possibility to explore all branches of the curve P (t, A(t)) = 0 in the neighbourhood of a given point t0 . This is based on Newton’s polygon method. All branches have a Puiseux expansion, that is, an expansion of the form: an (t − t0 )n/d A(t) = n≥n0

with n0 ∈ Z, d ∈ P. The coefficients an belong to C (in general, to an algebraic closure of the ground field). These expansions can be computed automatically using standard software. For instance, the Maple command puiseux of the algcurves package tells us that (14) has a unique solution that is a formal power series, the other three solutions starting with a term t −2 . Such Puiseux expansions are crucial for studying the asymptotic behaviour of the coefficients of an algebraic series A(t). As in the rational case, one has first to locate the singularities of A(t), considered as a function of a complex variable t. These singularities are found among the roots of the discriminant and of the leading coefficient of P (t, a) (seen as a polynomial in a). The singular expansion of A(t) near its singularities of smallest modulus can then be converted, using certain transfer theorems, into an asymptotic expansion of the coefficients [42], [43, VII.4]. Example 3.3 (Asymptotics of the coefficients of an algebraic series). Consider the series V (t) defined by (15). Its singularities lie among the roots of the discriminant (t) = −3 + 114t − 4635t 2 + 55296t 3 . Only one root is real. Denote it t0 ∼ 0.065. The modulus of the other two roots is smaller than t0 , so they could, in theory, be candidates for singularities. However,


807

V (t) has non-negative coefficients, and this implies, by Pringsheim’s theorem, that one of the roots of minimal modulus is real and positive. Hence V (t) has a unique singularity, lying at t0 . A Puiseux expansion at this point gives V (t) = c0 − c1 1 − t/t0 + O(t − t0 ), for some explicit (positive) algebraic numbers c0 and c1 , which translates into c1 [t n ]V (t) = √ t0−n n−3/2 (1 + o(1)) . 2 π The determination of asymptotic expansions for the coefficients of algebraic series is probably not far from being completely automated, at least in the case of series with non-negative coefficients [31], [43]. The “typical” behaviour is an ∼

κ ρ −n nd , (d + 1)

(16)

where κ is an algebraic number and d ∈ Q \ {−1, −2, −3, . . . }. Compare with the result (3) obtained for rational series. Again, the above statement is not exact, as the contribution of all dominant singularities must be taken into account. See [43, Thm. VII.6] for a complete statement. Let us add that, again, one can guess if a series A(t) given by its first coefficients satisfies an algebraic equation P (t, A(t)) = 0 of a given bi-degree (d, e). The guessing procedure requires to know at least (d + 1)(e + 1) coefficients, and amounts to solving a system of linear equations. It is implemented in the package Gfun of Maple [72]. For instance, given the 10 first coefficients of the series V (t) satisfying V (0) = 0 and (15), one automatically conjectures (15). 3.2. Plane trees. Our typical “algebraic” objects will be (plane) trees. Let us begin with their usual intuitive recursive definition. A tree is a graph formed of a distinguished vertex (called the root) to which are attached a certain number (possibly zero) of trees, ordered from left to right. The number of these trees is the degree of the root. The roots of these trees are the children of the root. A more rigorous definition describes a tree as a finite set of words on the alphabet P satisfying certain conditions [63]. We hope that our less formal definition and Figure 6 (a) suffice to understand what we mean. The vertices of a tree are often called nodes. Nodes of degree 0 are called leaves, the others are called inner nodes. The enumeration of classes of trees yields very often algebraic equations. Let us consider for instance the complete binary trees, that is, the trees in which all vertices have degree 0 or 2 (Figure 12). Let an be the number of such trees having n leaves. Then, by looking at the two (sub)trees of the root, one gets, for n > 1: an =

n−1 k=1

ak an−k .

808


(a)

(b)

(c)

Figure 6. (a) A plane tree. (b) A rooted planar map. (c) The corresponding 4-valent map (thick lines).

The initial condition is a1 = 1. In terms of GFs, this gives A(t) = t + A(t)2 , which is easily solved: √ 1 2n

1 − 1 − 4t A(t) = (17) = t n+1 . 2 n+1 n n≥0

More generally, many algebraic series obtained in enumeration are given as the first component of the solution of a system of the form Ai = Pi (t, A1 , . . . , Ak ),

(18)

for some polynomials Pi (t, x1 , . . . , xk ) having coefficients in Z. This system is said to be proper if Pi has no constant term (Pi (0, . . . , 0) = 0) and does not contain any linear term xi . It is positive if the coefficients of the Pi are non-negative. For instance, A1 = t 2 + A1 A2

and

A2 = 2tA31

is a proper positive system. The system is quadratic if every Pi (t, x1 , . . . , xk ) is a linear combination of the monomials t and x xm , for 1 ≤ ≤ m ≤ k. Theorem 3.4 ([75], Thm. 6.6.10 and [71], Thm. IV.2.2). A proper algebraic system has a unique solution (A1 , . . . , Ak ) in the set of formal power series in t with no constant term. This solution is called the canonical solution of the system. The series A1 is also the first component of the solution of – a proper quadratic system, – a proper system of the form Bi = tQi (t, B1 , . . . , B ), for 1 ≤ i ≤ . These two systems can be chosen to be positive if the original system is positive. Proof. Let us prove the last property, which we have not found in the above references. Assume A1 satisfies (18) and that this system is quadratic. The ith equation reads Ai = mit +ni Aσ (i) Aτ (i) . Rewrite each monomial Ai Aj as tUij and add theequations Uij = t mi mj + mi nj Uσ (j )τ (j ) + mj ni Uσ (i)τ (i) + ni nj Uσ (i)τ (i) Uσ (j )τ (j ) . The new system has the required properties. 2 Definition 3.5. A series A(t) is N-algebraic if it has coefficients in N and if A(t)−A(0) is the first component of the solution of a proper positive system.


809

Proper positive systems like (18) can always be given a combinatorial interpretation in terms of trees. Every vertex of these trees carries a label (i, c) where i ∈ [[k]] and c ∈ P. We say that i is the type of the vertex and that c is its colour. The type of a tree is the type of its root. Write A0 = t, so that Ai = Pi (A0 , A1 , . . . , Ak ). Let A0 be the set reduced to the tree with one node, labelled (0, 1). For i ∈ [[k]], let Ai be the set of trees such that – the root has type i, – the types of the subtrees of the root, visited from left to right, are 0, . . . , 0, 1, . . . , 1, . . . , k, . . . , k, in this order, – if exactly ej children of the root have type j , the colour of the root is any integer in the interval [1, m], where m is the coefficient of x0e0 . . . xkek in Pi (x0 , . . . , xk ). Then it is not hard to see that Ai (t) is the generating function of trees of type i, counted by the number of leaves. This explains why trees will be, in the rest of this paper, our typical “algebraic” objects. 3.3. Context-free languages. As in the case of rational (and, more precisely, Nrational) series, there exists a family of languages that is closely related to algebraic series. A context-free grammar G consists of – a set S = {S1 , . . . , Sk } of symbols, with one distinguished symbol, say, S1 , – a finite alphabet A of letters, disjoint from S, – a set of rewriting rules of the form Si → w where w is a non-empty word on the alphabet S ∪ A. The grammar is proper if there is no rule Si → Sj . The language L(G) generated by G is the set of words on the alphabet A that can be obtained from S1 by applying iteratively the rewriting rules. A language is context-free is there exists a context-free grammar that generates it. In this case there exists also a proper context-free grammar that generates it. Example 3.6 (Dyck words). Consider the grammar G having only one symbol, S, alphabet {a, b}, and rules S → ab + abS + aSb + aSbS (which is short for S → ab, S → abS, S → aSb, S → aSbS). It is easy to see that L(G) is the set of non-empty words u on {a, b} such that |u|a = |u|b and for every prefix v of u, |v|a ≥ |v|b . These words, called Dyck words, provide a simple encoding of the Dyck paths met in Example 2.9. A derivation tree associated with G is a plane tree in which all inner nodes are labelled by symbols, and all leaves by letters, in such a way that if a node is labelled Si and its children w1 , . . . , wk (from left to right), then the rewriting rule Si → w1 . . . wk is in the grammar. If the root is labelled S1 , then the word obtained by reading the labels of the leaves in prefix order (i.e., from left to right) belongs to the language generated by G. Conversely, for every word w in L(G), there exists at least one derivation tree with root labelled S1 that gives w. The grammar is said to be unambiguous if every word of L(G) admits a unique derivation tree.

810


Assume G is proper. For 1 ≤ i ≤ k, let Ai (t) be the generating function of derivation trees rooted at Si , counted by the number of leaves. With each rule r, associate the monomial M(r) = x0e0 . . . xkek where e0 [resp. ei , with i > 0] is the number of letters of A [resp. occurrences of Si ] in the right-hand side of r. Then the series A1 , . . . , Ak form the canonical solution of the proper positive system (18), with Pi (x0 , x1 , . . . , xk ) = M(r), r

where the sum runs over all rules r with left-hand side Si . Conversely, starting from a positive system Bi = tQi (t, B1 , . . . , Bk ) and its canonical solution, it is always possible to construct an unambiguous grammar with symbols S1 , . . . , Sk such that Bi is the generating function of derivation trees rooted at Si (the idea is to introduce a new letter ai for each occurrence of t). In view of Theorem 3.4 and Definition 3.5, this gives the following alternative characterization of N-algebraic series: Proposition 3.7. A series A(t) is N-algebraic if and only if only A(0) ∈ N and there exists an unambiguous context-free language having generating function A(t)−A(0). 3.4. The combinatorial intuition of algebraic generating functions. We have described two families of combinatorial objects that naturally yield algebraic GFs: plane trees and words of unambiguous context-free languages. We have, moreover, shown a close relationship between these two types of objects. These two families convey the standard intuition of what a family with an algebraic generating function looks like: the algebraicity suggests that it may (or should...) be possible to give a recursive description of the objects based on disjoint union of sets and concatenation of objects. Underlying such a description is a context-free grammar. This intuition is the basis of the so-called Schützenberger methodology, according to which the “right” combinatorial way of proving algebraicity is to describe a bijection between the objects one counts and the words of an unambiguous context-free language. This approach has led in the 80s and 90s to numerous satisfactory explanations of the algebraicity of certain series, and we describe some of them in this subsection. Let us, however, warn the reader that the similarities with the rational case will stop here. Indeed, it seems that the “context-free” intuition is far from explaining all algebraicity phenomena in enumerative combinatorics. In particular, (i) it is very likely that many families of objects have an algebraic, but not Nalgebraic generating function, (ii) there are many families of combinatorial objects with an algebraic GF that do not exhibit a clear “context-free” structure, based on union and concatenation. For several of these families, there is just no explanation of this type, be it clear or not. This will be discussed in the next subsections. For the moment, let us illustrate the “context-free” intuition.


811

3.4.1. Walks on a line. Let S be a finite subset of Z. Let W be the set of walks on the line Z that start from 0 and take their steps in S. The length of a walk is its number of steps. Let Wk be the set of walks ending at position k. For k ≥ 0, let Mk be the subset of Wk consisting of walks that never visit a negative position, and let M be the union of the sets Mk . In probabilistic terms, the walks in M would be called meanders and the walks of M0 excursions. Of course, a walk is simply a sequence of steps, hence a word on the alphabet S. Thus the sets of walks we have defined can be considered as languages on this alphabet. Theorem 3.8. The language W is simply S ∗ and is thus regular. The languages M, Wk and Mk are unambiguous context-free for all k. Proof. We only describe the (very simple) case S = {+1, −1}, to illustrate the ideas that are involved in the construction of the grammar. We encode the steps +1 by the letter a, the steps −1 by b, and introduce some auxiliary languages: • M0− , the subset of W0 formed of walks that never visit a positive position, • W0+ [resp. W0− ], the subset of W0 formed of walks that start with a [resp. b]. The language M0 will be generated from the symbol M0 , and similarly for the other languages. By looking at the first time a walk of M0 [resp. M0− ] reaches position 0 after its first step, one obtains M0 → a(1 + M0 )b(1 + M0 )

and

M0− → b(1 + M0− )a(1 + M0− ).

By considering the last visit to 0 of a walk of Mk , one obtains, for k > 0: Mk → (1 + M0 )a (1k=1 + Mk−1 ) . This is easily adapted to general meanders: M → M0 + (1 + M0 )a (1 + M) . Considering the first step of a walk of W0 gives W0 → W0+ + W0−

with

W0+ → M0 (1 + W0− ) and W0− → M0− (1 + W0+ ).

Finally, for k > 0, looking at the first visit at 1 [resp. −1] of a walk of Wk [resp. W−k ] yields Wk → (1 + M0− )a (1k=1 + Wk−1 ) [resp. W−k → (1 + M0 )b(1k=1 + W−(k−1) )]. For a general set of steps S, various grammars have been described for the languages Mk of meanders [38], [60], [59]. For Wk , we refer to [59, Section 4] where the (representative) case S = {−2, −1, 0, 1, 2} is treated. 2 Theorem 3.8 is often described in terms of walks in Z2 starting from (0, 0) and taking their steps in {(1, j ), j ∈ S}. The conditions on the positions of the walks

812


that lead to the definition of Mk and Wk are restated in terms of conditions on the ordinates of the vertices visited by the walk. A harmless generalization is obtained by taking steps in a finite subset S of P × Z. A walk is still encoded by a word on the alphabet S. The languages Wk remain unambiguous context-free. If each step (i, j ) is, moreover, weighted by a rational number wi,j , then the generating function of walks of W , counted by the coordinates of their endpoint, is W (t, s) =

1−

1

(i,j )∈S

wi,j t i s j

.

The generating function Wk (t) that counts (weighted) walks ending at ordinate k is the coefficient of s k in W (t, s). Since Wk is unambiguous context-free, the series Wk (t) is algebraic. This gives a combinatorial explanation of the following result [75, Thm. 6.3.3]. Theorem 3.9 (Diagonals of rational series). Let A(x, y) = m,n≥0 am,n x m y n be a series in two variables x and y, with coefficients in Q, that is rational. Then the diagonal of A, that is, the series A(t) = n≥0 an,n t n , is algebraic. Proof. By linearity, it suffices to consider the case A(x, y) =

1−

xa yb

0≤m,n≤d cm,n x

myn

,

with c0,0 = 0. Set x = ts and y = t/s. The diagonal of A satisfies A(t 2 ) = [s 0 ]A(ts, t/s) = t a+b [s b−a ]

1−

1 , m+n s m−n 0≤m,n≤d cm,n t

which is algebraic as it counts weighted paths in Wb−a , for a certain set of steps. Hence A(t) is algebraic too. 2 The converse of Theorem 3.9 holds: every series B(t) that is algebraic over Q(t) is the diagonal of a bivariate rational series A(x, t) [70]. Note. If one is simply interested in obtaining a set of algebraic equations defining the GFs of the sets Mk and Wk , a more straightforward approach is to use a partial fraction decomposition (for Wk ) and the kernel method (for Mk ). See [75, 6.3], and [17, Example 3]. 3.4.2. Directed animals. Let us move to an example where a neat context-free exists, but is uneasy to discover. We return to the directed animals defined in Section 2.5.1. As discussed there, there is no simple explanation as to why the number of compactsource animals is so simple (Theorem 2.8). Still, there is a convincing explanation for the algebraicity of the corresponding series: directed animals have, indeed, a context-free structure. This structure was discovered a few years after the proof of

813


Theorem 2.8, with the development by Viennot of the theory of heaps [81], a geometric version of partially commutative monoids [30]. Intuitively, a heap is obtained by dropping vertically some solid pieces, the one after the other. Thus, a piece lies either on the “floor” (then it is said to be minimal), or covers, at least partially, another piece. Directed animals are, in essence, heaps. To see this, replace every point of the animal by a dimer (Figure 7). Note that if the animal has a unique source, the associated heap has a unique minimal piece. Such heaps are named pyramids.

(a)

(b) Figure 7. (a) A directed animal and the associated pyramid. (b) A half-pyramid.

What makes heaps interesting here is that there exists a monoid structure on the set of heaps: The product of two heaps is obtained by putting one heap above the other and dropping its pieces. This product is the key in our context-free description of directed animals. Let us begin with the description of pyramids (one-source animals). A pyramid is either a half-pyramid (Figure 7 (b)), or the product of a half-pyramid and a pyramid (Figure 8, top). Let P (t) denote the GF of pyramids counted by the number of dimers, and H (t) denote the GF of half-pyramids. Then P (t) = H (t)(1 + P (t)). Now, a half-pyramid may be reduced to a single dimer. If it has several dimers, it is the product of a single dimer and of one or two half-pyramids (Figure 8, bottom), which implies H (t) = t + tH (t) + tH 2 (t).

P P

H

H

Figure 8. Decomposition of pyramids (top) and half-pyramids (bottom).

814


A trivial computation finally provides the GF of directed (single-source) animals:

√

1 1 − t − (1 + t)(1 − 3t) 1+t P (t) = −1 while H (t) = . 2 1 − 3t 2t The enumeration of compact-source directed animals is equivalent to the enumeration of heaps having a compact basis (the minimal dimers are adjacent). The generating function of heaps having a compact basis formed with k dimers is P (t)H (t)k−1 (Figure 9), which implies that the generating function of compact-source animals is P (t) t = . 1 − H (t) 1 − 3t

P

H

H

H

Figure 9. Decomposition of heaps having a compact basis.

3.5. The world of planar maps. We have seen in Section 3.2 that plane trees are the paradigm for objects with an algebraic generating function. A more general family of plane objects seems to be just as deeply associated with algebraic series, but for reasons that are far more mysterious: planar maps. A (planar) map is a proper embedding of a planar graph in the sphere (Figure 6 (b)). In order to avoid symmetries, all the maps we consider are rooted: this means that one edge is distinguished and oriented. Maps are only considered up to a continuous deformation of the sphere. A map induces a 2-cell decomposition of the sphere: the cells of dimension 0 [resp. 1, 2] are called vertices [resp. edges, faces]. Hence plane trees are maps with a single face. The interest for the enumeration of planar maps dates back to the early 60s, in connection with the 4-colour theorem. The first results are due to Tutte [77], [78], [79]. Ten to fifteen years later, maps started to be investigated independently in theoretical physics, as a model for 2-dimensional quantum gravity [28], [9]. However, neither the recursive approach used by Tutte and his disciples, nor the physics approach based on matrix integrals were able to explain in a combinatorially satisfactory way the following observations: – the generating functions of many classes of planar maps are algebraic, – the associated numbers are often irritatingly simple. Let us illustrate this with three examples. 1. General maps. The number of planar maps having n edges is [80]:

2.3n 2n . gn = (n + 1)(n + 2) n

(19)

815


The associated generating function G ≡ G(t) =

n≥0 gn t

n

satisfies:

−1 + 16t + (1 − 18t)G + 27t 2 G2 = 0.

(20)

2. Loopless triangulations. The number of loopless triangulations (maps in which all faces have degree 3) having 2n + 2 faces is [62]:

2n 3n . tn = (n + 1)(2n + 1) n The associated generating function T ≡ T (t) = n tn t n satisfies 1 − 27t + (−1 + 36t)T − 8tT 2 − 16t 2 T 3 = 0. 3. Three-connected triangulations. The number of 3-connected triangulations having 2n + 2 faces is [77]:

2 4n + 1 mn = . (n + 1)(3n + 2) n The associated generating function M ≡ M(t) = n tn t n satisfies −1 + 16t + (1 − 20t)M + (3t + 8t 2 )M 2 + 3t 2 M 3 + t 3 M 4 = 0. These maps are in bijection with rooted maximal planar simple graphs (graphs with no loop nor multiple edge that lose planarity as soon as one adds an edge). At last, in the past ten years, a general combinatorial picture has emerged, suggesting that maps are, in essence, unrooted plane trees. In what follows, we illustrate on the example of general maps the main three approaches that now exist, and give references for further developments of these methods. 3.5.1. The recursive approach. We leave to the reader to experience personally that maps do not have an obvious context-free structure. Still, maps do have a simple recursive structure, based on the deletion of the root-edge. However, in order to exploit this structure, one is forced to keep track of the degree of the root-face (the face lying to the right of the root edge). The decomposition illustrated in Figure 10 leads in a few lines to the following equation: G(u, t) = 1 + tu2 G(u, t)2 + tu

uG(u, t) − G(1, t) , u−1

(21)

where G(u, t) counts planar maps by the number of edges (t) and the degree of the root-face (u). It can be checked that the above equation defines G(u, t) uniquely as a formal power series in t (with polynomial coefficients in u). However, it is not clear on the equation why G(1, t) (and hence G(u, t)) are algebraic. In his original paper,

816


=

+

+

Figure 10. Tutte’s decomposition of rooted planar maps.

Tutte first guessed the value of G1 (t) := G(1, t), and then proved the existence of a series G(u, t) that fits with G1 (t) when u = 1, and satisfies the above equation. Still, a bit later, Brown came with a method for solving (21): the so-called quadratic method [29], [49, Sec. 2.9]. Write (21) in the form (2aG(u, t) + b)2 = δ, where a, b and δ are polynomials in t, u and G1 (t). That is, 2 2 2tu (u − 1)G(u, t) + tu2 − u + 1 = 4t 2 u3 (u − 1)G1 + (1 − u)2 − 4tu4 + 6tu3 + u4 t 2 − 2tu2 . It is not hard to see, even without knowing the value of G(u, t), that there exists a (unique) formal power series in t, say U ≡ U (t), that cancels the left-hand side of this equation. That is, U = 1 + tU 2 + 2tU 2 (U − 1)G(U, t). This implies that the series U is a double root of the polynomial δ that lies on the right-hand side. The discriminant of this polynomial (in u) thus vanishes: this gives the algebraic equation (20) satisfied by G(1, t). The enumeration of many other families of planar maps can also be attacked by a recursive description based on the deletion of an edge (or vertex, or face...). See for instance [62] for 2-connected triangulations, or [6] for maps with prescribed face degrees. (For maps with high connectivity, like 3-connected triangulations, an additional composition formula is often required [77], [3].) The resulting equations are usually of the form P (F (u), F1 , . . . , Fk , t, u) = 0,

(22)

where F (u) ≡ F (t, u), the main generating function, is a series in t with polynomial coefficients in u, and F1 , . . . , Fk are series in t only, independent of u. Brown’s quadratic method applies as long as the degree in F (u) is 2 (for the linear case, see the kernel method in [17], [2]). Recently, it was understood how these equations could be solved in full generality [22]. Moreover, the solution of any (well-founded) equation of the above type was shown to be algebraic. This provides two types of enumerative results: – the proof that many map generating functions are algebraic: it now suffices to exhibit an equation of the form (22), or to explain why such an equation exists, – the solution of previously unsolved map problems (like the enumeration of hardparticle configurations on maps, which led to (14), or that of triangulations with high vertex degrees [8]).

817


3.5.2. Matrix integrals. In the late 70s, it was understood by a group of physicists that certain matrix integral techniques coming from quantum field theory could be used to attack enumerative problems on maps [28], [9]. This approach proved to be extremely efficient (even if it is usually not fully rigorous). The first step is fairly automatized, and consists in converting the description of maps into a certain integral. For instance, the relevant integral for the enumeration of 4-valent maps (maps in which all vertices have degree 4) is Z(t, N) =

2N (N−1)/2 N 2 /2

(2π)

dH etr(−H

2 /2+tH 4 /N)

,

where the integration space is that of hermitian matrices H of size N, equipped with the Lebesgue measure dH = dxkk k< dxk dyk with hk = xk + iyk . As there is a classical bijection between 4-valent maps with n vertices and planar maps with n edges (Figure 6 (c)), we are still dealing with our reference problem: the enumeration of general planar maps. The connection between the above integral and maps is 1 log Z(t, N). N→∞ N 2

G(t) = tE (t) with E(t) = lim

Other map problems lead to integrals involving several hermitian matrices [55]. We refer to [83] for a neat explanation of the encoding of map problems by integrals, and to [45], [41] (and references therein) for the evaluation of integrals. 3.5.3. Planar maps and trees. We finally come to a combinatorial explanation of the formula/equation for gn and G(t). Take a plane binary tree with n (inner) nodes, planted at a leaf, and add to every inner node a new distinguished child, called a bud. At each node, we have three choices for the position of the bud (Figure 11 (a)). The new tree, called budding tree, has now n buds and n + 2 leaves. Now start from the root and walk around the tree in counterclockwise order, paying attention to the sequence of buds and leaves you meet. Each time a bud is immediately followed by root-leaf

node

bud

leaf (a)

(b)

(c)

The final edge

Figure 11. (a). A budding tree. (b) An intermediate step in the matching procedure. (c) The resulting 4-valent map, with its marked face.

818


a leaf in this sequence, match them by forming a new edge (Figure 11 (b)) and then go on walking around the plane figure thus obtained. At the end, exactly two leaves remain unmatched. Match them together and orient this final edge in one of the two possible ways. Also, mark the face to the left of the matching edge that ends at the root-leaf. Theorem 3.10 ([73]). The above correspondence is a bijection between pairs (T , ε) where T is a budding tree having n inner nodes and ε ∈ {0, 1}, and 4-valent maps with n vertices and a marked face. The value of ε tells how to orient the final matching edge. Schaeffer first used this bijection to explain combinatorially the formula (19). Indeed, the number of budding trees with n inner nodes is clearly 3n 2n /(n + 1) (see (17)), while the number of n 4-valent maps with n vertices and a marked face is (n + 2)gn . Eq. (19) follows. Later, it was realized that this construction could also be used to explain the algebraicity of the series G(t) [23]. Say that a budding tree is balanced if the rootleaf is not matched by a bud. Take such a tree, match all buds, and orient the final edge from the root-leaf to the other unmatched leaf. This gives a bijection between balanced budding trees and 4-valent maps. We thus have to count balanced trees, or, equivalently, the unbalanced ones. By re-rooting them at the bud that matches the root-leaf, one sees that they are in bijection with a node attached to three budding trees. This gives G(t) = B(t) − tB(t)3 ,

where B(t) = 3t (1 + B(t))2

counts budding trees by (inner) nodes. The above construction involves taking a difference of N-algebraic series, which needs not be N-algebraic. We actually conjecture that the series G(t) is not N-algebraic (see Section 3.6.4). There is little doubt that the above construction (once described in greater detail…) explains in a very satisfactory way both the simplicity of the formula giving gn and the algebraicity of G(t). Moreover, this is not an ad hoc, isolated magic trick: over the past ten years, it was realized that this construction is one in a family of constructions of the same type, which apply to numerous families of maps (Eulerian maps [73], maps with prescribed vertex degrees [23], constellations [18], bipartite maps with prescribed degrees [19], maps with higher connectivity [66], [47]). Definitely, these constructions reveal a lot about the combinatorial nature of planar maps. To conclude this section, let us mention that a different combinatorial construction for general planar maps, discovered in the early 80s [34], has recently been simplified [32] and adapted to other families of maps [36], [54], [25], [26]. It is a bit less easy to handle than the one based on trees with buds, but it allows one to keep track of the distances between some vertices of the map. This has led to remarkable connections with a random probability distribution called the Integrated SuperBrownian Excursion [32]. A third type of construction has emerged even more recently [7] for 2-connected triangulations, but no ones knows at the moment whether it will remain isolated or is just the tip of another iceberg.


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3.6. Algebraic series: some questions. We begin with three simple classes of objects that have an algebraic GF, but for reasons that remain mysterious. We then discuss a possible criterion (or necessary condition) for N-algebraicity, and finally the algebraicity of certain hypergeometric series. 3.6.1. Kreweras’ words and walks on the quarter plane. Let L be the set of words u on the alphabet {a, b, c} such that for every prefix v of u, |v|a ≥ |v|b and |v|a ≥ |v|c . These words encode certain walks on the plane: these walks start at (0, 0), are made of three types of steps, a = (1, 1), b = (−1, 0) and c = (0, −1), and never leave the first quadrant of the plane, defined by x, y ≥ 0. The pumping lemma [52, Thm. 4.7], applied to the word a n bn cn , shows that the language L is not context-free. However, its generating function is algebraic. More precisely, let us denote by i,j (n) the number of words u of L of length n such that |u|a − |u|b = i and |u|a − |u|c = j . They correspond to walks of length n ending at position (i, j ). Then the associated three-variable generating function is L(u, v; t) = i,j (n)ui v j t n i,j,n

=

√ √ 1 − uW 2 + 1/W − v 1 − vW 2 1 − 2 2 uv − t (u + v + u v ) uvt

1/W − u

where u = 1/u, v = 1/v and W ≡ W (t) is the unique power series in t satisfying W = t (2+W 3 ). Moreover, the number of walks ending at (i, 0) is remarkably simple:

4n (2i + 1) 2i 3n + 2i i,0 (3n + 2i) = . (n + i + 1)(2n + 2i + 1) i n The latter formula was proved in 1965 by Kreweras, in a fairly complicated way [57]. This rather mysterious result has attracted the attention of several combinatorialists since its publication [14], [48], [64]. The first combinatorial explanation of the above formula (in the case i = 0) has just been found by Bernardi [7]. Walks in the quarter plane do not always have an algebraic GF: for instance, the number of square lattice walks (with North, South, East and West steps) of size 2n that start and end at (0, 0) and remain in the quarter plane is

2n + 2 2 42n+1 1 ∼ , (2n + 1)(2n + 4) n + 1 π n3 and this asymptotic behaviour prevents the corresponding generating function from being algebraic (see (16)). The above formula is easily proved by looking at the projections of the walk onto the horizontal and vertical axes. 3.6.2. Walks on the slit plane. Take now any finite set of steps S ⊂ Z × {−1, 0, 1} (we say that these steps have small height variations). Let si,j (n) be the number of

820


walks of length n that start from the origin, consist of steps of S, never return to the non-positive horizontal axis {(−k, 0), k ≥ 0}, and end at (i, j ). Let S(u, v; t) be the associated generating function: S(u, v; t) = si,j (n)ui v j t n . i,j ∈Z,n≥0

Then this series is always algebraic, as well as the series Si,j (t) := n si,j (n)t n that counts walks ending at (i, j ) [13], [20]. For instance, when S is formed of the usual square lattice steps (North, South, West and East), then 1/2 1/2 √ √ 1 − 2t (1 + u) + 1 − 4t 1 + 2t (1 − u) + 1 + 4t S(u, v; t) = 1 − t (u + u + v + v) with u = 1/u and v = 1/v. Moreover, the number of walks ending at certain specific points is remarkably simple. For instance: s1,0 (2n + 1) = C2n+1 , s0,1 (2n + 1) = 4n Cn , s−1,1 (2n) = C2n , where Cn = 2n n /(n + 1) is the nth Catalan number, which counts binary trees (17), Dyck words, and numerous other combinatorial objects [75, Ch. 6]. The first of these three identities has been proved combinatorially [4]. The others still defeat our understanding. 3.6.3. Embedded binary trees. We consider again the complete binary trees met at the beginning of Section 3.2. Let us associate with each (inner) node of such a tree a label, equal to the difference between the number of right steps and the number of left steps one takes when going from the root to the node. In other words, the label of the node is its abscissa in the natural integer embedding of the tree (Figure 12). 0 1

−1 0

2 1

1

Figure 12. The integer embedding of a binary tree.

Let Sj ≡ Sj (t, u) be the generating function of binary trees counted by the number of nodes (variable t) and the number of nodes at abscissa j (variable u). Then for all j ∈ Z, this series is algebraic of degree (at most) 8 (while Sj (t, 1) is quadratic) [12]. Moreover, for j ≥ 0, Sj = T

(1 + μZ j )(1 + μZ j +5 ) , (1 + μZ j +2 )(1 + μZ j +3 )


821

where

(1 + Z 2 )2 , (1 − Z + Z 2 ) and μ ≡ μ(t, u) is the unique formal power series in t satisfying T = 1 + tT 2 ,

μ = (u − 1)

Z=t

Z(1 + μZ)2 (1 + μZ 2 )(1 + μZ 6 ) . (1 + Z)2 (1 + Z + Z 2 )(1 − Z)3 (1 − μ2 Z 5 )

Why is that so? This algebraicity property holds as well for other families of labelled trees [12], [24]. From these series, one can derive certain limit results on the distribution of the number of nodes at abscissa λn1/4 in a random tree with n nodes [12]. These results provide some information about the law of the integrated super-Brownian excursion [12], [21]. 3.6.4. N-algebraicity. N-algebraic series have been defined in Section 3.2 in terms of positive proper algebraic systems. The author has been unable to find in the literature a criterion, or even a necessary condition for an algebraic series with coefficients in N to be N-algebraic. Nor even an algebraic series with coefficients in N that would not be N-algebraic (together with a proof of this statement...). A partial answer could be provided by the study of the possible asymptotic behaviour of coefficients of N-algebraic series. It is very likely that not all behaviours of the form (16) are possible. An important result in this direction states that, if a proper positive system (18) is strongly connected, the nth coefficient of, say, A1 follows the general pattern (16), but with d = −3/2 [43, Thm. VII.7]. The system is strongly connected if, roughly speaking, the expression of every series Ai involves (possibly after a few iterations of the system) every other series Aj . For instance, the system defining the walks ending at 0 in Section 3.4.1 reads M0 = t 2 (1 + M0 )2

and

W0 = M0 (2 + W0 ).

involve W0 . Accordingly, the This system is not strongly connected, as M 2n0does not n −1/2 number of 2n-step walks returning to 0 is n ∼ κ4 n . If one can rule out the possibility that d = −5/2 for N-algebraic series, then this will prove that most map generating functions are not N-algebraic (see the examples in Section 3.5). 3.6.5. Some algebraic hypergeometric series. Consider the following series: d (ai n)! n ei=1 fn t = F (t) = t n, (b n)! j j =1 n≥0

n≥0

where a1 , . . . , ad , b1 , . . . , be are positive integers. This series is algebraic for some values of the ai ’s and bj ’s, as shown by the case (2n)! 1 . tn = √ 2 n! 1 − 4t n≥0

822


Can we describe all algebraic cases? Well, one can easily obtain some necessary conditions on the sequences a and b by looking at the asymptotics of fn . First, an algebraic power series has a finite, positive radius of convergence (unless it is a polynomial). This, combined with Stirling’s formula, gives at once a1 + · · · + ad = b1 + · · · + be .

(23)

Moreover, by looking at the dominant term in the asymptotic behaviour of fn , and comparing with (16), one obtains that either e = d, or e = d +1. The case d = e only gives the trivial solution F (t) = 1/(1 − t), and the complete answer to this problem is as follows [11], [69]: Theorem 3.11. Assume (23) holds and F (t) = 1/(1−t). The series F (t) is algebraic if and only if fn ∈ N for all n and e = d + 1. Here are some algebraic instances: fn =

(6n)!(n)! , (3n)!(2n)!2

fn =

(10n)!(n)! , (5n)!(4n)!(2n)!

fn =

(20n)!(n)! . (10n)!(7n)!(4n)!

The degree of these series is rather big: 12 [resp. 30] for the first [second] series above. This theorem provides a collection of algebraic series with nice integer coefficients: are these series N-algebraic? Do they count some interesting objects? Acknowledgements. The parts of this survey that do not deal exactly with the enumeration of combinatorial objects have often been influenced by discussions with some of my colleagues, including Frédérique Bassino, Henri Cohen, Philippe Flajolet, François Loeser, Géraud Sénizergues. Still, they should not be hold responsible for any of the flaws of this paper.

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Towards a structure theory for matrices and matroids Jim Geelen, Bert Gerards, and Geoff Whittle∗

Abstract. We survey recent work that is aimed at generalizing the results and techniques of the Graph Minors Project of Robertson and Seymour to matrices and matroids. Mathematics Subject Classification (2000). 05B35. Keywords. Matroids, minors, representability, well-quasi-ordering.

1. Introduction We are currently undertaking a program of research aimed at extending the results and techniques of the Graph Minors Project of Robertson and Seymour to matrices and matroids. Here we report on where we stand and where we expect to go. In particular, we discuss the structure of “minor-closed” classes of matrices over a fixed finite field. This requires a peculiar synthesis of graphs, topology, connectivity, and algebra. In addition to proving several long-standing conjectures in the area, we expect the structure theory will help to find efficient algorithms for a general class of problems on matrices and graphs. Most combinatorial computational problems are trivial in the sense that they are typically finite. However, even for modest size problems, enumerating the possibilities is practically infeasible; it often results in algorithms whose running time is exponential in the size of the problem. We seek smarter, more efficient, algorithms. In the theory of algorithms efficient typically means that the running time is polynomial in the size of the problem. Often the problems are modeled by graphs (networks) or matrices. The better picture we have of the model, the more likely it is that we can develop a quick algorithm for the problem. For instance, the problem at hand may be more tractable if the modeling graph can be drawn in the plane, or some other particular surface, without crossings. Then it is relevant that we can test efficiently if the graph has such advantageous appearance. Surface embeddability, and other related properties, are preserved when deleting an edge from the graph or contracting an edge (contracting means deleting the edge and identifying its ends). The result of any series of such deletions and contractions is called a minor of the graph. In this terminology, testing ∗ This research was partially supported by grants from the Natural Sciences and Engineering Research Council of Canada, the Nederlandse Organisatie voor Wetenschappelijk Onderzoek and the Marsden Fund of New Zealand.


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surface embeddability is testing a particular minor-closed graph property. So motivated by real-word computational problems, we end up with the fundamental question if minor-closed graph properties can be tested efficiently. That this is possible indeed for any fixed minor-closed graph property is one of the consequences of the ground-breaking work of Robertson and Seymour in their Graph Minors Project. One major outcome of this project is their proof of Wagner’s Conjecture that graphs are “well-quasi-ordered” under the minor-order, which is the following theorem. The Graph Minors Theorem ([31]). Any infinite family of graphs contains two members such that one is isomorphic to a minor of the other. This implies that for any minor-closed graph property there are only finitely many excluded minors, these are graphs that do not have the property but whose proper minors do have the property. For planarity, for instance, there are exactly two excluded minors: K5 and K3,3 ; this is Kuratowski’s famous characterization of planarity [22]. So, by the Graph Minors Theorem, to test a minor-closed graph property we only need to test containment of each of its excluded minors individually. That this is possible is another crucial outcome of Robertson and Seymour’s work. The Graph Minor Recognition Theorem ([29]). For each graph H , there exists a polynomial-time algorithm for testing if a graph has a minor isomorphic to H . This answered one of the twelve open problems in Garey and Johnson’s 1979 book on computational complexity [8]. So minor-closed graph properties can be tested efficiently. However, as noted earlier, also matrices are widely used as modeling tools, for example in integer programming models for operations research. Integer programming models are very general and powerful, but in a sense too general; they lead to “NP-hard” problems. However if the matrix in an integer programming model is totally unimodular, that means if all subdeterminants are 0, 1 or −1, then linear programming methods do solve the problem [20], and these methods are efficient. So this raises the issue of testing totally unimodularity, another open problem back in ’79, in Garey and Johnson’s book. Now, it turns out that a matrix being totally unimodular means that it is in a certain sense representable over any field, and also this embeddability property is closed under certain minor-operations. So also here the fundamental issue of testing minor-closed properties arises. For that we work at extending Robertson and Seymour’s graph minor theory to matrices. As the issues involved do not so much concern the actual matrices, but rather the underlying “matroids”, we work in that setting. A matroid consists of a finite set E, the ground set of the matroid, and a function r, the rank function of the matroid. This rank function is defined on the subsets of E and satisfies the following properties: 0 ≤ r(X) ≤ |X| for X ⊆ E; r(X) ≤ r(Y ) for all X ⊆ Y ⊆ E and r(X ∪ Y ) + r(X ∩ Y ) ≤ r(X) + r(Y ) for all X, Y ⊆ E. We call r(X) the rank of X and r(E) the rank of the matroid. The rank function of a matroid M is

Towards a structure theory for matrices and matroids

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denoted by rM . Two matroids are isomorphic if there is a rank-preserving bijection between their ground sets. Matrices yield matroids: If A = (ae : e ∈ E) is a matrix with columns ae over a field F, then the linear rank of the column submatrices (ae : e ∈ F ) with F ⊆ E is the rank function of a matroid, the vector matroid M(A) of A. If a matroid is isomorphic to a vector matroid of a matrix over F, we say that the matroid is representable over F or F-representable. (A vector matroid is often described as a configuration of points in a linear, affine or projective space instead of as the collection of columns of a matrix.) Also graphs yield matroids: Let G be a graph with edge set E and vertex set V . The rank of a graph is the number of its vertices minus the number of its components. If F ⊆ E, then the rank of F is the rank of the subgraph of G with edge set equal to F . This rank yields the rank function of a matroid, the cycle matroid M(G) of G. A matroid isomorphic to such a cycle matroid of a graph is called graphic. A graphic matroid is representable over any field by a matrix with two non-zero entries in every column, one equal to 1 and one equal to −1. We assume the reader to be familiar with the standard notions from graph theory. For matroid theory we refer to Oxley [25] or Welsh [43], but we will define the matroid terminology we use, as we go. Now we define matroid minors. Let e be an element of the ground set E of a matroid M. Deleting e from M is replacing M by the matroid with ground set E − {e} and with rank function equal to the restriction of rM to subsets of E − {e}. Contracting e from M is replacing M by the matroid with ground set E − {e} and with rank function rM (X ∪ {e}) − rM ({e}) for each X ⊆ E − {e}. A minor of a matroid is the result of any sequence of deletions and contractions. A minor of a vector matroid over a field F is representable over the field as well. Indeed, deleting an element amounts to just deleting the corresponding column whereas contracting an element f amounts to removing af from (ae : e ∈ E) and projecting all other columns in the direction of af on some arbitrary hyperplane not containing af . Deletions and contractions in a graph are in one-one correspondence with deletions and contractions in its cycle matroid. Thus the cycle matroid of a minor of a graph is a minor of the cycle matroid of the graph. So the notion of graph minors is in essence algebraic, or geometric, and in that sense it generalizes to matrices and matroids. This raises the question to what extent Robertson and Seymour’s graph minor theory extends to matroids. The following conjecture was by Robertson and Seymour, although to our knowledge not in print. The Well-Quasi-Ordering Conjecture. Let F be a finite field. Then any infinite set of F-representable matroids contains two matroids, one of which is isomorphic to a minor of the other. As yet, the Well-Quasi-Ordering Conjecture has not been resolved for any finite field. Note that it is equivalent to the conjecture that, for a finite field F, any minor-closed class of F-representable matroids has a finite number of F-representable excluded minors.

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The finiteness of the field in the Well-Quasi-Ordering Conjecture is essential. Indeed, suppose F is an infinite field and consider for each integer n ≥ 3, a 2n × 3 matrix with columns p1 , . . . , pn and q1 , . . . , qn , where p1 , . . . , pn are vectors in general position in F3 and each qi is spanned by pi and pi+1 , but is not spanned by any other pair among p1 , . . . , pn (where pn+1 = p1 ). As F is infinite, such matrices clearly exist. Among the vector matroids of these matrices none is a minor of another. Indeed, all members of the collection have rank 3, so all minors that use a contraction have too low rank to be in the collection; deleting an element from a member of the collection destroys the unique cyclic arrangement of linearly dependent triples pi , qi , pi+1 in a way that cannot be repaired by further deletions. We conjecture additionally that for matroids that are representable over a finite field minor-closed properties can be recognized in polynomial time, in other words we conjecture that also the Graph Minor Recognition Theorem extends. The Minor-Recognition Conjecture. For any finite field F and any F-representable matroid N, there is a polynomial-time algorithm for testing whether an F-representable matroid contains a minor isomorphic to N. At the heart of the Graph Minors Project is Robertson and Seymour’s Graph Minors Structure Theorem [30]. It describes constructively the graphs that do not contain a given graph as a minor. This constructive description enables techniques to establish the well-quasi-ordering and algorithmic consequences. For matroids, Seymour [36] used this approach successfully for characterizing total unimodularity (see Section 3). Our hope is to use the same strategy for general matroids that are representable over finite fields. Therefore we are developing a structure theory for such matroids. As a major role in the theory of graph minors is played by connectivity, we need an extension of graph connectivity to matroids. The basic ingredients of graph connectivity are separations, which tell where the connectivity is not that high, and Menger’s Theorem, which provides a way of certifying that the connectivity is not that low. A separation of a graph G is a pair (G1 , G2 ) of subgraphs of G such that G = G1 ∪ G2 ; the order of the separation (G1 , G2 ) is the number of vertices of G that lie in both G1 and G2 . A graph is k-connected if it has no separation (G1 , G2 ) of order l less than k such that G1 and G2 have at least l edges each. One of the fundamental theorems of graph theory is Menger’s Theorem. Menger’s Theorem ([24]). If G is a graph and S and T are two sets of vertices, then there either exist k disjoint paths, each connecting a vertex in S with a vertex in T , or (exclusively) G has a separation (G1 , G2 ) of order less than k such that S lies in G1 and T in G2 . As an illustration of how this theorem plays a role in finding minors consider the following easy result of Dirac [4]: A 3-connected graph with at least 4 vertices has a minor isomorphic to K4 . (Kn denotes the complete graph with n vertices; complete means that every pair of vertices is connected by an edge.) Here is a proof of Dirac’s


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result: we may assume the graph has two non-adjacent vertices s and t, otherwise the graph is complete and we are done. Apply Menger’s Theorem to the set S of neighbours of s and the set T of neighbours of t. This yields three P1 , P2 and P3 from s to t that only meet at their ends. As G is 3-connected, G − s − t is connected, so there exists a path Q that connects two of these paths and misses the third path. The union of P1 , P2 , P3 and Q clearly has a minor isomorphic to K4 . So Dirac’s result follows. This is a very easy result of course, but it may convince the reader of the need of a notion of matroid connectivity and a matroidal version of Menger’s Theorem. A separation of a matroid M is a partition (X, Y ) of the ground set E. The order of the separation (X, Y ) is rM (X) + rM (Y ) − rM (E) + 1; for a representable matroid this is the dimension of the intersection of the subspaces spanned by X and Y plus 1. A matroid is k-connected if it has no separation (X, Y ) of order l less than k such that X and Y have size l or more. If (G1 , G2 ) is a separation of G where E1 and E2 are the edge sets of G1 respectively G2 , then (E1 , E2 ) is a separation of M(G). When G1 , G2 , and G are connected graphs, the orders of these separations are the same. This is enough to consider matroid separations and matroid connectivity as genuine generalizations of these notions for graphs. And there is a matroidal generalization of Menger’s Theorem as well. Tutte’s Linking Theorem ([42]). Let M be a matroid and X and Y be disjoint subsets of its ground set. Then there exists a minor of M in which (X, Y ) is a separation of order at least k or (exclusively) M has a separation (A, B) of order less than k with X in A and Y in B. This follows quite easily from Edmonds’ Matroid Intersection Theorem [7], which is one of the fundamental theorems of matroid theory. So we see that the basic theory of graph connectivity does extend quite well to matroids. Besides the fact that a matroid structure theory will help proving the Well-QuasiOrdering Conjecture and Minor-Recognition Conjecture for matroids, we also expect that it will provide a handle on the following conjecture, probably the most famous open question in matroid theory. Rota’s Conjecture ([33]). Let F be a finite field. There are, up to isomorphism, only finitely many excluded minors for the class of F-representable matroids. This also has computational relevance, namely for the question how hard it is to decide if a matroid given by an oracle for the rank function is representable over a field F. Unfortunately, it does take exponentially many oracle calls to decide this, for any field [38]. But if Rota’s conjecture is true, then for every finite field F there exists, for every non-F-representable matroid, a polynomial-length certificate for this non-representability, that requires only a constant number of oracle calls. (It is known that this can be done by a quadratic number of oracle calls [14].) For the three smallest fields all excluded minors for representability over the field are known: there is one for GF(2) (Tutte [40]), there are four for GF(3) (Bixby [1]

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and Seymour [37], independently), and there are seven for GF(4) (Geelen, Gerards and Kapoor [9]). For all other finite fields, Rota’s conjecture is still open. A structure theory could well provide a way to prove it. Also in Rota’s conjecture finiteness of the field is essential. Lazarson [23] showed that there are an infinite number of excluded minors for representability over the reals and this is certainly true for all other infinite fields. Summarizing, en route for these three conjectures we are working at establishing the structure of minor-closed proper subclasses of matroids representable over a finite field. This work has already had some success. It turns out that excluding the cycle matroid of a planar graph as a minor imposes tangible structure on matroids over finite fields, so we begin with discussing that.

2. Excluding a planar graph Let F be a finite field and let H be a planar graph. We give a constructive structural description of F-representable matroids with no M(H )-minor and show that this description enables significant progress on the three conjectures in Section 1. Essentially the structure is to be decomposable into small pieces along low-order separations. We will first explain what that means. A branch-decomposition of matroid M is a tree T in which all vertices have degree 1 or 3, where the degree-1 vertices of T are in 1-1 correspondence with the elements of the ground set E of M. The width of an edge e in T is the order of the separation (X, Y ) of M where X contains the elements of E that correspond to the degree-1 vertices of T in one component of T − e and Y the elements of E that correspond to the degree-1 vertices of T in the other component of T − e. So a branch-decomposition is a data-structure for a collection of separations. The width of a branch-decomposition is the maximum of the widths of its edges and the branchwidth of a matroid is the minimum of the widths of all its branch-decompositions. So roughly low branch-width means to be decomposable into small pieces along low-order separations. Branch-width is a matroid generalization of branch-width for graphs defined by Robertson and Seymour [28]. For graphs it is, up to a constant multiplicative bound, the same as tree-width, also introduced by Robertson and Seymour. In the Graph Minors Project they mainly use tree-width and that notion does extend to matroids [19] as well. But branch-width is easier to work with for matroids, so here we will only use branch-width, also for graphs. Robertson and Seymour prove the following result. The Grid Theorem for graphs ([27]). For each planar graph H there is an integer k such that any graph with branch-width at least k has a minor isomorphic to H . This result is called the Grid Theorem because, as every planar graph is a minor of a grid, it suffices to prove it for the case that H is a grid. Here, a grid, or rather


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an n by n grid, refers to the graph with a vertex (i, j ) for each pair of integers i and j between 1 and n and an edge between any two pairs (i1 , j1 ) and (i2 , j2 ) with |i1 − i2 | + |j1 − j2 | = 1. If a matroid has a minor isomorphic to a cycle matroid of an n by n grid, we say it has a grid-minor. To convince oneself that each planar graph H is a minor of a sufficiently large grid, visualize H as drawn without crossings and with the edges and vertices as thick lines and dots on a piece of grid paper with a very fine grid. Consider a class of graphs that do not have a fixed planar graph as a minor. By the Grid Theorem for graphs, the members of that class have bounded branch-width. This constructive characterization provides considerable traction for both algorithmic and structural problems. For example, Robertson and Seymour [26] prove that any class of graphs of bounded branch-width is well-quasi-ordered. This extends to matroids over finite fields. Theorem ([12]). Let F be a finite field and k an integer. Then each infinite set of F-representable matroids with branch-width at most k has two members such that one is isomorphic to a minor of the other. Johnson, Robertson, and Seymour [21] conjectured that also the Grid Theorem for graphs extends to matroids over finite fields and this is indeed the case. The Grid Theorem for matroids ([13]). For each finite field F and each planar graph H , there exists an integer k such that each F-representable matroid with branchwidth at least k has a minor isomorphic to M(H ). As a consequence we obtain the following partial result towards the Well-QuasiOrdering Conjecture. Corollary. Let F be a finite field and H a planar graph. Then any infinite set of F-representable matroids with no minor isomorphic to M(H ) contains two matroids such that one is isomorphic to a minor of the other. In combination with results of Hlinˇený [18], we also obtain partial progress towards the Minor-Recognition Conjecture. Corollary. For each finite field F and each planar graph H , there is a polynomialtime algorithm for testing whether or not an F-representable matroid contains a minor isomorphic to M(H ). So for matroids over a fixed finite field we can efficiently test all minor-closed properties that do not hold for the cycle matroid of all planar graphs. Geelen and Whittle [17] show that for a finite field F and integer k, the number of excluded minors for F-representability that have branch width at most k is finite. In combination with the Grid Theorem for matroids this yields the following result. Corollary. For each finite field F and each planar graph H , there are only finitely many excluded minors for F-representability that do not have M(H ) as a minor.

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We see that the structure imposed on a class of matroids by excluding the matroid of a planar graph as a minor yields restricted solutions to the Well-Quasi-Ordering Conjecture, the Minor-Recognition Conjecture, and Rota’s Conjecture, and that is a promising beginning. The Grid Theorem for matroids is absolutely central in developing a structure theory for matroids. When specialized to graphs, the proof in [13] is different from the existing proofs in [3], [27], [32]. It is important to note that we had access to an extraordinary 150-page handwritten manuscript [21] of Johnson, Robertson, and Seymour describing their progress towards a grid theorem for matroids. The techniques we learned from their manuscript played a crucial role in parts of our proof. The proof also makes use of earlier results we obtained together with Neil Robertson [10], [11]. Regarding the result above on well-quasi-ordering of F-representable matroids of bounded branch-width it is interesting to note that the finiteness of F is essential there. This is illustrated by the sequence of matrices given in Section 1, below the Well-Quasi-Ordering Conjecture; they all have branch-width at most 3, as they all have rank 3. On the other hand, there are only finitely many excluded minors for the class of all matroids of branch width at most k, representable or not [11]. We conclude this section with a comment regarding Rota’s Conjecture. We have seen that for every finite field GF(q) there are a finite number of excluded minors for GF(q)-representability of any given branch width. In [15] it is proved that an excluded minor for GF(q)-representability of sufficiently large branch width cannot contain a PG(q + 6, q)-minor. (PG(n, q) is the matroid represented by points of the projective geometry of order n over GF(q).) So it follows that if Rota’s Conjecture fails for GF(q), then there must exist excluded minors with arbitrarily large grid-minors and no large projective geometry as a minor.

3. An example: the structure of regular matroids With the results in the previous section in hand we proceed towards a structure theory for matroids over finite fields. One of the prototypes of structural matroid theory and its algorithmic consequences concerns the totally unimodular matrices mentioned in Section 1. The question if a certain given matrix is totally unimodular can be translated into the question if a related, easy-to-construct, GF(2)-representable matroid is representable over all fields. Such matroids are called regular. Regularity is a minorclosed property. Tutte [40] proved that a GF(2)-representable matroid is regular if and only if it does not have a minor isomorphic to PG(2, 2), also called the Fano matroid, or to the dual of PG(2, 2). Here the dual of a matroid M is the matroid M ∗ with the same ground set E as M and with rank-function rM ∗ (X) = |X|−rM (E −X)+rM (E). Representability over a field F is closed under duality, hence so is regularity. Taking minors commutes with duality; although the roles of deletion and contraction swap.


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Tutte’s excluded minor characterization of regular matroids is one of the gems of matroid theory, but it does not tell how to decide if a given GF(2)-representable matroid is regular or not. That question was answered by a structural result, Seymour’s Regular Matroid Decomposition Theorem [36]: A matroid over GF(2) is regular if and only if it is the 1-, 2- or 3-sum of graphic matroids, duals of graphic matroids and copies of a particular 10-element matroid called R10 . Here a 1-, 2- or 3-sum of two (representable) matroids is carried out by embedding each of them in a distinct projective space and then combining these projective spaces by taking either their direct sum, in case of a 1-sum, or identifying single points or lines, in case of a 2-sum or a 3-sum. These “meeting” points or lines should be in both matroids and may or may not be deleted from the matroid after the composition. The sums as well as the reversed “decomposition” operations preserve regularity. So Seymour’s result gives a structural description of the class of regular matroids. Their “global structure” is that they are composed from smaller pieces along loworder separations. The pieces sit together in a tree-like fashion. The description of these pieces provides the “local structure”: each piece is either a graphic matroid, the dual of a graphic matroid or isomorphic to R10 . This combination of global and local structure is typical for all structural results in this paper. Seymour’s structural characterization of regular matroids is constructive, it can be used to design an algorithm for testing regularity in polynomial time. This goes as follows. First decide if the matroid is a 1-, 2- or 3-sum of smaller matroids. This can be done in polynomial time, as gluing two matrices together leaves a separation of order at most 3 in the composed matroid and Cunningham and Edmonds [2] observed that detecting these is a matroid intersection problem, which is solvable in polynomial time (Edmonds [7]). When the matroid is fully decomposed into “4-connected pieces”, each piece is tested for being isomorphic to R10 , which is trivial, or being a graphic matroid or the dual of a graphic matroid, which can be done by Tutte’s polynomialtime algorithm for testing graphicness [41]. If all pieces pass the test, the original matroid is regular, otherwise it is not. By the relation between regularity and total unimodularity this yields an algorithm for testing if a real matrix is totally unimodular or not (see Schrijver [35, Chapter 20] for a description of this algorithm in terms of the matrices). This is the only known polynomial-time algorithm for testing totally unimodularity. Thus the structure of matroids is crucial for the algorithmic aspects of this central property in operations research and combinatorial optimization. Actually matroids in general do play a major role in the theory of combinatorial optimization, see Schrijver [34]. A book on matroid decomposition is Truemper [39], it mainly concerns regular matroid decomposition and related topics. With Seymour’s regular matroid decomposition in mind we next discuss what we expect to be the structure of minor-closed classes of matroids that are representable over a finite field. It should be noted that the results will not be as “tight” as in Seymour’s decomposition theorem. Seymour provided a constructive description of all binary matroids that contain neither the Fano matroid nor its dual as a minor. More-

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over, none of the matroids obtained via that construction contain the Fano matroid or its dual. In contrast, the Grid Theorem for graphs provides a construction for the graphs that do not contain a given planar graph H as a minor, but some graphs obtained via the construction may contain H as a minor. The construction is, however, sufficiently restrictive that it does not build all planar graphs. For algorithmic and well-quasi-ordering purposes, this is good enough.

4. Global structure and local structure In Section 2 we discussed the structure of classes of matroids over finite fields that do not have a minor isomorphic to the cycle matroid of a particular planar graph: they can be decomposed into small pieces along low-order separations; they have low branch-width. So explorations beyond that concern matroids with high branch-width. The existence of large grid-minors in such matroids is useful in investigating their structure, but a matroid may have several high branch-width parts that are separated by low-order separations and we have to describe the structure of these parts separately. To get a handle on these parts, Robertson and Seymour [28] introduce tangles. A tangle really just indicates for each low-order separation on which side a particular high branch-width part lies. Formally, a tangle of order t assigns to each separation (X, Y ) of order less than t one of X and Y as the small side of (X, Y ) and the other side as the big side of (X, Y ). It is required that no three small sides of the tangle cover the ground set of the matroid and that no singletons are big. It turns out that the maximum order of a tangle in a matroid is the branch-width of the matroid; for graphs this was shown by Robertson and Seymour [28] and for matroids by Dhamatilike [5] (although this result was implicit in [28]). Combining this with the Grid Theorem, we see that grid-minors yield tangles. Indeed, if F ⊆ M is the set of elements of an n by n grid-minor of M, then F partitions naturally in the “horizontal” and “vertical” lines of the grid. If we consider for each separation (X, Y ) of order less than n the side that contains a line in F as big, then that yields a tangle of order n. It was shown in [16] that for any finite field F and any n there exists an integer t such that in any F-representable matroid any tangle of order t controls an n by n grid-minor. This generalizes a result from [32] for graphs. Here a tangle controls a minor N of M if no small side of a separation of order less than the rank of N contains all elements of N. So tangles “locate” highly connected areas of the matroid. If all small sides of one tangle are small in some other tangle, then they both seem to refer to the same highly connected part but the latter tangle does that more accurately. Therefore we are mainly interested in the maximal tangles, those for which the collection of small sets is inclusion-wise maximal. It turns out that matroids, like graphs [28], can be viewed as consisting of their maximal tangles put together in a tree-shaped structure, see [16] for details. This provides a global picture of a matroid. To complete that picture we have to describe the individual tangles, the local structure.


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To explain what we mean with that, we first explain what it means to reduce a set S in an F-represented matroid M. Consider M as a collection E of points in a projective geometry. Let X be the span of E − S in that projective geometry. For each S ⊆ S whose span meets X in a single point, we call that point xS . Let Y be the set of all points xS for all such sets S . Replacing M by the matroid represented by the union of Y and E − S is called reducing S. (For graphs, this more or less means to remove the edges in S and to add an edge between any pair of vertices that both lie in S and in the complement of S.) Let C be a class of matroids. A tangle has local structure in C if there exist separations (S1 , B1 ), . . . , (Sk , Bk ) in M with disjoint small sides S1 , . . . , Sk such that the matroid obtained from M by reducing each of S1 , . . . , Sk is in C. To describe the full structure we only need to characterize the minor-closed classes C that provide the local structure of tangles in matroids over the finite field that do not contain a particular minor.

5. The local structure of graph tangles The Graph Minor Structure Theorem says that for any n there exists a surface and integers m, d, k such that the tangles of a graph with no minor isomorphic to Kn have local structure in the class of graphs that lie on a surface with m vortices of depth at most d and k extra vertices. We explain what this means. A vortex with connectors v1 , . . . , vp is a graph H that is the union of graphs H1 , . . . , Hp such that vi is a vertex in Hi for each i = 1, . . . , p and such that if a vertex v of H occurs in Hi and Hj for some i, j = 1, . . . , p then v either occurs in all of Hi+1 , . . . , Hj −1 or in all of Hj +1 , . . . , Hi−1 (indices modulo n). The maximum size of the subgraphs H1 , H2 , . . . , Hp is the depth of the vortex. A graph is on a surface with m vortices of depth at most d if it can be constructed as follows: take a graph drawn on , select m faces and add to each of these faces a vortex of depth d to G that meets G and the other added vortices only in its connectors v1 , . . . , vn which lie in that order around the boundary of the face. If we additionally add k new vertices and new edges from these vertices to each other and to the rest of the graph, we obtain a graph that lies on a surface with m vortices of depth at most d and k extra vertices.

6. The local structure of matroid tangles What are the minor-closed classes needed to describe the local structure of matroids that are representable over a finite field? One natural minor-closed class is the class of graphic matroids. Also, if F is a subfield of F, then the class of F -representable matroids is a minor-closed class of F-representable matroids. There is another natural class, of Dowling matroids. They are like graphs and originally introduced by Dowling [6] and studied in greater depth by Zaslavsky [44], [45].

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A Dowling matroid is a matroid that can be represented over a field F by a matrix with the property that every column has at most two non-zero elements. We call such matrix a Dowling representation of the matroid. If the ratio between the nonzero elements in each column of a Dowling representation is in a subgroup of the multiplicative group of F, we call the matroid a Dowling matroid over . One can naturally associate a graph G(A) with a Dowling representation A. Each row of A is a vertex of G(A) and each column of A with two non-zeroes yields an edge in G(A) connecting the vertices corresponding to the rows that have the non-zeroes in that column. Thus we get for each surface and each subgroup of the multiplicative group of F the class of F-representable Dowling matroids that have Dowling representations over the subgroup and whose associated graphs embed on the surface. Obviously such a class is minor-closed. In fact, we can extend such minor-closed class by allowing a bounded number of “vortices” of bounded depth, these are obtained by adding matroid elements into bounded-rank subspaces arranged in a cyclic manner around a face in the embedding, similar to vortices in graphs. Finally we can extend a minor-closed class C of matroids by considering for some integer k the class of all rank-l perturbations of the members of C with l ≤ k. Here an F-representable matroid M is a rank-l perturbation of an F-representable matroid N if M and N have representations A and B, respectively, with the linear rank of A − B equal to l. Splitting a vertex in a graph amounts to a rank-1 perturbation of its cycle matroid. So adding k vertices to a graph amounts to adding a single vertex followed by a rank-(k − 1) perturbation of the cycle matroid. Adding a single vertex to a graph G does in general not correspond to a low-rank perturbation. However, fortunately, the cycle matroid of the resulting graph has a Dowling representation A with G(A) = G. Hence the Graph Minors Structure Theorem is captured by the matroid classes given above. Now we state our main results and conjectures on the structure of minor-closed classes over a finite field GF(q), where q = pk for some fixed prime p and some fixed integer k. We distinguish between three types of minor-classes. The first type are the classes that do not contain the cycle matroid of large complete graphs nor their duals. The second type are the classes that do not contain large projective geometries over the prime field GF(p) of GF(q). The third type are the classes that do not contain large projective geometries over GF(q). In each of the cases, T is a tangle in a GF(q)-representable matroid. Below n is a fixed integer and each of the qualitative bounds “low”, “bounded”, or “sufficiently large” indicates a bound only depending on q and n, so not on the particular tangles or matroids. Excluding M(Kn ) and M(Kn )∗ . We believe that we have proved that if T has sufficiently large order and does not control a minor isomorphic to M(Kn ) or to M(Kn )∗ , then T has local structure in the class of low-rank perturbations of GF(q)-


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representable matroids that can be obtained by adding a bounded number of vortices of bounded depth to a Dowling matroid whose associated graph is embedded in a surface of low genus, or of the duals of such matroids. This implies the Graph Minors Structure Theorem. With this result and duality, we can now restrict our attention to tangles that control the cycle matroid of a large complete graph. Excluding PG(n, p). We conjecture that if T controls a minor isomorphic to M(Km ) for a sufficiently large integer m but T does not control a minor isomorphic to PG(n, p), then T has local structure in the class of low-rank perturbations of GF(q)representable Dowling matroids. Roughly speaking the conjectures above state that if M is a GF(q)-representable matroid with no minor isomorphic to PG(n, p), then M admits a tree-like decomposition such that each part is either essentially a Dowling matroid or is essentially the dual of a Dowling matroid. For a field of prime order this would give the required constructive structural characterization of the minor-closed proper subclasses of matroids representable over the field. It is interesting to note here that a slight extension of Seymour’s regular matroid decomposition says that if a GF(2)-representable matroid has no minor isomorphic to PG(2, 2) then it can be constructed from graphic matroids, their duals, and copies of R10 and copies of the dual of PG(2, 2), by 1-, 2- and 3-sums [36]. As graphic matroids are GF(2)-representable Dowling matroids and as R10 and the dual of PG(2, 2) are low-rank perturbations of a trivial matroid, this result of Seymour’s implies the conjecture above for case that q = 2 and n = 2. Excluding PG(n, q). For the case that q is not prime, we conjecture that if T controls a minor isomorphic to PG(m, p) for a sufficiently large integer m but T does not control a minor isomorphic to PG(n, q), then T has local structure in the class of GF(q)-representable low-rank perturbations of matroids that are representable over a proper subfield of GF(q). Finally we can summarize all of the above into a single conjecture. For any minorclosed proper subclass M of GF(q)-representable matroids, each matroid in M admits a tree-like decomposition such that each part is either essentially a Dowling matroid, or is essentially the dual of a Dowling matroid, or is essentially represented over a proper subfield of GF(q).

References [1] Bixby, R. E., On Reid’s characterization of the ternary matroids. J. Combin. Theory Ser. B 26 (1979), 174–204. [2] Cunningham, W. H., and Edmonds, J., A combinatorial decomposition theory. Canad. J. Math. 32 (1980), 734–765.

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[3] Diestel, Reinhard, Gorbunov, Konstantin Yu., Jensen, Tommy R., and Thomassen, Carsten, Highly connected sets and the excluded grid theorem. J. Combin. Theory Ser. B 99 (1999), 61–73. [4] Dirac, G. A., A property in 4-chromatic graphs and some remarks on critical graphs. J. London Math. Soc. 27 (1952), 85–92. [5] Dharmatilake, J.,A min-max theorem using matroid separations. In Matroid Theory (Seattle WA 1995), Contemp. Math. 197, Amer. Math. Soc., Providence, RI, 1996, 333–342. [6] Dowling, T., A class of geometric lattices based on finite groups. J. Combin. Theory Ser. B 14 (1973), 61–86. [7] Edmonds, J., Submodular functions, matroids, and certain polyhedra. In Combinatorial Structures and Their Applications (Proceedings Calgary International Conference on Combinatorial Structures and Their Applications, Calgary, Alberta, 1969), ed. by R. Guy, H. Hanani, N. Sauer, and J. Sch˝onheim, Gordon and Breach, New York 1970, 69–87. [8] Garey, M. R., and Johnson, D. S., Computers and intractability: A guide to the theory of NP-completeness, Freeman, San Fransisco, 1979. [9] Geelen, J. F., Gerards, A. M. H., and Kapoor, A., The excluded minors for GF(4)representable matroids. J. Combin. Theory Ser. B 79 (2000), 247–299. [10] Geelen, J. F., Gerards, A. M. H., Robertson, N., and Whittle, G. P., Obstructions to branchdecomposition of matroids. Research Report 03-2, School of Mathematical and Computing Sciences, Victoria University of Wellington, 2004. [11] Geelen, J. F., Gerards, A. M. H., Robertson, N., and Whittle, G. P., On the excluded minors for the matroids of branch-width k. J. Combin. Theory Ser. B 88 (2003), 261–265. [12] Geelen, James F., Gerards, A. M. H., and Whittle, Geoff, Branch width and well-quasiordering in matroids and graphs. J. Combin. Theory Ser. B 84 (2002), 270–290. [13] Geelen, Jim, Gerards, Bert, and Whittle, Geoff, Excluding a planar graph from GF(q)representable matroids. Research Report 03-4, School of Mathematical and Computing Sciences, Victoria University of Wellington, 2003. [14] Geelen, Jim, Gerards, Bert, and Whittle, Geoff, Inequivalent representations of matroids I: An overview. In preparation, 2005. [15] Geelen, Jim, Gerards, Bert, and Whittle, Geoff, Inequivalent representations of matroids II: k-coherent matroids. In preparation, 2004. [16] Geelen, Jim, Gerards, Bert, and Whittle, Geoff, Tangles, tree-decompositions, and grids in matroids. Research Report 04-5, School of Mathematical and Computing Sciences, Victoria University of Wellington, 2004. [17] Geelen, J., and Whittle, G., Branch width and Rota’s conjecture. J. Combin. Theory Ser. B 86 (2002), 215–330. [18] Hlinˇený, P., Branch-width, parse trees, and monadic second-order logic for matroids. Preprint, 2002. [19] Hlinˇený, P., and Whittle, G., Matroid tree width. European J. Combin., to appear. [20] Hoffman, A. J., and Kruskal, J. B., Integral boundary points of convex polyhedra. In Linear Inequalities and Related Systems (ed. by H. W. Kuhn and A. W. Tucker), Ann. of Math. Stud. 38, Princeton University Press, Princeton, N.J., 1956, 223–246. [21] Johnson, Thor, Roberston, Neil, and Seymour, P. D., Connectivity in binary matroids. Manuscript.


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[22] Kuratowski, C., Sur le problème des courbes gauches en topologie. Fund. Math. 15 (1930), 271–283. [23] Lazarson, T., The representation problem for independence functions. J. London Math. Soc. 33 (1958), 21–25. [24] Menger, K., Zur allgemeinen Kurventheorie. Fund. Math. 10 (1927), 96–115. [25] Oxley, J. G., Matroid theory. Oxford University Press, New York, 1992. [26] Robertson, Neil, and Seymour, P. D., Graph Minors. IV. Tree-width and well-quasiordering. J. Combin. Theory Ser. B 48 (1990), 227–254. [27] Robertson, Neil, and Seymour, P. D., Graph Minors. V. Excluding a planar graph. J. Combin. Theory Ser. B 41 (1986), 92–114. [28] Robertson, Neil, and Seymour, P. D., Graph Minors. X. Obstructions to tree-decomposition. J. Combin. Theory Ser. B 52 (1991), 153–190. [29] Robertson, Neil, and Seymour, P. D., Graph Minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B 63 (1995), 65–110. [30] Robertson, Neil, and Seymour, P. D., Graph Minors. XVI. Excluding a non-planar graph. J. Combin. Theory Ser. B 89 (2003), 43–76. [31] Robertson, Neil, and Seymour, P. D., Graph Minors. XX. Wagner’s conjecture. J. Combin. Theory Ser. B 92 (2004), 325–357. [32] Robertson, Neil, Seymour, Paul, and Thomas, Robin, Quickly excluding a planar graph. J. Combin. Theory Ser. B 62 (1994), 323–348. [33] Rota, G.-C., Combinatorial theory, old and new. In Actes du Congrès International de Mathématiciens (Nice, 1970), vol. 3, Gauthier-Villars, Paris 1970, 229–233. [34] Schrijver, A., Combinatorial optimization — polyhedra and efficiency. Vol. A, Algorithms Combin. 24, Springer-Verlag, Berlin 2003. [35] Schrijver, A., Theory of linear and integer programming. John Wiley and Sons, Chichester 1986. [36] Seymour, P. D., Decomposition of regular matroids. J. Combin. Theory Ser. B 28 (1980), 305–359. [37] Seymour, P. D., Matroid representation over GF(3). J. Combin. Theory Ser. B 26 (1979), 159–173. [38] Seymour, P. D., Recognizing graphic matroids. Combinatorica 1 (1981), 75–78. [39] Truemper, K., Matroid decomposition. Academic Press, San Diego, 1992. [40] Tutte, W. T., A homotopy theorem for matroids I, II. Trans. Amer. Math. Soc. 88 (1958), 144–174. [41] Tutte, W. T., An algorithm for determining whether a given binary matroid is graphic. Proc. Amer. Math. Soc. 11 (1960), 905–917. [42] Tutte, W. T., Menger’s theorem for matroids. J. Res. Nat. Bur. Standards Sect. B. 69B (1965), 49–53. [43] Welsh, D. J. A., Matroid theory, Academic Press, London, 1976. [44] T. Zaslavsky, A mathematical bibliography of signed and gain graphs and allied areas. Manuscript prepared with Marge Pratt, Electron. J. Combin. 5 (1999), Dynamic Surveys 8, 124 pp.

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[45] Zaslavsky, T., Biased graphs. II. the three matroids. J. Combin. Theory Ser. B 51 (1991), 46–72. Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, Canada E-mail: [email protected] Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands and Technische Universiteit Eindhoven, Eindhoven, The Netherlands E-mail: [email protected] School of Mathematical and Computing Sciences, Victoria University, Wellington, New Zealand E-mail: [email protected]

Cherednik algebras, Macdonald polynomials and combinatorics Mark Haiman ∗

Abstract. In the first part of this article we review the general theory of Cherednik algebras and non-symmetric Macdonald polynomials, including a formulation and proof of the fundamental duality theorem in its proper general context. In the last section we summarize some of the combinatorial results in this area which we have recently obtained in collaboration with J. Haglund and N. Loehr. Mathematics Subject Classification (2000). Primary 33D52; Secondary 05E10. Keywords. Macdonald polynomials, affine Hecke algebras, Cherednik algebras.

1. Introduction The record is very long. The facts are few and may be briefly stated. —Miller v. San Francisco Methodist Episcopal (1932) This article consists of an overview of the theory of Cherednik algebras and nonsymmetric Macdonald polynomials, followed by the combinatorial formula for nonsymmetric Macdonald polynomials of type An−1 recently obtained by Haglund, Loehr and the author. The main points in the theory are duality (Theorems 4.10, 5.11), and its consequence, the intertwiner recurrence for Macdonald polynomials (Corollary 6.15), which is the key to the combinatorial study of non-symmetric Macdonald polynomials. The intertwiner recurrence can also be used to deduce other important results in the theory, such as the norm and evaluation formulas, but I have omitted those for lack of space. The theory of course did not spring into being in the tidy form in which I have attempted to package it here. Rather, it has been gradually clarified over almost twenty years through the efforts of many people, in a large literature which I will not attempt to cite in full. Let me only mention the origins of the theory in the works of Macdonald [13], [14], [15], Opdam [17], and Cherednik [1], [2] and remark that further important contributions were made by Ion, Knop, Koornwinder, Sahi, and van Diejen, among others. ∗ Work supported in part by NSF grant DMS-0301072.


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Mark Haiman

The overview given here necessarily has much in common with Macdonald’s monograph [16], which serves a similar purpose, but there are also several differences. I have systematically used the lattice formulation for root systems, because it is most natural from related points of view (algebraic groups, quantum groups), because it puts affine and other root systems on an equal footing, and because important elements of the theory (§2, 5.1–5.5, 5.13–5.15) apply to arbitrary root systems. I give a new and somewhat more general proof of the duality theorem; Macdonald’s proof, strictly speaking, applies to the root system of SLn , for instance, but not GLn or PGLn , although it can be adjusted to cover these cases. For the triangularity property of the Macdonald polynomials Eλ (Theorem 6.6), I use the affine Bruhat order on the weight lattice X, rather than the orbit-lexicographic order used by Macdonald. This simplifies some arguments, and is more natural in that the coefficient of x μ in Eλ is non-zero if and only if μ < λ in Bruhat order. I have also tried to use more transparent notation.

2. Root systems 2.1. We always consider root systems realized in a lattice. So, for us, a root system X, (αi ), (αi∨ ) consists of a finite-rank free abelian group X, whose dual lattice Hom(X, Z) is denoted X∨ , a finite set of vectors α1 , . . . , αn ∈ X, called simple roots, and a finite set of covectors α1∨ , . . . , αn∨ ∈ X∨ , called simple coroots. We denote by XQ (resp. XR ) the Q-vector space X ⊗Z Q (resp. R-vector space X ⊗Z R) spanned by X. The n × n matrix A with entries aij = αj , αi∨ is assumed to be a generalized Cartan matrix, satisfying the axioms (i) αi , αi∨ = 2, (ii) αj , αi∨ ≤ 0 for all j = i, (iii) αj , αi∨ = 0 if and only if αi , αj∨ = 0. The Dynkin diagram is the graph with nodes i = 1, . . . , n and an edge {i, j } for each aij = 0, usually with some decoration on the edges to indicate the values of aij , aj i . If the Dynkin diagram is connected, A is indecomposable. If there exist non-zero integers di such that αj , di αi∨ = αi , dj αj∨ for all i, j , then A is symmetrizable. The integers di can be assumed positive. If A is symmetrizable and indecomposable, the di are unique up to an overall common factor. Then di is length of the root αi . If there are only two root lengths, we call them long and short. If there is only one root length, every root is both long and short. 2.2. Let α ∈ X and α ∨ ∈ X∨ satisfy α, α ∨ = 2. The linear automorphism sαα ∨ (λ) = λ − λ, α ∨ α

Cherednik algebras, Macdonald polynomials and combinatorics

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of X is a reflection. It fixes the hyperplane λ, α ∨ = 0 pointwise, and sends α to −α. Thus (sα,α ∨ )2 = 1. The reflection on X∨ dual to sα,α ∨ is equal to sα ∨ ,α . If α ∨ is implicitly associated to α, we write sα for both sα,α ∨ and sα ∨ ,α . When α = αi and α ∨ = αi∨ are a simple root and corresponding coroot, we write si for sαi . The si are called simple reflections. 2.3. The root system X, (αi ), (αi∨ ) is non-degenerate if the simple roots αi are linearly independent. When the Cartan matrix A is non-singular, e.g., for any finite root system, then both X and its dual X ∨ , (αi∨ ), (αi ) are necessarily non-degenerate. When A is singular, for instance if the root system is affine (Definition 3.1), it is often convenient to take the simple roots to be a basis of XQ , in which case X is nondegenerate but its dual is degenerate. 2.4. Assume in what follows that X, (αi ), (αi∨ ) is non-degenerate. TheWeyl group W is the group of automorphisms of X (and of X∨ ) generated by the simple reflections si . The sets of roots and coroots are R= W (αi ), R ∨ = W (αi∨ ). i

i

The root and coroot lattices are Q = Z{α1 , . . . , αn } ⊆ X,

Q∨ = Z{α1∨ , . . . , αn∨ } ⊆ X∨ .

The set of positive roots is R+ = R ∩ Q+ , where Q+ = N{α1 , . . . , αn }. The dominant weights are the elements of the cone X+ = {λ ∈ X : λ, αi∨ ≥ 0 for all i}. The root system X, (αi ), (αi∨ ) is finite if W is a finite group, or equivalently, R is a finite set. The Cartan matrix A of a finite root system is symmetrizable, with positive definite symmetrization DA. Conversely, if A has a positive definite symmetrization, then R is finite. The finite root systems classify reductive algebraic groups G over any algebraically closed field k. Then X is the character group of a maximal torus in G, or weight lattice. Example 2.5. Let X = Zn , and identify X∨ with X using the standard inner product on Zn such that the unit vectors ei are orthogonal. Let αi = αi∨ = ei − ei+1 for i = 1, . . . , n − 1. This gives the root system of the group GLn . Replacing X with the root lattice Q and restricting the simple coroots to Q, we obtain the root system of the adjoint group PGLn (GLn modulo its center). The constant vector ε = e1 + · · · + en satisfies ε, αi∨ = 0 for all i. Let X = X/(Zε), with simple roots and coroots induced by those of X. This gives the root system of the simply connected group SLn . It is dual to the root system of PGLn . All three root systems have the same Cartan matrix, of type An−1 .

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2.6. We recall some standard facts. First, R = R+ ∪ −R+ , i.e., every root is positive or negative (note that R = −R, since si (αi ) = −αi for all i). The Weyl group W , with its generating set S of simple reflections si , is a Coxeter group with defining relations si2 = 1, si sj si . . . = sj si sj . . .

(mij factors on each side),

(1) (2)

where if aij aj i = 0, 1, 2 or 3, then mij = 2, 3, 4, or 6, respectively, and if aij aj i ≥ 4, there is no relation between si , sj . The length l(w) of w ∈ W is the minimal l such that w = si1 . . . sil . Such an expression is called a reduced factorization. More generally, if w = u1 u2 . . . ur with l(w) = l(u1 ) + · · · + l(ur ) we call u1 · u2 . . . ur a reduced factorization. If w = sj1 . . . sjl is a second reduced factorization, then the identity sj1 . . . sjl = si1 . . . sil holds in the monoid with generators si and the braid relations (2), that is, it does not depend on the relations si2 = 1. The length of w is equal to the number of positive roots carried into negative roots by w, i.e., l(w) = |R+ ∩ w−1 (−R+ )|. In particular, αi is the only positive root α such that si (α) ∈ −R+ . The following conditions are equivalent: (i) l(wsi ) < l(w); (ii) w(αi ) ∈ −R+ ; (iii) some reduced factorization of w ends with si . We abbreviate these conditions to wsi < w, and write si w < w when w−1 si < w−1 . If α = w(αi ) = w (αj ), then w(αi∨ ) = w (αj∨ ), so there is a well-defined coroot α ∨ = w(αi∨ ) associated to α and satisfying α, α ∨ = 2, and accordingly a welldefined reflection sα = sα,α ∨ = wsi w−1 . Warning: the correspondence α → α ∨ need not be bijective if the dual root system is degenerate. The map W → {±1}, w → (−1)l(w) is a group homomorphism. In particular, l(sα ) is always odd, and l(wsα ) = l(w). We put wsα < w if l(wsα ) < l(w). The Bruhat order is the partial order on W given by the transitive closure of these relations. 2.7. The braid group B(W ) is the group with generators Ti and the braid relations (2) with Ti in place of si . If w = si1 . . . sil is a reduced factorization, we set Tw = Ti1 . . . Til . These elements are well-defined and satisfy Tu Tv = Tuv

when uv = u · v is a reduced factorization.

(3)

There is a canonical homomorphism B(W ) → W , Ti → si . By the symmetry of the braid relations, there is an automorphism Ti ↔ Ti−1 of B(W ). 2.8. The affine Weyl group of X, (αi ), (αi∨ ) is the semidirect product W X. In this context, we use multiplicative notation for the group X, denoting λ ∈ X by x λ . Explicitly, W X is generated by its subgroups W and X with the additional relations si x λ si = x si (λ) .

(4)


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2.9. The (left) affine braid group B(W, X) of X, (αi ), (αi∨ ) is the group generated by B(W ) and X, with the additional relations Ti x λ = x λ Ti Ti x λ Ti = x si (λ)

if λ, αi∨ = 0 (i.e., if si (λ) = λ);

(5)

if λ, αi∨ = 1.

(6)

These two relations may be combined into the following analog of (4): Tia x λ Tib = x si (λ) ,

where a, b ∈ {±1} and λ, αi∨ = (a + b)/2

(7)

(the case a = b = −1 follows by taking inverses on both sides in (6)). The canonical homomorphism B(W ) → W extends to a homomorphism B(W, X) → W X which is the identity on X. For clarity when dealing with double affine braid groups later on, we define separately the right affine braid group B(X, W ), generated by W and X with additional relations Ti x λ = x λ Ti Ti−1 x λ Ti−1

=x

si (λ)

if λ, αi∨ = 0 ; if

λ, αi∨

= 1.

(8) (9)

There is an isomorphism B(X, W ) ∼ = B(W, X) which maps Ti → Ti−1 and is the identity on X. 2.10. If X, (αi ), (αi∨ ) is a non-degenerate root system, the root lattice Q is free ˆ ∨ with basis with basis (αi ). Identify Q∨ with a quotient of the free abelian group Q ∨ ∨ ˆ (αi ), and set P = Hom(Q , Z). The roots and coroots in X are then given by homomorphisms Q → X → P , where the matrix of the composite Q → P is the Cartan matrix A. Suppose that X → P factors through a second lattice X as Q → X → X → P . j

This induces a root system X , (αi ), (αi ∨ ) in X with the same Cartan matrix A and canonically isomorphic Weyl and braid groups W = W , B(W ) = B(W ). There is an induced homomorphism of affine braid groups jB : B(W, X) → B(W, X ) which restricts to j on X and to the canonical isomorphism on B(W ). Theorem 2.11. The image of jB : B(W, X) → B(W, X ) is normal in B(W, X ), and the induced maps ker(j ) → ker(jB ), coker(j ) → coker(jB ) are isomorphisms. Proof (outline). First suppose that X = X ⊕ Zν, where ν, αi∨ ∈ {0, 1} for all i. One proves that there exists an automorphism η of B(W, X) which fixes X, such that η(Ti ) = Ti if ν, αi∨ = 0, and η(Ti ) = Ti−1 x −αi if ν, αi∨ = 1. Then one checks that ηZ B(W, X) ∼ = B(W, X ), with η → x ν . Iterating this gives B(W, X ⊕ P ) ∼ = P B(W, X), and similarly, B(W, X ⊕ P ) ∼ = P B(W, X ).

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Replacing X, X with X ⊕ P , X ⊕ P , we may assume that X → P and X → P are surjective. Next one verifies that if X → X is surjective, with kernel Z, then B(W, X ) ∼ = B(W, X)/Z. Applying this to 0 → Z → X → P → 0 and 0 → Z → X → P → 0, we get surjections B(W, X) → B(W, P ), B(W, X ) → B(W, P ) with kernels Z, Z . The theorem then follows by some easy diagram chasing. 2 2.12. Let X, (αi ), (αi∨ ) be a root system. It may happen that for one or more of the simple roots αi , we have αi∨ ∈ 2X∨ . Then we can form another (degenerate) root system by adjoining a new simple root 2αi and coroot αi∨ /2. Note that s(2αi ),(αi∨ /2) = si , so this new root system has the same Weyl group as the original one, but a larger set of roots R = R ∪ W (2αi ). If a root system contains two simple roots αi , αi such that si = si and αi = ±αi , it is said to be non-reduced, otherwise it is reduced. We remark that si = si implies αi = dαi , αi∨ = (1/d)αi∨ , where d ∈ {±1, ±2, ±1/2}. Hence every non-reduced root system is constructed by extensions as above from a reduced root system with the same Weyl group.

3. Affine root systems and affine Weyl groups Definition 3.1. A root system X, (αi ), (αi∨ ) is affine if its Cartan matrix A is singular, and for every proper subset J of the indices, the root system X, (αi )i∈J , (αi∨ )i∈J is finite. 3.2. The definition implies that the nullspace of A is one-dimensional. If X is nondegenerate, then {λ ∈ Q : λ, αi∨ = 0 for all i} is a sublattice of rank 1. It always has a (unique) generator δ ∈ Q+ , called the nullroot. We index the simple roots by i = 0, 1, . . . , n. We always assume that i = 0 is an affine node, meaning that α0 ∈ Q α + Qδ for some root α of the finite root system X, (α1 , . . . , αn ), (α1∨ , . . . , αn∨ ) . This condition is equivalent to s1 , . . . , sn generating the finite Weyl group W0 = W/Q 0 , where W is the Weyl group and Q 0 is the kernel of its induced action on X/(X ∩ Qδ). Every affine root system has at least one affine node. 3.3. The affine Cartan matrices are classified in Kac [8] and Macdonald [16]. They are symmetrizable and indecomposable. We refer to them using Macdonald’s nomenclature, but with a tilde over the names to distinguish them from finite types. Those n , or Xn(1) in Kac, are the untwisted types, where Xn = An , Bn , Cn , Dn , denoted X n∨ , C n∨ , E6,7,8 , F4 , or G2 is a Cartan matrix of finite type. Their duals (if different) B (2) (2) (2) (3) ∨ ∨ are the dual untwisted types, denoted A ,G F 4 2 2n−1 , Dn+1 , E6 , and D4 in Kac. (2)

The remaining mixed types, denoted A2n in Kac, are exceptional in that they have three root lengths. Although the mixed types are isomorphic to their duals, we prefer


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n when the distinguished to distinguish between them, denoting a mixed type as BC ∨ affine root α0 is the longest simple root, and BC n when α0 is the shortest simple root. ∨ n , C n∨ , BC n , BC ∨ Types B n contain one or more simple roots αi such that αj , αi ∨ is even for all j . There exist affine root systems X of these types such that αi ∈ 2X∨ . A non-reduced affine root system is a non-reduced extension (§2.12) of such a root system X. 3.4. The Weyl group Wa of any affine root system X, (αi ), (αi∨ ) is to isomorphic

∨ the affine Weyl group W = Q0 W0 of some finite root system Y, (αi ), (αi ) . Conversely, the affine Weyl group Y W0 of any finite root system is a semidirect extension Wa of the Weyl group of a corresponding affine root system. We now fix precise notation and explain how this correspondence comes about. 3.5. Let Y, (αi ), (αi ∨ ) , i = 1, . . . , n, be a finite root system, with Weyl group W0 and root lattice Q 0 . Let φ be the (unique) dominant short root. Let We = Y W0 be the affine Weyl group of Y , and set Wa = Q 0 W0 ⊆ We . Write y λ for λ ∈ Y regarded as an element of We . The orbit W0 (φ ) consists of all the short roots, and

spans Q0 . Defining s0 = y φ sφ , it follows that s0 and s1 , . . . , sn ∈ W0 generate Wa . We will construct an affine root system whose Weyl group W is isomorphic to Wa , with simple reflections corresponding to the generators s0 , . . . , sn . 3.6. Let X = Y ∨ ⊕ Z, and fix a non-zero element δ in the second summand. We need not assume that δ is a generator, so in general we have X = Y ∨ ⊕ Zδ/m for some positive integer m. Define the pairing X, Y → Z, extending the canonical pairing Y ∨ , Y → Z, with δ, Y = 0. Let θ = φ ∨ be the highest coroot. For i = 0, set αi = αi ∨ and αi∨ = αi (regarded as a linear functional on X via · , ·). Put α0 = δ − θ and α0∨ = −φ . The subgroup W0 ⊆ Wa acts via its original action on Y ∨ , fixing δ. The subgroup Q 0 ⊆ Wa acts by translations, given by the formula

y β (μ∨ ) = μ∨ − μ∨ , β δ,

(10)

One checks that the element y φ sφ ∈ Wa acts as the simple reflection s0 , identifying Wa with the Weyl group W of X. For Y of type Zn (Z = A, B, . . . , G), the affine root system X just constructed is n , with nullroot δ. In this case the affine roots are of untwisted type Z R = R0 ∨ + Zδ,

(11)

and the positive roots are R+ = (R0 ∨ + Z>0 δ) ∪ (R0 ∨ )+ . 3.7. Let X, (α0 , . . . , αn ), (α0∨ , . . . , αn∨ ) be any affine root system, W its Weyl group. Let Q0 , W0∨be the root lattice and Weyl group of the finite root system X, (α1 , . . . , αn ), (α1 , . . . , αn∨ ) . If X is of untwisted type, we have just seen that ∨ ∨ ∼ ∼ W ∼ = Q 0 W0 , where Q 0 = Q∨ 0 . If X is of untwisted type, then W = W (X ) =

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∨ Q 0 W0 , where Q 0 = (Q∨ 0 ) = Q0 . If X is of mixed type, its Weyl group is of

∼ type Cn , so W = Q0 W0 where Q 0 is of type Cn , hence Q 0 = Q∨ 0 for BC n , and ∨ n . Q 0 = Q0 for BC

3.8. Twisted affine root systems can also be constructed in the manner of §3.6, by taking θ to be any dominant coroot of Y or of a non-reduced finite root system containing Y . This yields dual untwisted types when θ is short, and mixed types when θ is one-half of a long coroot or twice a short coroot. However, when θ = φ ∨ , we no longer have W = Q 0 W0 . 3.9. We now return to the situation of §3.5, fixing the finite root system Y and untwisted affine root system X = Y ∨ ⊕ Zδ/m in what follows. The affine Weyl group We = Y W0 of Y is called the extended affine Weyl group. The action of Q 0 on X given by (10) extends to an action of Y , hence the action of Wa = Q 0 W0 extends to We . By (11), We preserves the set of affine roots R. 3.10. The further properties of Wa and We are best understood in terms of the following “alcove picture.” Let H = {x ∈ XR∨ : δ, x = 1} be the level 1 plane, and let ∨ 0 ∈H ∨ ) = 0, δ, ∨ = 1. The group W fixes δ, hence acts be the linear functional ∨ (Y e 0 0 on H . The translations Y ⊂ We act on H by y λ (μ) = μ + λ,

(12)

and the finite Weyl group W0 is generated by reflections fixing ∨ 0 . In particular, the ∨ ) ⊂ H , equivariantly ∼ W map y λ → ∨ + λ identifies Y /W with the orbit W ( = e 0 e 0 0 with respect to the original action of W0 on Y , and the action of Q 0 ⊆ Y by translations. Each affine root α ∈ R induces an affine-linear functional α(x) = α, x on H . Its zero set hα = {x ∈ H : α(x) = 0} is an affine hyperplane in H , and sα ∈ W = Wa fixes hα pointwise. The space H is tessellated by affine alcoves bounded by the root ∨ ) = {x ∈ hyperplanes hα . We distinguish the dominant alcove A0 = H ∩ (R+ X+ H : α(x) ≥ 0 for all α ∈ R+ }. The alcove A0 is a fundamental domain for the action of Wa on H . Its walls are the root hyperplanes hαi for the simple affine roots α0 , . . . , αn . Let ⊆ We be the stabilizer of A0 , or equivalently, = {π ∈ We : π(R+ ) = R+ }. Since preserves the set of simple roots, it normalizes the subgroup Wa ⊆ We and the set of Coxeter generators S = {s0 , . . . , sn } ⊆ Wa . The following are immediate. Corollary 3.11. With the notation above, we have We = Wa . Moreover, is the normalizer in We of the set of Coxeter generators S = {s0 , . . . , sn }. Corollary 3.12. The canonical homomorphism Y ⊂ We → We /Wa = induces an isomorphism Y/Q 0 ∼ = . In particular, is abelian. To make this explicit, write π ∈ uniquely as π = y λπ · vπ ∈ Y W0 .

(13)


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Then π maps to the coset of λπ in Y/Q 0 . In the notation of §3.10, we have ∨ 0 +λπ = ∨ ∨ λ π y ( 0 ) = π( 0 ) ∈ A0 . Equivalently, λπ ∈ Y is a dominant weight such that λ, φ ∨ ≤ 1, or minuscule weight. Conversely, if λ ∈ Y is minuscule, there is a unique π ∈ such that y λ−λπ ∈ Wa . Then λ = λπ , because both weights are minuscule and A0 is a fundamental domain for Wa . The minuscule weights λπ (including λ1 = 0) are thereby in bijection with . 3.13. The distinguished elements

y φ = s0 sφ ,

y λπ = π vπ−1 ,

(14)

where φ is the dominant short root and λπ are the minuscule weights, are characterized

as the unique translations such that s0 ∈ y φ W0 , π ∈ y λπ W0 , consistent with our having written We = Y W0 . If we write We = W0 Y , we instead distinguish the translations

y −φ = sφ s0 , y −λπ = vπ π −1 (15) corresponding to the anti-dominant short root and the “anti-minuscule” weights. Of course (14) and (15) are equivalent, but the corresponding formulas for the left and right affine braid groups will not be (see Theorem 4.2, Corollary 4.3).

4. Double affine braid groups 4.1. Let We = Y W0 = Wa be an extended affine Weyl group (§3.9–3.13). By Corollary 3.11, acts on Wa by Coxeter group automorphisms. Hence also acts on B(Wa ), and we can form the extended affine braid group B(We ) = B(Wa ). Define the length function on We = Wa by l(π w) = l(w). Note that l(wπ) = l(πwπ ) = l(wπ ) = l(w). The length of v = π w is again equal to |R+ ∩ v −1 (−R+ )|, or to the number of affine hyperplanes hα separating v(A0 ) from A0 in the alcove picture (§3.10). Identity (3) continues to hold in B(We ). The counterpart to Corollary 3.11 is the following theorem of Bernstein (see [9, (4.4)]). Theorem 4.2. The identification Wa = Y W0 lifts to an isomorphism B(We ) ∼ = B(Y, W0 ) between the extended affine braid group defined above, and the (right) affine braid group (§2.9) of the finite root system Y . The isomorphism is the identity

on B(W0 ) and given on the remaining generators by y φ ↔ T0 Tsφ , y λπ ↔ π Tvπ−1 , in the notation of §3.5 and (13). We describe the restriction of the isomorphism to Y ⊆ B(Y, W0 ) more explicitly. If λ, μ ∈ Y+ are dominant, the alcove picture shows that l(y λ+μ ) = l(y λ ) + l(y μ ). Hence Ty λ+μ = Ty λ Ty μ in B(We ). It follows that there is a well-defined group homomorphism φ : Y → B(We ) such that y λ−μ → Ty λ Ty−1 μ for λ, μ ∈ Y+ . In

particular, this yields the formulas y φ → Ty φ = T0 Tsφ , y λπ → Ty λπ = π Tvπ−1 .

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One verifies using the alcove picture that the elements φ(y λ ) and the generators Ti of B(W0 ) satisfy the defining relations of B(Y, W0 ). Hence φ extends to a homomorphism B(Y, W0 ) → B(We ). Next one verifies (with the help of Lemma 4.20,

∈ B(Q 0 , W0 ) satisfies braid relations with the genbelow) that the element y φ Ts−1 φ erators Ti , giving a homomorphism B(Wa ) → B(Q 0 , W0 ) inverse to φ. Hence φ maps B(Q 0 , W0 ) isomorphically onto B(Wa ), and by Theorem 2.11, it follows that φ is an isomorphism. Corollary 4.3. For a (left) extended affine Weyl group Wa = W0 X, there is an isomorphism B(We ) ∼ = B(W0 , X) between the extended affine braid group and the left affine braid group of X, which is the identity on B(W0 ), and satisfies x −φ ↔ Tsφ T0 , x −λπ ↔ Tvπ π −1 . 4.4. We come now in the theory. Fix two finite root systems to the key construction X, (αi ), (αi∨ ) , Y, (αi ), (αi ∨ ) with the same Weyl group W0 . More accurately, assume given an isomorphism of Coxeter groups W0 = (W (X), S) ∼ = (W (Y ), S ), and label the simple roots so that si corresponds to si for each i = 1, . . . , n. Let φ ∈ Q0 ⊆ X, φ ∈ Q 0 ⊆ Y be the dominant short roots. Let θ ∈ Q0 , θ ∈ Q 0 be the dominant roots such that sθ = sφ , sθ = sφ . There are unique W0 -equivariant pairings (X, Q 0 ) → Z, (Q0 , Y ) → Z such that (β, φ ) = β, θ ∨ for all β ∈ X and (φ, β ) = β , θ ∨ for all β ∈ Y . One checks that (φ, φ ) = (φ, φ ) = 2 if sφ = sφ , and (φ, φ ) = (φ, φ ) = 1 if sφ = sφ . By W0 -equivariance, the two pairings therefore agree on Q0 × Q 0 . Fix a W0 -invariant pairing (X, Y ) → Q extending the two pairings (· , ·) and (· , ·) , and choose m such that (X, Y ) ⊆ Z/m. Remark 4.5. The Cartan matrices of X and Y are clearly either of the same type (Zn , Zn ), or of dual types (Zn , Zn∨ ). In the symmetric case (Zn , Zn ), the roots θ = φ, θ = φ are short, and the pairing (· , ·) restricts on Q0 = Q 0 to the W0 -equivariant pairing such that (α, α) = 2 for short roots α. In the dual case (Zn , Zn∨ ), θ and θ are long, and the pairing restricts to the canonical pairing between Q0 and Q 0 = Q∨ 0. Types G2 and F4 are isomorphic to their duals, but only after relabelling the simple roots. Thus there is a genuine difference between types (G2 , G2 ) and (G2 , G∨ 2 ), for instance. In particular, θ = φ in the first case, and θ = φ in the second. = X ⊕ Zδ/m, Y = Y ⊕ Zδ /m. Extend the linear 4.6. Given the data in §4.4, set X so that δ, α ∨ = 0. Define α0 = δ − θ, and let α ∨ be the functionals αi∨ on X to X 0 i ∨ , we get two extension of −θ such that δ, α0∨ = 0. Making similar definitions in Y affine root systems (α0 , . . . , αn ), (α ∨ , . . . , αn∨ ) , , (α , . . . , αn ), (α ∨ , . . . , αn ∨ ) . X, Y 0 0 0 and X on Y by Let Y act on X y λ (μ) = μ − (μ, λ)δ,

x μ (λ) = λ − (μ, λ)δ .


853

Since (· , ·) is W0 -invariant, this extends to actions of the extended affine Weyl groups We = Y W0 ,

We = W0 X

Y We are the (left, and Y , respectively. The semidirect products We X, on X right) extended double affine Weyl groups. We have the following easy counterpart of Corollary 3.11. We , which is the ∼ Corollary 4.7. There is a canonical isomorphism We X =Y

identity on X, Y and W0 , and maps q = x δ to y −δ . In fact, both groups are identified with W0 (X Y ), where X Y is the Heisenberg group generated by X, Y and central element q 1/m , with relations x μ y λ = q (μ,λ) y λ x μ .

Remarks 4.8. (a) For consistency, set q = y −δ in the “right” double affine Weyl We . Then the isomorphism maps q to q. group Y Y are of untwisted (b) When X and Y are of dual types, the affine root systems X, type (§3.6). When X and Y are of the same type, then X, Y are of dual untwisted type (§3.8). (c) The requirement that (· , ·) extend the pairings (X, Q 0 ) → Z and (Q0 , Y ) → Z in §4.4 ensures that

We y φ sφ = s0 ∈ W (X),

). We sφ x −φ = s0 ∈ W (Y

the subgroup Wa = Q W0 Under the action of We = Y W0 = Wa on X, 0 By Corollary 3.11, ⊂ We acts is therefore identified with the Weyl group of X. by automorphisms of the root system, i.e. it permutes the affine simple roots on X as the semi-direct product of the Weyl group Wa and and coroots. So We acts on X the group of automorphisms . In particular, the extended double affine Weyl group is the semidirect product We X (Wa X) Similar remarks of with the affine Weyl group (§2.8) of the affine root system X.

apply to Y We . it also acts naturally 4.9. Since acts by automorphisms of the affine root system X, (§2.9), and we can form the semidirect product B(Wa , X), which on B(Wa , X) of the affine root we may regard as an extended (left) affine braid group B(We , X) Similarly, we can define B(Y , We ) = B(Y , Wa ) . Define q = x δ system X.

−δ in B(We , X), and q = y ∈ B(Y , We ), as in Remark 4.8(a). We come now to the fundamental theorem. ∼ Y We lifts to an isomorphism = Theorem 4.10. The isomorphism We X , We ), which is the identity on X, Y , and B(W0 ), and maps ∼ B(We , X) = B(Y

854

Mark Haiman

q = x δ to q = y −δ . (Here X, Y are identified with their images under B(W0 , X) ∼ = , We ) and B(Y, W0 ) ∼ using Theorem 4.2 B(We ) → B(Y = B(We ) → B(We , X), and Corollary 4.3)

= B(Y , We ) is the (extended) double affine braid group. The group B(We , X) ) given by , We ) → B(We , Y 4.11. By §2.9, there is an isomorphism : B(Y (T0 ) = T0 −1 ,

(y λ ) = y λ , (π) = π

(π ∈ ),

(Ti ) = Ti−1

(i = 1, . . . , n).

, We ) is defined by T0 = y φ Ts−1 , whereas T0 in B(We , Y ) The element T0 in B(Y φ

, We ) is given by π = y λπ vπ → is given by T0 = Ts−1 y −φ . Similarly, → B(Y φ

−1 → T −1 y −λπ , and therefore y λπ T −1 −1 , whereas → B(We , Y ) is given by π v π

vπ

, We ) via the identification B(We ) = Moreover, X is embedded in B(Y π →

−φ B(W0 , X), which is characterized by x → Tsφ T0 and x −λπ → Tvπ π −1 ,

) is given via B(We ) = B(X, W0 ) by x φ → T0 Tsφ , whereas X ⊂ B(We , Y ), finally, q denotes y δ . Taking into account that x λπ → π Tv −1 . In B(We , Y y λπ Tvπ .

π

(Tw ) = Tw−1 −1 for all w ∈ W0 , all this implies (T0 ) = T0−1 ,

(x μ ) = x μ , (π) = π

(π ∈ ),

(q) = q −1 .

Theorem 4.10 therefore has the following equivalent alternate formulation. ∼ B(W , Y = ), which is the Corollary 4.12. There is an isomorphism B(We , X) e

δ −1 −δ identity on X, Y , and , maps q = x to q = y , and maps the generators Ti of B(W0 ) to Ti−1 . 4.13. Cherednik [1] announced Theorem 4.10 in the case X = Y , and suggested a possible topological proof, which was completed by Ion [7]. Macdonald [16, 3.5–3.7] gave an elementary proof, which however involves quite a bit of case-checking and

∨ only applies when X = Hom(Q∨ 0 , Z), Y = Hom(Q0 , Z). We now outline a different elementary proof. First assume that the theorem holds in the “unextended” case, X = Q0 , Y = Q 0 , We = Wa , We = Wa . We will deduce the general case. , Wa ) as a normal subgroup, with , Wa ) embeds in B(Y By Theorem 2.11, B(Q 0 /Q = Y/Q ∼ quotient Y 0 0 = . Moreover, ⊆ B(We ) = B(Y, W0 ) is a subgroup

, Wa ). , Wa ) ∼ , Wa ), giving the semidirect decomposition B(Y of B(Y = B(Q 0 , Wa ), hence B(Y 0 ) ∼ , Wa ) ∼ By assumption, we have B(Wa , Q = B(Q = 0 0 ) = B(We , Q0 ). This establishes the case where X = Q0 and Y is B(Wa , Q general. Exchanging X and Y , we also get the case Y = Q 0 , We = Wa , where now X and We are general.


855

, Wa ) and B(Y , We ) = B(Y , Wa ) , , We ) = B(Q By definition, B(Q 0 0

∼ with = X/Q0 the same for both groups. Again, Theorem 2.11 implies that the first group is a normal subgroup of the second, with quotient . So we can repeat the preceding argument to get the general case. 4.14. Now fix X = Q0 , Y = Q 0 , so We = Wa , We = Wa . Using Theorem 4.2 and Corollary 4.3, we identify B(Wa ) = B(W0 , X), B(Wa ) = B(Y, W0 ). Then each B(Y , Wa ) has generators T0 , T , T1 , . . . , Tn , q 1/m . In both groups, group B(Wa , X), 0 1/m q is central, the generators T0 , T1 , . . . , Tn satisfy the braid relations of B(Wa ), and T0 , T1 , . . . , Tn satisfy those of B(Wa ). The additional relations (7) for λ ∈ Q0 and i = 0 complete a presentation of since those for i = 0 already hold in B(W0 , X) = B(Wa ). For conB(Wa , X), venience, we write down these extra relations again here, after applying the identity λ, α0∨ = −λ, θ ∨ : T0a x λ T0b = x s0 (λ) ,

where a, b ∈ {±1} and −λ, θ ∨ = (a + b)/2.

(16)

In view of Corollary 4.12, to prove the theorem it suffices to express (16) in a “selfdual” form, in the sense that the substitutions T0 ↔ T0 −1 , Ti ↔ Ti−1 , q ↔ q −1 (i = 0) should transform (16) into its counterpart with the roles of X and Y interchanged. Lemma 4.15. Relations (16) reduce to the case when λ is a short positive root α = θ (i.e., α = φ if θ = φ is short). Proof. The short roots β = ±θ span Q0 . Hence we can always write λ = β1 + · · · + βm , where βi ∈ (R0 )short \ {±θ}. In particular, βi , θ ∨ ∈ {0, ±1} for all i. Given that λ, θ ∨ ∈ {0, ±1}, we can always order the βi so that those with βi , θ ∨ = 1 and those with βi , θ ∨ = −1 alternate. Writing (16) in the form T0a x λ = x s0 (λ) T0−b , it is easy to see that it follows from the same relation for each βi . This reduces us to the case that α = ±θ is a short root. The case of (16) for λ, θ ∨ = 1 implies the case for λ, θ ∨ = −1, so positive roots α suffice. 2 4.16. A parabolic subgroup of W0 is a subgroup of the form WJ = si : i ∈ J , where J ⊆ {1, . . . , n}. Since φ and φ are dominant, their stabilizers are parabolic subgroups WJ , WJ respectively, where J = {i : φ, αi∨ = 0}, and J = {i : φ , αi ∨ = 0}. Recall that each left, right and double coset vWJ , WJ v, WJ vWJ has a unique representative of minimal length, which is also minimal in the Bruhat order. Proposition 4.17. Relations (16) for λ = α = θ a short positive root reduce to relations of the following two forms: (a) For v such that (v(φ), φ ) = 0 and v minimal in WJ vWJ , the relation T0 Tv T0 −1 Tv−1 = Tv T0 −1 Tv−1 T0 .

856

Mark Haiman

(b) For v = v1 such that (v(φ), φ ) = 1 and v minimal in WJ vWJ , define v2 , v3 , v4 minimal respectively in WJ vsφ WJ , WJ sθ vsφ WJ , WJ sθ vWJ ; this given, the relation T0−1 Tv1 T0 −1 Tv−1 T0−1 Tv3 T0 −1 Tv−1 = q. 2 4 Proof. We can always write α = v(φ) with v minimal in vWJ . If i ∈ J , then Ti

commutes with T0 . In B(Wa ) = B(W0 , Q0 ) we have x si (α) = Ti x α Ti , , = ±1 for every positive short root α. These facts imply that relations (16) are invariant under replacement of α with w(α) ∈ WJ α. Hence we can assume v minimal in WJ vWJ . We show that when α, θ ∨ = (v(φ), φ ) = 0, relation (16), which in this case reads T0 x α = x α T0 , is equivalent to (a). The minimality of v in vWJ implies that if v = si1 . . . sil is a reduced factorization, then sik+1 . . . sil (φ), αi∨k = 1 for all k. Hence x α = Tv x φ Tv −1 = Tv T0 −1 Ts−1 Tv −1 . The minimality also implies that sφ = v −1 sα v is φ

Tv −1 = Tv−1 Ts−1 , and x α = Tv T0 −1 Tv−1 Ts−1 . a reduced factorization. Therefore Ts−1 α α φ ∨ Now, since α, θ = 0, we have s0 sα = sα s0 , and both sides of this equation are reduced factorizations. Hence T0 commutes with Tsα , so (16) is equivalent to T0 commuting with Tv T0 −1 Tv−1 . For α, θ ∨ = (v(φ), φ ) = 1, we have s0 (α) = α + α0 = α − θ + δ, and thus relation (16) in this case reads T0−1 x α T0−1 = qx −β , or T0−1 x α T0−1 x β = q, where β = −sθ (α) satisfies α + β = θ. Let u be the minimal representative of sθ vsφ WJ . Then β = u(φ), and the same reasoning as in the previous paragraph gives −1 , x β = T T −1 T −1 . Our relation now takes the form x α = Tv T0 −1 Tvs u 0 usφ φ −1 −1 −1 T0−1 Tv T0 −1 Tvs T Tu T0 −1 Tus = q. φ 0 φ

(17)

Using §2.6 and the fact that sφ (αi ) = αi for all i ∈ J , we deduce (for any J ) (*) if x, y are minimal in WJ x, WJ y = WJ xsφ , respectively, and xw is minimal in WJ xWJ , then yw is minimal in WJ yWJ . By construction, u and v are minimal in their left WJ cosets, and (*) implies the same for usφ and vsφ . Hence the elements v1 = v, v2 , v3 , v4 defined in (b) are the minimal representatives of WJ v, WJ vsφ , WJ u, WJ usφ respectively. By the analog of (*) for sθ (operating on the left), we see that v1 = v implies v4 = usφ , and if we set v2 = wvsφ , then v3 = wu. Now w ∈ WJ commutes with T0 , and the factorizations vsφ = w−1 v2 , u = w−1 v3 are reduced, so (17) reduces to (b). 2 0 ), where Corollary 4.18. The (unextended) double affine braid group B(Wa , Q B(Wa ) = B(Q 0 , W0 ), has a presentation with generators T0 , T0 , T1 , . . . , Tn , q 1/m and the following (manifestly self-dual) relations: q 1/m is central; braid relations for T0 , T1 , . . . , Tn ∈ B(Wa ) and for T0 , T1 , . . . , Tn ∈ B(Wa ); and the relations in Proposition 4.17.


857

Example 4.19. Let X = Y be of type An−1 , with αi = αi∨ = ei − ei+1 as in Example 2.5. Then φ = θ = φ = θ = e1 − en , and WJ = WJ = s2 , . . . , sn−2 . 0 ) is given by q central, braid relations and The presentation of B(Wa , Q (a)

T0 commutes with T1 Tn−1 T0 −1 (T1 Tn−1 )−1 ,

(b)

−1 −1 −1 −1 T0−1 T1 T0 −1 T1−1 T2−1 . . . Tn−1 T0 Tn−1 T0 −1 Tn−1 Tn−2 . . . T1−1 = q.

There are seven double cosets WJ vWJ . Two have v(φ) = ±φ, one yields (a), and the other four provide the elements v1 , . . . , v4 in (b). In fact, in every type there turns 4 , out to be only one relation of type (b) and at most two of type (a), except for D which has three of type (a). Lemma 4.20. If φ is the dominant short root of a finite root system X, and v ∈ W0 is such that α = v(φ) ∈ (R0 )+ , then in B(W0 , X) we have Tv x φ Tsφ Tv−1 = x α Tsα . Proof. This reduces to the case that v is minimal in vWJ (in the notation of §4.16). As α in the proof of Proposition 4.17 we then have Tv x φ Tsφ = x α Tv−1 2 −1 Tsφ = x Tsα Tv . Lemma 4.20 will be used in the proof of Theorem 5.11. Its variant for B(Y, W0 ) φ −1 α −1 is Tv−1 −1 y Tsφ Tv −1 = y Tsα , which is useful for verifying the braid relations in the proof of Theorem 4.2.

5. Hecke algebras and Cherednik algebras 5.1. Let X, (αi ), (αi∨ ) be a non-degenerate root system, with Cartan matrix A, Weyl group W , and roots R. To each W -orbit in R we associate a parameter uα , uα = uβ if β = w(α). Set ui = uαi . The ui are assumed to be invertible elements of some commutative ground ring A. If αi∨ ∈ 2X∨ , we also introduce a second parameter u i . Lemma 5.2. Let H be an A-algebra containing the group algebra AX, and Ti ∈ H. (i) If αi∨ ∈ 2X∨ , then commutation relations (5)–(6) and the quadratic relation (Ti − ui )(Ti + u−1 i )=0

(18)

imply the more general commutation relations, for all λ ∈ X, Ti x λ − x si (λ) Ti =

(ui − u−1 i ) λ (x − x si (λ) ). α 1−x i

(19)

(ii) If αi∨ ∈ 2X∨ , then (5)–(6), (18) and the additional quadratic relation (Ti−1 x −αi − u i )(Ti−1 x −αi + u −1 i )

(20)

858

Mark Haiman

imply Ti x λ − x si (λ) Ti =

−1 αi

(ui − u−1 i ) + (ui − ui )x (x λ − x si (λ) ) 1 − x 2αi

(21)

(iii) Given (18), relation (21) implies (20), and (19) implies that (20) holds with u i = ui . Note that the denominators in (19), (21) divide x λ − x si (λi ) . For the well-known proof, observe that each side of (19), (21), viewed as an operator on x λ , satisfies F (x λ x μ ) = F (x λ )x μ + x si (λ) F (x μ ). Hence (19), (21) for x λ , x μ , imply the same for x λ±μ . This reduces (i) to the special cases λ, αi∨ ∈ {0, 1}, which in turn reduce to (5)–(6), using the identity Ti−1 = Ti − ui + u−1 i , which is equivalent to (18). Similarly, (ii) reduces to the special cases λ, αi∨ = 0, which is (5) ((6) is vacuous if αi∨ ∈ 2X∨ ), and λ = αi (since αi , αi∨ = 2). Modulo (18), this last case is equivalent to (20), which also gives (iii) in case (ii). For (iii) in case (i), observe that (19) is just (21) with u i = ui . Definition 5.3. The affine Hecke algebra H(W, X) is the quotient (AB(W, X))/j, where j is the 2-sided ideal generated by the quadratic relations (18) for all i, plus (20) for each i such that αi∨ ∈ 2X∨ . Equivalently, H(W, X) is generated by elements x λ (λ ∈ X) and Ti satisfying the braid relations of B(W ), quadratic relations (18), and relations (19) or (21) depending on whether or not αi∨ ∈ 2X∨ . Proposition 5.4. The subalgebra of H(W, X) generated by the elements Ti is isomorphic to the ordinary Hecke algebra H(W ), with basis {Tw : w ∈ W }, and H(W, X) has basis {Tw x λ }. Proof. The commutation relations (19), (21) imply that the elements Tw x λ span; they are independent because the specialization ui = u i = 1 collapses H(W, X) to the group algebra A · (W X). (More precisely, specialization implies the result for

±1 A = Z[u±1 2 i , ui ], and the general case follows by extension of scalars.) 5.5. Let be a group acting by automorphisms of the root system X, (αi ), (αi∨ ) , and assume that ui = uj , u i = uj for αj ∈ (αi ). Then acts on H(W, X), and we define the extended affine Hecke algebra to be the twisted group algebra · H(W, X) generated by and H(W, X) with relations πf = π(f )π for π ∈ , f ∈ H(W, X). Up to now the root system X was arbitrary. If X is finite, with W0 X = Wa as in Corollary 4.3, then H(W0 , X) is isomorphic to the twisted group algebra H(Wa )· of the ordinary Hecke algebra of Wa . The most interesting case is when X is affine; as constructed in §4.6. specifically when X = X Y , We = Wa , We = Wa as in Definition 5.6. Given X, Y , (· , ·), X, is the extended affine Hecke algebra §4.4–4.9, the (left) Cherednik algebra H(We , X) · H(Wa , X).


859

is generated by x λ ∈ X, π ∈ , T0 , . . . , Tn and q ±1/m , Equivalently, H(We , X) and the quadratic satisfying the relations of the double affine braid group B(We , X) ∨ ∨ relations (18), plus (20) if αi ∈ 2X . , We ), but first we must re5.7. We will also define a right Cherednik algebra H(Y

∨ . Define index the parameters. For convenience, we define uj = uj if αj∨ ∈ 2X ui = ui for i = 0, and set u0 = uj , where αj is a short simple root of the finite root ∨ for i = 0 (there is at most one such index i), set u = u0 . system X. If αi ∨ ∈ 2Y i ∨ , set u = u . If α0 ∨ ∈ 2Y 0 0 , We ) to be the algebra with generators y μ (μ ∈ Y ), π ∈ , We now define H(Y , We ), T0 , T1 , . . . , Tn , q ±1/m satisfying the relations of the right affine braid group B(Y

∨ ∨ , the relations relations (18) with ui in place of ui , and for αi ∈ 2Y

(Ti −1 y αi − u i )(Ti −1 y αi + u −1 i ),

(22)

where we define Ti = Ti if i = 0. Corollary 5.8. The elements {y μ Tw x λ } (μ ∈ Y , λ ∈ X, w ∈ W0 ) form an A[q ±1/m ] H(Y , We ). basis of the Cherednik algebras H(We , X), and by symmetry for This follows easily from Proposition 5.4 for H(We , X)

μ λ , We ). We remark that the factors y Tw x can be taken in any order. H(Y ∨ if and only if X, Y are both of type Bn and acts Lemma 5.9. We have α0 ∨ ∈ 2Y trivially on the simple roots of X. Proof. By definition, α0 ∨ = −θ ∨ . We can only have θ ∨ ∈ 2Y ∨ if Y is of type Bn and θ = φ is short, hence X is also of type Bn . Let P0 be the image of the canonical

homomorphism Y → Hom(Q ∨ 0 , Z). For type Bn we have either Q0 = P0 or P0 /Q 0 ∼ = Z/2Z, with Q 0 = P0 iff the short roots α satisfy α ∨ ∈ 2Y ∨ . The isomorphism ∼ = Y/Q 0 (Corollary 3.12) identifies P0 /Q 0 with the quotient of 2 by the kernel of its action on the simple roots of X. Y are of type C n∨ . Label the Dynkin Remark 5.10. If X, Y are of type Bn , then X, diagram •⇐•—•— · · · —•⇒• . (23) 0 1 2 n−1 n ∨ , then acts non-trivially, exchanging nodes 0 and n, and similarly for If α0 ∨ ∈ 2Y ∨

α0 and . The four associated parameters are related by the diagram (u 0 = u 0 ) = (u n = u0 ) (u0 = u n ) = (un = un ) ,

(24)

where the horizontal equalities hold if α ∨ ∈ 2Y ∨ for short roots α ∈ Y , and the vertical ones hold if α ∨ ∈ 2X∨ for short roots α ∈ X.

860

Mark Haiman

, We ), which is the ∼ Theorem 5.11. There is an isomorphism H(We , X) = H(Y

identity on all the generators X, Y , q, Ti , T0 , T0 , π , π . Proof. For the most part, this is Theorem 4.10, but we must prove that relations (22) By definition, T = Ts−1 x −φ . By and the case of (18) for T0 hold in H(We , X). 0 φ

Lemma 4.20, this is conjugate to Tj−1 x −αj for a short simple root αj . Then (20) for

Tj implies (18) for T0 . Similarly, if i = 0 in (22), then αi is short, and Ti −1 y αi

is conjugate to y αi Ti −1 and in turn to T0 = y φ Ts−1 . By Lemma 5.9, we only φ have i = 0 in (22) when X, Y are both of type Bn , so θ = φ, θ = φ . Then

T0 −1 y α0 = q −1 x φ Tsφ y −φ = x −α0 T0−1 , which is conjugate to T0−1 x −α0 . 2 Corollary 5.12. Assume given an automorphism ε : A → A such that ε(ui ) = u−1 i ,

, Y ∼ ) H(W ε(u i ) = u −1 . Then there is an ε-linear isomorphism H(W , X) = e e i −1

−1 which is the identity on X, Y , , , maps q to q , and maps Ti to Ti for all ) are as in §5.7. i = 0 , 0, 1, . . . , n, where the parameters ui , u i for H(We , Y Proof. The map in §4.11, composed with ε, preserves (18) and interchanges (22) in place of X. with the version of (20) for Y 2 5.13. Let H = · H(W, X) be an extended affine Hecke algebra. The ordinary (extended) Hecke algebra · H(W ) has a one-dimensional representation 1 = Ae such that πe = e, Ti e = ui e. The induced representation IndH H(W ) (1) is the polynomial representation. Proposition 5.4 implies that it is isomorphic to the left regular representation AX of X, with acting via its action on X, and T0 , . . . , Tn acting as the operators Ti = ui si +

(ui − u−1 i ) (1 − si ) α 1−x i

= ui − u−1 i

1 − u2i x αi (1 − si ) 1 − x αi

= −u−1 i + ui (1 + si )

αi 1 − u−2 i x 1 − x αi

(25) (26) (27)

or, if αi∨ ∈ 2X∨ , Ti = ui si +

−1 αi

(ui − u−1 i ) + (ui − ui )x (1 − si ) 1 − x 2αi

= ui − u−1 i

(1 − ui u i x αi )(1 + (ui /u i )x αi ) (1 − si ) 1 − x 2αi

= −u−1 i + ui (1 + si )

(1 − (ui u i )−1 x αi )(1 + (u i /ui )x αi ) . 1 − x 2αi

(28) (29) (30)


861

In particular, these operators satisfy braid relations. The quadratic relations can be seen directly from (26)–(27) and (29)–(30). The polynomial representation specializes at ui = u i = 1 to the A-linearization of the action of (W X) on X. It is faithful if acts faithfully. 5.14. For any root α ∈ R, define a partial ordering on X by μ 0. Each root string λ + Zα is totally ordered by 0. He also proved that λ3 ≥ 2.88 if it exists, and claimed that

880

Jeong Han Kim

λ4 ≥ 4.52 and λ5 ≥ 6.06 etc. could be proven by the same method. Pittel, Spencer and Wormald [38] proved a general theorem which implies that λt exists for fixed t ≥ 3 and identified their values. We present a slightly weaker version of this result. For a Poisson random variable Poi(ρ) with mean ρ let P (ρ, i) = Pr[Poi(ρ) = i] and Q(ρ, i) = Pr[Poi(ρ) ≥ i], i.e., P (ρ, i) = e−ρ

ρi i!

and Q(ρ, i) :=

∞

P (ρ, j ) = e−ρ

j =i

and let λt = min ρ>0

∞ ρj j =i

j!

,

ρ . Q(ρ, t − 1)

Theorem 1.6. Let t ≥ 3, λ(n, p) = p(n − 1). Then 0 if λ(n, p) < λt − n−δ , Pr[G(n, p) has a non-empty t-core] −→ 1 if λ(n, p) > λt + n−δ for any δ ∈ (0, 1/2), and the t-core when λ(n, p) > λt +n−δ has (1+o(1))Q(θλ λ, t)n vertices, whp, where θλ is the largest solution for the equation θ − Q(θλ, t − 1) = 0. There are many studies about the t-cores of various types of random graphs and random hypergraphs. Fernholz and Ramachandran [21], [22] studied random graph conditions on given degree sequences. Cooper [13] found the critical values for t-cores of a uniform multihypergraph with given degree sequences that includes the random k-uniform hypergraph H (n, p ; k). Molloy [37] considered cores for random hypergraphs and random satisfiability problems for Boolean formulas. Recently, S. Janson and M. J. Luczak [26] also gave seemingly simpler proofs for t-core problems that contain the result of Pittel, Spencer and Wormald. For more information and techniques used in the above mentioned papers readers are referred to [26]. Using the Poisson cloning model for random hypergraphs together with the cut-off line algorithm we are able to completely analyze the t-core problem for the random uniform hypergraph. We also believe that the cut-off line algorithm can be used to analyze the t-core problem for random hypergraphs conditioned on certain degree sequences as in [13], [21], [22], [26]. As the 2-core of G(n, p) behaves quite differently from the other t-cores of H (n, p ; k), we exclude the case k = t = 2, which will be studied in a subsequent paper. The critical value for the problem turns out to be the minimum λ such that there is a positive solution for the equation θ − Q(θ k−1 λ, t − 1) = 0.

(1.1)

It is not difficult to check that the minimum is λcrt (k, t) := min ρ>0

ρ . Q(ρ, t − 1)k−1

(1.2)

881

Poisson cloning model for random graphs 1

For λ > λcrt (k, t), let θλ be the largest solution of the equation θ k−1 −Q(θλ, t −1) = 0. Theorem 1.7. Let k, t ≥ 2, excluding k = t = 2, and σ n−1/2 . Subcritical phase: If λ(n, p ; k) := p n−1 k−1 = λcrt− σ is uniformly bounded from below by 0 and i0 (k, t) is the minimum i such that ki ≥ ti/k, then Pr[Vt (H (n, p ; k)) = ∅] = e− (σ

2 n)

+ O n−(t−1−t/k)i0 (k,t) ,

and for any δ > 0, Pr[|Vt (H (n, p ; k))| ≥ δn] = e− (σ

2 n)

+ e− (δ

2k/(k−1)) n)

.

(1.3)

Supercritical phase: If λ = λ(n, p ; k) = λcrt + σ is uniformly bounded from above, then for all α in the range 1 α σ n1/2 ,

2 Pr |Vt (n, p ; k)| − Q(θλ λ, t)n ≥ α(n/σ )1/2 = e− (α ) ,

(1.4)

and, for any i ≥ t and the sets Vt (i) (resp. Wt (i)) of vertices of degree i (resp. larger than or equal to i) in the t-core,

2 2 Pr |Vt (i)| − P (θλ λ, i)n ≥ δn ≤ 2e− (min{δ σ n,σ n}) , and

2 2 Pr |Wt (i)| − Q(θλ λ, i)n ≥ δn ≤ 2e− (min{δ σ n,σ n}) .

In particular, for λ = λcrt + σ and ρcrt := θλcrt (k,t) λcrt (k, t), i |Vt (i)| = 1 + O(σ 1/2 ) P (ρcrt , i)n + O (n/σ )1/2 log n , with probability 1 − 2e− (min{log

2 n,σ 2 n})

.

As one might guess, we will prove a stronger theorem (Theorem 6.2) for the Poisson cloning model HPC (n, p ; k), from which Theorem 1.7 easily follows. In the next section the Poisson cloning model is defined in detail. The cut-off line algorithm and the cut-off line lemma are presented in Section 3. In Section 4 we study Chernoff type large derivation inequalities that will be used in most of our proofs. In Section 5, a generalized core is defined and the main lemma is presented. Section 6 is devoted to the proof of Theorem 1.7. As the proof of Theorem 1.4 is more sophisticated, we only give the proof ideas in Section 7. We conclude this paper with final remarks in Section 8. Due to the space limitation, many proofs are omitted. They can be found on the author’s web site.

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2. The Poisson cloning model To construct the Poisson cloning model GPC (n, p) for random graphs, let V be a set of n vertices. We take i.i.d. Poisson λ = p(n − 1) random variables d(v), v ∈ V , and then take d(v) copies of each vertex v ∈ V . The copies of v are called clones of v, or simply v-clones. Since the sum ofPoisson random variables is also a Poisson random variable, the total number Nλ := v∈V d(v) of clones is a Poisson λn random variable. It is sometimes convenient to take a reverse, but equivalent, construction. We first take a Poisson λn = 2p n2 random variables Nλ and then take Nλ unlabelled clones. Each clone is independently labelled as v-clone uniformly at random, in the sense that v is chosen uniformly at random from V . It is well known that the numbers d(v) of v-clones are i.i.d. Poisson random variables with mean λ. If Nλ is even, the multigraph GPC (n, p) is defined by generating a (uniform) random perfect matching of those Nλ clones, and contracting clones of the same vertex. That is, if a v-clone and a w-clone are matched, then the edge {v, w} is in GPC (n, p) with multiplicity. In the case that v = w, it produces a loop that contributes 2 to the degree of v. If Nλ is odd, we may define GPC (n, p) to be any graph with a special loop that, unlike other loops, contributes only 1 to the degree of the corresponding vertex. In particular, if Nλ is odd, GPC (n, p) is not a simple graph. Strictly speaking, GPC (n, p) varies depending on how to define it when Nλ is odd. However, if only simple graphs are concerned, the case of Nλ being odd would not matter. For example, the probability that GPC (n, p) is a simple graph with a component larger than 0.1n does not depend on how GPC (n, p) is defined when Nλ is odd, as it is not a simple graph anyway. Generally, for any collection G of simple graphs, the probability that GPC (n, p) is in G is totally independent of how GPC (n, p) is defined when Nλ is odd. Notice that properties of simple graphs are actually mean collections of simple graphs. Therefore, when properties of simple graphs are concerned, it is not necessary to describe GPC (n, p) for odd Nλ . Here are two specific ways to generate the uniform random matching. Example 2.1. One may keep matching two clones chosen uniformly at random among all unmatched clones. Example 2.2. One may keep choosing his or her favorite unmatched clone, and matching it to a clone selected uniformly at random from all other unmatched clones. If Nλ is even both examples would yield uniform random perfect matchings. If Nλ is odd, then each of them would yield a matching and an unmatched clone. We may create the special loop consisting of the vertex for which the unmatched clone is labelled. More specific ways to choose random clones will be described in the next section. Generally for k ≥ 3, the Poisson cloning model HPC (n, p ; k) for k-uniform hy pergraphs may be defined in the same way: We take i.i.d. Poisson λ = p n−1 random k−1 variables d(v), v ∈ V , and then take d(v) clones of each v. If Nλ := v∈V d(v) is

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Poisson cloning model for random graphs

divisible by k, the multihypergraph HPC (n, p; k) is defined by generating a uniform random perfect matching consisting of k-tuples of those Nλ clones, and contracting clones of the same vertex. That is, if v1 -clone, v2 -clone, …, vk -clone are matched in the perfect matching, then the edge {v1 , v2 , . . . , vk } is in HPC (n, p ; k) with multiplicity. If Nλ is not divisible by k, HPC (n, p ; k) may be any hypergraph with a special edge consisting of Nλ − kNλ /k vertices. In particular, HPC (n, p ; k) is not k-uniform when Nλ is not divisible by k. Therefore, as long as properties of k-uniform hypergraphs are concerned, we do not have to describe HPC (n, p ; k) when Nλ is not divisible by k. We show that the Poisson cloning model HPC (n, p ; k), k ≥ 2, is contiguous to the classical model H (n, p ; k) when the expected average degree is a constant. Theorem 1.1 (restated). Suppose k ≥ 2 and p = (n1−k ). Then for any collection H of k-uniform simple hypergraphs, c1 Pr[HPC (n, p ; k) ∈ H] ≤ Pr[H (n, p ; k) ∈ H] 1 ≤ c2 Pr[HPC (n, p ; k) ∈ H] k + e−n , where 2

p k n p n c1 = k 1/2 e n (2)(k )+ 2 (k ) + O(n−1/2 ),

c2 =

k 1/k + o(1), c1 (k − 1) k−1

and o(1) goes to 0 as n goes to infinity. Proof. See [30].

2

3. The λ-cell and the cut-off line algorithm To generate a uniform random perfect matching of Nλ clones, we may keep matching k unmatched clones uniformly at random (cf. Example 2.1). Another way is to choose the first clone as we like and match it to k − 1 clones selected uniformly at random among all other unmatched clones (cf. Example 2.2). As there are many ways to choose the first clone, we may take a way that makes the given problem easier to analyze. Formally, a sequence S = (Si ) of choice functions determines how to choose the first clone at each step, where Si tells us which unmatched clone is to be the first clone for the i th edge in the random perfect matching. A choice function may be deterministic or random. If less than k clones remain unmatched, the edge consisting of those clones will be added. The clone chosen by Si is called the i th chosen clone, or simply a chosen clone. We also present a more specific way to select the k − 1 random clones to be matched to the chosen clone. The way introduced here will be useful to solve problems mentioned in the introduction. First, independently assign to each clone a uniform

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random real number between 0 and λ = p n−1 k−1 . For the sake of convenience, a clone is called the largest, the smallest, etc. if so is the number assigned to it. In addition, a clone is called θλ-large (resp. θλ-small) if its assigned number is larger than or equal to (resp. smaller than) θλ. To visualize the labelled clones with assigned numbers, one may consider n horizontal line segments from (0, j ) to (λ, j ), j = 0, . . . , n−1 in the two-dimensional plane R2 . The vj -clone with assigned number x can be regarded as the point (x, j ) in the corresponding line segment. Then each line segment with the points corresponding to clones with assigned numbers is an independent Poisson arrival process with density 1, up to time λ. The set of these Poisson arrival processes is called a Poisson (λ, n)-cell or simply a λ-cell. We will consider sequences of choice functions that choose an unmatched clone without changing the joint distribution of the numbers assigned to all other unmatched clones. Such a choice function is called oblivious. A sequence of oblivious choice functions is also called oblivious. The choice function that chooses the largest unmatched clone is not oblivious, as the numbers assigned to the other clones must be smaller than the largest assigned number. As an example of an oblivious choice function one may consider the choice function that chooses a v-clone for a vertex v with fewer than 3 unmatched clones. For a more general example, let a vertex v and its clones be called t-light if there are fewer than t unmatched v-clones. Example 3.1. Suppose there is an order of all clones which is independent of the assigned numbers. The sequence of the choice functions that choose the first t-light clone is oblivious. A cut-off line algorithm is determined by an oblivious sequence of choice functions. Once a clone is obliviously chosen, the largest k−1 clones among all unmatched clones are to be matched to the chosen clone. This may be further implemented by moving the cut-off line to the left until k − 1 vertices are found: Initially, the cut-off line of the λ-cell is the vertical line in R2 containing the point (λ, 0). The initial cut-off value, or cut-off number, is λ. At the first step, once the chosen clone is given, move the cut-off line to the left until exactly k − 1 unmatched clones, excluding the chosen clone, are on or in the right side of the line. The new cut-off value, which is denoted by 1 , is to be the number assigned to the (k − 1)th largest clone. The new cut-off line is, of course, the vertical line containing ( 1 , 0). Repeating this procedure, one may obtain the i th cut-off value i and the corresponding cut-off line. Notice that, after the i th step ends with the cut-off value i , all numbers assigned to unmatched clones are i.i.d. uniform random numbers between 0 to i , as the choice functions are oblivious. Let Ni be the number of unmatched clones after step i. That is, Ni = Nλ − ik. Since the (i + 1)th choice function tells how to choose the first clone to form the (i + 1)th edge without changing the distribution of the assigned numbers, the distribution of i+1 is the distribution of the (k − 1)th largest number among Ni − 1 independent uniform random numbers between 0 and i . Let 1 − Tj be the random variable representing the largest number among j independent uniform random numbers between 0 and 1. Or equivalently, Tj is the random variable

885


representing the smallest number among the random numbers. Then the largest number among the Ni − 1 random numbers has the same distribution as i (1 − TNi −1 ). Repeating this k − 1 times, we have i+1 = i (1 − TNi −1 )(1 − TNi −2 ) . . . (1 − TNi −k+1 ), and hence i+1 = i (1 − TNi −1 ) . . . (1 − TNi −k+1 ) = i−1 (1 − TNi−1 −1 ) . . . (1 − TNi−1 −k+1 ) · (1 − TNi −1 ) . . . (1 − TNi −k+1 ) =λ

Nλ −(i+1)k+1

(1 − Tj ).

j =Nλ −1 kNλ −j

It is crucial to observe that, once Nλ is given, all Ti are mutually independent random variables. This makes the random variable i highly concentrated near its mean, which enables us to develop theories as if i were a constant. The cutoff value i will provide enough information to resolve some otherwise difficult problems. k For θ in the range 0 ≤ θ ≤ 1, let (θ) be the cut-off value when 1 − θ k−1 λn or more clones are matched for the first time. Conversely, let N(θ) be the number of matched clones until the cut-off line reaches θλ. Lemma 3.2 (Cut-off line lemma). Let k ≥ 2 and λ > 0 be fixed. Then for θ1 < 1 uniformly bounded below from 0 and 0 < ≤ n, Pr and Pr

max | (θ) − θλ| ≥

θ :θ1 ≤θ ≤1

n

2

≤ 2e

− (min{ , (1−θ

1 )n

})

2 k − (min{ , (1−θ }) 1 )n . max |N(θ) − (1 − θ k−1 )λn| ≥ ≤ 2e

θ:θ1 ≤θ ≤1

Proof. See [30].

2

4. Large deviation inequalities In this section a generalized Chernoff bound and an inequality for random process is given. Let X1 , . . . , Xm be a sequence of random variables such that the distribution of Xi is determined if all the values of X1 , . . . , Xi−1 are known. For example, Xi = (θi ) with 1 ≥ θ1 ≥ · · · ≥ θm ≥ 0 in a Poisson λ-cell. If the upper and/or lower bounds are known for the conditional means E[Xi |X1 , . . . , Xi−1 ] and for the conditional second and third monuments, then Chernoff type large deviation inequalities may be obtained not only for jm=1 Xj but for min1≤i≤m ji =1 Xj and/or

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max1≤i≤m ji =1 Xj . Large deviation inequalities for such minima or maxima are especially useful in various situations. Lemma 3.2 can be shown using such inequalities too. Lemma 4.1. Let X1 , . . . , Xm be a sequence of random variables. Suppose that E[Xi |X1 , . . . , Xi−1 ] ≤ μi ,

(4.1)

and that there are positive constants ai and bi such that E[(Xi − μi )2 |X1 , . . . , Xi−1 ] ≤ ai ,

(4.2)

and

(4.3) E[(Xi − μi )3 eξ(Xi −μi ) |X1 , . . . , Xi−1 ] ≤ bi for all 0 ≤ ξ ≤ ξ0 . m 1/2 Then for any α with 0 < α ≤ ξ0 , i=1 ai m m m 1/2

α m α2 i=1 bi ≤ exp − Xi ≥ μi + α ai . 1 + m Pr 3/2 2 3 i=1 ai i=1

i=1

i=1

Similarly, E[Xi |X1 , . . . , Xi−1 ] ≥ μi

(4.4)

E[(Xi − μi )3 eξ(Xi −μi ) |X1 , . . . , Xi−1 ] ≥ bi for all ξ0 ≤ ξ < 0

(4.5)

together with (4.2) and

implies that m 1/2 α m α2 i=1 bi Xi ≤ μi − α ai . 1 − m ≤ exp − Pr 3/2 2 3 i=1 ai i=1 i=1 i=1 m

m

2

Proof. See [30].

As it is sometimes tedious to point out the value of α and to check the required bounds for it, the following forms of inequalities are often more convenient. Corollary 4.2 (Generalized Chernoff bound). If δξ0 bi ≤ ai for some 0 < δ ≤ 1, then (4.1)–(4.3) imply Pr

m

Xi ≥

i=1

m i=1

for all R > 0. Similarly, if −δξ0 and (4.5) yield Pr

m

i=1

for all R > 0.

Xi ≤

m i=1

1 2 m μi + R ≤ e− 3 min{δξ0 R, R / i=1 ai }

bi ≤

ai for some 0 < δ ≤ 1, then (4.2), (4.4)

1 2 m μi − R ≤ e− 3 min{δξ0 R, R / i=1 ai }

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Let Xθ , θ ≥ 0, be random variables which are possibly set-valued. Here θ may be integers as well as real numbers. Suppose that (θ) is a random variable depending on {Xθ }θ ≤θ and θ, and ψ = ψ({Xθ }θ ≤θ1 ; θ0 , θ1 )

and

ψθ = ψθ ({Xθ }θ ≤θ1 ; θ0 , θ, θ1 ).

The random variables ψ and ψθ are used to bound (θ). Example 4.3. Let X1 , X2 , . . . be i.i.d. Bernoulli random variables with mean p and Si = ji =1 Xj . Set (i) = |Si − ip| and ψ = (n) and

ψi = |Sn − Si − (n − i)p|.

Then, since Si − ip = Sn − np − (Sn − Si − (n − i)p) we have (i) ≤ ψ + ψi . Example 4.4. Consider the (λ, n)-cell defined in the previous section. Let vθ be the vertex that has its largest clone at (1 − θ)λ. If such a vertex does not exist, vθ is defined to be ℵ, assuming ℵ ∈ V . As there is no possibility that two distinct clones are assigned the same number, vθ is well-defined. Let Xθ = vθ and V (θ) be the set of vertices that contain no clone larger than or equal to (1 − θ)λ. That is, V (θ ) = V \ {vθ : 0 ≤ θ ≤ θ}. Clearly, E[|V (θ)|] = e−θλ n. Observing that for θ0 ≤ θ ≤ θ1 one has e−(θ1 −θ )λ |V (θ)| − e−θ λ n ≤ |V (θ1 )| − e−θ1 λ n + |V (θ1 )| − e−(θ1 −θ)λ |V (θ)| , we may set (θ) = ||V (θ)| − e−θ λ n|, ψ = e(θ1 −θ0 ) (θ1 )

and ψθ = e(θ1 −θ0 ) |V (θ1 )| − e−(θ1 −θ)λ |V (θ)| .

We bound the probabilities maxθ0 ≤θ ≤θ1 (θ) ≥ R and minθ0 ≤θ≤θ1 (θ) ≤ R under some conditions. Lemma 4.5. Let 0 ≤ θ0 < θ1 , R = R1 +R2 , R1 , R2 > 0 and θ be events depending on {Xθ }θ ≤θ . If (θ) ≤ ψ + ψθ for all θ0 ≤ θ ≤ θ1 , then Pr

max (θ) ≥ R ≤ Pr ψ ≥ R1 + Pr

θ0 ≤θ ≤θ1

θ:θ0 ≤θ≤θ1

+

max

θ

max 1(θ ) Pr ψθ ≥ R2 | {Xθ }θ ≤θ .

θ :θ0 ≤θ ≤θ1 {Xθ }θ ≤θ

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Jeong Han Kim

Similarly, if (θ) ≥ ψ + ψθ

for all θ0 ≤ θ ≤ θ1 ,

then

Pr min (θ) ≤ −R ≤ Pr ψ ≤ −R1 + Pr θ0 ≤θ ≤θ1

θ

θ:θ0 ≤θ≤θ1

+

max

max 1(θ ) Pr ψθ ≤ −R2 | {Xθ }θ ≤θ .

θ :θ0 ≤θ ≤θ1 {Xθ }θ ≤θ

2

Proof. See [30]. Example 4.3 (continued). As Pr[ψ ≥ R1 ] ≤ e

R2

1 − (min{R1 , p(1−p)n })

and Pr[ψi ≥ R2 |X1 , . . . , Xi ] = Pr[ψi ≥ R2 ] ≤ e

R2

2 − (min{R2 , p(1−p)(n−i) })

,

Lemma 4.5 for R1 = R2 = R/2 and θ = ∅ gives Pr[ max |Si − pi| ≥ R] ≤ e

2

R − (min{R, p(1−p)n })

i:0≤i≤n

.

Example 4.4 (continued). Since 1(v has no (1 − θ)λ-large clone) |V (θ)| = v∈V

is a sum of i.i.d. Bernoulli random variables with mean e−θλ ,

R2 Pr |V (θ)| − e−θ λ n ≥ R ≤ e− (min{R, θn }) , especially

2

− (min{R, θR n }) 1 Pr ψ ≥ R/2 ≤ e .

Once {Xθ }θ ≤θ is given, V (θ) is determined and V (θ1 ) = 1(v has no (1 − θ1 )λ-large clone) v∈V (θ )

is a sum of i.i.d. Bernoulli random variables with mean e−(θ1 −θ)λ . Thus R2

R2 − (min{R, (θ −θ)|V (θ)| }) ≤ 2e − (min{R, θn }) , 1 Pr ψθ ≥ R/2 | {Xθ }θ ≤θ ≤ 2e and Lemma 4.5 for θ0 = 0 and θ = ∅ yields Pr

R2 max |V (θ) − e−θ λ n| ≥ R ≤ 2e− (min{R, θn }) .

θ :0≤θ ≤θ1


889

5. Generalized cores and the main lemma In this section we introduce generalized cores and the main lemma. The main lemma will play a crucial roles in the proofs of the theorems mentioned in the introduction. We start with some terminology. A generalized degree is an ordered pair (d1 , d2 ) of non-negative integers. The inequality between two generalized degrees is determined by the inequality between the first coordinates and the reverse inequality between the second coordinates. That is, (d1 , d2 ) ≥ (d1 , d2 ) if and only if d1 ≥ d1 and d2 ≤ d2 . A property for generalized degrees is simply a set of generalized degrees. A property P is increasing if generalized degrees larger than an element in P are also in P . When a property P depends only on the first coordinate of generalized degrees, it is simply a property of degrees. For the t-core problem, we will use Pt-core = {(d1 , d2 ) : d1 ≥ t}. To estimate the size of the largest component, we will set Pcomp = {(d1 , d2 ) : d2 = 0}. Given the Poisson λ-cell on the set V of n vertices and θ with 0 ≤ θ ≤ 1, let dv (θ) be the number of v-clones smaller than θλ. Similarly, d¯v (θ) is the number of v-clones larger than or equal to θλ. Then Dv (θ) := (dv (θ), d¯v (θ)) are i.i.d. random variables. In particular, for any property P the events Dv (θ) ∈ P are independent and occur with the same probability, say p(θ, λ; P ), or simply p(θ ). For an increasing property P , the P -process is defined as follows. Construct n−1 the Poisson λ-cell as described in Section 3, where λ = p k−1 . The vertex set V = {v0 , . . . , vn−1 } will be regarded as an ordered set so that the i th vertex is vi−1 . The P -process is a generalization of Example 2.2 for which choice functions choose t-light clones. The P -process. Initially, the cut-off value = λ. Activate all vertices v with Dv (1) ∈ P . All clones of the activated vertices are activated too. Put those clones in a stack in an arbitrary order. However, this does not mean that the clones are removed from the λ-cell. (a) If the stack is empty, go to (b). If the stack is nonempty, choose the first clone in the stack and move the cut-off line to the left until the largest k − 1 unmatched clones, excluding the chosen clone, are found. (So, the cut-off value keeps decreasing.) Then match the k − 1 clones to the chosen clone. Remove all matched clones from the stack and repeat the process. A vertex that has not been activated is to be activated as soon as Dv ( /λ) ∈ P . This can be done even before all k − 1 clones are found. Its unmatched clones are to be activated too and put into the stack immediately. Clones found while moving the cut-off line are also in the stack until they are matched. (b) Activate the first vertex in V which has not been activated. Its clones are activated too. Put those clones into the stack. Then go to (a). Clones in the stack are called active. The steps carried out by the instruction described in (b) are called forced steps as it is necessary to artificially activate a vertex. When the cut-off line is at θλ, all θλ-large clones are matched or will be matched at the end of the step and all vertices v with Dv (θ) ∈ P have been activated. All other

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Jeong Han Kim

vertices can have been activated only by forced steps. Let V (θ) = VP (θ) be the set of vertices v with Dv (θ) ∈ P , and let M(θ) = MP (θ) be the number of θλ-large clones plus the number of θλ-small clones of vertices v not in V (θ). That is, M(θ) = d¯v (θ) + dv (θ)1(v ∈ V (θ)) = d¯v (θ) + dv (θ)1(Dv (θ) ∈ P ). v∈V

v∈V

Recalling that N(θ) is the number of matched clones until the cut-off line reaches θ λ, the number A(θ) of active clones (when the cut-off value is) at θλ is at least as large as M(θ) − N(θ). On the other hand, the difference A(θ) − (M(θ) − N(θ)) is at most the number F (θ) of clones activated in forced steps until θλ, i.e., M(θ) − N(θ) ≤ A(θ) ≤ M(θ) − N(θ) + F (θ).

(5.1)

As the cut-off lemma gives a concentration inequality for N(θ), Pr

2 k − (min{ , (1−θ } 1 )n , max |N(θ) − (1 − θ k−1 )λn| ≥ ≤ 2e

θ :θ1 ≤θ ≤1

a concentration inequality for M(θ) will be enough to obtain a similar inequality for B(θ ) := M(θ) − N(θ). More precisely, we will show that under appropriate hypotheses Pr

2 − (min{ , (1−θ }) 1 )n , max |M(θ) − (λ − q(θ))n| ≤ ≤ 2e

θ :θ1 ≤θ ≤1

where

q(θ) = q(θ, λ; P ) = E dv (θ)1(Dv (θ) ∈ P ) .

As the dv (θ)’s and Dv (θ)’s are identically distributed, q(θ ) does not depend on v. Also, recall that p(θ) = Pr[Dv (θ) ∈ P ]. As we will see later, B(θ) is very close to A(θ). Hence a concentration inequality for B(θ ) plays a very important roles in all of our proofs. Lemma 5.1 (Main lemma). In the P -process, if θ1 < 1 uniformly bounded from below by 0, 1 − p(θ1 ) = O(1 − θ1 ) and p(θ1 ) = (1), then for all in the range 0 < ≤ n we have Pr

2 − (min{ , (1−θ }) 1 )n max |V (θ)| − p(θ)n ≤ ≤ 2e

θ :θ1 ≤θ ≤1

and Pr

2 k − (min{ , (1−θ }) 1 )n . max B(θ) − (λθ k−1 − q(θ))n ≤ ≤ 2e

θ :θ1 ≤θ ≤1

Proof. See [30].

2

891


6. Cores of random hypergraphs In thissection we prove Theorem 1.7. Let λ > 0 and H (λ) = HPC (n, p), where n−1 λ = p r−1 . Let the property P = {(d1 , d2 ) : d1 ≥ t}. Then p(θ) = Q(θλ, t)

and q(θ ) = θλQ(θλ, t − 1).

The main lemma gives Corollary 6.1. For θ1 ≤ 1 uniformly bounded from below by 0 and in the range 0 < ≤ n, Pr

2 max |V (θ)| − Q(θλ, t)n ≥ ≤ 2e− (min{ , n })

θ :θ1 ≤θ ≤1

and Pr

1

2 max B(θ) − (θ k−1 − Q(θλ, t − 1))θλn ≥ ≤ 2e− (min{ , n }) .

θ :θ1 ≤θ≤1

Subcritical Region: For λ = λcrt − σ , σ > 0 and θ1 = δ/λcrt with δ = 0.1 it is easy to see that there is a constant c > 0 such that 1

(θ k−1 − Q(θλ, t − 1))θλn ≥ cσ n for all θ with θ1 ≤ θ ≤ 1. Let τ be the first time the number A(θ) of active clones at θλ becomes 0. Then the second part of Corollary 6.1 gives Pr[τ ≥ θ1 ] ≤ Pr[B(θ) = 0 for some θ with θ1 ≤ θ ≤ 1]

1 ≤ Pr max B(θ) − (θ k−1 − Q(θλ, t − 1))θλn ≥ cσ n θ :θ1 ≤θ ≤1

≤ 2e

− (σ 2 n)

.

As θ1 λ ≤ θ1 λcrt = δ, and hence Q(θ1 λ, t) ≤ δ/2 for t ≥ 2, the first part of Corollary 6.1 yields Pr[|Vt (HPC (n, p ; k))| ≥ δn] ≤ Pr[τ ≥ θ1 ] + Pr[|V (θ1 )| ≥ δn] ≤ 2e− (σ

2 n)

.

Therefore Theorem 1.1 implies that Pr[|Vt (H (n, p ; k))| ≥ δn] ≤ 2e− (σ

2 n)

.

To complete the proof, we observe that the t-core of size i has at least ti/k at least ti/k edges, edges. Let Zi be the number of subgraphs on i vertices with i = i0 , . . . , δn, where i0 = i0 (k, t) is the least i such that ki ≥ ti/k. Then i n ni i ti k (6.1) pti/k ≤ pti/k =: Li , E[Zi ] ≤ i ti/k i! (ti/k)!

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Jeong Han Kim

where ti/k actually means ti/k. Hence k kt n i −(k−1)t i (k−1)t−k Li+k =O k t n =O = O(δ (k−1)(t−1)−1 ). Li i i n

That is, Li+k /Li exponentially decreases. For i = i0 , . . . , i0 + k − 1, Li = O(ni n−i(k−1)t/k ) = O(n−i(t−1−t/k) ) implies that Pr[Vt (H (n, p ; k)) = ∅] ≤ 2e− (σ

2 n)

+ O(n−i0 (t−1−t/k) ),

2

as desired.

Supercritical region: We will prove the following theorem. Theorem 6.2. Suppose that p n−1 k−1 ≥ λcrt +σ and 0 < δ ≤ 1. Then, with probability 1 − 2e− (min{δ

2 σ n,σ 2 n})

, Vt = Vt (HPC (n, p ; k)) satisfies

Q(θλ λ, t)n − δn ≤ |Vt | ≤ Q(θλ λ, t)n + δn,

(6.2)

and the degrees of vertices of the t-core are i.i.d. t-truncated Poisson random variables with parameter t := θλ λ + β for some β with |β| ≤ δ. Moreover, the distribution of the t-core is the same as that of the t-truncated Poisson cloning model with parameters |Vt | and t . Proof. Let λ = λcrt + σ , σ > 0 and θλ be the largest solution for the equation 1

θ k−1 − Q(θλ, t − 1) = 0. Then it is not hard to check that there are constants c1 , c2 > 0 such that for θ in the range θλ ≤ θ ≤ 1, 1

θ k−1 − Q(θλ, t − 1) ≥ c1 σ 1/2 (θ − θλ ), and for θ in the range θλ − c2 σ 1/2 ≤ θ ≤ θλ , 1

θ k−1 − Q(θλ, t − 1) ≤ −c1 σ 1/2 (θλ − θ). Let τ be the largest θ with A(θ) = 0. Then V (τ ) is the t-core of HPC (n, p; k). For θ1 = θλ + δ and θ2 = θλ − min{δ, c2 σ 1/2 } with 0 < δ ≤ 1, Corollary 6.1 gives Pr[τ ≥ θ1 ] ≤ Pr[B(θ) = 0 for some θ with θ1 ≤ θ ≤ 1]

1 ≤ Pr max |B(θ) − (θ k−1 − Q(θλ, t − 1))θλn| ≥ c1 σ 1/2 δn θ :θ1 ≤θ ≤1

≤ 2e

− (δ 2 σ n)

893


and Pr[τ < θ2 ] ≤ Pr[B(θ2 ) > 0] 1

≤ Pr |B(θ2 ) − (θ2k−1 − Q(θ2 λ, t − 1))θ2 λn| ≥ c1 σ 1/2 min{δ, c2 σ 1/2 }n ≤ 2e− (min{δ d dθ Q(θλ, t)

Since

2 σ n,σ 2 n})

.

= λP (θλ, t − 1) ≤ λ, we have

Q(θ1 λ, t) ≤ Q(θλ λ, t) + λδ, and Q(θ2 λ, t) ≥ Q(θλ λ, t) − λδ, and Corollary 6.1 implies that Pr[V (θ1 ) − Q(θλ λ, t)n ≥ 2λδn] ≤ 2e− (δ and

2 n)

Pr[V (θ2 ) − Q(θλ λ, t)n ≤ −2λδn] ≤ 2e− (δ

2 n)

.

Therefore Pr[|τ − θλ | > δ] ≤ Pr[τ ≥ θ1 ] + Pr[τ ≤ θ2 ] ≤ 2e− (min{δ and, replacing δ by

2 σ n, σ 2 n})

δ 2λ ,

Pr[|V (τ ) − Q(θλ λ, t)n| ≥ δn] ≤ Pr[τ ≥ θ1 ] + Pr[τ ≤ θ2 ] + 2e− (δ ≤ 2e− (min{δ

2 σ n, σ 2 n})

2 n)

.

Clearly, once V (τ ) and t := τ λ are given, the residual degrees dv (τ ), v ∈ V (τ ), are i.i.d. t-truncated Poisson random variables with parameter t . 2 Once Vt and t are given, |Vt (i)|, i ≥ t, is the sum of i.i.d. Bernoulli random P ( t ,i) . Similarly, the size of Wt (i) = j ≥i Vt (j ) is variables with mean pi ( t ) := Q( t ,t) t ,i) the sum of i.i.d. Bernoulli random variables with mean qi ( t ) := Q( Q( t ,t) . Applying the generalized Chernoff bound (Lemma 4.2), we have

2 Pr |Vt (i)| − pi( t )|Vt | ≥ δ|Vt ||Vt , t ≤ 2e− (δ |Vt |)

and

2 Pr |Wt (i)| − qi( t )|Vt | ≥ δ|Vt ||Vt , t ≤ 2e− (δ |Vt |) .

Combining this with Lemma 6.2 and using |P (ρ, i) − P (ρ , i)| ≤ |ρ − ρ |, and |Q(ρ, i) − Q(ρ , i)| ≤ |ρ − ρ |, we obtain, for any i,

2 2 Pr |Vt (i)| − P (θλ λ, i)n ≥ δn ≤ 2e− (min{δ σ n,σ n}) ,

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Jeong Han Kim

and

2 2 Pr |Wt (i)| − Q(θλ λ, i)n ≥ δn ≤ 2e− (min{δ σ n,σ n}) .

In particular, as θλ = θcrt + (σ 1/2 ) for uniformly bounded σ it follows that for λ = λcrt + σ , |Vt (i)| = (1 + O(σ 1/2 ))i P (θcrt λcrt , i)n + O (n/σ )1/2 log n , with probability 1 − 2e− (min{log

2 n,σ 2 n})

.

7. The emergence of the giant component In this section we just give ideas for the proof of Theorem 1.4. Let the property P be {(d1 , d2 ) : d2 = 0}. Then p(θ) = e−(1−θ)λ and q(θ ) = θλe−(1−θ)λ , and the main lemma gives Corollary 7.1. For θ1 ≥ 1 uniformly bounded from above by 1 and in the range 0 < ≤ n, Pr

2 − (min{ , (1−θ }) 1 )n max |V (θ)| − e−(1−θ )λ n ≥ ≤ 2e

θ :0≤θ ≤θ1

and Pr

2 − (min{ , (1−θ }) 1 )n . max B(θ) − (θ − e−(1−θ )λ )θλn ≥ ≤ 2e

θ :0≤θ ≤θ1

To estimate A(θ) it is now enough for us to estimate F (θ) by (5.1). Once good estimations for F (θ) are established, we may take similar (but slightly more complicated) approaches used in the previous section. It is convenient to consider an (imaginary) secondary stack with parameter ρ, or simply ρ-secondary stack. Initially, the secondary stack with parameter ρ consists of the first ρn vertices v0 , . . . , vρn−1 of V . The set of those ρn vertices is denoted by Vρ . Whenever the primary stack is empty, the first vertex in the secondary stack that has not been activated must be activated. Its clones are activated too and put into the primary stack. The activated vertex as well as vertices activated by other means are no longer in the secondary stack. If the secondary stack is empty, go back to the regular procedure. This does not change the P -process at all, but will be used just for the analysis. Let τρ be the largest τ such that, at τ λ, the primary stack becomes empty after the secondary stack is empty. Thus, once the cut-off line reaches τρ λ, no active clones are provided from the secondary stack. Denote by C(ρ) the union of the components containing any vertex in Vρ . The following lemma is useful to predict how large τρ is.

895


Lemma 7.2. Suppose 0 < δ, ρ < 1 and θ1 , θ2 ≤ 1 are uniformly bounded from below by 0. Then Pr[τρ ≥ θ1 ] ≤ Pr

min

θ :θ1 ≤θ ≤1

2 B(θ) ≤ −(1 − δ)θ1 λe−(1−θ1 )λ ρn + 2e− (δ ρn) ,

and conversely, Pr[τρ ≤ θ2 ] ≤ Pr[B(θ2 ) ≥ −(1 + δ)θ2 λe−(1−θ2 )λ ρn] + 2e− (δ Proof. See [30].

2 ρn)

.

2

Once the value of τρ is known quite precisely, a good estimation of F (θ) is possible. Using similar (but slightly more complicated) arguments used in the previous section, estimation of A(θ) is also possible. Due to space limitation, the proof of Theorem 1.4 is omitted.

8. Closing remarks The Poisson λ-cell is introduced to analyze those properties of GPC (n, p), for which the degrees are i.i.d. Poisson random variables with mean λ = p(n−1). Then various nice properties of Poisson random variables are used to analyze sizes of the largest component and the t-core of GPC (n, p). We believe that the approaches presented in this paper are useful to analyze problems with similar flavors, especially problems related to branching processes. For example, we can easily modify the proofs of Theorem 1.7 to analyze the pure literal rule for the random k-SAT problems, k ≥ 3. Another example may be the Karp–Sipser Algorithm to find a large matching of the random graph. (See [29], [3].) In a subsequent paper, we will analyze the structure of the 2-core of G(n, p) and the largest strong component of the random directed graph as well as the pure literal rule for the random 2-SAT problem. For the random (hyper)graph with a given sequence (di ), we may also introduce the (di )-cell, in which the vertex vi has di clones and each clone is assigned a uniform random real number between 0 and the average degree n1 n−1 i=0 di . Though it is not possible to use all of the nice properties of Poisson random variables any more, we believe that the (di )-cell equipped the cut-off line algorithm can be used to prove stronger results for the t-core problems considered in various papers including [13], [21], [22], [26], [37]. Recall that the degrees in G(n, p) has the binomial distribution with parameters n − 1 and p. By introducing the Poisson cloning model, we somehow first take the limit of the binomial distribution, which is the Poisson distribution. In general, many limiting distributions like Poisson and Gaussian ones have nice properties. In our opinion this is because various small differences are eliminated by taking the limits, and limiting distributions have some symmetric and/or invariant properties. Thus

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one may wonder whether there is an infinite graph that shares most properties of the random graphs G(n, p) with large enough n. So, in a sense, the infinite graph, if it exists, can be regarded as the limit of G(n, p). An infinite graph which Aldous [1] considered to solve the linear assignment problem may or may not be a (primitive) version of such an infinity graph. Though it may be impossible to construct such a graph, the approaches taken in this paper might be useful to find one, if any. Acknowledgement. The author thanks C. Borgs, J. Chayes, B. Bollobás and Y. Peres for helpful discussions.

References [1] Aldous, D., The zeta(2) limit in the random assignment problem. Random Structures Algorithms 18 (2001), 381–418. [2] Alon, N., Spencer, J., The Probabilistic Method. 2nd ed., Wiley-Interscience, New York, NY, 2000. [3] Aronson, J., Frieze, A., Pittel, B., Maximum matchings in sparse random graphs: KarpSipser revisited. Random Structures Algorithms 12 (1998), 111–178. [4] Athreya, K. B., Ney, P. E., Branching processes. Grundlehren Math. Wiss. 196, SpringerVerlag, Berlin 1972. [5] Bollobás, B., The evolution of random graphs. Trans. Amer. Mat. Soc. 286 (1984), 257–274. [6] Bollobás, B., The evolution of sparse graphs. In Graph Theory and Combinatorics (ed. by B. Bollobás), Academic Press, London 1984, 35–57. [7] Bollobás, B., Random graphs. Academic Press, London 1985. [8] Bollobás, B., The chromatic number of random graphs. Combinatorica 8 (1988), 49–56. [9] Bollobás, B., Erd˝os, P., Cliques in random graphs. Math. Proc. Cambridge Philos. Soc. 80 (1976), 419–427. [10] Bollobás, B., Frieze A. M., On matchings and Hamiltonian cycles. In Random Graphs ’83 (ed. by M. Karoński and A. Ruciński), North-Holland, Amsterdam, NewYork 1985, 23–46. [11] Bollobás, B., Thomason A., Random graphs of small order. In Random Graphs ’83 (ed. by M. Karoński and A. Ruciński), North-Holland, Amsterdam, New York 1985, 47–97. [12] Chvátal, V., Almost all graphs with 1.44 edges are 3-colorable. Random Structures Algorithms 2 (1991), 11–28. [13] Cooper, C., The cores of random hypergraphs with a given degree sequence. Random Structures Algorithms 25 (2004), 353–375. [14] Erd˝os, P., Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53 (1947), 292–294. [15] Erd˝os, P., Rényi, A., On random graphs I. Publ. Math. Debrecen 6 (1959), 290–297. [16] Erd˝os, P., Rényi, A., On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5 (1960), 17–61. [17] Erd˝os, P., Rényi, A., On the strength of connectedness of a random graph. Acta Math. Acad. Sci. Hung. 12 (1961), 261–267.


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[18] Erd˝os, P., Rényi, A., On random matrices. Publ. Math. Inst. Hung. Acad. Sci. 8 (1964), 455–461. [19] Erd˝os, P., Rényi, A., On the existence of a factor of degree one of a connected random graph. Acta Math. Acad. Sci. Hung. 17 (1966), 359–368. [20] Erd˝os, P., Rényi, A., On random matrices II. Stud. Sci. Math. Hung. 3 (1968), 459–464. [21] Fernholz D., Ramachandran V., The giant k-core of a random graph with a specified degree sequence. Manuscript, 2003. [22] Fernholz D., Ramachandran V., Cores and connectivity in sparse random graphs. Tech. Report TR-04-13, University of Texas at Austin, Department of Computer Science, Austin, 2004. [23] Flajolet, D., Knuth D. E., Pittel, B., The first cycle in an evolving graph. Discrete Math. 75 (1989), 167–215. [24] Janson, S., Poisson convergence and Poisson processes with applications to random graphs. Stochastic Process. Appl. 26 (1988), 1–30. [25] Janson, S., Knuth D. E., Łuczak, T., Pittel, B., The birth of the giant component. Random Structures Algorithms 3 (1993), 233–358. [26] Janson, S., Łuczak, M. J., A simple solution to the k-core problem. Manuscript, 2005. [27] Janson, S., Łuczak, T., Ruciński, A., Random graphs. Wiley-Interscience, New York, NY, 2000. [28] Karp, R. M., The transitive closure of a random digraph. Random Structures Algorithms 1 (1990), 73–93. [29] Karp, R. M., Sipser, M., Maximum matchings in sparse random graphs. In Proceedings of the 22nd IEEE Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 1981, 364–375. [30] Kim., J.-H., Poisson cloning model for random graph. Manuscript, 2005. [31] Koml´s, J., Szemerédi E., Limit distributions for the existence of Hamilton cycles in a random graph. Discrete Math. 43 (1983), 55–63. [32] Łuczak, T., Component behavior near the critical point of the random graph process. Random Structures Algorithms 1 (1990), 287–310. [33] Łuczak, T., The chromatic number of random graphs. Combinatorica 11 (1991), 45–54. [34] Łuczak, T., Size and connectivity of the k-core of a random graph. Discrete Math. 91 (1991), 61–68. [35] Łuczak, T., Pittel, B., Wierman, J., The birth of the giant component. Trans. Amer. Mat. Soc. 341 (1994), 721–748. [36] Matula, D., The largest clique size in a random graph. Tech. Rep., Dept. Comp. Sci., Southern Methodist University, Dallas, 1976. [37] Molloy, M., Cores in random random hypergraphs and Boolean formulas. Random Structures Algorithms 27 (2005), 124–135. [38] Pittel, B., Spencer J., Wormald, N., Sudden emergence of a giant k-core in a random graph. J. Combin. Theory Ser. B 67 (1996), 111–151. [39] Shamir, E., Spencer, J., Sharp concentration of the chromatic number on random graph Gn,p . Combinatorica 7 (1987), 124–129. Microsoft Research, One Microsoft Way, Redmond, WA 98052, U.S.A E-mail: [email protected]

Randomness and regularity Tomasz Łuczak

Abstract. For the last ten years the theory of random structures has been one of the most rapidly evolving fields of discrete mathematics. The existence of sparse structures with good ‘global’ properties has been vital for many applications in mathematics and computer science, and studying properties of such objects led to many challenging mathematical problems. In the paper we report on recent progress on this subject related to some variants of Szemerédi’s Regularity Lemma. Mathematics Subject Classification (2000). Primary 05C80, 05D05; Secondary 05C35, 05C65, 05D40. Keywords. Random graphs, extremal graph theory, regularity lemma, removal lemma, density theorems.

1. Introduction In the paper we consider ‘extremal’properties of families of sets, i.e., we study the size of maximal subfamilies of a given family with certain property. Let [A]r be the family of all r-sets (i.e. sets of r elements) contained in A; if A = [n] = {1, 2, . . . , n} we put [n]r = [[n]]r . Two classical examples of extremal results for [n]r are Szemerédi’s and Turán’s theorems. Let us recall that Szemerédi’s density theorem [17] states that if rk (n) denote the maximum size of a subset of [n] = [n]1 which contains no non-trivial arithmetic progression of length k, then, rk (n) = o(n). In order to formulate Turán’s theorem, we need some notation. For given r, s, n, n ≥ s > r ≥ 2, let ex([s]r , [n]r ) denote the size of the largest family A ⊆ [n]r such that for no set B ⊆ [n], |B| = s, we have [B]r ⊆ A. Furthermore, let α(m, r) = lim sup n→∞

ex([m]r , [n]r ) n .

(1)

r

Turán’s theorem [21] states that α(m, 2) = m−2 m−1 for m ≥ 2. Let us remark that we do not know the value of α(m, r) for any pair (m, r) with m > r > 2; e.g., the question whether α(4, 3) = 5/9 is a well known open problem of extremal set theory. The main problem we are concerned in this paper is the existence of families of r-sets which are ‘sparse’, or, at least, ‘locally sparse’, yet preserve some of the properties of [n]r stated in the theorems above. In the following section, we state a few specific problems on the existence of locally sparse structures with good ‘global’ Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Tomasz Łuczak

properties. Then, we explain why the standard probabilistic method cannot be directly used to study extremal properties of graphs and hypergraphs. Next we recall another important result of modern combinatorics: Szemerédi’s Regularity Lemma and show how it could help in dealing with such problems. We conclude with a few remarks on possible generalizations of known results and some speculation on developments which are still to come.

2. Locally sparse structures with good extremal properties Let us introduce first some notation. An r-uniform hypergraph is a pair H = (V , E), where V is the set of vertices of H and E ⊆ [V ]r denotes the set of its edges. We say that a hypergraph H = (V , E ) is a subhypergraph of a hypergraph H = (V , E ), if V ⊆ V and E ⊆ E . The complete r-uniform hypergraph ([m], [m]r ) we denote (r) (2) by Km , and set Km = Km . A 2-uniform hypergraph is called a graph. Let H = (V , E) be an r-uniform hypergraph, and let C = {W1 , . . . , Wt } be a family of s-subsets of V such that [Wi ]r ⊆ E, for i = 1, . . . , , t. We say that C is a loose (s, t)-circuit if t ≥ 3, (s, t) = (3, 3), Wi ∩ Wi+1 = ∅, for i = 1, 2, . . . , t − 1, and W1 ∩Wt = ∅. We call C a tight (s, t)-circuit if either t = 2 and |W1 ∩W2 | ≥ r +1, or t ≥ 3 and |Wi ∩ Wi+1 | ≥ r, for i = 1, 2, . . . , t − 1, as well as |W1 ∩ Wt | ≥ r. Finally, for r-uniform hypergraphs H = (V , E) and H = (V , E ), let ex(H , H ) be the number of edges in the largest subhypergraph of H which contains no copies of H , and ex(H , H ) = ex(H , H )/|E|. It is easy to see (e.g., [8], Prop.8.4.), that for a given H the function ex(H , H ) is maximized for complete H , i.e., ex(H , H ) ≤ ex(H , [V ]r ). One of the first results on the existence of locally sparse structures with good extremal properties was proved by Erd˝os [1] nearly fifty years ago. It states that there are graphs with large girth and no large independent sets (and so with large chromatic number). Theorem 2.1. For each and ε > 0 there exists a graph G(, ε) = (V , E) such that G(, ε) contains no (2, t)-circuits with t ≤ , but each subset W ⊆ V such that |W | ≥ ε|V | contains an edge of G(, ε). In the following section we present Erd˝os’ elegant non-constructive proof of this fact. Then we shall try to use a similar idea to get the following sparse version of Turan’s theorem. Conjecture 2.2. For any r, s, ε > 0, and , there exist an n = n(r, m, , ε) and an r-uniform hypergraph G(r) (s, , ε) = (V , E) such that (i) G(r) (s, , ε) contains no tight (s, t)-cycles with 2 ≤ t ≤ ; (ii) each subhypergraph H (r) ⊆ G(r) (s, , ε) with at least (α(s, r) + ε)|E| edges contains a subset B, |B| = s, such that [B]r ⊆ H (r) , i.e., ex([n]s , G(r) (s, , ε)) ≤ α(s, r) + ε .

Randomness and regularity

901

In Sections 3-5 below we describe how to approach Conjecture 2.2 using a special version of the Regularity Lemma. Here we remark only that the existence of G(r) (s, , ε) has been shown only for r = 2, s = 3 (Frankl and Rödl [2] and Haxell et al. [7]), r = 2, s = 4 (Kohayakawa et al. [10]), and recently for r = 2, s = 5 (Gerke et al. [4]). We conclude this section with a conjecture on a sparse version of Szemerédi’s density theorem. Here a (k, t)-arithmetic circuit is a family of t non-trivial arithmetic progressions A1 , . . . , At of length k such that Ai ∩ Ai+1 = ∅ for i = 1, 2 . . . , t − 1, and A1 ∩ At = ∅. Conjecture 2.3. For any k, , and α > 0, there exist an ε = ε(α, k, ) > 0, n = n(k, α, ), and a set A = A(k, , α, n) ⊆ [n] such that (i) A contains no (k, t)-arithmetic circuits for t ≤ ; (ii) any non-trivial arithmetic progression of length nε in [n] contains at most k elements of A; (iii) each subset B of A with at least α|A| elements contains a non-trivial arithmetic progression of length k. Kohayakawa et al. [9] showed the existence of A = A(3, , α, n) for any α > 0 and . Their proof was based on the idea used by Ruzsa and Szemerédi [16] to show that r3 (n) = o(n). Let us also mention that since Szemerédi’s density theorem can be deduced from some extremal results for hypergraphs (see Frankl and Rödl [3], Nagle et al. [13], and Rödl and Skokan [15]) it is in principle possible, although somewhat unlikely, that one can imitate the argument from [9] and verify Conjecture 2.3 for all k ≥ 4 (cf., Conjecture 6.2 below).

3. Random structures For 0 ≤ p ≤ 1 and natural numbers n, r, let G(r) (n, p) denote the random r-uniform hypergraph with vertex set [n], where edges G(r) (n, p) are chosen from [n]r independently with probability p. Thus, the numberof edges of G(r) (n, p) is a binomially distributed random variable with parameters nr and p. Typically, we are interested only in the asymptotic behavior of G(r) (n, p) when n → ∞ and the probability p may depend on n. In particular, we say that for a given function p = p(n) the hypergraph G(r) (n, p) has a property A a.a.s. if the probability that G(r) (n, p) has A tends to 1 as n → ∞. Since in this note we deal mainly with graphs, instead of G(2) (n, p) we write briefly G(n, p). Let us recall Erd˝os’ proof of Theorem 2.1. Fix and ε > 0. Let n be very large and p be the probability which is neither too small (so G(n, p) contains no large independent sets) nor too large (so G(n, p) is locally sparse). A good choice for p is, say, p = p(n) = n−1+1/2 , but, for n large enough, any p = p(n) such that 10/ε ≤ np ≤ n1/ /10 would do.

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Let X = X(n, ) be the random variable which counts (2, t)-circuits with t ≤ in G(n, p). Then, for n large enough, we have EX ≤

2i i i i=3

2

2

pi ≤ 2+3 (np) ≤ n1/2 .

(2)

Thus, from Markov’s inequality, Pr(X ≥ n/2) ≤ 2n−1/2 , and so, for large enough n, with probability at least 2/3 > 1 − 2n−1/2 , we have X ≤ n/2. On the other hand, for the number Y = Y (n, k) of independent sets of size k in G(n, p) we have k n k n ) ( (1 − p) 2 ≤ 2 exp − p . (3) Pr(Y > 0) ≤ EX = k 2 If k = εn/2 then Pr(Y > 0) tends to 0 as n → ∞, i.e., for n large enough we have Pr(Y > 0) < 2/3. Now, let G(, ε) be a graph obtained from G(n, p) by removing one vertex from each (2, t)-circuit with t ≤ . Then, with probability at least 1/3, G(, ε) fulfills the assertion of Theorem 2.1. The main goal of this paper is to discuss how one can verify Conjecture 2.2 using a modified version of Erd˝os’ approach. We shall concentrate on the simplest non-trivial case of Conjecture 2.2, when s = 3 and r = 2. In order to deduce the existence of G(2) (3, , ε) from appropriate properties of the random graph G(n, p) first we need to guess what value of p we are to use. More specifically, we should find the smallest possible value of p0 = p0 (n) such that a.a.s. in each subgraph of G(n, p0 ) which contains, say, 51% of its edges one can find a triangle. Note that if a graph G contains m edges and t triangles, then there is a triangle-free subgraph of G with at least m − t edges. n 3 Thus, it seems that in G(n, p0 ) the expected number of triangles (equal n to ) must be at least of the order of the expected number of edges (equal p 3 0 2 p0 ), −1/2 ). It turns out that this necessary i.e., p0 = p0 (n) should be at least as large as (n condition is also sufficient and the following holds (see Frankl and Rödl [2], Haxell et al. [7]). −1/2 , Theorem 3.1. For every δ > 0 there exists c = c(δ) such that if = p(n) ≥ cn np then a.a.s. each subgraph of G(n, p) with at least (1/2 + δ) 2 p edges contains a triangle.

Let us try to prove Theorem 3.1 using Erd˝os’ argument. To this end one has to bound the expected number of triangle-free subgraphs H of G(n, p), containing 51% of edges of G(n, p), using a formula similar to (3). In order to do that one needs to estimate the probability that such a large subgraph H of G(n, p) contains no triangles. The first problem which immediately emerges is the fact that our argument must depend strongly on the fact that H has more than 51% of edges of G(n, p), since in every graph G one can find a large bipartite subgraph which contains more than half of its edges. Thus, we have to use some property of H shared by all subgraphs of G(n, p) with more than half of its edges, and does not hold for, say, bipartite


903

subgraphs of G(n, p); i.e., we should consider only graphs H which are ‘essentially non-bipartite’. Then, we need to show that a.a.s. each subgraph of G(n, p) containing at least 51% of its edges is ‘essentially non-bipartite’, and estimate the probability that a ’random essentially non-bipartite’ graph is triangle-free. However, now we face another, more serious obstacle. The number of subsets of the set of edges of G(n, p) is much larger then the number of subsets of the set of vertices of G(n, p). Consequently, the factor 2n in (3) should be replaced by exp((n2 p)). Hence, we should estimate the probability that a ‘random essentially non-bipartite’ subgraph H of G(n, p) is triangle-free by a quantity which is much smaller than the probability that G(n, p) contains no edges at all! This is the crucial and most difficult part of the whole argument. It is also precisely the reason why we can show Conjecture 2.2 only for r = 2 and s = 3, 4, 5; for all other cases the proof breaks at this point. Finally, it is easy to check that if p = p(n) = n−1/2+o(1) , then for any given a.a.s. G(n, p) contains fewer than o(n2 p) tight (3, t)-circuits for t ≤ which can be removed from G(n, p) without affecting much its extremal properties. Unfortunately, one cannot deal in the same way with loose (3, t)-circuits. The reason is quite simple: for t ≥ 4 the number of loose (3, t)-circuits grows much faster than the number of triangles, because, roughly speaking, two triangles of G(n, p) are much more likely to share a vertex than an edge. Clearly, the same is true if instead of G(n, p) we consider a shadow of G(3) (n, p), i.e., we randomly generate triples of vertices and then replace each of them by a triangle. Still, it is not inconceivable that Conjecture 2.2 can be settled in the affirmative by a non-constructive method using more sophisticated models of random hypergraphs; there have been a fair amount of attempts in this direction but so far all of them have failed miserably.

4. Regularity Lemma One of the main ingredients of Szemerédi’s ingenious proof of the density theorem was the Regularity Lemma which for the last thirty years has become one of the most efficient tools of modern graph theory. In order to formulate it rigorously we need a few technical definitions. For a graph G = (V , E) and W, W ⊆ V let e(W, W ) denote the number of edges joining W and W . A pair (A, B) of disjoint subsets of vertices of G is called an ε-regular pair, if for every subsets A ⊆ A, |A | ≥ ε|A|, B ⊆ B, |B | ≥ ε|B|, e(A , B ) e(A, B) (4) |A ||B | − |A||B| ≤ ε. An ε-regular pair behaves in many respects as the bipartite random graph G(A, B, ρ), in which edges between A and B appear independently with probability ρ = e(A, B)/|A||B|. In particular, it is easy to check, that if a pair (A, B)

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is ε-regular then the number of subgraphs of a given size in the bipartite subgraph G[A, B] induced by A ∪ B in G is close to the expected number of such subgraphs in G(A, B, ρ). For instance, if (A, B) is ε-regular, then the number of cycles of length four in G[A, B] is equal (ρ 4 /4 ± h(ε))|A|2 |B|2 , where h(ε) is a function which tends to 0 as ε → 0. The implication in the other direction holds as well: if the number of cycles of length four in G[A, B] is smaller than (ρ 4 /4 + ε)|A|2 |B|2 , then the pair (A, B) is h (ε)-regular for some function h (ε) which tend to 0 as ε → 0. Let us also mention that ε-regularity implies the correct number of small subgraphs even if we consider more than one ε-regular pair. For instance, if three disjoint sets A1 , A2 , A3 ⊆ V are such that each of the pairs (A1 , A2 ), (A2 , A3 ), (A1 , A3 ) is ε-regular with density ρ, the number of triangles in the tripartite graph induced in G by these sets is (ρ 3 ± h (ε))|A1 ||A2 ||A3 |, where h (ε) → 0 as ε → 0. A partition V = V1 ∪ · · · ∪ Vk of the vertex set of a graph G = (V , E) is called a (k, ε)-partition if for all i, j = 1, 2, . . . , k we have |Vi | − |Vj | ≤ 1 and all except at most εk 2 pairs (Vi , Vj ), 1 ≤ i < j ≤ k, are ε-regular. Now Szemerédi’s Regularity Lemma (see [17] and [18]) can be stated as follows. Lemma 4.1 (Szemerédi’s Regularity Lemma). For every ε > 0 there exists K such that each graph G with more than 1/ε vertices admits a (k, ε)-regular partition for some k, 1/ε < k < K. Note that if k > 1/ε, then for a (k, ε)-regular partition there are at most n2 /k + εn2 ≤ 2εn2 edges of G which are either contained inside sets Vi or join pairs which are not ε-regular. Thus, Szemerédi’s Regularity Lemma says that all but 2εn2 edges of any graph G can be partitioned into at most k ε-regular pairs, for some 1/ε ≤ k ≤ K, where K does not depend on the number of vertices in G. Unfortunately, K = K(ε) grows very fast to infinity as ε → 0 (see Gowers [5]), so most of the applications of the Regularity Lemma give very poor bounds of estimated quantities. The Regularity Lemma can be reformulated and generalized in several ways. For instance, one can view it as a statement on the compactness of certain metric space (Lovász and Szegedy [12]); an information-theoretic approach to it can be found in Tao [19]. Another versions of the Regularity Lemma ensure the existence of ‘weak’ (k, ε)-partitions, or ‘partial ε-covers’ consisting of reasonably large ε-pairs. However the two most important developments in this area are, in my opinion, generalizations of the Regularity Lemma to sparse graphs and to hypergraphs. In the following sections we discuss how the Regularity Lemma can be modified to work efficiently for sparse graphs; here we say a few words on a much harder (both to state and to prove) version of the Regularity Lemma for hypergraphs. Several years ago Frankl and Rödl [3] generalized the Regularity Lemma to r-uniform hypergraphs and proved it, together with a supplementary ‘counting lemma’, for r = 3. The case r ≥ 4 has been dealt with by Rödl and Skokan [15] and Nagle et al. [13], and, independently, by Gowers [6]. As was noticed by Frankl and Rödl [3], their version of the Regularity Lemma implies the following Removal Lemma which, in turn, can be used to show Szemerédi’s density theorem (for details see Rödl et al. [14]).


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Lemma 4.2 (Removal Lemma). For m ≥ r ≥ 2 and every δ > 0 there exist η > 0 and n0 so that for every r-uniform hypergraph F on m vertices and r-uniform hypergraph H on n, n ≥ n0 , vertices the following holds. If H contains at most ηnm copies of F , then one can delete δnr edges of H to destroy all copies of F . Thus, the Removal Lemma states that if the number of copies of F is large enough, then they must be, in some sense, uniformly distributed in H , i.e., the large number of copies of F makes a hypergraph H , or at least parts of it, close to a random graph. In fact all known proofs of Lemma 4.2 are based on this idea. We should apply the Regularity Lemma to H , and then show that, if the number of copies of F in H is large, then there exists a big random-like subgraph H of H which contains an anticipated number of copies of F .

5. Regularity Lemma: sparse graphs Note that Lemma 4.1 is basically meaningless for sparse graphs since the definition (4) of ε-regular pair (A, B) does not say much on the distribution of edges between A and B if the density ρ = e(A, B)/|A||B| is smaller than ε. Thus, let us modify the definition of an ε-regular pair by ‘scaling’ the density of the pair by d which, typically, is the density of the graph G = (V , E). Hence, we say that a pair (A, B) of disjoint subsets of vertices of a graph G = (V , E) is (d, ε)-regular, if for each pair of subsets A ⊆ A, |A | ≥ ε|A|, B ⊆ B, |B | ≥ ε|B|, we have e(A , B ) e(A, B) ≤ dε. − (5) |A ||B | |A||B| If dG = |E|/ |V2 | we call a (dG , ε)-regular pair strongly ε-regular. A strongly (k, ε)regular partition of vertices of G is defined in a similar way as (k, ε)-regular |V | partition. Moreover, we say that a graph G = (V , E) with density dG = |E|/ 2 is (η, b)bounded if each subgraph H of G with r ≥ η|V | vertices contains not more than br 2 edges. As was observed independently by Kohayakawa and by Rödl (see Kohayakawa and Rödl [11] and references therein), one can mimic the proof of Szemerédi’s Regularity Lemma to get the following result. Lemma 5.1. For every ε > 0 and b there exist η and K such that each (η, b)-bounded graph G with more than 1/ε vertices admits a strongly (k, ε)-regular partition for some k, 1/ε < k < K. The assumption that G is (η, b)-bounded is typically not very restrictive. For instance, if η > 0 and np → ∞ as n → ∞, then the random graph G(n, p) is a.a.s. (η, 2)-bounded. Consequently, a.a.s. each subgraph of such G(n, p) which contains at least half of its edges is (η, 4)-bounded. A more serious problem is that, unlike in the dense case, from the fact that a sparse pair is strongly ε-regular it does not follow that the number of cycles of length four

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in that pair is close to the number of cycles of length four in the random bipartite graph of the same density. In a similar way, for every ε > 0 there exists δ > 0 and a tripartite graph G with vertex set V1 ∪ V2 ∪ V3 , |V1 | = |V2 | = |V3 | = n such that all three pairs (V1 , V2 ), (V2 , V3 ), and (V1 , V3 ) are strongly ε-regular pairs with densities larger than δ, yet G is triangle-free. Nevertheless, Kohayakawa, Łuczak, and Rödl conjectured in [10] that such triangle-free tripartite graphs consisting of dense ε-regular triples are so rare that a.a.s. the random graph G(n, p) contains none of them as a subgraph. In order to state the conjecture rigorously we need one more definition. Let G(n, p; ε, s) be a graph chosen at random from the family of all s-partite graphs with vertex set V1 ∪ V2 ∪ · · · ∪ Vs , |V1 | = |V2 | = · · · = |Vs | = n, such that for each i, j , 1 ≤ i < j ≤ s, the pair (Vi , Vj ) spans a bipartite strongly ε-regular graph with pn2 edges. Then the conjecture of Kohayakawa, Łuczak and Rödl for complete graphs goes as follows (for a more general statement see [10]). s Conjecture 5.2. For every s and δ > 0 there exist ε > 0 and C such that if ns p(2) > Cn2 p, then the probability that G(n, p; ε, s) contains no copies of Ks is smaller 2 than δ n p .

A stronger ‘counting’ version of Conjecture 5.2 goes as follows. s Conjecture 5.3. For every s and δ > 0 there exist ε > 0 and C such that if ns p(2) > s Cn2 p, then the probability that G(n, p; ε, s) contains fewer than ns p(2) /2 copies 2 of Ks is smaller than δ n p .

So far Conjectures 5.2 and 5.3 have been shown only for s = 3, 4, 5 (see Gerke et al. [4] and the references therein). Let us observe that Theorem 3.1 follows immediately from the fact √ that Conjecture 5.2 holds for s = 3. Indeed, let us fix δ > 0 and let p = C/ n, where C is a large constant. Take a subgraph H of G(n, p) with at least (1/2 + δ) n2 p edges. Choose ε > 0 much smaller than δ and apply Lemma 5.1 to H to find in it a strong (k, ε)-partition with 1/ε < k < K (as we have already pointed out for every η > 0, a.a.s. G(n, p) is (2, η)-bounded and so H is (4, η)-bounded and fulfills assumptions of the lemma). Since H contains more than half of the edges of G(n, p), and edges in G(n, p) are uniformly distributed around the graph, there exist three sets V , V , V of the partition such that each of the pairs (V , V ), (V , V ), (V , V ), is strongly ε-regular and has density at least δp/10. (Let us remark that now a vague notion of an ‘essentially non-bipartite’ subgraph H we have used in Section 3 can be made precise: a graph H is essentially non-bipartite if it contains a balanced tripartite graph on (n) vertices which consists of three dense strongly ε-regular pairs.) Now, one can use Conjecture 5.2 and argue as in (3) that a.a.s. each tripartite subgraph of G(n, p) of such a type contains a triangle. Thus, H contains a triangle and Theorem 3.1 follows. √ Finally, let us also note that if, say, p = log n/ n, then elementary calculations similar to that used by Erd˝os (cf. (3)) reveal that for every fixed a.a.s. the number


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of tight (3, t)-circuits in G(n, p) with t ≤ is o(n2 p). Thus, one can obtain a graph G(2) (3, , δ) with all the properties specified in Conjecture 2.2 by deleting from G(n, p) all edges which belong to tight (3, t)-circuits, t ≤ .

6. Final remarks It is easy to see that, arguing as in the proof of Theorem 3.1 above, one can show the existence of a graph G(2) (s, , δ) (see Conjecture 2.2) for every s for which Conjecture 5.2 holds. A precise formulation of analogs of Conjectures 5.2 and 5.3 for hypergraphs would become very technical, thus we only mention that if appropriately stated hypergraph version of Conjecture 5.3 is true then the following straightforward ‘probabilistic’ generalization of the Removal Lemma holds. Conjecture 6.1. For s > r ≥ 2 and every δ > 0 there exist η > 0 such that a.a.s. in each subhypergraph H of the random r-uniform hypergraph G(r) (n, p) which contains s (r) fewer than ηns p(r ) copies of Ks one can destroy all these copies by removing fewer r than δn p hyperedges of H . An analogous question on the validity of a probabilistic version of Szemerédi’s density theorem can be stated as follows. Conjecture 6.2. For every δ > 0 and k there exists η > 0 such that a.a.s. in each subset A of G(1) (n, p) with fewer than ηn2 pk non-trivial arithmetic progressions of length k all these progressions can be destroyed by removing fewer than δnp elements from A. Conjecture 6.1 states that a.a.s. a random hypergraphs G(r) (n, p) has a property A s such that if a hypergraph G has A each subgraph H of G with fewer than ηns p(r ) (r) (r) copies of Ks can be made Ks -free by deleting fewer than δnr p hyperedges. One can ask if A follows from some simple property A , i.e., whether there is a compact characterization of ‘pseudorandom’ sparse hypergraphs. A natural candidate for A is the property that the number of some special subhypergraphs in G is close to the expected value of the number of such subhypergraphs in the random hypergraph with the same density. In the case of graphs a good choice for ‘probing’ graphs seem to be cycles of length four. It is known (see Thomasson [20]) that if the number of cycles of length four in a graph G is close to the anticipated one, then edges in G are ‘uniformly distributed’ around G. Nonetheless we do not know if the ‘correct’ number of cycles of length four, possibly matched with some additional requirements ensuring that G is locally sparse, can guarantee that G has good ‘extremal’ properties like those described in Conjecture 6.1. Another challenging problem is to strengthen the definition of a strongly ε-regular pair to, say, a ‘super ε-regular pair’ such that the analog of Lemma 5.1 remains valid in this setting (i.e., each dense subgraph of a random-like graph G admits a ‘super (k, ε)-partition’) and furthermore, each

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tripartite graph which consists of three dense super ε-regular pairs contains a triangle. Similar questions can be asked for hypergraphs, as well as for the subsets of [n] (or, in somewhat more natural setting, for subsets of Zn , where n is a prime).

References [1] Erd˝os, P. Graph theory and probability. Canad. J. Math. 11 (1959), 34–38. [2] Frankl, P., and Rödl, V., Large triangle-free subgraphs in graphs without K4 . Graphs Combin. 2 (1986), 135–144. [3] Frankl, P., and Rödl, V., Extremal problems on set systems. Random Structures Algorithms 20 (2002), 131–164. [4] Gerke, S., Schickinger, T., Steger, A., K5 -free subgraphs of random graphs. Random Structures Algorithms 24 (2004), 194–232. [5] Gowers, W. T., Lower bounds of tower type for Szemerédi’s uniformity lemma. Geom. Funct. Anal. 7 (1997), 322–337. [6] Gowers, W. T., Quasirandomness, counting and regularity for 3-uniform hypergraphs. Combin. Probab. Comput. 15 (2006), 143–184. [7] Haxell, P. E., Kohayakawa, Y., Łuczak, T., Turán’s extremal problem graphs: forbidding odd cycles. Combinatorica 16 (1996), 107–122. [8] Janson, S., Łuczak, T., Ruciński, A., Random Graphs. Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York 2000. [9] Kohayakawa,Y., Łuczak, T., and V. Rödl, Arithmetic progressions of length three in subsets of a random set. Acta Arith. 75 (1996), 133–163. [10] Kohayakawa, Y., Łuczak, T., and Rödl, V., On K4 -free subgraphs of random graphs. Combinatorica 17 (1997), 173–213. [11] Kohayakawa, Y., and Rödl, V., Regular pairs in sparse random graphs. Random Structures Algorithms 22 (2003), 359–434. [12] Lovasz, L., Szegedy, B., Szemerédi’s Lemma for the analyst. Preprint. [13] Nagle, B., Rödl, V., Schacht, M., The counting lemma for regular k-uniform hypergraphs. Random Structures Algorithms 28 (2006), 113–179. [14] Rödl, V., Nagle, B., Skokan, J., Schacht, M., Kohayakawa, Y., The hypergraph regularity method and its applications. Proc. Nat. Acad. Sci. USA 102 (2005) 8109–8113. [15] Rödl, V., Skokan, J., Regularity lemma for k-uniform hypergraphs. Random Structures Algorithms 25 (2004), 1–42. [16] Ruzsa, I., Szemerédi, E., Triple systems with no six points carrying three triangles. In Combinatorics (Proc. Fifth Hungarian Colloq. Keszthely, 1976), Vol. II, Colloq. Math. Soc. János Bolyai 18, North-Holland, Amsterdam, New York 1978, 939–945. [17] Szemerédi, E., On sets of integers containing no k elements in arithmetic progression. Acta Arith. 28 (1975), 299–345. [18] Szemerédi, E., Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes (ed. by J. Bermond et al.), Colloq. Internat. CNRS 260, CNRS, Paris 1978, 399–401.


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[19] Tao, T., An information-theoretic proof of Szemerédi’s regularity lemma. Preprint [20] Thomason, A., Pseudorandom graphs. In Random graphs ’85 (Poznań, 1985), NorthHolland Math. Stud. 144, North-Holland, Amsterdam 1987, 307–331. [21] Turán, P., Egy gráfelméleti szélsöérték feladotról. Mat. Fiz. Lapok 48 (1941), 436–452; see also On the theory of graphs. Colloq. Math. 3 (1954), 19–30. Adam Mickiewicz University, Faculty of Mathematics and CS, 61-614 Poznań, Poland E-mail: [email protected]

Additive combinatorics and geometry of numbers Imre Z. Ruzsa ∗

Abstract. We meditate on the following questions. What are the best analogs of measure and dimension for discrete sets? How should a discrete analogue of the Brunn–Minkowski inequality look like? And back to the continuous case, are we happy with the usual concepts of measure and dimension for studying the addition of sets? Mathematics Subject Classification (2000). Primary 11B75; Secondary 05-99, 11H06, 52C07.

Keywords. Sumsets, additive combinatorics, lattice points, volume.

1. Introduction “Additive combinatorics” is a name coined by (I think) Tao and Van for the title of their book in preparation to denote the study of additive properties of general sets – mainly of integers, but also in other structures. Works on this topics are generally classified as additive or combinatorial number theory. The first result that connects additive properties to geometrical position is perhaps the following theorem of Freiman. Theorem 1.1 (Freiman [3], Lemma 1.14). Let A ⊂ Rd be a finite set, |A| = m. Assume that A is proper d-dimensional, that is, it is not contained in any affine hyperplane. Then d(d + 1) |A + A| ≥ (d + 1)m − . 2 This theorem is exact, equality can occur, namely it holds when A is a “long simplex”, a set of the form Ldm = {0, e1 , 2e1 , . . . , (m − d)e1 , e2 , e3 , . . . , ed }.

(1.1)

In particular, if no assumption is made on the dimension, then the minimal possible cardinality of the sumset is 2m − 1, with equality for arithmetic progressions. This result can be extended to sums of different sets. This extension is problematic from the beginning, namely the assumption “d-dimensional” can be interpreted in ∗ Supported by Hungarian National Foundation for Scientific Research (OTKA), Grants No. T 38396, T 43623, T 42750.


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different ways. We can stipulate that both sets be d-dimensional, or only one, or, in the weakest form, make this assumption on the sumset only. An immediate extension of Freiman’s above result goes as follows. Theorem 1.2 ([11], Corollary 1.1). If A, B ⊂ Rd , |A| ≤ |B| and dim(A + B) = d, then we have d(d + 1) |A + B| ≥ |B| + d|A| − . 2 We can compare these results to the continuous case. Let A, B be Borel sets in Rd ; μ will denote the Lebesgue measure. The celebrated Brunn–Minkowski inequality asserts that (1.2) μ(A + B)1/d ≥ μ(A)1/d + μ(B)1/d , and here equality holds if A and B are homothetic convex sets, and under mild and natural assumptions this is the only case of equality. It can also be observed that the case A = B is completely obvious here: we have μ(A + A) ≥ μ(2 · A) = 2d μ(A). Also the constant 2d is much larger than the constant d + 1 in Theorem 1.1. This is necessary, as there are examples of equality, however, one feels that this is an exceptional phenomenon and better estimations should hold for “typical” sets. A further difference is the asymmetrical nature of the discrete result and the symmetry of the continuous one. Finally, when |A| is fixed, Theorem 1.2 gives a linear increment, while (1.2) yields μ(A + B) ≥ μ(B) + dμ(A)1/d μ(B)1−1/d . In the next section we tell what can be said if we use cardinality as the discrete analog of measure, and prescribe only the dimension of the sets. Later we try to find other spatial properties that may be used to study sumsets. We meditate on the following questions (without being able to even conjecture a definitive answer). What are the best analogs of measure and dimension for discrete sets? How should a discrete analogue of the Brunn–Minkowski inequality look like? The partial answers also suggest questions in the continuous case. Should we be satisfied with the usual concepts of measure and dimension for studying the addition of sets? Most of the paper is a survey, however, there are some new results in Sections 4 and 6. We end the introduction by fixing some notations, which were tacitly used above. For two sets A, B (in any structure with an operation called addition) by their sum we mean the set A + B = {a + b : a ∈ A, b ∈ B}. We use A − B similarly. For repeated addition we write kA = A + · · · + A

(k times),

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in contrast to k · A = {ka : k ∈ A}. Mostly our sets will be in an Euclidean space Rd , and e1 , . . . , ed will be the system of unit vectors. We define initially the dimension dim A of a set A ⊂ Rd as the dimension of the smallest affine hyperplane containing A. (This definition will be modified in Section 3).

2. Results using cardinality and dimension We consider finite sets in an Euclidean space Rd . Put Fd (m, n) = min{|A + B| : |A| = m, |B| = n, dim(A + B) = d}, Fd (m, n) = min{|A + B| : |A| = m, |B| = n, dim B = d}, Fd (m, n) = min{|A + B| : |A| = m, |B| = n, dim A = dim B = d}. Fd is defined for m+n ≥ d +2, Fd for n ≥ d +1 and Fd for m ≥ d +1, n ≥ d +1. Fd and Fd are obviously symmetric, while Fd may not be (and, in fact, we will see that for certain values of m, n it is not), and they are connected by the obvious inequalities Fd (m, n) ≤ Fd (m, n) ≤ Fd (m, n). I determined the behaviour of Fd and of Fd for m ≤ n. The more difficult problem of describing Fd and Fd for m > n was solved by Gardner and Gronchi [4]; we shall quote their results later. To describe Fd define another function Gd as follows: Gd (m, n) = n +

m−1

min(d, n − j ),

n≥m≥1

j =1

and for m > n extend it symmetrically, putting Gd (m, n) = Gd (n, m). In other words, if n − m ≥ d, then we have Gd (m, n) = n + d(m − 1). If 0 ≤ t = n − m < d, then for n > d we have Gd (m, n) = n + d(m − 1) −

(d − t)(d − t − 1) d(d + 1) t (t + 1) = n(d + 1) − − , 2 2 2

and for n ≤ d

(m − 1)(2n − m) . 2 With this notation we have the following result. Gd (m, n) = n +

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Theorem 2.1 ([11], Theorem 1). For all positive integers m, n and d satisfying m + n ≥ d + 2 we have Fd (m, n) ≥ Gd (m, n). Theorem 1.2 is an immediate consequence. Theorem 2.1 is typically exact; the next theorem summarizes the cases when we have examples of equality. Theorem 2.2 ([11], Theorem 2). Assume 1 ≤ m ≤ n. We have Fd (m, n) = Fd (m, n) = Gd (m, n) unless either n < d + 1 or m ≤ n − m ≤ d (in this case n ≤ 2d). The construction goes as follows. Assume 1 ≤ m ≤ n, n ≥ d + 1. Let B be a long simplex, B = Ldn as defined in (1.1). If n − m ≥ d, we put A = {0e1 , 1e1 , . . . , (m − 1)e1 }. This set satisfies |A| = m. The set A + B consists of the vectors ie1 , 0 ≤ i ≤ n + m − d − 1 and the vectors ie1 + ej , 0 ≤ i ≤ m − 1, 2 ≤ j ≤ d, consequently |A + B| = n + d(m − 1) = Gd (m, n). If n − m = t < d, write t = d − k and assume k ≤ m. Now A is defined by A = {0e1 , 1e1 , . . . , (m − k)e1 } ∪ {e2 , . . . , ek }. This set satisfies |A| = m. The set A+B consists of the vectors ie1 , 0 ≤ i ≤ 2(n−d), the vectors ie1 + ej , 0 ≤ i ≤ n − d, 2 ≤ j ≤ d, finally ei + ej , 2 ≤ i, j ≤ k, hence |A + B| = 2(n − d) + 1 + (d − 1)(n − d + 1) + = n(d + 1) −

k(k − 1) 2

d(d + 1) t (t + 1) − = Gd (m, n). 2 2

These constructions cover all pairs m, n except those listed in Theorem 2.2. Observe that A is also a long simplex of lower dimension. For a few small values the exact bounds are yet to be determined. We now describe Gardner and Gronchi’s [4] bound for Fd (m, n). Informally their main result (Theorem 5.1) asserts that the |A + B| is minimalized when B = Ldn , a long simplex, and A is as near to the set of points inside a homothetic simplex as possible. More exactly the define (for a fixed value of n) the weight of a point x = (x1 , . . . , xd ) as x1 w(x) = + x2 + · · · + xd . n−d

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This defines an ordering by writing x < y if either w(x) < w(y) or w(x) = w(y) and for some j we have xj > yj and xi = yi for i < j . Let Ddmn be the collection of the first m vectors with nonnegative integer coordinates in this ordering. We have Ddnn = Ldn = B, and, more generally, Ddmn = rB for any integer m such that r +d −1 r +d −1 m = |rB| = (n − d) + . d d −1 For such values of m we also have

r +d r +d |A + B| = |(r + 1)B| = (n − d) + . d d −1

With this notation their result sounds as follows. Theorem 2.3 (Gardner and Gronchi [4], Theorem 5.1). If A, B ⊂ Rd , |A| = m, |B| = n and dim B = d, then we have |A + B| ≥ |Ddmn + Ldn |. For m < n this reproves Theorem 2.2. For m ≥ n the extremal set Ddmn is also d-dimensional, thus this result also gives the value of Fd . Corollary 2.4. For m ≥ n > d we have Fd (m, n) = Fd (m, n) = |Ddmn + Ldn |. A formula for the value of this function is given in [4], Section 6. We quote some interesting consequences. Theorem 2.5 (Gardner and Gronchi [4], Theorem 6.5). If A, B ⊂ Rd , |A| = m ≥ |B| = n and dim B = d, then we have |A + B| ≥ m + (d − 1)n + (n − d)1−1/d (m − d)1/d −

d(d − 1) . 2

Theorem 2.6 (Gardner and Gronchi [4], Theorem 6.6). If A, B ⊂ Rd , |A| = m, |B| = n and dim B = d, then we have |A + B|

1/n

≥m

1/d

+

n−d d!

1/d .

This result is as close to the Brunn–Minkowski inequality as we can get by using only the cardinality of the summands.

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3. The impact function and the hull volume While we will focus our attention to sets in Euclidean spaces, some definitions and results can be formulated more clearly in a more general setting. So let now G be a commutative group. For a fixed finite set B ⊂ G we define its impact function by ξB (m) = ξB (m, G) = min{|A + B| : A ⊂ G, |A| = m}. This is defined for all positive integers if G is infinite, and for m ≤ |G| if G is finite. This function embodies what can be told about cardinality of sumsets if one of the set is unrestricted up to cardinality. The name is a translation of Plünnecke’s “Wirkungsfunktion”, who first studied this concept systematically for density [9]. We will be interested mainly in the infinite case, and in this case the dependence on G can be omitted. Lemma 3.1. Let G, G be infinite commutative groups, G ⊂ G, and let B ⊂ G be a finite set. We have (3.1) ξB (m, G) = ξB (m, G ) for all m. Proof. Take an A ⊂ G, |A| = m with |A + B| = ξB (m, G). Let A = A1 ∪ · · · ∪ Ak be its decomposition according to cosets of G . For each 1 ≤ i ≤ k take an element xi from the coset containing Ai so that the sets Ai − xi are pairwise disjoint; this is easily done as long as G is infinite. The set (Ai − xi ) A = satisfies A ⊂ G , |A | = n and |Ai − xi + B| = |Ai + B| = |A + B| = ξB (m, G), |A + B| ≤ hence ξB (m, G ) ≤ ξB (m, G). The inequality in the other direction is obvious.

2

In the case of finite groups the connection between ξB (m, G) and ξB (m, G ) can also be described by arguments like in chapters 3 and 4 of Plünnecke’s above mentioned book [9]. We restrict our attention to infinite groups, and henceforth omit the reference to G and write just ξB (m) instead. Let G be a torsionfree group. Take a finite B ⊂ G, and let G be the subgroup generated by B − B, that is, the smallest subgroup such that B is contained in a single coset. Let B = B − a with some a ∈ B, so that B ⊂ G . The group G , as any finitely generated torsionfree group, is isomorphic to the additive group Zd for some d. Let ϕ : G → Zd be such an isomorphism and B = ϕ(B ). By Lemma 3.1 we have ξB = ξB = ξB ,


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so when studying the impact function we can restrict our attention to sets in Zd that contain the origin and generate the whole lattice; we then study the set “in its natural habitat”. Definition 3.2. Let B be a finite set in a torsionfree group G. By the dimension of B we mean the number d defined above, and denote it by dim B. By the hull volume of B we mean the volume of the convex hull of the set B described above and denote it by hv B. The set B is determined up to an automorphism of Zd . These automorphisms are exactly linear maps of determinant ±1, hence the hull volume is uniquely defined. Observe that this dimension is not the same as the dimension described in the Introduction; in the case when B ⊂ Rk with some k, this is its dimension over the field of rationals. Theorem 3.3. Let B be a finite set in a torsionfree group G, d = dim B, v = hv B. We have lim |kB|k −d = v. A proof can be found in [12], Section 11, though this form is not explicitly stated there. An outline is as follows. By using the arguments above we may assume that B ⊂ Zd , 0 ∈ B and B generates Zd . Let B ∗ be the convex hull of B. Then kB is contained in k · B ∗ . The number of lattice points in k · B is asymptotically μ(k ·B ∗ ) = k d v; this yields an upper estimate. To get a lower estimate one proves that with some constant p, kB contains all the lattice points inside translate of (k −p)·B ∗ ; this is Lemma 11.2 of [12]. This means that the hull volume can be defined without any reference to convexity and measure, and this definition can even be extended to commutative semigroups. This follows from the following result of Khovanskii [5], [6]; for a simple proof see [8]. Theorem 3.4 (Khovanskii). Let B be a finite set in a commutative semigroup. There is a k0 , depending on the set B, such that |kB| is a polynomial function of k for k > k0 . Definition 3.5. Let B be a finite set in a commutative semigroup, and let vk d be the leading term of the polynomial which coincides with |kB| for large k. By the dimension of B we mean the degree d of this polynomial, and by the hull volume we mean the leading coefficient v. It turns out that in Zd , hence in any torsionfree group, the dimension and hull volume determine the asymptotic behaviour of the impact function. Theorem 3.6. Let B be a finite set in a torsionfree commutative group G, d = dim B, v = hv B. We have lim(ξB (m)1/d − m1/d ) = v 1/d .

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This is the main result (Theorem 3.1) of [12]. In the same paper I announce the same result for non necessarily torsionfree commutative groups without proof (Theorem 3.4). In a general semigroup A + B may consist of a single element, so an attempt to an immediate generalization fails. Problem 3.7. Does the limit lim ξB (m)1/d − m1/d exist in general commutative semigroups? Is there a condition weaker than cancellativity to guarantee its positivity? Theorem 3.6 can be effectivized as follows (Theorems 3.2 and 3.3 of [12]). Theorem 3.8. With the notations of the previous theorem, if d ≥ 2 and m ≥ v, we have ξB (m) ≤ m + dv 1/d m1−1/d + c1 v 2/d m1−2/d , ξB (m)1/d − m1/d ≤ v 1/d + c2 v 2/d m−1/d . (c1 , c2 depend on d.) With n = |B| for large m we have ξB (m) ≥ m + dv 1/d m1−1/d − c3 v ξB (m)1/d − m1/d ≥ v 1/d − c4 v

d+3 2d

d+3 2d

3

n−1/2 m1− 2d ,

n−1/2 m−1/(2d) .

Probably the real error terms are much smaller than these estimates. For d = 1 we have the obvious inequality ξB (m) ≤ m + v, with equality for large√m because√the integers ξB (m)−m √ cannot converge to v otherwise. For d = 2 already ξB (m)− m can converge to v from both directions. Theorem 3.9. The impact function of the set B = {0, e1 , e2 } ⊂ Z2 satisfies √ √ ξB (m) − m > v

(3.2)

for all m. The impact function of the set B = {0, e1 , e2 , −(e1 + e2 )} ⊂ Z2 satisfies √ √ ξB (m) − m < v

(3.3)

for infinitely many m. Inequality (3.2) was announced in [12] without proof as Theorem 4.1, and it is a special case of Gardner and Gronchi’s Theorem 2.6. Inequality 3.3 is Theorem 4.3 of [12]. √ √ √ I cannot decide whether there is a set such that ξB (m) − m < v for all m.

4. The impact volume Besides cardinality we saw the hull volume as a contender for the title “discrete volume”. For both we had something resembling the Brunn–Minkowski inequality;


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for cardinality we had Gardner and Gronchi’s Theorem 2.6, which has the (necessary) factor d!, and for the hull volume we have Theorem 3.6, which only holds asymptotically. There is an easy way to find a quantity for which the analogue of the Brunn– Minkowski inequality hods exactly: we can make it a definition. Definition 4.1. The d-dimensional impact volume of a set B (in an arbitrarily commutative group) is the quantity d ivd (B) = inf ξB (m)1/d − m1/d . m∈N

Note that the d above may differ from the dimension of B, in fact, it need not be an integer. It seems, however, that the only really interesting case is d = dim B. The following statement list some immediate consequences of this definition. Statement 4.2. Let B be a finite set in a commutative torsionfree group. (a) ivd (B) is a decreasing function of d. (b) If |B| = n, then

iv1 (B) = n − 1

and

d ivd (B) ≤ n1/d − 1

(4.1)

for every d. (c) ivd (B) = 0 for d > dim B. (d) For every pair A, B of finite sets in the same group and every d we have ivd (A + B)1/d ≥ ivd (A)1/d + ivd (B)1/d .

(4.2)

The price we have to pay for the discrete Brunn–Minkowski inequality (4.2) is that there is no easy way to compute the impact volume for a general set. We have the following estimates. Theorem 4.3. Let B be a finite set in a commutative torsionfree group, dim B = d, |B| = n. We have n−d ≤ ivd (B) ≤ hv B, (4.3) d! with equality in both places if B is a long simplex. The first inequality follows form Theorem 2.6 of Gardner and Gronchi, the second from Theorem 3.6. Problem 4.4. What is the maximal possible value of ivd (B) for n-element d-dimensional sets? Is perhaps the bound in (4.1) exact?

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We now describe the impact volume for another important class of sets, namely cubes. Theorem 4.5. Let n1 , . . . , nd be positive integers and let B = {(x1 , . . . , xd ) ∈ Zd : 0 ≤ xi ≤ ni }.

(4.4)

We have ivd (B) = hv B = v = n1 . . . nd . Problem 4.6. Is is true that when B is the set of lattice points within a convex lattice polytope, then hv B and ivd (B) are very near? They may differ, as the second example in Theorem 3.9 shows. We shall deduce Theorem 4.5 from the following one. Theorem 4.7. Let G = G1 × G2 be a commutative group represented as the direct product of the groups G1 and G2 . Let B = B1 ×B2 ⊂ G be a finite set with B1 ⊂ G1 , B2 ⊂ G2 . We have (4.5) ivd (B) ≥ ivd−1 (B1 )iv1 (B2 ). Proof. Write ivd (B) = v, ivd−1 (B1 ) = v1 , iv1 (B2 ) = v2 (which is = |B| − 1 if G2 is torsionfree). We want to estimate |A + B| from below for a general set A ⊂ G with |A| = m. Firs we transform them to some standard form; this will be the procedure what Gardner and Gronchi call compression. Let A1 be the projection of A to G1 , and for an x ∈ A1 write A(x) = {y ∈ G2 : (x, y) ∈ A}. Let and

A = {(x, i) : x ∈ A1 , i ∈ Z, 0 ≤ i ≤ |A(x)| − 1} B = {(x, i) : x ∈ B1 , i ∈ Z, 0 ≤ i ≤ v2 }.

We have A , B ⊂ G = G1 × Z. Lemma 4.8. We have |A | = |A|,

|A + B | ≤ |A + B|.

(4.6)

Proof. The equality is clear. To prove the inequality, write S = A + B, S = A + B . With the obvious notation, we will show that |S (x)| ≤ |S(x)| for each x. To this end observe that (A(x ) + B(x )) = S(x) = x +x =x

x ∈x−B1

A(x ) + B2 ,

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hence

|S(x)| ≥ max |A(x ) + B2 | ≥ max |A(x )| + v2 . x ∈x−B1

Similarly S (x) =

x ∈x−B1

(A (x ) + B (x )) =

x +x =x

[0, |A(x )| + v2 − 1],

x ∈x−B1

and so

|S (x)| = max |A(x )| + v2 .

2

x ∈x−B1

Now we continue the proof of the theorem. Decompose A into layers according to the value of the second component; write A =

k

Li × {i},

i=0

where k = max |A(x)|, Li ⊂ G1 . Write |Li | = mi . We have L0 ⊃ L1 ⊃ · · · ⊃ Lk , consequently m0 ≥ m1 ≥ · · · ≥ mk . The set S is the union of the sets (Li + B1 ) × {i + j }, 0 ≤ i ≤ v2 . By the above inclusion it is sufficient to consider the Li with the smallest possible i, that is, S = (L0 + B1 ) × {0, 1, . . . , v2 } ∪

k

(Li + B1 ) × {i + v2 }.

i=1

We obtain that |S | = v2 |L0 + B1 | +

k

|Li + B1 |.

(4.7)

i=0

To estimate the summands we use the d − 1-dimensional impact of B1 , and we get

d−1

d−1 1 1 mi ≥ m0d−1 + v1d−1 ; |Li + B1 | ≥ mi + v1 m0 the second inequality follows from mi ≤ m0 . By substituting this into (4.7) and recalling that mi = m we obtain 1 d−1

1 d−1

d−1 1 1 m m0d−1 + v1d−1 . (4.8) |S| ≥ v2 + m0 Consider the right side as a function of the real variable m0 . By differentiating we find that it assumes its minimum at 1/d

m0 = v1 (m/v2 )1−1/d . (This minimum typically is not attained; this m0 may be < 1 or > m, and it is generally not integer). Substituting this value of m0 into (4.8) we obtain the desired bound d 2 |S| ≥ m1/d + (v1 v2 )1/d .

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Problem 4.9. Does equality always hold in Theorem 4.7? I expect a negative answer. Problem 4.10. Can Theorem 4.7 be extended to an inequality of the form ivd1 +d2 (B1 × B2 ) ≥ ivd1 (B1 )ivd2 (B2 )? Proof of Theorem 4.5. To prove ≥ we use induction on d. The case d = 1 is obvious, and Theorem 4.7 provides the inductive step. This means that with the cube B defined in (4.4) we have d |A + B| ≥ |A|1/d + v 1/d . Equality can occur for infinitely many values of |A|, namely it holds whenever A is also a cube of the form A = {(x1 , . . . , xd ) ∈ Zd : 0 ≤ xi ≤ kni − 1} with some integer k; we have |A| = k d v, |A + B| = (k + 1)d v. It may be difficult to describe ξB (m) for values of m which are not of the form k d v. Possibly an argument like Gardner and Gronchi’s for the simplex may work. Observe that these special sets A are not homothetic to B; in particular, A = B may not yield a case of equality. 2 As Theorem 4.3 shows, the impact volume can be d! times smaller than cardinality. The example we have of this phenomenon, the long simplex, is, however, “barely” d-dimensional, and I expect that a better estimates hold for a “substantially” d-dimensional set. Definition 4.11. The thickness ϑ(B) of a set B ⊂ Rd is the smallest integer k d d with the

k property that there is a hyperplane P of R and x1 , . . . , xk ∈ R such that B ⊂ i=1 P + xj . Conjecture 4.12. For every ε > 0 and d there is a k such that for every B ⊂ Rd with ϑ(B) > k we have ivd (B) > (1 − ε)|B|. This conjecture would yield a discrete Brunn–Minkowski inequality of the form |A + B|1/d ≥ |A|1/d + (1 − ε)|B|1/d assuming a bound on the thickness of B. Such an inequality is true at least in the special case A = B. This can be deduced from a result of Freiman ( [3], Lemma 2.12; see also Bilu [1]), which sounds as follows. If A ⊂ Rd and |2A| < (2d − ε)|A|, then there is a hyperplane P such that |P ∩ A| > δ|A|, with δ = δ(d, ε) > 0.


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5. Meditation on the continuous case Let A, B be Borel sets in Rd . The Brunn–Minkowski inequality (1.2) estimates μ(A + B) in a natural way, with equality if A and B are homothetic convex sets. Like in the discrete case, we can define the impact function of the set B by ξB (a) = inf{μ(A + B) : μ(A) = a}. Thus (1.2) is equivalent to d ξB (a) ≥ a 1/d + μ(B)1/d , and this is the best possible estimate in terms of μ(B) only. To measure the degree of nonconvexity we propose to use the measure of the convex hull beside the measure of the set. This is analogous to the hull volume, and it is sufficient to describe the asymptotic behaviour of ξ . Theorem 5.1 ([13], Theorem 1.). For every bounded Borel set B ⊂ Rd of positive measure we have lim ξB (a)1/d − a 1/d = μ(conv B)1/d . a→∞

This is the continuous analogue of Theorem 3.6, and there is an analogue to the effective version Theorem 3.8 as well. Note that by considering sets homothetic to conv B we immediately obtain ξB (a)1/d ≤ a 1/d + μ(conv B)1/d , thus we need only to give a lower estimate. This is as follows. Theorem 5.2 ([13], Theorem 2.). Let μ(B) = b, μ(conv B) = v. We have ξB (a)1/d ≥ a 1/d + v 1/d 1 − c(v/b)1/2 (v/a)1/(2d) ξB (a) ≥ a + dv 1/d a 1−1/d 1 − c(v/b)1/2 (v/a)1/(2d) with a suitable positive constant c depending on d. If v > b, we get a nontrivial improvement over the Brunn–Minkowski inequality for a > a0 (b, v). It would be desirable to find an improvement also for small values of a, or, even more, to find the best estimate in terms of μ(B) and μ(conv B). The exact bound and the structure of the extremal set may be complicated. This is already so in the case d = 1, which was solved in [10]. Observe that in one dimension μ(conv B) is the diameter of B. Theorem 5.3 ([10], Theorem 2). Let B ⊂ R, and write μ(B) = b, μ(conv B) = v. If v(v − b) b{v/b}(1 − {v/b}) a≥ + , (5.1) 2b 2

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Imre Z. Ruzsa

then ξB (a) = a + v. If (5.1) does not hold, then let k be the unique positive integer satisfying k(k − 1) a k(k + 1) ≤ < 2 b 2 and define δ by

a k(k − 1) = + δk. b 2

We have ξB (a) ≥ a + (k + δ)b, and equality holds if B = [0, b] ∪ {v}. A set A such that ξB (a) = μ(A + B) for the above set B is given by A = [0, (k − 1 + δ)b] ∪ [v, v + (k − 2 + δ)b] ∪ · · · ∪ [(k − 1)v, (k − 1)v + δb]. A less exact, but simple and still quite good lower bound sounds as follows. Corollary 5.4 ([10], Theorem 1). Let B ⊂ R, and write μ(B) = b, μ(conv B) = v. We have √ ξB (a) ≥ min a + v, ( a + b/2)2 . A comparison with the 2-dimensional Brunn–Minkowski inequality gives the following interpretation: initially a long one-dimensional set B tries to behave as if it were a two-dimensional set of area b/2. It can be observed that (5.4) is weaker than the obvious inequality μ(A + B) ≥ μ(A) + μ(B)

(5.2)

for small a. For small values of a Theorem 5.3 yields the following improvement of (5.2). Corollary 5.5 ([10], Corollary 3.1). If a ≤ b, then we have μ(A + B) ≥ min(2a + b, a + v). If b < a ≤ 3b, then we have

3 μ(A + B) ≥ min (a + b), a + v . 2 Problem 5.6. How large must μ(A+B) be if μ(A), μ(B), μ(conv A) and μ(conv B) are given? What are the minima of μ(A + A) and μ(A − A) for fixed μ(A) and μ(conv A)?

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The results above show that for d = 1 (like in the discrete case, but for less obvious reasons) the limit relation becomes an equality for a > a0 . Again, this is no longer the case for d = 2. An example of a set B ⊂ R2 such that ξB (a)1/2 < a 1/2 + v 1/2 will hold for certain arbitrarily large values of a is as follows. Let 0 < c < 1 and let B consist of the square [0, c] × [0, c] and the points (0, 1), (1, 0) and (1, 1). Hence b = c2 and v = 1. For an integer n ≥ 1 put An = [0, n] × [0, n] ∪

n

[j, j + c] × [n, n + c] ∪

j =0

n−1

[n, n + c] × [j, j + c].

j =0

Thus An consists of a square of side n and 2n + 1 small squares of side c, hence μ(An ) = n2 + (2n + 1)b. We can easily see that An + B = An+1 . Hence by considering the set A = An we see that for a number a of the form a = n2 + (2n + 1)b we have √ 2 a+1 . ξB (a) ≤ μ(An+1 ) = (n + 1)2 + (2n + 3)b < A more detailed calculation leads to ξB (a)1/2 ≤ a 1/2 + 1 − ca −1 (for these special values of a). If we tried to define an impact volume in the continuous case, we would recover the volume, at least for compact sets. Still, the above results and questions suggest that ordinary volume is not the best tool to understand additive properties. Perhaps one could try to modify the definition of impact volume by requiring μ(A) ≥ μ(B). So put d iv∗ (B) = inf ξB (a)1/d − a 1/d . a≥μ(B)

Problem 5.7. Find a lower estimate for iv∗ (B) in terms of μ(B) and μ(conv B).

6. Back to one dimension The results in the previous section, Theorem 5.3 and Corollaries 5.4 and 5.5 show that one can have nontrivial results in the seemingly uninteresting one-dimensional case. We now try to do the same, and will find bounds on |A + B| using the cardinality and

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Imre Z. Ruzsa

hull volume of B. Observe that in one dimension the hull volume is the smallest l such that B is contained in an arithmetic progression {b, b + q, . . . , b + lq}: the reduced diameter of B. It is possible to give bounds using nothing else than the hull volume. Theorem 6.1. Let B be a one-dimensional set in a torsionfree commutative group, hv B = v ≥ 3. (a) For m>

(v − 1)(v − 2) 2

we have ξB (m) = m + v. (b) If (k − 1)(k − 2) k(k − 1) <m≤ 2 2 with some integer 2 ≤ k < v, then ξB (m) ≥ m + k. Equality holds for the set B = {0, 1, v} ⊂ Z. For v ≤ 2 we have obviously ξB (m) = m + v for all m (such a set cannot be anything else than a v + 1-term arithmetic progression). This will be deduced from the following result, where the cardinality of B is also taken into account. Theorem 6.2. Let B be a one-dimensional set in a torsionfree commutative group, hv B = v ≥ 3, |B| = n. Define w by

w=

min

d|v, d≤n−2

d

n−2 . d

(6.1)

For every m we have

w m tw ξB (m) ≥ m + min v, + min + t∈ N 2 t 2

.

(6.2)

√ The minimum is attained either at the floor or at the ceiling of 2m/w. Unlike the previous theorem, typically we do not have examples of equality, and the extremal value and the structure of extremal sets is probably complicated. Also the value of w depends on divisibility properties of v and n. After the proof we give a less exact but simpler corollary. Proof. By Lemma 3.1 we may assume that B ⊂ Z, its smallest element is 0 and it generates Z; then its largest element is just v. Lemma 6.3. Let B be the set of residues of elements of B modulo v. For every nonempty X ⊂ Zv we have |X + B | ≥ min(|X| + w, v).

(6.3)

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Proof. By Kneser’s theorem we have |X + B | ≥ |X + H | + |B + H | − |H | with some subgroup H of the additive group Zv . Write |H | = d; clearly d|v. If d = v, we have |X + H | = v and we are ready. Assume d < v. B contains 0 and it generates Zv , hence it cannot be contained in H so we have |B + H | ≥ 2|H | = 2d. This gives the desired bound if d > n − 2. Assume d ≤ n − 2. Since |B + H | is a multiple of d and it is at least |B | = n − 1, we obtain

n−1 n−2 |B + H | ≥ d 2 =d 1+ ≥ d + w. d d We resume the proof of Theorem 6.2. Take a set A ⊂ Z, |A| = m. We are going to estimate |A + B| from below. For j ∈ Zv let u(j ) be the number of integers a ∈ A, a ≡ j (mod v) and let U (j ) be the corresponding number for the sumset A + B. We have U (j ) ≥ u(j ) + 1

(6.4)

whenever U (j ) > 0; this follows by adding the numbers 0, v to each element of A in this residue class if u(j ) > 0, and holds obviously for u(j ) = 0. We also have U (j ) ≥ u(j − b)

(6.5)

for every b ∈ B . Write r(k) = {j : u(j ) ≥ k}, R(k) = {j : U (j ) ≥ k}. Inequality (6.4) implies R(k) ⊃ r(k − 1)

(k ≥ 2),

(6.6)

R(k) ⊃ r(k) + B

(k ≥ 1).

(6.7)

and inequality (6.5) implies

First case. U (j ) > 0 for all j . In this case by summing (6.4) we get |A + B| = U (j ) ≥ v + u(j ) = |A| + v. Second case. There is a j with U (j ) = 0. Then we have |R(k)| < v for every k > 0. An application of Lemma 6.3 to the sets r(k) yields, in view of (6.7), |R(k)| ≥ |r(k)| + w

(6.8)

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as long as r(k) = ∅. Let t be the largest integer with r(t) = ∅. We have (6.8) for 1 ≤ k ≤ t, and (6.6) yields |R(k)| ≥ |r(k − 1)| (6.9) for all k ≥ 2. Consequently for 1 ≤ k ≤ t + 1 we have

k−1 k−1 |R(k)| ≥ (|r(k)| + w). |r(k − 1)| + 1 − t t

(6.10)

Indeed, for k = 1 (6.10) is identical with (6.8), for k = t + 1 it is identical with (6.9) and for 2 ≤ k ≤ t it is a linear combination of the two. By summing (6.10) we obtain |A + B| =

k≥1

|R(k)| ≥

t+1 k=1

t t +1 1 w+ 1+ |R(k)| ≥ |r(k)| 2 t

k=1

1 t +1 |A|, w+ 1+ = 2 t

2

as claimed in (6.2).

Corollary 6.4. With the assumptions and notations of Theorem 6.2 we have √ 2 (6.11) ξB (m) ≥ min m + v, m + w/2 . Proof. This follows from (6.2) and the inequality of arithmetic and geometric means. 2 This can be interpreted as that the set tries to imitate a two-dimensional set of area w/2. Proof of Theorem 6.1. Parts (a)–(b) of the theorem can be reformulated as follows: if ξB (n) ≤ m + k with some k < v, then m ≤ k(k − 1)/2. Theorem 6.2 yields (using only that w ≥ 1) the existence of a positive integer t such that m t +1 + ≤ k, t 2 hence

t (t + 1) . 2 The right side, as a function of t, is increasing up to k −1/2 and decreasing afterwards; the minimal values at integers are assumed at t = k − 1 and k, and both are equal to k(k − 1)/2. To show the case of equality in case (b), write m = k(k − 1)/2 − l with 0 ≤ l ≤ k − 2. The set A will contain the integers in the intervals [iv, iv + k − 3 − i] for 0 ≤ i ≤ l − 1 and [iv, iv + k − 2 − i] for l ≤ i ≤ k − 2. 2 m ≤ kt +


929

We illustrate the strength of Theorem 6.2 by deducing from it the two-dimensional estimate √ 2 ξL (m) > m + (n − 2)/2 for the long triangle L = L2n . Indeed, a suitable linear mapping maps this set L onto the set B = {0, 1, . . . , n − 2, v} with arbitrary v. If we choose v to be prime, then in √ √ 2 (6.1) we have w = n − 2, and if v is so large that m + l > m + w , then from Corollary 6.4 we obtain √ 2 m + w/2 . ξL (m) ≥ ξB (m) ≥ This is essentially the two-dimensional case of Theorem 2.6 of Gardner and Gronchi. On the other hand, for small values of m this inequality is weak, can even be worse than the obvious bound |A + B| ≥ |A| + |B| − 1. There are results that are especially suited to the study of small values; we quote two of them. In both let A, B ⊂ Z, A = {a1 , . . . , am }, B = {b1 , . . . , bn } with 0 = a1 < · · · < am = u, 0 = b1 < · · · < bn = v. Theorem 6.5 (Freiman [2]). If gcd(a1 , . . . , am , b1 , . . . , bn ) = 1 and u ≤ v, then |A + B| ≥ min(m + v, m + n + min(m, n) − 3). This bears a remarkable similarity to the two-dimensional case of Theorem 1.2 (and it can be deduced like Theorem 2.6) Theorem 6.6 (Lev and Smelianski [7]). If gcd(b1 , . . . , bn ) = 1 and u ≤ v, then |A + B| ≥ min(m + v, n + 2m − δ), where δ = 3 if u = v and δ = 2 if u < v. Observe that the above theorems cannot be directly compared to ours because of the somewhat different structure of the assumptions. Problem 6.7. Find a common generalization of Theorems 6.2 and 6.6.

References [1] Bilu, Y., Structure of sets with small sumset. Structure theory of set addition. Astérisque 258 (1999), 77–108. [2] Freiman, G., Inverse problems of additive number theory. VI. On the addition of finite sets. III. Izv. Vyss. Ucebn. Zaved. Matematika 3 (28) (1962), 151–157 (in Russian). [3] —, Foundations of a structural theory of set addition. Transl. Math. Monogr. 37, Amer. Math. Soc., Providence, RI, 1973. [4] Gardner, R. J., and Gronchi, P., A Brunn-Minkowski inequality for the integer lattice. Trans. Amer. Math. Soc. 353 (2001), 3995–4024.

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[5] Khovanskii, A. G., Newton polyhedron, Hilbert polynomial, and sums of finite sets. Funct. Anal. Appl. 26 (1992), 276–281. [6] —, Sums of finite sets, orbits of commutative semigroups, and hilbert functions. Funct. Anal. Appl. 29 (1995), 102–112. [7] Lev, V. F., and Smeliansky, P., On addition of two distinct sets of integers. Acta Arith. 70 (1995), 85–91. [8] Nathanson, M. B., and Ruzsa, I. Z., Polynomial growth of sumsets in abelian semigroups. J. Théor. Nombres Bordeaux 14 (2002), 553–560. [9] Plünnecke, H., Eigenschaften und Abschätzungen von Wirkungsfunktionen. Gesellschaft für Mathematik und Darenverarbeitung, Bonn, 1969. [10] Ruzsa, I. Z., Diameter of sets and measure of sumsets. Monatsh. Math. 112 (1991), 323–328. [11] —, Sum of sets in several dimensions. Combinatorica 14 (1994), 485–490. [12] —, Sets of sums and commutative graphs. Studia Sci. Math. Hungar. 30 (1–2) (1995), 127–148. [13] —, The Brunn-Minkowski inequality and nonconvex sets. Geom. Dedicata 67 (1997), 337–348. Alfréd Rényi Institute of Mathematics, 1364 Budapest, Pf. 127, Hungary E-mail: [email protected]

Geometric bistellar flips: the setting, the context and a construction Francisco Santos ∗

Abstract. We give a self-contained introduction to the theory of secondary polytopes and geometric bistellar flips in triangulations of polytopes and point sets, as well as a review of some of the known results and connections to algebraic geometry, topological combinatorics, and other areas. As a new result, we announce the construction of a point set in general position with a disconnected space of triangulations. This shows, for the first time, that the poset of strict polyhedral subdivisions of a point set is not always connected. Mathematics Subject Classification (2000). Primary 52B11; Secondary 52B20. Keywords. Triangulation, point configuration, bistellar flip, polyhedral subdivision, disconnected flip-graph.

Introduction Geometric bistellar flips are “elementary moves”, that is, minimal changes, between triangulations of a point set in affine space Rd . In their present form they were introduced around 1990 by Gel’fand, Kapranov and Zelevinskii during their study of discriminants and resultants for sparse polynomials [28], [29]. Not surprisingly, then, these bistellar flips have several connections to algebraic geometry. For example, the author’s previous constructions of point sets with a disconnected graph of triangulations in dimensions five and six [64], [67] imply that certain algebraic schemes considered in the literature [4], [13], [33], [57], including the so-called toric Hilbert scheme, are sometimes not connected. Triangulations of point sets play also an obvious role in applied areas such as computational geometry or computer aided geometric design, where a region of the plane or 3-space is triangulated in order to approximate a surface, answer proximity or visibility questions, etc. See, for example, the survey articles [8], [10], or [25]. In these fields, flips between triangulations have also been considered since long [40]. Among other things, they are used as the basic step to compute an optimal triangulation of a point set incrementally, that is, adding the points one by one. This incremental flipping algorithm is the one usually preferred for, for example, computing the Delaunay ∗ Partially supported by the Spanish Ministry of Education and Science, grant number MTM2005-08618C02-02.


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triangulation, as “the most intuitive and easy to implement” [8], and yet as efficient as any other. In both the applied and the theoretical framework, the situation is the same: a fixed set of points A ⊂ Rd is given to us (the “sites” for a Delaunay triangulation computation, the test points for a surface reconstruction, or a set of monomials, represented as points in Zd , in the algebro-geometric context) and we need to either explore the collection of all possible triangulations of this set A or search for a particular one that satisfies certain optimality properties. Geometric bistellar flips are the natural way to do this. For this reason, it was considered one of the main open questions in polytope theory ten years ago whether point sets exist with triangulations that cannot be connected via these flips [80]. As we have mentioned above, this question was answered positively by the author of this paper, starting in dimension five. The question is still open in dimensions three and four. This paper intends to be an introduction to this topic, organized in three parts. The first section is a self-contained introduction to the theory of geometric bistellar flips and secondary polytopes in triangulations of point sets, aimed at the non-expert. The results in it are certainly not new (most come from the original work of Gel’fand, Kapranov and Zelevinskii mentioned above) but the author wants to think that this section has some expository novelty; several examples that illustrate the theory are given, and our introduction of geometric bistellar flips first as certain polyhedral subdivisions and only afterwards as transformations between triangulations is designed to show that the definition is as natural as can be. This section finishes with an account of the state-of-the-art regarding knowledge of the graph of flips for sets with “few” points or “small” dimension, with an emphasis on the differences between dimensions two and three. The second section develops in more detail the two contexts in which we have mentioned that flips are interesting (computational geometry and algebraic geometry) together with other two, that we call “combinatorial topology” and “topological combinatorics”. Combinatorial topology refers to the study of topological manifolds via triangulations of them. Bistellar flips have been proposed as a tool for manifold recognition [18], [46], and triangulations of the 3-sphere without bistellar flips other than “insertion of new vertices” are known [24]. Topological combinatorics refers to topological methods in combinatorics, particularly to the topology of partially ordered sets (posets) via their order complexes. The graph of triangulations of a point set A consists of the first two levels in the poset of polyhedral subdivisions of A, which in turn is just an instance of several similar posets studied in combinatorics with motivations and applications ranging from oriented matroid theory to bundle theories in differential geometry. The third section announces for the first time the construction of a point set in general position whose graph of triangulations is not connected. The details of the proof appear in [68]. The point set is also the smallest one known so far to have a disconnected graph of flips.

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Theorem. There is a set of 17 points in general position in R6 whose graph of triangulations is not connected. As usual in geometric combinatorics, a finite point set A ⊂ Rd is said to be in general position |A| if no d + 2 of the points lie in an affine hyperplane. Equivalently, if determinants defined by the point set vanish. Point sets in general none of the d+1 position form an open dense subset in the space Rn×d of sets of dimension d with n elements. That is to say, “random point sets” are in general position. Point sets that are not in general position are said to be in special position. The connectivity question has received special attention in general position even before disconnected examples in special position were found. For example, Challenge 3 in [80] and Problem 28 in [50] specifically ask whether disconnected graphs of flips exist for point sets in special position (the latter asks this only for dimension 3). Although it was clear (at least to the author of this paper) from the previous examples of disconnected graphs of flips that examples in general position should also exist, modifying those particular examples to general position and proving that their flip-graphs are still not connected is not an easy task for quite intrinsic reasons: the proofs of non-connectednes in [64], [67] are based on the fact that the point sets considered there are cartesian products of lower dimensional ones. In our opinion, an example of a disconnected graph of flips in general position is interesting for the following three reasons: 1. The definition of flip that is most common in computational geometry coincides with ours (which is the standard one in algebraic geometry and polytope combinatorics) only for point sets in general position. In special position, the computational geometric definition is far more restrictive and, in particular, taking it makes disconnected graphs of flips in special position be “no surprise”. For example, Edelsbrunner [25] says that the flip-graph among the (three) triangulations of a regular octahedron is not connected; see Section 2.1. 2. Leaving aside the question of definition, in engineering applications the coordinates of points are usually approximate and there is no loss in perturbing them into general position. That is, the general position case is sometimes the only case. 3. Even in a purely theoretical framework, point sets in general position have somehow simpler properties than those in special position. If a point set A in special position has a non-connected graph of flips then automatically some subset of A (perhaps A itself) has a disconnected poset of subdivisions. This poset is sometimes called the Baues poset of A and its study is (part of) the so-called generalized Baues problem. See Section 2.3, or [61] for more precise information on this. In particular, the present example is the first one (proven) to have a disconnected Baues poset. Corollary. There is a set of at most 17 points in R6 whose poset of proper polyhedral subdivisions is not connected.

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1. The setting 1.1. Triangulations. Regular triangulations and subdivisions Triangulations and polyhedral subdivisions. A (convex) polytope P is the convex hull of a finite set of points in the affine space Rd . A face of P is its intersection with any hyperplane that does not cross the relative interior of P . (Here, the relative interior of S ⊆ Rd is the interior of S regarded as a subset of its affine span). We remind the reader that the faces of dimensions 0, 1, d − 2 and d − 1 of a d-polytope are called vertices, edges, ridges and facets, respectively. Vertices of P form the minimal S such that P = conv(S). A k-simplex is a polytope whose vertices (necessarily k + 1) are affinely independent. It has k+1 i+1 faces of each dimension i = 0, . . . , k, which are all simplices. Definition 1.1. Let A be a finite point set in Rd . A triangulation of A is any collection T of affinely spanning and affinely independent subsets of A with the following properties: 1. if σ and σ are in T , then conv(σ ) ∩ conv(σ ) is a face of both conv(σ ) and conv(σ ). That is, T induces a geometric simplicial complex in Rk ; 2. σ ∈T conv(σ ) = conv(A). That is, T covers the convex hull of A. Note that our definition allows for some points of A not to be used at all in a particular triangulation. Extremal points (vertices of conv(A)) are used in every triangulation. The elements of a triangulation T are called cells. We can define polyhedral subdivisions of A by removing the requirement of the sets σ to be affinely independent in Definition 1.1. Since a general subset σ of A may contain points which are not vertices of conv(σ ), now the fact that the elements of a subdivision are subsets of A rather than “subpolytopes” is not just a formality: points which are not vertices of any “cell” in the subdivision may still be considered “used” as elements of some cells. In order to get a nicer concept of polyhedral subdivision, we also modify part 1 in Definition 1.1, adding the following (redundant for affinely independent sets) condition: conv(σ ∩ σ ) ∩ σ = conv(σ ∩ σ ) ∩ σ

for all σ, σ ∈ T .

That is, if A contains some point in the common face conv(σ ∩ σ ) of conv(σ ) and conv(σ ) but not a vertex of it, that point is either in both or in none of σ and σ . Polyhedral subdivisions of A form a partially ordered set (or poset) with respect to the following refinement relation: S refines S :⇔ for all σ ∈ S there exists σ ∈ S such that σ ⊆ σ . Triangulations are, of course, the minimal elements in this poset. The poset has a unique maximal element, namely the trivial suvdivision {A}.

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Example 1.2. Let A be the following set of five points a1 , . . . , a5 in the plane. We take the convention that points are displayed as columns in a matrix, and that an extra homogenization coordinate (the row of 1’s in the following matrix) is added so that linear algebra, rather than affine geometry, can be used for computations: a1

a2

a3

a4

0 ⎜ A= ⎝ 0 1

3 0 1

0 3 1

3 3 1

⎛

a5

⎞ 1 ⎟ 1⎠ 1

(1)

The following are the nine polyhedral subdivisions of A. Arrows represent the refinement relation, pointing from the coarser to the finer subdivision. For clarity, we write “125” meaning {a1 , a2 , a5 }, and so on. Figure 1 shows pictures of the subdivisions. In the corners are the four triangulations of A and in the middle is the trivial subdivision. {125, 135, 235, 234} ← {1235, 234} → {135, 234} ↑ ↑ ↑ {125, 135, 2345} ← {12345} → {1234} ↓ ↓ ↓ {125, 135, 245, 345} ← {1245, 1345} → {124, 134} The last two columns of subdivisions geometrically induce the same decomposition 3

4

3

5

4

3

4

5

1

2

1

2

1

2

3

4

3

4

3

4

5

5

1

2

1

2

1

2

3

4

3

4

3

4

2

1

2

5 1

5 2

1

Figure 1. The nine polyhedral subdivisions of a certain point set.

of conv(A) into subpolygons. Still, we consider them different subdivisions since the middle column “uses” the interior point 5 while the right column does not.

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Regular subdivisions. Let a point set A be given, and choose a function w : A → R to lift A to Rd+1 as the point set Aw := {(a, w(a) : a ∈ A}. A lower facet of conv(Aw ) is a facet whose supporting hyperplane lies below the interior of conv(Aw ). The following is a polyhedral subdivision of A, where π : Rd+1 → Rd is the projection that forgets the last coordinate: Tw := {π(F ∩ Aw ) : F is a lower facet of conv(Aw )}. Geometrically, we are projecting down onto A the lower envelope of Aw , keeping track of points that lie in the lower boundary even if they are not vertices of a facet. Definition 1.3. The polyhedral subdivisions and triangulations that can be obtained in this way are called regular. If w is sufficiently generic then Tw is clearly a triangulation. Regular triangulations are particularly simple and yet quite versatile. They appear in different contexts under different names such as coherent [29], convex [36], [77], Gale [49], or generalized (or, weighted ) Delaunay [25] triangulations. The latter refers to the fact that the Delaunay triangulation of A, probably the most used triangulation in applications, is the regular triangulation obtained with w(a) = a 2 , where · is the euclidean norm. Example 1.4. Let a1 4 A= ⎝ 0 0 ⎛

a2 0 4 0

a3 0 0 4

a4 2 1 1

a5 1 2 1

a6 ⎞ 1 1 ⎠. 2

This is a configuration of six points in the affine plane with equation x1 + x2 + x3 = 4 in R3 . Since the matrix is already homogeneous (meaning precisely that columns lie in an affine hyperplane) we do not need the extra homogenization row. The configuration consists of two parallel equilateral triangles, one inside the other. We leave it to the reader to check that the following are two non-regular triangulations (see Figure 2): T1 := {124, 235, 136, 245, 356, 146, 456}, T2 := {125, 236, 134, 145, 256, 346, 456}. This example is the smallest possible, since 1-dimensional point configurations and point configurations with at most d + 3 points in any dimension d only have regular triangulations. The former is easy to prove and the latter was first shown in [44]. The earliest appearance of these two non-regular triangulations that we know of is in [20], although they are closely related to Schönhardt’s classical example of a non-convex 3-polytope that cannot be triangulated [69].1 1 We describe Schönhardt’s polyhedron and its relation to this example in Example 1.21.

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Geometric bistellar flips: the setting, the context and a construction a1

a4 a5 a2

a6 a3

T1

T2

Figure 2. A point configuration with two non-regular triangulations.

Remark 1.5. Suppose that two point sets A = {a1 , . . . , an } and B = {b1 , . . . , bn } have the same oriented matroid [17], or order type. This means that for every subset I ⊂ {1, . . . , n} of labels, the determinants of the point sets {ai : i ∈ I } and {bi : i ∈ I } have the same sign.2 It is an easy exercise to check that then A and B have the same triangulations and subdivisions.3 However, they do not necessarily have the same regular subdivisions. For example, the points of example 1.4 are in general position and, hence, their oriented matroid does not change by a small perturbation of coordinates. But any sufficiently generic perturbation makes one of the two nonregular triangulations T1 and T2 become regular. Still, the following is true [65]: the existence of non-regular triangulations of A depends only on the oriented matroid of A. The secondary polytope. Let LA denote the space of all lifting functions w : A → R on a certain point set A ⊂ Rd with n elements. In principle LA is isomorphic to Rn in an obvious way; but we mod-out functions that lift all of A to a hyperplane, because adding one of them to a given lifting function w does not (combinatorially) change the lower envelope of Aw . We call these particular lifting functions affine. They form a linear subspace of dimension d + 1 of Rn . Hence, after we mod-out affine functions we have LA ∼ = Rn−d−1 . For a given polyhedral subdivision T of A, the subset of LA consisting of functions w that produce T = Tw , is a (relatively open) polyhedral cone; that is, it is defined by a finite set of linear homogeneous equalities and strict inequalities. Equalities appear only if T is not a triangulation and express the fact that if σ ∈ T is not affinely independent then w must lift all σ to lie in a hyperplane. Inequalities express the fact that for each σ ∈ T and point a ∈ A \ σ , a is lifted above the hyperplane spanned by the lifting of σ . The polyhedral cones obtained for different choices of T are glued together forming a polyhedral fan, that is, a “cone over a polyhedral complex”, called the secondary fan of A. The prototypical example of a fan is the normal fan of a polytope, whose 2 Observe that the bijection between A and B implicit by the labels is part of the definition. 3 More precisely, the implicit bijection between A and B induces a bijection between their polyhedral subdi-

visions.

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cones are the exterior normal cones of different faces of P . A seminal result in the theory of triangulations of polytopes is that the secondary fan is actually polytopal; that is, it is the normal fan of a certain polytope: Theorem 1.6 (Gel’fand–Kapranov–Zelevinskii [28], [29]). For every point set A of n points affinely spanning Rd there is a polytope (A) in LA ∼ = Rn−d−1 whose normal fan is the secondary fan of A. In particular, the poset of regular subdivisions of A is isomorphic to the poset of faces of (A). Vertices correspond to regular triangulations and (A) itself (which is, by convention, considered a face) corresponds to the trivial subdivision. The polytope (A) is called the secondary polytope of A. There are two standard ways to construct the secondary polytope (A) of a point set A.4 The original one, by Gel’fand, Kapranov and Zelevinskii [28], [29] gives, for each regular triangulation T of A, coordinates of the corresponding vertex vT of (A) in terms of the volumes of simplices incident in T to each point of A. The second one, by Billera and Sturmfels [14], describes the whole polytope σ (A) as the Minkowski integral of the fibers of the affine projection π : A → conv(A), where A is a simplex with |A| vertices (hence, of dimension |A| − 1) and π bijects the vertices of A to A (see Theorem 2.8). Example 1.7 (Example 1.2 continued). Figure 3 shows the secondary fan of the five points. To mod-out affine functions we have taken w(a1 ) = w(a2 ) = w(a3 ) = 0, and the horizontal and vertical coordinates in the figure give the values of w(a4 ) and w(a5 ), respectively. The triangulation corresponding to each two-dimensional cone is displayed. In this example all nine polyhedral subdivisions are regular (in agreement

Figure 3. The secondary fan of Example 1.2.

with the result of [44] mentioned in Example 1.4) and the secondary polytope is a quadrilateral. 4 Polytopality of a fan is equivalent to the feasibility of a certain system of linear equalities and strict inequalities. But here we mean more direct and intrinsic constructions of the secondary polytope.


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Example 1.8 (Example 1.4 continued). The secondary polytope of this point set is 3-dimensional, and contains a hexagonal face corresponding to the regular subdivision {1245, 2356, 1346, 456}. This regular subdivision can be refined to a triangulation in eight ways, by independently inserting a diagonal in the quadrilaterals 1245, 2356 and 1346. Six of these triangulations are regular, and correspond to the vertices of the hexagon. The other two, T1 and T2 , are non-regular and they “lie” in the center of the hexagon. We have mentioned that if the point set is perturbed slightly then one of the triangulations becomes regular. What happens in the secondary polytope is the following: the perturbation “inflates” the hexagon so that the eight points (the vertices of the hexagon and the two interior points representing T1 and T2 ) become, combinatorially, the vertices of a cube. The points corresponding to T1 and T2 move in opposite directions, one of them going to the interior of the secondary polytope and the other becoming a new vertex of it. The hexagonal face gets refined into three quadrilaterals. Of course, the vertices of the hexagon also move in the process, and are no longer coplanar. Example 1.9 (The convex n-gon and the associahedron). All triangulations of a convex n-gon are regular and their number is the n − 2nd Catalan number 1 2n − 4 . Cn−2 := n−1 n−2 The corresponding secondary polytope is called the associahedron. The name comes from the fact that there is a bijection between triangulations of the n-gon and the ways to put the n − 2 parentheses in an associative product of n − 1 factors. The associahedron is a classical object in combinatorics, first studied5 by Stasheff and Tamari [76], [72]. It was shown to be polytopal by Haiman (unpublished) and Lee [43]. That its diameter equals 2n−10 “for every sufficiently big n”6 was shown by Sleator, Tarjan and Thurston [71], with motivations coming from theoretical computer science and tools from hyperbolic geometry. Remark 1.10. In Sections 2.3 and 2.4 we will mention triangulations of a set of vectors rather than points. They are defined exactly as triangulations of point sets, just changing the word “affinely” to “linearly” and the operator “conv” to “pos” (“positive span”) in Definition 1.1. Put differently, a triangulation of a vector set A ⊂ Rd+1 is a simplicial fan covering pos(A) and whose rays are in the positive directions of (not necessarily all) the elements of A. Equivalently, and perhaps closer to readers familiar with classical geometry, we can, without loss of generality, normalize all 5As a combinatorial cell complex, without an explicit polytopal realization. 6 Sleator et al. do not say “how big” is “sufficiently big” in their statement, but conjecture that n ≥ 13 is enough. We consider this an interesting and somehow shameful open question.

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vectors of A to lie in the unit sphere S d . Then, triangulations of A are the geodesic triangulations, with vertices contained in A, of the spherical convex hull of A. The existence and properties of regular subdivisions and secondary fans (and of the bistellar flips introduced in the next section) generalize almost without change to vector configurations.7 1.2. Geometric bistellar flips Flips as polyhedral subdivisions. In order to introduce the notion of local move (flip) between triangulations of A, we use the secondary fan as a guiding light: whatever our definition is, restricted to regular triangulations a flip should correspond to crossing a “wall” between two full-dimensional cones in the secondary fan; that is, a flip between two regular triangulations T1 and T2 can be regarded as certain regular subdivision T0 with the property that its only two regular refinements are precisely T1 and T2 . Some thought will convince the reader that the necessary and sufficient condition for a lifting function w : A → R to produce a Tw with this property is that there is a unique minimal affinely dependent subset in A whose lifting is contained in some lower facet of the lifted point set Aw . This leads to the following simple, although perhaps not very practical, definition. Definition 1.11. Let T be a (not-necessarily regular) subdivision of a point set A. We say that T is a flip if there is a unique affinely dependent subset C ∈ A contained in some cell of T . Lemma 1.12. If T is a flip, then there are exactly two proper refinements of T , which are both triangulations. Proof. Let T1 be a refinement of T . Let C be the unique affinely dependent subset of A contained in some cell of T . Each cell of T containing C gets refined in T1 , while each cell not containing C is also a cell in T1 . The statement then follows from the understanding of the combinatorics of point sets with a unique affinely dependent subset C. Let S be such a point set. Each point in S \ C is affinely independent of the rest, so S is an “iterated cone” over C. In particular, there is a face F of S such that S ∩ F = C and every refinement of S consists of a refinement of F coned to the points of S \ C. Moreover, all cells of T containing C must have F refined the same way, so that there is a bijection between the refinements of T and the polyhedral subdivisions of C, as a point set. The result then follows from the fact (see below) that a minimal affinely dependent set C has 2 exactly three subdivisions: the trivial one and two triangulations. 7Although with one notable difference. For a general vector configuration not every function w : A → R produces a lift with a well-defined “lower envelope”. Only the functions that do, namely those for which a linear hyperplane exists containing or lying below all the lifted vectors, define a regular polyhedral subdivision. These functions form a cone in LA . The secondary fan is still well-defined but, of course, it cannot be the normal fan of a polytope. It is, however, the normal fan of an unbounded convex polyhedron, called the secondary polyhedron of A [12].


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This lemma allows us to understand a flip, even in the non-regular case, as a relation or a transformation between its two refinements. This is the usual usage of the word “flip”, and our next topic. Flips as elementary changes. A minimal affinely dependent set C is called a circuit in geometric combinatorics. The points in a circuit C = {c1 , . . . , ck } satisfy

a unique (up to a constant) affine dependence equation λ1 c1 + · · · + λk ck = 0 with λi = 0, and all the λi must be non zero (or otherwise C is not minimally dependent). This affine dependence implicitly decomposes C into two subsets C+ = {ci : λi > 0},

C− = {ci : λi < 0}.

The pair (C+ , C− ) is usually called a signed or oriented circuit. We will slightly abuse notation and speak of “the circuit C = (C+ , C− )”, unless we need to emphasize the distinction between the set C (the support of the circuit) and its partition. A more geometric description is that (C+ , C− ) is the only partition of C into two subsets whose convex hulls intersect, and that they intersect in their relative interiors. This is usually called Radon’s property [58] and the oriented circuit a Radon partition. Spanning and affinely independent subsets of C are all the sets of the form C \{ci }. Moreover, by Radon’s property two such sets C \ {ci } and C \ {cj } can be cells in the same triangulation of C if and only if ci and cj lie in the same side of the partition. In other words: Lemma 1.13. A circuit C = (C+ , C− ) has exactly two triangulations: T+C := {C \ {ci } : ci ∈ C+ },

T−C := {C \ {ci } : ci ∈ C− }.

This leads to a second definition of flip, equivalent to Definition 1.11, but more operational. This is the definition originally devised by Gel’fand, Kapranov and Zelevinskii [29]. The link of a set τ ⊆ A in a triangulation T of A is defined as link T (τ ) := {ρ ⊆ A : ρ ∩ τ = ∅, ρ ∪ τ ∈ T }. Definition 1.14. Let T1 be a triangulation of a point set A. Suppose that T1 contains one of the triangulations, say T+C , of a circuit C = (C+ , C− ). Suppose also that all cells τ ∈ T+C have the same link in T1 , and call it L. Then, we say that C supports a geometric bistellar flip (or a flip, for short) in T1 and that the following triangulation T2 of A is obtained from T1 by this flip: T2 := T1 \ {ρ ∩ τ : ρ ∈ L, τ ∈ T+C } ∪ {ρ ∩ τ : ρ ∈ L, τ ∈ T−C }. If i = |C+ | and j = |C− | we say that the flip is of type (i, j ). Flips of types (1, j ) and (i, 1) are called, insertion and deletion flips, since they add or remove a vertex in the triangulation. The graph of flips of A has as vertices all the triangulations of A and as edges the geometric bistellar flips between them.

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Of course, an (i, j ) flip can always be reversed, giving a (j, i) flip. The reason for the words “geometric bistellar” in our flips can be found in Section 2.2. Example 1.15 (Examples 1.2 and 1.7 continued). The change between the two top triangulations in Figure 3 is a (2, 2) flip, as is the change between the two bottom ones. The flip from the top-right to the bottom-right is a (1, 3) flip (“1 triangle disappears and 3 are inserted”) and the flip from the top-left to the bottom-left is a (1, 2) flip (“one edge is removed, together with its link, and two are inserted, with the same link”). The latter is supported in the circuit formed by the three collinear points. We omit the proof of the following natural statement. Theorem 1.16. Definitions 1.11 and 1.14 are equivalent: two triangulations T1 and T2 of a point set A are connected by a flip in the sense of 1.14 if and only if they are the two proper refinements of a flip in the sense of 1.11. The following two facts are proved in [65]: Remark 1.17. 1. If all proper refinements of a subdivision T are triangulations, then T has exactly two of them and T is a flip. That is to say, flips are exactly the “next-to-minimal” elements in the refinement poset of all subdivisions of A. 2. Every non-regular subdivision can be refined to a non-regular triangulation. In particular, not only edges of the secondary polytope correspond to flips between two regular triangulations, but also every flip between two regular triangulations corresponds to an edge. Detecting flips. Definitions 1.11 and 1.14 are both based on the existence of a flippable circuit C with certain properties. But in order to detect flips only some circuits need to be checked: Lemma 1.18. Every flip in a triangulation T other than an insertion flip is supported in a circuit contained in the union of two adjacent cells of T . Observe that the circuit contained in two adjacent cells always exists and is unique. Also, that the insertion flips left aside in this statement are easy to detect:8 There is one for each point a ∈ A not used in T , that inserts the point a by subdividing the minimum (perhaps not full-dimensional) simplex τ ⊆ σ ∈ T such that a ∈ conv(τ ). The flippable circuit is ({a}, τ ). Proof. Let C = (C+ , C− ) be a circuit that supports a flip in T , with |C+ | ≥ 2. Observe that |C+ | is also the number of many maximal simplices in T+C , so let τ1 and τ2 be two of them, which differ in a single element, and let ρ be an element of link T (τ1 ) = link T (τ2 ). Then, ρ ∪ τ1 and ρ ∪ τ2 are adjacent cells in T and C is the unique circuit contained in τ1 ∪ τ2 ∪ ρ. 2 8 We mean, theoretically. Algorithmically, insertion flips are far from trivial since they imply locating the simplex of T that contains the point a to be inserted, which takes about the logarithm of the number of simplices in T . This is very expensive, since algorithms in computational geometry that use flipping in triangulations usually are designed to take constant time per flip other than an insertion flip. See Section 2.1.

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Monotone sequences of flips. The graph of flips among regular triangulations of a point set A of dimension d is connected, since it is the graph of a polytope.9 A fundamental fact exploited in computational geometry is that one can actually flip between regular triangulations monotonically, in the following sense. Let w : A → R be a certain generic lifting function. We can use w to lift every triangulation T of A as a function HT ,w : conv(A) → R, by affinely interpolating w in each cell of T . We say that T1 <w T2 (“T1 is below T2 , with respect to w”) if HT1 ,w ≤ HT2 ,w pointwise and HT1 ,w = HT2 ,w globally. This defines a partial order <w on the set of all triangulations, whose global minimum and maximum are, respectively, Tw and T−w .10 Definition 1.19. A sequence of flips is monotone with respect to w if every flip goes from a triangulation T to a triangulation T <w T . By definition of the secondary polytope (A) as having the secondary fan as its normal fan, lifting functions are linear functionals on it. Then, it is no surprise that for the regular triangulations T1 and T2 corresponding to vertices vT1 and vT2 of the secondary polytope one has11 : T1 <w T2 ⇒ w, vT1 < w, vT1 . In fact, w, vT equals the volume between the graphs of the functions HTw ,w and HT ,w . Since the converse implication holds whenever T1 and T2 are related by a flip, we have: Lemma 1.20. For every lifting function w and every regular triangulation T there is a w-monotone sequence of flips from T to the regular triangulation Tw . If T is not regular this may be false, even in dimension 2: Example 1.21 (Examples 1.4 and 1.8 continued). Let A be the point configuration of Example 1.4 (see Figure 2), except perturbed by slightly rotating the interior triangle “123” counter-clockwise. That is, a1 4−ε A= ⎝ ε 0 ⎛

a2 0 4−ε ε

a3 ε 0 4−ε

a4 2 1 1

a5 1 2 1

a6 ⎞ 1 1 ⎠, 2

9 Even more, it is (|A| − d − 1)-connected. Remember that a graph is called k-connected if removing less than k vertices from it it stays connected. A classical theorem of Balinski [79] says that the graph of a k-polytope is k-connected. 10 In case they are triangulations. If not, every triangulation that refines T or T w −w is, respectively, minimal or maximal. 11 The same is true for non-regular triangulations. The point v is well-defined, via the Gel’fand-KapranovT Zelevinskii coordinates for the secondary polytope, even if T is not regular. The only difference is that if T is not regular then vT is not a vertex of the secondary polytope.

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for a small ε > 0. This perturbation keeps the triangulation T1 non-regular and makes T2 regular. Let w : A → R lift the exterior triangle 123 to height zero and the interior triangle 456 to height one. The graph of HT2 ,w is a strictly concave surface (that is, T2 = T−w ) and there is no w-monotone flip in T1 , since its only three flips are the diagonal-edge flips on “16”, “24” and “35”, which are “towards HT2 ,w ”. This example appeared in [26]. Another explanation of why no w-monotone flip exists in T1 is that when we close the graph of the function HT1 ,w by adding to it the triangle 123, it becomes a nonconvex polyhedron P with the property that no tetrahedron (with vertices contained in those of P ) is completely contained in the region enclosed by P . This polyhedron is affinely equivalent to Schönhardt’s [69] classical example of a non-convex polyhedron in R3 that cannot be triangulated without additional vertices. 1.3. The cases of small dimension or few points. Throughout this section A denotes a point set with n elements and dimension d. Sets with few points. If n = d +1, then A is independent and the trivial subdivision is its unique triangulation. If n = d + 2 then A has a unique circuit and exactly two triangulations, connected by a flip. If n = d + 3, it was proved by Lee [44] that all triangulations are regular. Since the secondary fan is 2-dimensional, the secondary polytope is a polygon, whose graph (a cycle) is the graph of flips. If n = d +4, then A can have non-regular triangulations (see Example 1.4). Still, it is proven in [7] that every triangulation has at least three flips and that the flip-graph is 3-connected. For point sets with n = d + 5 the flip-graph is not known to be always connected. Dimension 1. Triangulating a one-dimensional point set is just choosing which of the interior points are used. That is, n points in dimension 1 have 2n−2 triangulations. The flip-graph is the graph of an (n − 2)-dimensional cube and all triangulations are regular. The secondary polytope is the same cube. Dimension 2. In dimension two the graph of (2, 2)-flips among triangulations using all points of A 12 is known to be connected since long [40], and connectivity of the whole graph—including the triangulations that do not use all points and the insertion or deletion flips—is straightforward from that. Even more, one can always flip monotonically13 from any triangulation to the Delaunay triangulation using only (2, 2) flips. Quadratically many (with respect to the number of points) flips are sometimes necessary and always suffice (see, e.g., [25, p. 11]). However, with general flips: Proposition 1.22. The flip-graph of any A ⊆ R2 has diameter smaller than 4n. 12 This is the graph usually considered in two-dimensional computational geometry literature. 13 With respect to the lift w(a) := a 2 .


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Proof. Let a be an extremal point of A and T an arbitrary triangulation. If T has triangles not incident to a then there is at least a flip that decreases the number of them (proof left to the reader). Since the number of triangles in a planar triangulation with vi interior vertices and vb boundary vertices is exactly 2vi + vb − 2 (by Euler’s formula) we can flip from any triangulation to one with every triangle incident to a in at most 2vi + vb − 3 < 2n − nb flips. Now, exactly as in the 1-dimensional case, the graph of flips between triangulations in which every triangle is incident to a is the graph of a cube of dimension equal to the number of “boundary but non-extremal” points of A. Hence, we can flip between 2 any two triangulations in (2n − nb ) + nb + (2n − nb ) < 4n flips. Remark 1.23. The preceding proof is another example of monotone flipping, this time with respect to any lifting function w : A → R with w(a) 0 be a family of polynomials with pn a polynomial of degree d(n) in n variables. A circuit family {Cn }n>0 is said to compute {pn } if for every n, the polynomial computed by Cn equals pn . In the following we shall simply write {pn } for the family {pn }n>0 . Arithmetic branching programs are a restricted form of arithmetic circuits in which every multiplication gate has fanin exactly two, and in addition at least one of the two gates, from which edges are incident to the multiplication gate, is an input gate. Such circuits are also called skew circuits. The class VPF , the arithmetic analog of class P, is defined to be the set of polynomial families {pn } over a field F such that (1) each pn is of degree nO(1) , and (2) there exists a circuit family {Cn } computing {pn } such that Cn is of size nO(1) .2 The class VNPF , the arithmetic analog of class NP, is defined to be the set of polynomial families {pn } over a field F such that (1) each pn is of degree nO(1) , and (2) there exists a family of polynomials {qn } ∈ VPF such that for every n, pn (x1 , x2 , . . . , xn ) =

1 1 y1 =0 y2 =0

···

1

qn+m (x1 , x2 , . . . , xn , y1 , y2 , . . . , ym )

ym =0

with m = nO(1) .3 Given two polynomials p(x1 , x2 , . . . , xn ) and q(y1 , y2 , · · · , ym ) over a field F , we say that p is a projection of q if p(x1 , x2 , . . . , xn ) = q(α1 , α2 , · · · , αm ) where each αi ∈ F ∪ {x1 , x2 , . . . , xn }. Given two polynomial families {pn } and {qn }, we say that {pn } is a p-projection of {qn } if for every n there exists an m = nO(1) such that pn is a projection of qm .4 Let per F = {per F,n } and detF = {det F,n } denote the families of permanent and determinant polynomials over a field F respectively. Both these families contain polynomials in n2 variables for each n. Valiant [24] proved that: Theorem 2.1 ([24]). For any F , per F ∈ VNPF . In addition, for any F , char(F ) = 2, for any polynomial family {pn } in VNPF , {pn } is a p-projection of per F . So permanent is as hard to compute as any polynomial family in VNP. In contrast, determinant can be computed efficiently. A nice characterization of determinant was shown in [4], [21], [25]: Theorem 2.2 ([4], [21], [25]). For any F , detF can be computed by polynomialsized arithmetic branching programs. In addition, for any F and for any polynomial 2 In addition, circuit C must be efficiently computable given 1n . This property does not seem to play any n role in obtaining lower bounds. 3 The class #P is the ‘functional’ version of the class VNP : a polynomial family {p } ∈ VNP belongs to n F F F #PF when for each n, pn is viewed as a map from F n to F . 4Again, given 1n , the projection specified by (α , α , . . . , α ) should be efficiently computable. m 1 2

Determinant versus permanent

989

family {pn } computed by polynomial-sized arithmetic branching programs, {pn } is a p-projection of detF . In fact, all families in VP are almost p-projections of determinant. Theorem 2.3 ([23]). Let C be a circuit of size s computing a polynomial of degree d. There exists another circuit computing the same polynomial of size s O(1) and depth O(log s + log d). Corollary 2.4. Any circuit family in VPF can be computed by circuit families of polynomial size and logarithmic depth. Corollary 2.5. For every circuit family {pn } ∈ VPF and for every n, pn is a projection of det F,m where m = nO(log n) . The above characterizations of complexities of determinant and permanent imply that, in order to separate VPF from VNPF , it is enough to show that per F is not an almost p-projection of detF (in the sense above).

3. Known lower bounds on permanent Lower bounds are known on permanent only for restricted circuits. In this section, we describe important lower bounds of this kind. Three major restrictions have been considered for proving such lower bounds: monotone circuits, constant depth circuits, and multilinear formulas. 3.1. Monotone circuits. A circuit over Q or R is monotone if all the constants in the circuit are non-negative. This is a very restricted class of circuits since no cancellations can occur in it. Jerrum and Snir [8] showed that any monotone circuit family that computes permanent must have exponential size. 3.2. Constant depth circuits. In this restriction, the depth of a circuit family is fixed, i.e., it is independent of n. Permanent (or any polynomial of degree nO(1) for that matter) can be computed by an exponential size depth two circuit family. Conversely, it is easy to see that any depth two circuit family computing permanent must have exponential size. Depth three circuit families are, however, non-trivial. A depth three circuit can be of the form “sum-of-products-of-sums” or “product-of-sums-of-products.” The latter form can easily be seen to require exponential size to compute permanent (the topmost multiplication gate can be shown to be redundant transforming the circuit to a depth two circuit). Circuit families of the first form are powerful: Ben Or observed that they can efficiently compute all symmetric polynomials of degree nO(1) over fields of characteristic zero.

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The best known lower bound in fields of characteristic zero is by Shpilka and Wigderson [19] who proved that permanent (and determinant) requires a circuit family of size (n2 ). Their idea is to consider the space spanned by all partial derivatives of the polynomials computed at each gate of a given circuit. They show that for a depth three circuit with small size, the space spanned by the derivatives of the output polynomial would be of small dimension while the space spanned by derivatives of permanent is of large dimension. Over finite fields, the situation is better. Grigoriev and Razborov [6] showed an exponential lower bound on the size of depth three circuit families computing determinant and permanent. Their approach was to show that polynomial computed by a depth three circuit of small size can be ‘approximated’ by a low-degree polynomial (approximated in the sense that the two polynomials agree on a large set of points from the field). Then they observed that determinant and permanent cannot be approximated by low-degree polynomials. 3.3. Multilinear formulas. Multilinear formulas are circuits such that (1) the outdegree of every gate is at most one, and (2) the polynomial computed at every gate is multilinear. Such circuits have severely limited multiplication gates – the polynomials input to a multiplication gate must be over disjoint sets of variables. Using a combination of partial derivative technique and random restrictions (setting some input variables to random values), Raz [15] proved a lower bound of n(log n) on the size of families of multilinear formulas computing permanent and determinant.

4. The algebraic geometry approach Mulmuley and Sohoni [11] have offered a completely new approach to the problem of proving a lower bound on permanent for unrestricted circuits by transforming the problem to geometric settings. In this section, we give a brief overview of their approach. F,m = Suppose, for F = Q, per F,n is a projection of detF,m , m > n. Define per m−n xm · per . It follows that per is also a projection of det (just multiply all F,m 2 F,n F,m constants of the projection by xm2 ). This can be written as F,m (x1 , x2 , . . . , xm2 ) = A · detF,m = det F,m ((x1 , x2 , . . . , xm2 ) · A), per where A is an m2 ×m2 matrix over Q. The matrix A is singular whenever m > n since F,m . Let Aε¯ be a slight ‘perturbation’ the variables xn2 +1 , . . ., xm2 −1 do not occur in per of A obtained by adding εi,j to the (i, j )th entry of A. For nearly all values of ε¯ close to zero, Aε¯ is non-singular and the polynomial Aε¯ · detF,m approximates the polynomial F,m very well (all the coefficients of two polynomials are close to each other). Now per 2 consider the space V = CM with M = m +m−1 . Every homogeneous polynomial of m degree m in m2 variables can be viewed as a point in this space (degree m monomials


991

F,m are points in V (since F = Q and both forming the basis). So both detF,m and per 2 polynomials are of degree m in m variables). Let O be the orbit of detF,m under the action of GLm2 (C), i.e., O = {B · detF,m | B is an invertible matrix over C}. Set O can be viewed as a set of points in V . The above argument shows the following: Lemma 4.1 ([11]). If per F,n is a projection of detF,m then the point corresponding F,m in V lies in the closure of the set O in V . Conversely, if per F,m lies in the to per closure of O then per F,n can be approximated by projections of det F,m to any desired accuracy. This (near) characterization is the starting point of their approach. Instead of V , we can work in the projective space P (V ) too since both the polynomials are homogeneous. The same near characterization holds in P (V ) as well with GLm2 (C) replaced by SLm2 (C), the group of all matrices with determinant 1. The advantage of working in P (V ) is that the closure of O (under the classical Euclidean topology) coincides with the closure of O under Zariski topology [12]. In Zariski topology, there is the F,m -stable well-studied notion of stability that captures this problem: detF,m is per F,m lies in the closure of the orbit O (we abuse notation here under SLm2 (C) if per by using the same names for polynomials and sets in P (V ) as for the corresponding ones in V ). Points in the orbit O have a useful property. For any point p ∈ P (V ), let Gp = {A ∈ SLm2 (C) | A · p = p}. The group Gp is called the stabilizer of p. Lemma 4.2. For any point p ∈ O, Gp is a conjugate of GdetF,m . Proof. Let p = B · det F,m ∈ O. Then Gp = B · GdetF,m · B −1 .

2

F,m under SLm2 (C) is a closed set (such Suppose the orbit of the polynomial per F,m under SLm2 (C). By polynomials are called stable). Let Q be the orbit of per Luna’s slice theorem, there is a neighborhood N of Q such that for any point p ∈ N, F,m , Gp is a conjugate of a subgroup of Gper F,m . Since the closure of O contains per there is a point in N, say q, such that q = B · detF,m . This means Gq is a conjugate of GdetF,m . Therefore, GdetF,m is a conjugate of a subgroup of Gper F,m . On the other hand, it is well known that GdetF,m is ‘larger’ than Gper F,m : GdetF,m is characterized by the transformations of the kind X → A · X · B −1 where A, B ∈ GLm (C) while −1 where Gper F,m is characterized by the transformations of the kind X → A · X · B A, B ∈ GLm (C) and both A and B are either diagonal or permutation matrices. Therefore, GdetF,m cannot be a conjugate of a subgroup of Gper F,m . (This is a rough argument; to make it precise, more work is needed.)

992

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F,m is not stable (interestingly, per F,n is stable in the smaller Unfortunately, per dimensional space defined by degree n homogeneous polynomials in n2 variables; the translation to higher dimensional space ruins the stability). Mulmuley and Sohoni F,m is partially stable. Now define the notion of partial stability and show that per their aim is to make the above argument work even for partially stable points. A more detailed explanation of their approach is in [16].

5. The derandomization approach Kabanets and Impagliazzo [9] have discovered another new approach for proving lower bounds on permanent. Unlike the previous one, this approach is based on arithmetic circuits. In this section we outline their approach and its variation in [1]. The Identity Testing problem is defined as follows: given an arithmetic circuit C over a field F as input, decide if the polynomial computed by the circuit is the zero polynomial. This is a classical problem in computational algebra and there exist several randomized polynomial-time algorithms for it. Perhaps the simplest one is by Schwartz and Zippel [17], [26]: randomly choose values for variables of C from a set in F of size 2d, here d is the degree of C (if |F | < 2d then extend F slightly); output ZERO if C evaluates to zero, otherwise NON-ZERO. An easy argument shows that this test is correct with probability at least 21 when C computes a non-zero polynomial and always correct when C computes a zero polynomial. Kabanets and Impagliazzo show that if there exists a deterministic subexponential o(1) (= 2n ) time algorithm for solving Identity Testing problem then at least one of the following two lower bounds hold: 1. NEXP requires superpolynomial sized boolean circuits. 2. Permanent requires superpolynomial sized arithmetic circuits. To see this, suppose that permanent has polynomial sized arithmetic circuits for some field F of characteristic different from two. Consider a non-deterministic machine that, on input 1n , guesses the circuit that computes per F,n and verifies it to be correct. It does this by inductively verifying that the circuit, under appropriate settings of its inputs, computes per F,n−1 correctly and then verifying the equation for per F,n that expresses it in terms of per F,n−1 . Verifying the equation is an instance of Identity Testing problem and so can be done in subexponential time by assumption. 2 Therefore, given any matrix A ∈ F n , per A can be computed in non-deterministic subexponential time. Now assume that NEXP has polynomial sized boolean circuits. By [3], [22], it follows that NEXP ⊆ P#P . Since the complexity of #P is exactly the complexity of computing permanent, it follows that NEXP is in non-deterministic subexponential time contradicting the non-deterministic time hierarchy theorem [18]. This result falls short of pointing a way for proving lower bounds on permanent – besides finding a deterministic algorithm for Identity Testing, one needs to assume

993


NEXP has polynomial sized boolean circuits which is very unlikely to be true. However, it does point to a connection between Identity Testing problem and permanent lower bounds. This connection was strengthened in [1] by defining pseudo-random generators for arithmetic circuits. Pseudo-random generators in the boolean settings have been studied intensively (see, e.g., [5], [13], [7], [20]). It is known that constructing pseudo-random generators is equivalent to proving lower bounds in the boolean settings. In [1], pseudo-random generators are defined in arithmetic settings and a similar equivalence is observed. Let AC F be the class of all arithmetic circuits over F and AF ⊆ AC F . Definition 5.1. A function f : N → (F [y])∗ is called an ((n), n)-pseudo-random generator against AF if the following holds: • f (n) ∈ (F [y])n+1 for every n > 0. • Let f (n) = (f1 (y), . . . , fn (y), g(y)). Then each fi (y) as well as g(y) is a polynomial of degree at most 2(n) . • For any circuit C ∈ AF of size n with m ≤ n inputs: C(x1 , x2 , . . . , xm ) = 0 iff C(f1 (y), f2 (y), . . . , fm (y)) = 0 (mod g(y)). A direct application of Schwartz–Zippel lemma [17], [26] shows that there always exist (O(log n), n)-pseudo-random generators against AC F . Call such generators optimal pseudo-random generators. Pseudo-random generators that can be efficiently computed are of special interest. Definition 5.2. An ((n), n)-pseudo-random generator f against AF is efficiently computable if f (n) is computable in time 2O((n)) . An easy argument shows that if there exists an efficiently computable ((n), n)pseudo-random generator against AC F then the Identity Testing problem can be solved deterministically in time 2O((n)) : evaluate the given circuit C of size n modulo g(y) after substituting for the i th input variable the polynomial fi (y) where f (n) = (f1 (y), . . . , fn (y), g(y)). In particular, if there exists an efficiently computable optimal pseudo-random generator against AC F then Identity Testing can be solved in polynomial time. An efficiently computable pseudo-random generator also results in a lower bound. Theorem 5.3 ([1]). Let f be an efficiently computable ((n), n)-pseudo-random generator against AF . Then there is a multilinear polynomial in 2(n) variables, computable in time 2O((n)) , that cannot be computed by any circuit in AF of size n. Proof. For any m = (n), define the polynomial qf (x1 , x2 , . . . , x2m ) by qf (x1 , x2 , . . . , x2m ) = cS · xi . S⊆[1,2m]

i∈S

994

Manindra Agrawal

The coefficients cS satisfy the condition cS · fi (y) = 0 S⊆[1,2m]

i∈S

where f (n) = (f1 (y), f2 (y), . . . , fn (y), g(y)). Such a qf always exists as the following argument shows. The number of coefficients of qf are exactly 22m . These need to satisfy a polynomial equation of degree at most 2m · 2m . So the equation gives rise to at most 2m · 2m + 1 homogeneous constraints on the coefficients. Since (2m · 2m + 1) < 22m for m ≥ 3, there is always a non-trivial polynomial qf satisfying all the conditions. The polynomial qf can be computed by solving a system of 2O(m) linear equations in 2O(m) variables over the field F . Each of these equations can be computed in time 2O(m) using computability of f . Therefore, qf can be computed in time 2O(m) . Now suppose qf can be computed by a circuit C ∈ AF of size n. By the definition of the polynomial qf , it follows that C(f1 (y), f2 (y), . . . , f2m (y)) = 0. The size of circuit C is n and it computes a non-zero polynomial. This contradicts the pseudo-randomness 2 of f . A partial converse of this theorem can also be shown: if there exists a polynomial family computable in time 2O((n)) that cannot be computed by any size n circuit family in AF then there exists an efficiently computable (2 (n), n)-pseudo-random generator against AF , when the degree of every size n circuit in AF is bounded by nO(1) . An efficient optimal pseudo-random generator against AC F yields a polynomial that requires exponential (in the number of variables) sized circuits. However, it is not clear whether the polynomial qf can be computed as permanent of a matrix of size mO(1) . To get this, one needs to show that all the coefficients cS of qf are themselves efficiently computable. If this is done, then using the VNP characterization of permanent, it follows that qf equals the permanent of a matrix of size mO(1) . This results in an exponential lower bound on permanent. For a superpolynomial lower bound one needs either an (no(1) , n)-pseudo random generator against AC F or an optimal pseudo-random generators against a much smaller class of circuits. Theorem 5.4 ([1]). Let f be an efficiently computable optimal pseudo-random generator against the class of circuits of depth ω(1) such that the associated polynomial qf is in VNP. Then permanent cannot computed by any polynomial sized circuit. Proof. From the previous theorem, it follows that the polynomial qf cannot be computed by exponential sized circuits of depth ω(1). A size nd , depth d log n arithmetic circuit with fanin two multiplication gates can be translated to a subexponential sized

995


depth d circuit by “cutting” the circuit into log n layers of depth d each, and then “flattening” each layer to a subexponential sized circuit of depth two. Since every polynomial sized circuit computing permanent can be transformed to a depth O(log n), 2 size nO(1) circuit with fanin two multiplication gates [23], the theorem follows. It is not clear at the moment how to construct optimal pseudo-random generators against constant depth circuits. In [1] a generator is conjectured. Unconditionally, we only know generators against depth two, polynomial sized circuits (the proof is easy, see [1]). We know an optimal generator against the following very special class of circuits too: A = {Cn (x) | Cn (x) = (1 + x)n − 1 − x n over the ring Zn }. Notice that the circuits in the class A are not over a fixed field (or ring), and the size of the circuit Cn is O(log n) and the degree is n. In [2], the following optimal generator was constructed against A:

f (m) = x, 0, . . . , 0, x

16m5

·

5 4 16m 4m

((x − a)r − 1) .

r=1 a=1

6. Concluding remarks The problem of proving that the permanent of a size n matrix cannot be expressed as determinant of size nO(log n) matrix is of great importance in complexity theory. While the existing approaches have failed to shed light on this, one hopes that at least one of the two new approaches will eventually lead to a solution of the problem. Acknowledgements. I wish to thank Somenath Biswas for enjoyable discussions and help in preparing this article.

References [1] Agrawal, M., Proving lower bounds via pseudo-random generators. In Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Comput. Sci. 3821, Springer-Verlag, Berlin 2005, 92–105. [2] Agrawal, M., On derandomizing tests for certain polynomial identities. In Proceedings of 18th Annual IEEE Conference on Computational Complexity, IEEE Computer Society, Los Alamitos, CA, 2003, 355–362. [3] Babai, L., Fortnow, L., Nisan, N., and Wigderson, A., BPP has subexponential time simulations unless EXPTIME has publishable proofs. Comput. Complexity 3 (4) (1963), 307–318. [4] Damm, C., DET=L#L . Technical Report Informatik, Preprint 8, Fachbereich Informatik der Humboldt Universität zu Berlin, 1991.

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[5] Goldreich, O., Foundation of Cryptography I: Basic Tools. Cambridge University Press, Cambridge 2001. [6] Grigoriev, D., and Razborov, A:, Exponential lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields. Appl. Algebra Engrg. Comm. Comput. 10 (6) (2000), 467–487, 2000. [7] Impagliazzo, R., and Wigderson, A., P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Proceedings of the 29th Annual ACM Symposium on Theory of Computing, ACM Press, New York 1997, 220–229. [8] Jerrum, M., and Snir, M., Some exact complexity results for straight-line computations over semirings. J. ACM 29 (3) (1982), 874–897. [9] Kabanets, Valentine, and Impagliazzo, Russell, Derandomizing polyonmial identity tests means proving circuit lower bounds. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, ACM Press, New York 2003, 355–364. [10] Minc, H., Permanents. Addision-Wesley, 1978. [11] Mulmulay, K., and Sohoni, M., Geometric complexity theory, P vs. NP, and explicit obstructions. SIAM J. Comput. 31 (2) (2002), 496–526. [12] D. Mumford, D., Algebraic Geometry I: Complex Projective Varieties. Grundlehren Math. Wiss. 221, Springer-Verlag, Berlin 1976. [13] Nisan, N., and Wigderson, A., Hardness vs. randomness. J. Comput. System Sci. 49 (2) (1994), 149–167. [14] Nisan, N., and Wigderson, A., Lower bounds on arithmetic circuits via partial derivatives. Comput. Complexity 6 (3) (1996/97), 217–234. [15] Raz, Ran, Multi-linear formulas for permanent and determinant and of super-polynomial size. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM Press, New York 2004, 633–641. [16] Regan, K., Understanding the Mulmuley-Sohoni approach to P vs. NP. Bulletin of the European Association for Theoretical Computer Science 78 (2002), 86–97. Lance Fortnow’s Computational Complexity Column. [17] Schwartz, J. T., Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27 (4) (1980), 701–717. [18] Seiferas, J., Fischer, M., and Meyer, A., Separating nondeterministic time complexity classes. J. ACM 25 (1) (1987), 146–167. [19] Shpilka, A., and Wigderson, A., Depth-3 arithmetic circuits over fields of characteristic zero. Comput. Complexity 10 (1) (2001), 1–27. [20] Sudan, M., Trevisan, L., and Vadhan, S., Pseudorandom generators without the XOR lemma. In Proceedings of the 31th Annual ACM Symposium on Theory of Computing, ACM Press, New York 1999, 537–546. [21] Toda, S., Counting problems computationally equivalent to the determinant. Manuscript, 1991. [22] Toda, S., PP is as hard as the polyonmial-time hierarchy. SIAM J. Comput. 20 (1991), 865–877. [23] Valiant, L., Skyum, S., Berkowitz, S., and Rackoff, C., Fast parallel computation of polynnomials using few processors. SIAM J. Comput. 12 (1983), 641–644.


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[24] Valiant, L., Completeness classes in algebra. In Proceedings of the 11th Annual ACM Symposium on Theory of Computing, ACM Press, New York 1979, 249–261. [25] Vinay, V., Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits. In Proceedings of the Structure in Complexity Theory Conference, Lecture Notes in Comput. Sci. 223, Springer-Verlag, Berlin 1991, 270–284. [26] Zippel, R. E., Probabilistic algorithms for sparse polynomials. In EUROSCAM’79, Lecture Notes in Comput. Sci. 72, Springer-Verlag, Berlin 1979, 216–226. Department of Computer Science and Engineering, Indian Institute of Technology, Kanpur 208016, India E-mail: [email protected]

The additivity problem in quantum information theory Alexander S. Holevo∗

Abstract. In this lecture we survey the present status of the additivity problem for the classical capacity and related characteristics of quantum channels – one of the most profound mathematical problems of quantum information theory. Mathematics Subject Classification (2000). Primary 94A15; Secondary 81P68. Keywords. Quantum information theory, quantum channel, classical capacity, entanglement, additivity problem.

1. Introduction The problems of data transmission and storage by quantum information carriers received increasing attention during past decade, owing to the burst of activity in the field of quantum information and computation [42], [22]. At present we are witnessing emergence of theoretical and experimental foundations of the quantum information science. It represents a new exciting research field addressing a number of fundamental issues both in quantum physics and in information and computer sciences. On the other hand, it provides a rich source of well-motivated mathematical problems, often having simple formulations but hard solutions. A central result in the classical information theory is the coding theorem, establishing the possibility of reliable data transmission and processing at rates lower than the capacity of the communication channel. The issue of the information capacity of quantum channels arose soon after publication of Shannon’s pioneering paper and goes back to the works of Gabor, Brillouin and Gordon, asking for fundamental limits on the rate and quality of information transmission. These works laid a physical foundation and raised the question of consistent mathematical treatment of the problem. Important steps in this direction were made in the seventies when quantum statistical decision theory was created, making a noncommutative probability frame for this circle of problems, see [21] for a survey. A dramatic progress has been achieved during the past decade [42], [6], [22]. In particular, a number of coding theorems was discovered, moreover, it was realized ∗ This work was done partially when the author was LeverhulmeVisiting Professor at CQC, DAMTP, University of Cambridge. The work was also supported by the Program “Theoretical Problems of Modern Mathematics” of the Mathematics Division of RAS. The author is grateful to A. Ekert, M. B. Ruskai, M. Shirokov, Yu. M. Suhov and R. F. Werner for fruitful discussions.


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that the quantum channel is characterized by the whole spectrum of capacities depending on the nature of the information resources and the specific protocols used for the transmission, see [6], [14]. This new age of quantum information science is characterized by emphasis onto the new possibilities (rather than mere restrictions) inherent in the quantum nature of the information processing agent. On the other hand, the questions of information capacities turned out to be relevant to the theory of quantum computations, particularly in connection with quantum error-correction, communication protocols, algorithmic complexity and a number of other important issues. The quantum information processing systems have a specifically novel resource, entanglement, a kind of non-classical correlation between parts of the composite quantum system. Among many other unusual features it underlies the strict superadditivity of Shannon information due to entangled decodings in a situation formally similar to the classical memoryless channels [20], [22], [6]. Namely, for independent quantum systems there are entangled measurements which can bear more information than the arithmetic sum of information from these systems. This property has profound consequences for the theory of quantum communication channels and their capacities. A closely related issue is the additivity of the capacity-related quantities for the memoryless quantum channels with respect to entangled encodings. Should the additivity fail, this would mean that applying entangled inputs to several independent uses of a quantum channel may result in superadditive increase of its capacity for transmission of classical information. However so far there is neither a single evidence of such a non-additivity, nor a general proof for the additivity. In this lecture we survey the present status of this problem. We start in Section 2 with the classical case, where the additivity holds for almost obvious reasons. We then describe the problem in the finite-dimensional quantum setting in Section 3, discussing also the various formulations of the additivity conjecture and connections between them. Positive results for several concrete classes of channels are briefly surveyed in Section 4, where also an important counterexample to the additivity of the minimal output quantum Rényi entropy is discussed. Since quantum communication channels are described mathematically as completely positive maps, we devote Subsection 4.1 to the description of their structure paying attention to the notion of complementary maps which leads to new examples of additivity. Section 5 is devoted to different formulations of the additivity conjecture using tools from convex analysis. In Subsection 5.4 we present an argument, essentially due to P. Shor, implying the global equivalence of different forms of the additivity conjecture. We conclude with Section 6, where we briefly outline the works treating the infinite-dimensional case.

The additivity problem in quantum information theory

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2. Additivity in the classical information theory Let X, Y be two finite sets (alphabets), and let [(x, y)]x∈X,y∈Y be a stochastic matrix, i.e., 1. (x, y) ≥ 0, x ∈ X, y ∈ Y; 2. y∈Y (x, y) = 1, x ∈ X. In information theory a stochastic matrix describes a (noisy) channel from X to Y. It transforms an input probability distribution π on X into the output probability distribution π = π on Y. Denote by P (X) = π : π(x) ≥ 0, π(x) = 1 x∈X

the simplex of all probability distributions π on X. Extreme points of P (X) are the degenerate probability distributions δx on X. Notice the following obvious property: For a direct product X1 × X2 of two alphabets, extreme points of P (X1 × X2 ) are precisely the products of extreme points of P (Xj ): extP (X1 × X2 ) = extP (X1 ) × extP (X2 ). The most important characteristic of a channel is its capacity C() = max H (π) − π(x)H (δx ) , π ∈P (X)

(1)

(2)

x

where the expression in curly brackets is equal to the Shannon mutual information between the input and the output of the channel. Here H (π) = − π(x) log π(x) x

is the entropy of the probability distribution π. One of the main results of information theory – the coding theorem for memoryless channels, see e.g. [10] – says that the quantity (2) is the ultimate rate of asymptotically perfect transmission of information by n independent uses of the channel , when n → ∞. The capacity has the fundamental additivity property C(1 ⊗ 2 ) = C(1 ) + C(2 ).

(3)

Here the inequality ≥ (superadditivity) follows by restricting to the independent inputs, while the opposite inequality can be proved by using subadditivity of the output entropy H (π) and the property (1) for the second term in the Shannon information (which is equal to minus the conditional output entropy). The additivity is an important ingredient of the proof of the coding theorem, implying C(⊗n ) = nC(),

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where ⊗n = · · ⊗ . It expresses the “memoryless” character of the informa ⊗ · n

tion transmission scheme based on the independent uses of the channel. For schemes with memory the capacity can be strictly superadditive. In what follows we are going to describe the noncommutative analog of the quantity C(), as well as several other related quantities playing a basic role in quantum information theory. The corresponding additivity property was conjectured to hold also in the noncommutative case, although so far there is neither a general proof, nor a counterexample; moreover the additivity is no longer “natural” since an analog of the underlying basic fact (1) breaks dramatically in the noncommutative case.

3. Quantum channels 3.1. The χ-capacity. Let H be a unitary space and let M(H) be the algebra of all linear operators in H. By choosing an orthonormal basis, H can be identified with the space Hd of d-dimensional complex vectors and M(H) with the algebra Md of complex d × d-matrices. We shall consider linear maps which take operators F in d-dimensional unitary space H to operators F = (F ) in a d -dimensional space H . Sometimes these are called “superoperators” or “supermatrices” because they can be described as matrices with d 2 × d 2 entries [9]. Let j : M(Hj ) → M(Hj ); j = 1, 2, be two such maps, and let 1 ⊗ 2 : M(H1 ) ⊗ M(H2 ) → M(H1 ) ⊗ M(H2 ) be their tensor product defined by the natural action on product operators and then extended by linearity. An operator F ∈ M(H) is called positive, F ≥ 0, if the corresponding matrix is positive semidefinite and the map : M(H) → M(H ) is called positive if F ≥ 0 implies (F ) ≥ 0. Especially important for us will be the class of completely positive (CP) maps [51], [9], [43]. The map : M(H) → M(H ) is completely positive, if for d = 1, 2, . . . the maps ⊗ Idd are all positive, where Idd : Md → Md is the identity map of the algebra of d × d- matrices. It follows that the tensor product of CP maps is again CP, since 1 ⊗ 2 = (Idd1 ⊗ 2 ) (1 ⊗ Idd2 ). There are positive maps that are not CP, a basic example being provided by the matrix transposition F → F in a fixed basis. Finite quantum system is described by a unitary space H. The convex subset S(H) = {ρ : ρ ≥ 0, Tr ρ = 1} of M(H) is called the quantum state space. The operators ρ ∈ S(H) are called density operators or quantum states. The state space is a compact convex set with the

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extreme boundary P(H) = extS(H) = {ρ : ρ ≥ 0, Tr ρ = 1, ρ 2 = ρ}. Thus extreme points of S(H), which are also called pure states, are one-dimensional projectors, ρ = Pψ for a vector ψ ∈ H with unit norm, see, e.g. [42], [22]. Instead of the classical relation (1), one has the following relation for a tensor product H1 ⊗ H2 of two unitary spaces extS(H1 ⊗ H2 ) extS(H1 ) × extS(H2 ),

(4)

since apparently there are continually many pure states Pψ in H1 ⊗ H2 , given by vectors ψ not representable as a tensor product ψ1 ⊗ ψ2 . In quantum theory the tensor product H1 ⊗ H2 describes the composite (bipartite) system. Vectors that are not of the form ψ1 ⊗ψ2 , as well as the corresponding pure states, are called entangled. In an entangled pure state of a bipartite quantum system, neither of the parts is in a pure state, in a sharp contrast to the classical systems. A CP map is called a (quantum) channel, if it is trace preserving, i.e. if it maps quantum states into quantum states (possibly in another unitary space H ). A channel is called unital if d = d and (I ) = I , where I (I ) is the identity operator in H resp. H . The von Neumann entropy of a density operator ρ H (ρ) = − Tr ρ log ρ is nonnegative concave continuous function on S(H) vanishing on P(H) and taking the maximal value log d on the chaotic state ρ = dI . The noncommutative analog of the quantity (2) is the χ -capacity [20], [22] of the channel , Cχ () = max H π(x)ρ(x) − π(x)H ((ρ(x))) , (5) π

x

x

where the maximum is taken over all state ensembles i.e. finite probability distributions π on the quantum state space S(H) ascribing probabilities π(x) to density operators ρ(x)1 . The additivity conjecture is whether the analog of the property (3) holds for quantum channels, i.e. ?

Cχ (1 ⊗ 2 ) = Cχ (1 ) + Cχ (2 ).

(6)

Here again ⊗ is the tensor product of the two channels describing independent uses of the channels on the states of the composite system. This is the earliest additivity conjecture in quantum information theory which can be traced back to [5], see also [20], [6]. 1 In the finite dimensional case we are considering the maximum is indeed attained on π with support having at most d 2 states, where d = dimH [50].

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In physical terms this problem can also be formulated as: “Can entanglement between input states help to send classical information through quantum channels?” The classical capacity of a quantum channel is defined as the maximal transmission rate per use of the channel, with coding and decoding chosen for an increasing number n of independent uses of the channel ⊗n = · · ⊗ ⊗ · n

such that the error probability goes to zero as n → ∞. A basic result of quantum information theory – the quantum coding theorem [19], [49] – implies that the classical capacity C() and the χ-capacity Cχ () are connected by the formula C() = limn→∞ (1/n)Cχ (⊗n ). Since Cχ () is easily seen to be superadditive, i.e. Cχ (1 ⊗ 2 ) ≥ Cχ (1 ) + Cχ (2 ), one has C() ≥ Cχ (). If the additivity (6) holds, then Cχ (⊗n ) = nCχ (), and this would imply C() = Cχ (). Such a result would be very much welcome from a mathematical point of view, giving a relatively easily computable “single-letter” expression for the classical capacity of a quantum channel. On the other hand, such an equality is rather counter-intuitive in view of the relation (4) and existence of waste variety of pure entangled states. In fact, there are several quantities that are nonadditive under the tensor product of quantum channels such as: a) the Shannon information maximized over entangled outputs [20]; b) the quantum capacity [6]; c) the minimal output Rényi entropy [53] and some others, the classical counterparts of which are additive. In the following we shall consider the case c) which is most relevant to our main problem (6). 3.2. Entropic characteristics of CP maps and channels. The quantum Rényi entropy of order p > 1 of a density operator ρ is defined as Rp (ρ) =

1 log Tr ρ p , 1−p

(7)

so that the minimal output Rényi entropy of the channel is Rˇp () = min Rp ((ρ)) = ρ∈S(H )

p log νp (), 1−p

where νp () = max [Tr (ρ)p ]1/p ρ∈S(H)

(8)


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is a “measure of output purity” of the channel introduced in [3]2 . In the limit p ↓ 1 the quantum Rényi entropies monotonically increase and uniformly converge to the entropy of a density operator ρ, lim Rp (ρ) = H (ρ),

p↓1

so that introducing the minimal output entropy Hˇ () = min H ((ρ)) ρ∈S(H)

(9)

of the quantum channel , one has limp↓1 Rˇp () = Hˇ (). The classical analog of the quantity (8) is νp () = max π p , π ∈P (X)

p 1/p is the l -norm of f = (f (x)) where f p = p x∈X . The function x∈X |f (x)| π → π p is convex continuous and hence attains the maximum at an extreme point of P (X), i.e. on a degenerate probability distribution δx . Hence the basic property (1) implies the multiplicativity relation νp (1 ⊗ 2 ) = νp (1 )νp (2 ),

(10)

which is equivalent to the additivity property of the minimal output Rényi entropies

implying in turn

Rˇp (1 ⊗ 2 ) = Rˇp (1 ) + Rˇp (2 ),

(11)

Hˇ (1 ⊗ 2 ) = Hˇ (1 ) + Hˇ (2 ),

(12)

in the limit p ↓ 1. Notice that the inequality ≤ is obvious in (11), (12). Unlike the classical case, there is no apparent reason for these multiplicativity/additivity properties to hold in the case of quantum channels. Nevertheless there are several important classes of channels for which the multiplicativity (10) can be proved for all p > 1, although there is also an example where it breaks for sufficiently large p. This, however, does not preclude that it can hold for p close to 1, and the validity of (10) for p ∈ (1, 1 + ε), with ε > 0, implies validity of the additivity property (12), which, as we shall see, is closely related to the additivity of the χ-capacity (6). Here we would also like to mention that multiplicativity of more general (q → p)norms was studied for the cases where at least some of the maps 1 , 2 is not CP, see [38], [37], [35]. Basing on the advanced theory of the operator Lp -spaces [44], [43], there is an interesting study concerning the multiplicativity of completely bounded p-norms, which however is related to the additivity of a completely different entropic quantity [15]. 2 In the finite-dimensional case all the functions of the state we are considering are easily seen to be continuous and their extrema on the state space are attained. However it is not so in infinite-dimensional case, and then the attainability of the extrema requires separate study, see Section 6.

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4. Some classes of CP maps and channels 4.1. Representations of CP maps. Here we recollect some facts concerning the structure of CP maps and channels. Given three unitary spaces HA , HB , HC and a linear operator V : HA → HB ⊗ HC , the relation (ρ) = Tr HC VρV ∗ ,

˜ (ρ) = Tr HB VρV ∗ ;

ρ ∈ M(HA )

(13)

˜ : M(HA ) → M(HC ), which are defines two CP maps : M(HA ) → M(HB ), called mutually complementary [25] (or conjugate [36]). If V is an isometry then both maps are channels. For any linear map : M(H) → M(H ) the dual map ∗ : M(H ) → M(H) is defined by the formula Tr (ρ)X = Tr ρ∗ (X);

ρ ∈ M(H), X ∈ M(H ).

If is CP, then ∗ is also CP. The relations (13) are equivalent to ∗ (X) = V ∗ (X ⊗ IC )V ;

X ∈ M(HB ),

(14)

˜ ∗ (X) = V ∗ (IB ⊗ X)V ;

X ∈ M(HC ).

(15)

The Stinespring dilation theorem [51] concerning CP maps on arbitrary C ∗ -algebras, for the particular case in question amounts to the statement that for a given CP map there are a space HC and an operator V satisfying (14). This implies that given a CP ˜ always exists. map , a complementary map By introducing a basis {ejC } in HC and operators Vj : HA → HB defined by

(ϕ, Vj ψ) = ϕ ⊗ ejC , V ψ ; ϕ ∈ HB , ψ ∈ HA , the first relation in (13) can be rewritten as (ρ) =

dC

Vj ρVj∗ ;

ρ ∈ M(HA ).

(16)

j =1

C The map (16) is a channel if and only if jd=1 Vj∗ Vj = I . The relation (16) is usually called the Kraus representation (see also Choi [9]). Of course, there are similar ˜ and the dual maps. representations for the complementary map Theorem 4.1 ([25], [36]). If one of the relations (11), (12) holds for the CP maps (channels) 1 , 2 , then similar relations holds for the pair of their complementary ˜ 1, ˜ 2 . If one of these relations holds for a given 1 and arbitrary 2 , then a maps ˜ 1 and arbitrary 2 . similar relation holds for the complementary map Validity of the multiplicativity conjecture (10) for all p ≥ 1 and of the additivity conjectures (12), (6) was established in a number of cases where one channel is arbitrary and the other belongs to one of the classes we are going to discuss.

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4.2. Entanglement-breaking maps and their complementary maps. Any linear map : M(H) → M(H ) admits a representation (X) = Nj Tr XMj , (17) j

where {Mj }, {Nj } are finite collections of operators in H and H , respectively. This simply follows from the finite dimensionality of H, H and the fact that any linear functional on M(H) has the form X → TrXM, where M ∈ M(H). Proposition 4.2. For a linear map : M(H) → M(H ) the following conditions are equivalent: (i) There is a representation (17) such that Mj ≥ 0, Nj ≥ 0. (ii) The map is CP and has the representation (16) with rank one operators Vj . (iii) For d = 2, 3, . . . and any ρ12 ∈ S(H ⊗ Hd ), Aα ⊗ Bα (18) ( ⊗ Idd )(ρ12 ) = α

where Aα ≥ 0 and Bα ≥ 0. Channels satisfying the condition (i) were introduced in [20], and the above characterization was obtained in [29] where such maps were termed entanglement-breaking. In the case of channels, (18) means that the output state ( ⊗ Idd )(ρ12 ) is always separable, i.e. a convex combination of (unentangled) product states. Entanglementbreaking channels can be written in the form (ρ) = ρj Tr ρMj , (19) j

where {ρj } is a finite collection of states in H , and {Mj } a resolution of the identity in H, i.e. a collection of operators satisfying Mj ≥ 0, Mj = I. j

Resolutions of the identity describe quantum observables [22], and the channel (19) corresponds to a measurement of the observable {Mj } over an input state ρ resulting in a probability distribution {Tr ρMj }, which is followed by preparation of the output state ρj . Thus, there is a classical information processing stage inside the channel which is responsible for the entanglement-breaking. The simplest example is the completely depolarizing channel (ρ) =

I Tr ρ d

which maps an arbitrary state to the chaotic state dI .

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As shown in [25], [36], the complementary maps to the entanglement-breaking maps have the form ˜ (ρ) =

dC

cj k ψj |ρ|ψk Ej k ,

ρ ∈ M(HA ),

(20)

j,k=1

where [cj k ] is a nonnegative definite matrix, {ψj }j =1,dC a system of vectors in HA , and the Ej k ’s are the matrix units in HC . In the special case where {ψj }j =1,dC is an orthonormal basis, (20) is the diagonal CP map in the sense of [31]. Diagonal channels are characterized by the additional property cjj ≡ 1. A simplest example of the diagonal channel is the ideal channel Id, which is complementary to the completely depolarizing channel. For general entanglement-breaking channels the additivity property (12) with arbitrary second channel was established by Shor [47], preceded by results in [20] on special subclasses of such channels. The multiplicativity property (10) for all p > 1 was established by King [32], basing on the Lieb–Thirring inequality [40]: for A, B ∈ M(H), A, B ≥ 0, and p ≥ 1 Tr(AB)p ≤ Tr Ap B p .

(21)

By Theorem 4.1 this implies the corresponding properties for the complementary maps and channels of the form (20). 4.3. Covariant channels. Let G be a group (either finite or continuous) and let g → UgA , UgB , g ∈ G, be two projective (unitary) representations of G in HA , HB . The CP map : M(HA ) → M(HB ) is covariant if (UgA ρUgA∗ ) = UgB (ρ)UgB ∗

(22)

for all g ∈ G and all ρ. For a covariant CP map there exists a covariant Stinespring dilation: namely, there is a projective representation g → UgC in HB , such that (UgB ⊗ UgC )V = V UgA , see e.g. [24]. It follows that the complementary map is also covariant: C∗ ˜ gA ρUgA∗ ) = UgC (ρ)U ˜ (U g .

Lemma 4.3. If the representation UgA is irreducible, then IA Cχ () = H − Hˇ (). dA

(23)

Since the tensor product of irreducible representations of possibly different groups G1 , G2 is an irreducible representation of the group G1 × G2 , it follows that the additivity properties (12) and (6) are equivalent for channels satisfying the condition of Lemma 4.3. Symmetry considerations also help to compute explicitly the entropic characteristics of covariant channels. Then, in the case of additivity, C = Cχ gives an explicit expression for the classical capacity of the channel.

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4.4. The unital qubit channels. The simplest and yet fundamental quantum system is the qubit (quantum bit), where dimH = 2. A convenient basis in M2 is formed by the Pauli matrices 1 0 0 1 0 −i 1 0 , σx = , σy = , σz = . σ0 ≡ I = 0 1 1 0 i 0 0 −1 It is known [45] that an arbitrary unital channel : M2 → M2 can be decomposed as (24) (ρ) = U2 (U1 ρU1∗ )U2∗ , where U1 , U2 are unitary matrices and has the following canonical Kraus representation:

(ρ) = μγ σγ ρσγ , (25) γ =0,x,y,z

where {μγ } is a probability distribution. The unital qubit channels (25) are covariant with respect to the projective representation of the group Z2 × Z2 defined by U00 = σ0 ,

U01 = σz ,

U10 = σx ,

U11 = −iσy .

Therefore the relation (23) holds for this class of channels. By using a convex decomposition into diagonal channels of special form and applying to these the Lieb–Thirring inequality (21), King [33] established (10) for all p > 1, (12) and (6) for the case where 1 is an arbitrary unital qubit channel and 2 is an arbitrary channel. There are recent positive results concerning nonunital qubit channels [35]. 4.5. Depolarizing channel. The depolarizing channel in Hd is I (ρ) = (1 − p)ρ + p Tr ρ, d

0≤p≤

d2 . d2 − 1

(26)

If p ≤ 1 this describes a mixture of the ideal channel Id and the completely depo2 larizing channel. For the whole range 0 ≤ p ≤ d 2d−1 complete positivity can be proven by using the Kraus decomposition, see e.g. [42]. The depolarizing channel is characterized by the property of unitary covariance, (UρU ∗ ) = U (ρ)U ∗ , for an arbitrary unitary operator U in H. The properties (10) for all p > 1, (12) and (6) were proved in [34] for the case where 1 is a depolarizing channel and 2 is arbitrary, using a method similar to the case of the unital qubit channels. Complementarity for depolarizing channels is computed in [11].

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4.6. A transpose-depolarizing channel. Let us consider in some detail the extreme transpose-depolarizing channel 1 (ρ) = (27) I Tr ρ − ρ , d −1 where ρ is the transpose of ρ in an orthonormal basis {ej } in Hd . Complete positivity of the map (27) follows from the representation (ρ) =

d 1 (Ej k − Ekj )ρ(Ej k − Ekj )∗ . 2(d − 1)

(28)

j,k=1

It has the covariance property (UρU ∗ ) = U (ρ)U ∗ for an arbitrary unitary U , where U is complex conjugate in the basis {ej }. It follows that the relation (23) holds for this channel. This channel is interesting in that it breaks the additivity of the minimal Rényi entropy (11) with 1 = 2 = for d > 3 and large enough p [53]. At the same time it fulfills (12), see [41], [12], and even (11) for 1 ≤ p ≤ 2 [13]. For generalizations to broader classes of channels as well as to the more general forms of additivity, see [41], [1], [54]. This example also shows that although the Lieb–Thirring inequality can be used in several cases to prove the additivity conjecture (11) for all p > 1, it cannot serve for a general proof. Moreover, there is even no general proof covering all these cases, since each time application of the Lieb–Thirring inequality is supplied with an argument specific to the case under consideration. The complementary channel which shares the multiplicativity/additivity properties with the channel (27) is ˜ (ρ) =

2 P− (ρ ⊗ I2 )P− (d − 1)

(29)

(see [25] for more details). Here P− is the projector onto the antisymmetric subspace of H ⊗ H of dimension d(d−1) 2 . The covariance property of the channel (29) is ∗ ∗ ˜ ˜ (UρU ) = (U ⊗ U )(ρ)(U ⊗ U ∗ ),

as follows from the fact that P− (U ⊗ U ) = (U ⊗ U )P− .

5. A hierarchy of the additivity conjectures 5.1. Convex closure. To find out the intrinsic connection between the output entropy and the χ -capacity, let us define the average ρ¯π = x π(x)ρ(x) of the ensemble π and rewrite the expression (5) in the form Cχ () = max H ((ρ)) − Hˆ (ρ) , (30) ρ∈S(H)


where Hˆ (ρ) = min

π :ρ¯π =ρ

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π(x)H ((ρ(x)))

x

is the convex closure [30] of the output entropy H ((ρ))3 . The function Hˆ (ρ) is a natural generalization of another important quantity in quantum information theory, namely the “entanglement of formation” [6] and reduces to it when the channel is a partial trace. This quantity has the conjectured superadditivity property: for an arbitrary state ρ12 ∈ S(H1 ⊗ H2 ) and arbitrary channels 1 , 2 , Hˆ 1 ⊗2 (ρ12 ) ≥ Hˆ 1 (ρ1 ) + Hˆ 2 (ρ2 ),

(31)

where ρ1 , ρ2 are the partial traces of ρ12 in H1 , H2 . It is not difficult to see that this property implies additivity of both the minimal output entropy and the χ -capacity: Proposition 5.1. The superadditivity property (31) implies the additivity properties (12) and (6) for given channels 1 , 2 . In the spirit of Theorem 4.1, one can prove ([25]) that if the relation (31) holds for the pair of CP maps (channels) 1 , 2 , then similar ˜ 1, ˜ 2 . If one of these relation holds for the pair of their complementary maps relations holds for given 1 and arbitrary 2 , then a similar relation holds for the ˜ 1 and arbitrary 2 . complementary map Let {pj } be a finite probability distribution and let j : M(H) → M(Hj ) be a col

lection of channels. The channel : M(H) → M ⊕Hj is called orthogonal j convex sum of the channels j , if (ρ) = j ⊕pj j (ρ) for all ρ ∈ S(H). Proposition 5.2 ([26]). Let 2 be an arbitrary channel. The properties (11), (12), (31) hold if 1 is an orthogonal convex sum of either an ideal channel or completely depolarizing channel and a channel (0) such that the corresponding property holds for (0) and 2 . It follows that such a 1 fulfils the additivity of χ-capacity (6). In this way, for example, one obtains all the additivity properties for the important case of the erasure channel pρ 0 , (ρ) = 0 (1 − p) Tr ρ as it is the orthogonal convex sum of an ideal and a completely depolarizing channel. 3 Here the same comment applies as to the attainability of the maximum in (5).

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5.2. Additivity for constrained channels. In this section we consider several equivalent formulations of the additivity conjecture for channels with arbitrarily constrained inputs [26], which formally is substantially stronger than additivity of the unconstrained χ-capacity. Let us denote χ (ρ) = H ((ρ)) − Hˆ (ρ),

(32)

then the function χ (ρ) is continuous and concave on the set S(H) of all states in H. Consider the constraint on the ensemble π with the average ρ¯π = x π(x)ρ(x), defined by the requirement ρ¯π ∈ A, where A is a closed subset of states. A particular case is the linear constraint A = {ρ : Tr ρA ≤ α} for a positive operator A and a number α ≥ 0. Define the χ -capacity of the A-constrained channel by Cχ (; A) = max χ (ρ). ρ∈A

(33)

Note that the χ -capacity for the unconstrained channel is Cχ () = C(; S(H)). On the other hand, χ (ρ) = Cχ (; {ρ}). Let 1 , 2 be two channels with the constraints A1 , A2 . For the channel 1 ⊗ 2 we introduce the constraint A1 ⊗ A2 ≡ {ρ : Tr H2 ρ ∈ A1 , Tr H1 ρ ∈ A2 } and consider the conjecture Cχ (1 ⊗ 2 ; A1 ⊗ A2 ) = Cχ (1 ; A1 ) + Cχ (2 ; A2 ),

(34)

which apparently implies (6). Theorem 5.3. Let 1 and 2 be two fixed channels. The following properties are equivalent: (i) Equality (34) holds for arbitrary linear constraints A1 , A2 . (ii) Equality (34) holds for arbitrary closed A1 , A2 . (iii) For arbitrary ρ12 ∈ S(H1 ⊗ H2 ), χ1 ⊗2 (ρ12 ) ≤ χ1 (ρ1 ) + χ2 (ρ2 ).

(35)

(iv) Inequality (31) holds for arbitrary ρ12 ∈ S(H1 ⊗ H2 ). Here each property is easily seen to imply the preceding one, while the implication (i) ⇒ (iv) is nontrivial. By Proposition 5.1 any of these properties imply the additivity properties (12), (6). 5.3. The convex duality formulation. In [4], tools from convex analysis were applied to study the relation of the additivity problem to superadditivity of entanglement

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of formation. Here we apply a similar approach to the conjecture (31). Given a channel , its output entropy H ((ρ)) is a continuous concave function on the state space S(H). Consider its modified Legendre transform H∗ (X) = min {Tr ρX + H ((ρ))} ρ∈S(H )

= min {Tr ρX + Hˆ (ρ)}, ρ∈S(H )

X ∈ Mh (H),

(36)

where Mh (H) is a real normed space of Hermitian operators in H. Now let 1 , 2 be two channels. Lemma 5.4. The superadditivity (31) of the convex closure Hˆ (ρ) is equivalent to the following additivity property of H∗ (X): H∗ 1 ⊗2 (X1 ⊗ I2 + I1 ⊗ X2 ) = H∗ 1 (X1 ) + H∗ 2 (X2 ),

(37)

for all X1 ∈ Mh (H1 ), X2 ∈ Mh (H2 ). Since H∗ (0) = Hˇ (), by letting X1 = X2 = 0, the relation (37) implies additivity of the minimal output entropy (12). 5.4. The global equivalence. A remarkable result was obtained by Shor [48] who showed that different forms of the additivity conjecture become equivalent if one considers their validity for all channels. Here we describe a basic construction from [48] which in combination with Proposition 5.1 and Theorem 5.3 suffices for the proof of the following result. Theorem 5.5. The conjectures (6), (12), (31), (34) are globally equivalent in the sense that if one of them holds true for all channels 1 , 2 , then any of the others is also true for all channels. Let us argue that if additivity of the minimal output entropy (12) holds for all channels, then (37) holds for all channels. By Lemma 5.4 this will imply (31) and hence, by Theorem 5.3, all the other properties. First of all we observe that H∗ (X + λI ) = H∗ (X) + λ, which implies that it is sufficient to establish (37) only for X1 , X2 ≥ 0. The idea of proof is to build, for any channel and X ≥ 0, a sequence of channels X,n such that min H (X,n (ρ)) ≡ Hˇ (X,n ) = min[H ((ρ)) + Tr ρX] + o(1) ≡ H∗ (X) + o(1). ρ

ρ

One can then apply the convex duality argument to deduce for the original channels the additivity property (37), which is equivalent to (31), from the additivity of the minimal output entropy for channels X,n . Given a channel : M(H) → M(H ) and a positive X ∈ H, the new channel X,n is constructed as follows. Choose a constant c ≥ X , then E = c−1 X satisfies 0 ≤ E ≤ I . Let qn ∈ (0, 1) be such that (1 − qn ) log n = c,

n = 2, 3, . . . .

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Then X,n : M(H) → M(Hn ), where Hn = H ⊕H n ⊕ C, acts on ρ ∈ M(H) as follows: ⎤ ⎡ qn (ρ) 0 0 ⎦. X,n (ρ) = ⎣ 0 (38) 0 (1 − qn )(Tr ρE) Inn 0 0 (1 − qn ) Tr ρ(I − E) This is an orthogonal convex sum of CP maps, preserving trace, and hence is a channel. The intuition is that the action of X,n (ρ) can be described as follows. With probability qn (which tends to 1 as n → ∞) it acts as the channel , resulting in the state (ρ). With probability (1 − qn ), however, a quantum measurement described by the resolution of the identity (the quantum observable) {E, I − E} is made, so that the first outcome appears with probability Tr ρE, while the second appears with probability Tr ρ(I − E). In the first case the output is the chaotic state Inn in the n-dimensional unitary space Hn ; in the second case the output is a pure state orthogonal to H ⊕H n . In this way the channel X,n (ρ) with high probability qn acts as the initial channel, while with a small probability (1 − qn )(Tr ρE) outputs a high dimensional chaotic state Inn , providing the knowledge about the value of Tr ρE = c−1 Tr ρX involved in the definition of H∗ (X). This is formalized by proving the uniform estimate

H (X,n ⊗ 2 )(ρ12 ) = qn H ( ⊗ 2 )(ρ12 ) + Tr ρ1 X + o(1), double application of which reduces the property (37) for initial channels to the additivity of the minimal output entropy for the channels X,n . A modification of this construction can be also used to show that if the unconstrained additivity (6) holds for all channels, then additivity (34) for all channels with arbitrary constraints holds as well [26]. This completes the global equivalence.

6. Infinite-dimensional channels We have seen that the additivity problem is not completely solved even for the minimal dimension 2. Nevertheless there are several good reasons to consider the problem in infinite dimensions. There is an important and interesting class of Bosonic Gaussian channels, see [28], which act in infinite dimensional Hilbert space. Analysis of continuity properties of the entropic characteristics of an infinite-dimensional channel becomes important since, as is well known, the entropy may then have a rather pathological behavior. It is only lower semicontinuous and “almost everywhere” infinite in the infinitedimensional case [52]. Another issue is the study of conditions for compactness of subsets of quantum states and ensembles, giving a key for attainability of extrema in expressions for the capacity and the convex closure of the output entropy. The proof of global equivalence of different forms of the additivity conjecture for finite dimensional channels (Section 5.4), using infinitely growing channel extensions


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in fact relies upon the discontinuity of the χ-capacity as a function of the channel in infinite dimensions. This also calls for a study of continuity properties of the entropic quantities related to the classical capacity of infinite dimensional channels. Such a study was undertaken in a series of works [23], [27], [46]. In particular it was shown that in spite of the aforementioned discontinuities, additivity for all finite-dimensional channels implies additivity of the χ-capacity of infinite-dimensional channels with arbitrary constraints [46]. There are two important features essential for channels in infinite dimensions. One is the necessity of the input constraints (such as mean energy constraint for Gaussian channels) to prevent from infinite capacities (although considering input constraints was shown quite useful also in the study of the additivity conjecture for channels in finite dimensions [26]). The other is the natural appearance of infinite, and, in general, “continuous” state ensembles understood as probability measures on the set of all quantum states. By using compactness criteria from probability and operator theory one can show that the set of all such generalized ensembles with the barycenter in a compact set of states is itself weakly compact. With this in hand a sufficient condition for existence of an optimal generalized ensemble for a constrained quantum channel can be given. This condition can be efficiently verified in the case of Bosonic Gaussian channels with constrained mean energy [27]. However apart from mere existence one would like to have an explicit description of the optimal states and ensembles in the case of quantum Gaussian channels. In classical information theory Gaussian channels have Gaussian maximizers, and there is an analytical counterpart of this phenomenon for (q → p)-norms of integral operators with Gaussian kernels, see [39]. Whether a similar description holds true for Bosonic Gaussian channels is another open question (for some partial results in this direction see [28], [17], [18], [55]). We only mention here that a positive solution of this question may also depend on the validity of the multiplicativity/ additivity conjecture [39], [55].

References [1] Alicki, R., Fannes, M., Note on multiple additivity of minimal Rényi entropy output of the Werner-Holevo channels. quant-ph/0407033. [2] Amosov, G. G., Holevo, A. S., On the multiplicativity conjecture for quantum channels. Theor. Probab. Appl. 47 (1) (2002), 143–146. [3] Amosov, G. G., Holevo, A. S., and Werner, R. F., On some additivity problems in quantum information theory. Probl. Inform. Transm. 36 (4) (2000), 25–34. [4] Audenaert, K. M. R., Braunstein, S. L., On strong superadditivity of the entanglement of formation. Commun. Math. Phys. 246 (2004), 443–452. [5] Bennett, C. H., Fuchs, C. A., Smolin, J. A., Entanglement-enhanced classical communication on a noisy quantum channel. In Quantum Communication, Computing and Measurement, Proc. QCM96 (ed. by O. Hirota, A. S. Holevo and C. M. Caves), Plenum, New York 1997, 79–88.

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[6] Bennett, C. H., Shor, P. W., Quantum information theory. IEEE Trans. Inform. Theory 44 (1998), 2724–2742. [7] Bhatia, R., Matrix Analysis. Grad. Texts in Math. 169, Springer-Verlag, New York 1997. [8] Carlen, E. A., Lieb, E. H., A Minkowski type trace inequality and strong subadditivity of quantum entropy. In Differential operators and spectral theory (ed. by V. Buslaev, M. Solomyak and D. Yafaev), Amer. Math. Soc. Transl. Ser. 2 189, Amer. Math. Soc., Providence, RI, 1999, 59–68. [9] Choi, M.-D., Completely positive maps on complex matrices. Linear Algebra Appl. 10 (1975), 285–290. [10] Cover, T. M., Thomas, J. A., Elements of Information Theory. J. Wiley and Sons, New York 1991. [11] Datta, N., Holevo, A. S., Complementarity and additivity for depolarizing channels. quantph/0510145. [12] Datta, N., Holevo, A. S., Suhov, Y. M., A quantum channel with additive minimum output entropy. quant-ph/0408176. [13] Datta, N., Multiplicativity of maximal p-norms in Werner-Holevo channels for 1 ≤ p ≤ 2. quant-ph/0410063. [14] Devetak, I., Hayden, A. W., Winter, A., A resource framework for quantum Shannon theory. quant-ph/0512015. [15] Devetak, I., Junge, M., King, C., Ruskai, M. B., Multiplicativity of completely bounded p-norms implies a new additivity result. quant-ph/0506196. [16] Fukuda, M., Holevo, A. S., On Weyl-covariant channels. quant-ph/0510148. [17] Giovannetti, V., Lloyd, S., Maccone, L., Shapiro, J. H., Yen, B. J., Minimum Rényi and Wehrl entropies at the output of bosonic channels. quant-ph/0404037. [18] Giovannetti, V., Lloyd, S., Additivity properties of a Gaussian channel. quant-ph/0403075. [19] Holevo, A. S., The capacity of quantum communication channel with general signal states. IEEE Trans. Inform. Theory 44 (1) (1998), 269–272. [20] Holevo, A. S., Quantum coding theorems. Russ. Math. Surveys 53 (1998), 1295–1331. [21] Holevo, A. S., Statistical structure of quantum theory. Lect. Notes Phys. Monogr. 67, Springer-Verlag, Berlin 2001. [22] Holevo, A. S., An introduction to quantum information theory. MCCME (Moscow Independent University), Moscow 2002. [23] Holevo, A. S., Entanglement-assisted capacities of constrained quantum channels. Theory Probab. Appl. 48 (2003), 243–255. [24] Holevo, A. S., Additivity conjecture and covariant channels. Int. J. Quant. Inform. 3 (1) (2005), 41–48. [25] Holevo, A. S., On complementary channels and the additivity problem. quant-ph/0509101. [26] Holevo, A. S., Shirokov, M. E., On Shor’s channel extension and constrained channels. Commun. Math. Phys. 249 (2004), 417–430. [27] Holevo, A. S., Shirokov, M. E., Continuous ensembles and the χ-capacity of infinitedimensional channels. quant-ph/0403072. [28] Holevo, A. S., Werner, R. F., Evaluating capacities of Bosonic Gaussian channels. Phys. Rev. A. 63 (2001), 032312.


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[29] Horodecki, M., Shor, P. W., Ruskai, M. B., General entanglement breaking channels. Rev. Math. Phys. 15 (2003), 629–641. [30] Magaril-Il’yaev, G. G., Tikhomirov, B. M., Convex analysis: theory and applications. Editorial URSS, Moscow 2000, Transl. Math. Monogr. 222, Amer. Math. Soc, Providence, RI, 2003. [31] King, C., An application of the matrix inequality in quantum information theory. quantph/0412046. [32] King, C., Maximal p-norms of entanglement breaking channels. quant-ph/0212057. [33] King, C., Additivity for a class of unital qubit channels. quant-ph/0103156. [34] King, C., The capacity of the quantum depolarizing channel. quant-ph/0204172. [35] King, C., Koldan, N., New multiplicativity results for qubit maps. quant-ph/0512185. [36] King, C., Matsumoto, K., Natanson, M., and Ruskai, M. B., Properties of conjugate channels with applications to additivity and multiplicativity. quant-ph/0509126. [37] King, C., Nathanson, M., Ruskai, M.-B., Multiplicativity properties of entrywise positive maps on matrix algebras. quant-ph/0409181. [38] King, C., Ruskai, M.-B., Comments on multiplicativity of maximal p-norms when p = 2. Quantum Inf. Comput. 4 (2004), 500–512. [39] Lieb, E. H., Gaussian kernels have only Gaussian maximizers, Invent. Math. 102 (1990), 179–208. [40] Lieb, E. H., Thirring, W. E., Inequalities for the moments of the eigenvalues of the Schrödinger Hamiltonian and their relation to Sobolev inequalities. In Essays in Honor of Valentine Bargmann (ed. by E. H. Lieb, B. Simon, A. Wightman), Stud. Math.Phys., Princeton University Press, 1976, 269–297. [41] Matsumoto, K., Yura, F., Entanglement cost of antisymmetric states and additivity of capacity of some channels. quant-ph/0306009. [42] Nielsen, M. A., Chuang, I., Quantum Computation and Quantum Information. Cambridge University Press, Cambridge 2000. [43] Paulsen, V., Completely bounded maps and operator algebras. Cambridge University Press, Cambridge 2002. [44] Pisier, G., Non-Commutative Vector Valued Lp -spaces and Completely p-summing Maps. Astérisque 247 (1998). [45] Ruskai, M. B., Szarek, S., Werner, E., An analysis of completely-positive trace-preserving maps on 2 × 2 matrices. Linear Algebra Appl. 347 (2002), 159–187. [46] Shirokov, M. E., The Holevo capacity of infinite dimensional channels. Commun. Math. Phys. 262 (2006), 137–159. [47] Shor, P. W., Additivity of the classical capacity of entanglement-breaking quantum channels. J. Math. Phys. 43 (2003), 4334–4340. [48] Shor, P. W., Equivalence of additivity questions in quantum information theory. Commun. Math. Phys. 246 (2004), 453–472. [49] Schumacher, B., Westmoreland,M. D. , Sending classical information via noisy quantum channel. Phys. Rev. A. 56 (1) (1997), 131–138. [50] Schumacher, B., Westmoreland, M. D. , Optimal signal ensembles. Phys. Rev. A 63 (2001), 022308.

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[51] Stinespring, W. F., Positive functions on C ∗ -algebras. Proc. Amer. Math. Soc. 6 (1955), 211–316. [52] Wehrl, A., General properties of entropy. Rev. Mod. Phys. 50 (1978), 221–250. [53] Werner, R. F., Holevo, A. S., Counterexample to an additivity conjecture for output purity of quantum channels. J. Math. Phys. 43 (2002), 4353. [54] Wolf, M. M., Eisert, J., Classical information capacity of a class of quantum channels. quant-ph/0412133. [55] Wolf, M. M., Giedke, G., Cirac, J. I., Extremality of Gaussian quantum states. quantph/0509154. Steklov Mathematical Institute, Gubkina 8, 119991 Moscow, Russian Federation E-mail: [email protected]

Complex networks and decentralized search algorithms Jon Kleinberg∗

Abstract. The study of complex networks has emerged over the past several years as a theme spanning many disciplines, ranging from mathematics and computer science to the social and biological sciences. A significant amount of recent work in this area has focused on the development of random graph models that capture some of the qualitative properties observed in large-scale network data; such models have the potential to help us reason, at a general level, about the ways in which real-world networks are organized. We survey one particular line of network research, concerned with small-world phenomena and decentralized search algorithms, that illustrates this style of analysis. We begin by describing a well-known experiment that provided the first empirical basis for the “six degrees of separation” phenomenon in social networks; we then discuss some probabilistic network models motivated by this work, illustrating how these models lead to novel algorithmic and graph-theoretic questions, and how they are supported by recent empirical studies of large social networks. Mathematics Subject Classification (2000). Primary 68R10; Secondary 05C80, 91D30. Keywords. Random graphs, complex networks, search algorithms, social network analysis.

1. Introduction Over the past decade, the study of complex networks has emerged as a theme running through research in a wide range of areas. The growth of the Internet and the World Wide Web has led computer scientists to seek ways to manage the complexity of these networks, and to help users navigate their vast information content. Social scientists have been confronted by social network data on a scale previously unimagined: datasets on communication within organizations, on collaboration in professional communities, and on relationships in financial domains. Biologists have delved into the interactions that define the pathways of a cell’s metabolism, discovering that the network structure of these interactions can provide insight into fundamental biological processes. The drive to understand all these issues has resulted in what some have called a “new science of networks” – a phenomenological study of networks as they arise in the physical world, in the virtual world, and in society. At a mathematical level, much of this work has been rooted in the study of random graphs [14], an area at the intersection of combinatorics and discrete probability that ∗ Supported in part by a David and Lucile Packard Foundation Fellowship, a John D. and Catherine T. MacArthur Foundation Fellowship, and NSF grants CCF-0325453, IIS-0329064, CNS-0403340, and BCS0537606.


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is concerned with the properties of graphs generated by random processes. While this has been an active topic of study since the work of Erdös and Rényi in the 1950s [26], the appearance of rich, large-scale network data in the 1990s stimulated a tremendous influx of researchers from many different communities. Much of this recent cross-disciplinary work has sought to develop random graph models that more tightly capture the qualitative properties found in large social, technological, and information networks; in many cases, these models are closely related to earlier work in the random graphs literature, but the issues arising in the motivating applications lead to new types of mathematical questions. For surveys covering different aspects of this general area, and in particular reflecting the various techniques of some of the different disciplines that have contributed to it, we refer the reader to recent review papers by Albert and Barabási [4], Bollobás [15], Kleinberg and Lawrence [39], Newman [52], and Strogatz [60], the volume of articles edited by Ben-Naim et al. [10], and the monographs by Dorogovtsev and Mendes [23] and Durrett [25], as well as books by Barabási [8] and Watts [62] aimed at more general audiences. What does one hope to achieve from a probabilistic model of a complex network arising in the natural or social world? A basic strategy pursued in much of this research is to define a stylized network model, produced by a random mechanism that reflects the processes shaping the real network, and to show that this stylized model reproduces properties observed in the real network. Clearly the full range of factors that contribute to the observed structure will be too intricate to be fully captured by any simple model. But a finding based on a random graph formulation can help argue that the observed properties may have a simple underlying basis, even if their specifics are very complex. While it is crucial to realize the limitations of this type of activity – and not to read too much into the detailed conclusions drawn from a simple model – the development of such models has been a valuable means of proposing concrete, mathematically precise hypotheses about network structure and evolution that can then serve as starting points for further empirical investigation. And at its most effective, this process of modeling via random graphs can suggest novel types of qualitative network features – structures that people had not thought to define previously, and which become patterns to look for in new network datasets. In the remainder of the present paper, we survey one line of work, motivated by the “small-world phenomenon” and some related search problems, that illustrates this style of analysis. We begin with a striking experiment by the social psychologist Stanley Milgram that frames the empirical issues very clearly [50], [61]; we describe a sequence of models based on random graphs that capture aspects of this phenomenon [64], [36], [37], [38], [63]; and we then discuss recent work that has identified some of the qualitative aspects of these models in large-scale network data [1], [43], [49]. We conclude with some further extensions to these random graph models, discussing the results and questions that they lead to.

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2. The small-world phenomenon The small-world phenomenon – the principle that we are all linked by short chains of acquaintances, or “six degrees of separation” [29] – has long been the subject of anecdotal fascination among the general public, and more recently has become the subject of both experimental and theoretical research. At its most basic level, it is a statement about networks, and human social networks in particular; it concerns the graph with one node corresponding to each person in the world, and an edge joining two people if they know each other on a first-name basis. When we say that this graph is a “small world,” we mean, informally, that almost every pair of nodes is connected by a path with an extremely small number of steps. One could worry about whether this graph is precisely specified – for example, what exactly it means to know someone on a first-name basis – but however one fixes a working definition for this, it is clear that the resulting graph encodes an enormous amount of information about society in general. It is also clear that it would be essentially impossible to determine its structure precisely. How then could one hope to test, empirically, the claim that most pairs of nodes in this graph are connected by short paths? The social psychologist Stanley Milgram [50], [61] took up this challenge in the 1960s, conducting an experiment to test the small-world property by having people explicitly construct paths through the social network defined by acquaintanceship. To this end, he chose a target person in the network, a stockbroker living in a suburb of Boston, and asked a collection of randomly chosen “starter” individuals each to forward a letter to the target. He provided the target’s name, address, occupation, and some personal information, but stipulated that the participants could not mail the letter directly to the target; rather, each participant could only advance the letter by forwarding it to a single acquaintance that he or she knew on a first-name basis, with the goal of reaching the target as rapidly as possible. Each letter thus passed successively from one acquaintance to another, closing in on the stockbroker outside Boston. The letters thus acted as virtual “tracers,” mapping out paths through the social network. Milgram found that the median length among the completed paths was six, providing the first concrete evidence for the abundance of short paths connecting far-flung pairs of individuals in society, as well as supplying the basis for the number “six” in the resulting pop-cultural mantra. One needs to be careful in interpreting this finding, of course: many of the chains never reached the target, and the target himself was a relatively “high-status” individual who may have been easier to reach than an arbitrary person (see e.g. the recent critique by Kleinfeld [41]). But since Milgram’s work, the overall conclusion has been accepted at least at a qualitative level: social networks tend to exhibit very short paths between essentially arbitrary pairs of nodes.

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3. Basic models of small-world networks Why should social networks exhibit this type of a small-world property? Earlier we suggested that interesting empirical findings about networks often motivate the development of new random graph models, but we have to be careful in framing the issue here: a simple abundance of short paths is in fact something that most basic models of random graphs already “get right.” As a paradigmatic example of such a result, consider the following theorem of Bollobás and de la Vega [14], [17]. Theorem 3.1 ([17]). Fix a constant k ≥ 3. If we choose uniformly at random from the set of all n-node graphs in which each node has degree exactly k, then with high probability every pair of nodes will be joined by a path of length O(log n). (Following standard notation and terminology, we say that the degree of a node is the number of edges incident to it. We say that a function is O(f (n)) if there is a constant c so that for all sufficiently large n, the function is bounded by cf (n).) In fact, [17] states a much more detailed result concerning the dependence on n, but this will not be crucial for our purposes here. Path lengths that are logarithmic in n – or more generally polylogarithmic, bounded by a polynomial function of log n – will be our “gold standard” in most of this discussion. We will keep the term “small world” itself informal; but we will consider a graph to be a small world, roughly, when all (or most) pairs of nodes are connected by paths of length polylogarithmic in n, since in such a case the path lengths are exponentially smaller than the number of nodes. Watts and Strogatz [64] argued that there is something crucial missing from the picture provided by Theorem 3.1. A standard random graph (for example, as in Theorem 3.1) is locally very sparse; with reasonably high probability, none of the neighbors of a given node v are themselves neighbors of one another. But this is far from true in most naturally occurring networks: in real network data, many of a node’s neighbors are joined to each other by edges. (For example, in a social network, many of our friends know each other.) Indeed, at an implicit level, this is a large part of what makes the small-world phenomenon surprising to many people when they first hear it: the social network appears from the local perspective of any one node to be highly “clustered,” rather than the kind of branching tree-like structure that would more obviously reach many nodes along very short paths. Thus, Watts and Strogatz proposed thinking about small-world networks as a kind of superposition: a structured, high-diameter network with a relatively small number of “random” links added in. As a model for social networks, the structured underlying network represents the “typical” social links that we form with the people who live near us, or who work with us; the additional random links are the chance, long-range connections that play a large role in creating short paths through the network as a whole. This kind of hybrid random graph model had been studied earlier by Bollobás and Chung [16]; they showed that a small density of random links can indeed produce


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short paths very effectively. In particular they proved the following, among other results. Theorem 3.2 ([16]). Consider a graph G formed by adding a random matching to an n-node cycle. (In other words, we assume n is even, pair up the nodes on the cycle uniformly at random, and add edges between each of these node pairs.) With high probability, every pair of nodes will be joined by a path of length O(log n). Here too, Bollobás and Chung in fact proved a much more detailed bound on the path lengths; see [16] for further details. This is quite close to the setting of the Watts-Strogatz work, who also considered cycles with random matchings as a model system for analysis. For our purposes here, we will begin with the following grid-based model, which is qualitatively very similar. We start with a two-dimensional n × n grid graph, and then for each node v, we add one extra directed edge to some other node w chosen uniformly at random. (We will refer to w as the long-range contact of v; to distinguish this, we will refer to the other neighbors of v, defined by the edges of the grid, as its local contacts.) Following the Watts-Strogatz framework, one can interpret this model as a metaphor for a social network embedded in an underlying physical space – people tend to know their geographic neighbors, as well as having friendships that span long distances. It is also closely related to long-range percolation models, though the questions we consider are fairly different; we discuss these connections in Section 7. For the present discussion, though, the essential feature of this model is its superposition of structured and random links, and it is important to note that the results to follow carry over directly to a wide range of variations on the model. Indeed, a significant part of what follows will be focused on a search for the most general framework in which to formulate these results.

4. Decentralized search in small-world networks Thus far we have been discussing purely structural issues; but if one thinks about it, the original Milgram experiment contains a striking algorithmic discovery as well: not only did short paths exist in the social network, but people, using knowledge only of their own acquaintances, were able to collectively construct paths to the target. This was a necessary consequence of the way Milgram formulated the task for his participants; if one really wanted the shortest path from a starting person to the target, one should have instructed the starter to forward a letter to all of his or her friends, who in turn should have forwarded the letter to all of their friends, and so forth. This “flooding” of the network would have reached the target as rapidly as possible; but for obvious reasons, such an experiment was not a feasible option. As a result, Milgram was forced to embark on the much more interesting experiment of constructing paths by “tunneling” through the network, with the letter advancing just one person at a

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time – a process that could well have failed to reach the target, even if a short path existed. This algorithmic aspect of the small-world phenomenon raises fundamental questions – why should the social network have been structured so as to make this type of decentralized routing so effective? Clearly the network contained some type of “gradient” that helped participants guide messages toward the target, and this is something that we can try to model; the goal would be to see whether decentralized routing can be proved to work in a simple random-graph model, and if so, to try extracting from this model some qualitative properties that distinguish networks in which this type of routing can succeed. It is worth noting that these issues reach far beyond the Milgram experiment or even social networks; routing with limited information is something that takes place in communication networks, in browsing behavior on the World Wide Web, in neurological networks, and in a number of other settings – so an understanding of the structural underpinnings of efficient decentralized routing is a question that spans all these domains. To begin with, we need to be precise about what we mean by a decentralized algorithm. In the context of the grid-based model in the previous section, we will consider algorithms that seek to pass a message from a starting node s to a target node t, by advancing the message along edges. In each step of this process, the current message-holder v has knowledge of the underlying grid structure, the location of the target t on the grid, and its own long-range contact. The crucial point is that it does not know the long-range contacts of any other nodes. (Optionally, we can choose to have v know the path taken by the message thus far, but this will not be crucial in any of the results to follow.) Using this information, v must choose one of its network neighbors w to pass the message to; the process then continues from w. We will evaluate decentralized algorithms according to their delivery time – the expected number of steps required to reach the target, over a randomly generated set of longrange contacts, and randomly chosen starting and target nodes. Our goal will be to find algorithms with delivery times that are polylogarithmic in n. It is interesting that while Watts and Strogatz proposed their model without the algorithmic aspect in mind, it is remarkably effective as a simple system in which to study the effectiveness of decentralized routing. Indeed, to be able to pose the question in a non-trivial way, one wants a network that is partially known to the algorithm and partially unknown – clearly in the Milgram experiment, as well as in other settings, individual nodes use knowledge not just of their own local connections, but also of certain global “reference frames” (comparable to the grid structure in our setting) in which the network is embedded. Furthermore, for the problem to be interesting, the “known” part of the network should be likely to contain no short path from the source to the target, but there should be a short path in the full network. The Watts-Strogatz model combines all these features in a minimal way, and thus allows us to consider how nodes can use what they know about the network structure to construct short paths. Despite all this, the first result here is negative.


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Theorem 4.1 ([36], [37]). The delivery time of any decentralized algorithm in the grid-based model is (n2/3 ). (We say that a function is (f (n)) if there is a constant c so that for infinitely many n, the function is at least cf (n).) This shows that there are simple models in which there can be an exponential separation between the lengths of paths and the delivery times of decentralized algorithms to find these paths. However, it is clearly not the end of the story; rather, it says that the random links in the Watts-Strogatz model are somehow too “unstructured” to support the kind of decentralized routing that one found in the Milgram experiment. It also raises the question of finding a simple extension of the model in which efficient decentralized routing becomes possible. To extend the model, we introduce one additional parameter α ≥ 0 that controls the extent to which the long-range links are correlated with the geometry of the underlying grid. First, for two nodes v and w, we define their grid distance ρ(v, w) to be the number of edges in a shortest path between them on the grid. The idea behind the extended model is to have the long-range contacts favor nodes at smaller grid distance, where the bias is determined by α. Specifically, we define the grid-based model with exponent α as follows. We start with a two-dimensional n × n grid graph, and then for each node v, we add one extra directed edge to some other long-range contact; we choose w as the long-range contact for v with probability proportional to ρ(v, w)−α . Note that α = 0 corresponds to the original Watts-Strogatz model, while large values of α produce networks in which essentially no edges span long distances on the grid. We now have a continuum of models that can be studied, parameterized by α. When α is very small, the long-range links are “too random,” and can’t be used effectively by a decentralized algorithm; when α is large, the long-range links appear to be “not random enough,” since they simply don’t provide enough of the long-distance jumps that are needed to create a small world. Is there an optimal operating point for the network, where the distribution of long-range links is sufficiently balanced between these extremes to be of use to a decentralized routing algorithm? In fact there is; as the following theorem shows, there is a unique value of α in the grid-based model for which a polylogarithmic delivery time is achievable. Theorem 4.2 ([36], [37]). (a) For 0 ≤ α < 2, the delivery time of any decentralized algorithm in the grid-based model is (n(2−α)/3 ). (b) For α = 2, there is a decentralized algorithm with delivery time O(log2 n). (c) For α > 2, the delivery time of any decentralized algorithm in the grid-based model is (n(α−2)/(α−1) ). (We note that the lower bounds in (a) and (c) hold even if each node has an arbitrary constant number of long-range contacts, rather than just one.) The decentralized algorithm achieving the bound in (b) is very simple: each node simply forwards the message to a neighbor – long-range or local – whose grid distance to the target is as small as possible. (In other words, each node uses its long-range

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contact if this gets the message closer to the target on the grid; otherwise, it uses a local contact in the direction of the target.) The analysis of this algorithm proceeds by showing that, for a constant ε > 0, there is a probability of at least ε/ log n in every step that the grid distance to the target will be halved. It is also worth noting that the proof can be directly adapted to a grid in any constant number of dimensions; an analogous trichotomy arises, with polylogarithmic delivery time achievable only when α is equal to the dimension. At a more general level, the proof of Theorem 4.2(b) shows that the crucial property of exponent α = 2 is the following: rather than producing long-range contacts that are uniformly distributed over the grid (as one gets from exponent α = 0), it produces long-range contacts that are approximately uniformly distributed over “distance scales”: the probability that the long-range contact of v is at a grid distance between 2j −1 and 2j away from v is approximately the same for all values of j from 1 to log n. From this property, one sees that there is a reasonable chance of halving the message’s grid distance to the target, independent of how far away it currently is. The property also has an intuitively natural meaning in the context of the original Milgram experiment; subject to all the other simplifications made in the grid model, it says very roughly that decentralized routing can be effective when people have approximately the same density of acquaintances at many different levels of distance resolution. And finally, this approximate uniformity over distance scales is the type of qualitative property that we mentioned as a goal at the outset. It is something that we can search for in other models and in real network data – tasks that we undertake in the next two sections.

5. Decentralized search in other models Hierarchical models. A natural variation on the model of the previous section is to suppose that the network is embedded in a hierarchy rather than a grid – in other words, that the nodes reside at the leaves of a complete b-ary tree, and the underlying “distance” between two nodes is based on the height of their lowest common ancestor in this tree. There are a number of settings where such a model suggests itself. To begin with, follow-up work on the Milgram experiment found that most decisions made by participants on how to forward the letter were based on one of two kinds of cues: geographical and occupational [35]. And if a two-dimensional grid is natural as a simple abstraction for the role of geography, then a hierarchy is a reasonable, also simple, approximation of the way in which people categorize occupations. Another domain in which hierarchies arise naturally is in the relationships among Web pages: for example, a Web page about sequence analysis of the yeast genome could be classified as being about genetics, more generally about biology, and more generally still about science, while a Web page reviewing performances of Verdi’s Aida could


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be classified as being about opera, more generally about music, and more generally still about the arts. A natural assumption is that the density of links is lower for node pairs that are more widely separated in the underlying hierarchy, and this forms the basis for the following hierarchical model with exponent β. We begin with a complete b-ary tree having n leaves (and hence of height h = logb n). For two leaves v and w, let us define their tree distance h(v, w) to be the height of their lowest common ancestor in the underlying tree. We now define the following random directed graph G on the set V of leaves: for a value k and for each node v in V , we construct k edges out of v, choosing w as the endpoint of the i th edge independently with probability proportional to b−βh(v,w) . (We will refer to k as the out-degree of the model.) Thus, β works much like α did in the grid-based model; when β = 0, we get uniform random selection, while larger values of β bias the selection more toward “nearby” nodes. Now, in this case, a decentralized search algorithm is given the locations of a starting node s and a target node t in the hierarchy, and it must construct a path from s to t, knowing only the edges out of nodes that it explicitly visits. Note that in defining the performance metric for a decentralized search algorithm in this model, we face a problem that we didn’t encounter in the grid-based model: the graph G may not contain a path from s to t. Thus, we say that a decentralized algorithm here has delivery time f (n) if, on a randomly generated n-node network, and with s and t chosen uniformly at random, the algorithm produces a path of length O(f (n)) with probability at least 1 − ε(n), where ε(·) is a function going to 0 as n increases. We now have the following analogue of Theorem 4.2, establishing that there is a unique value of β for which polylogarithmic delivery time can be achieved when the network has polylogarithmic out-degree. This is achieved at β = 1, when the probability that v links to a node at tree distance h is almost uniform over choices of h. Also by analogy with the grid-based model, it suffices to use the simple “greedy” algorithm that always seeks to reduce the tree distance to the target by as much as possible. Theorem 5.1 ([38]). (a) In the hierarchical model with exponent β = 1 and out-degree k = c log2 n, for a sufficiently large constant c, there is a decentralized algorithm with polylogarithmic delivery time. (b) For every β = 1 and every polylogarithmic function k(n), there is no decentralized algorithm in the hierarchical model with exponent β and out-degree k(n) that achieves polylogarithmic delivery time. Watts, Dodds, and Newman [63] independently proposed a model in which each node resides in several distinct hierarchies, reflecting the notion that participants in the small-world experiment were simultaneously taking into account several different notions of “proximity” to the target. Concretely, their model constructs a random graph G as follows. We begin with q distinct complete b-ary trees, for a constant q, and in each of these trees, we independently choose a random one-to-one mapping of the nodes onto the leaves. We then apply a version of the hierarchical model

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above, separately in each of the trees; the result is that each node of G acquires edges independently through its participation in each tree. (There are a few minor differences between their procedure within each hierarchy and the hierarchical model described above; in particular, they map multiple nodes to the same leaf in each hierarchy, and they generate each edge by choosing the tail uniformly at random, and then the head according to the hierarchical model. The result is that nodes will not in general all have the same out-degree.) Precisely characterizing the power of decentralized search in this model, at an analytical level, is an open question, but Watts et al. describe a number of interesting findings obtained through simulation [63]. They study what is perhaps the most natural search algorithm, in which the current message-holder forwards the message to its neighbor who is closest (in the sense of tree distance) to the target in any of the hierarchies. Using an empirical definition of efficiency on networks of several hundred thousand nodes, they examined the set of (β, q) pairs for which the search algorithm was efficient; they found that this “searchable region” was centered around values of β ≥ 1 (but relatively close to 1), and on small constant values of q. (Setting q equal to 2 or 3 yielded the widest range of β for which efficient search was possible.) The resulting claim, at a qualitative level, is that efficient search is facilitated by having a small number of different ways to measure proximity of nodes, and by having a small bias toward nearby nodes in the construction of random edges. Models based on set systems. One can imagine many other ways to construct networks in this general style – for example, placing nodes on both a hierarchy and a lattice simultaneously – and so it becomes natural to consider more general frameworks in which a range of these bounds on searchability might follow simultaneously from a single result. One such approach is based on constructing a random graph from an underlying set system, following the intuition that individuals in a social network often form connections because they are both members of the same small group [38]. In other words, two people might be more likely to form a link because they live in the same town, work in the same profession, have the same religious affiliation, or follow the work of the same obscure novelist. Concretely, we start with a set of nodes V , and a collection of subsets S = {S1 , S2 , . . . , Sm } of V , which we will call the set of groups. It is hard to say much of interest for arbitrary set systems, but we would like our framework to include at least the collection of balls or subsquares in a grid, and the collection of rooted sub-trees in a hierarchy. Thus we consider set systems that satisfy some simple combinatorial properties shared by these two types of collections. Specifically, for constants λ < 1 and κ > 1, we impose the following three properties. (i) The full set V is one of the groups. (ii) If Si is a group of size g ≥ 2 containing a node v, then there is a group Sj ⊆ Si containing v that is strictly smaller than Si , but has size at least min(λg, g − 1).


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(iii) If Si1 , Si2 , Si3 , . . . are groups that all have size at most g and all contain a common node v, then their union has size at most κg. The most interesting property here is (iii), which can viewed as a type of “bounded growth” requirement; one can easily verify that it (along with (i) and (ii)) holds for the set of balls in a grid and the set of rooted sub-trees in a hierarchy. Given a collection of groups, we construct a random graph as follows. For nodes v and w, we define g(v, w) to be the size of the smallest group containing both of them – this will serve as a notion of “distance” between v and w. For a fixed exponent γ and out-degree value k, we construct k edges out of each node v, choosing w as the endpoint of the i th edge from v independently with probability proportional to g(v, w)−γ . We will refer to this as the group-based model with set system S, exponent γ , and outdegree k. A decentralized search algorithm in such a random graph is given knowledge of the full set system, and the identity of a target node; but it only learns the links out of a node v when it reaches v. We now have the following theorem. Theorem 5.2 ([38]). (a) Given an arbitrary set system S satisfying properties (i), (ii), and (iii), there is a decentralized algorithm with polylogarithmic delivery time in the group-based model with set system S, exponent γ = 1, and out-degree k = c log2 n, for a sufficiently large constant c. (b) For every set system S satisfying properties (i), (ii), and (iii), every γ < 1, and every polylogarithmic function k(n), there is no decentralized algorithm achieving polylogarithmic delivery time in the group-based model with set system S, exponent γ and out-degree k(n). In other words, efficient decentralized search is possible when nodes link to each other with probability inversely proportional to the size of the smallest group containing both of them. As a simple concrete example, if the groups are the balls in a two-dimensional grid, then the size of the smallest group containing two nodes at distance ρ is proportional to ρ 2 , and so the link probability indicated by Theorem 5.2 (a) is proportional to ρ −2 ; this yields an analogue of Theorem 4.2 (b), the inverse-square result for grids. (The present setting is not exactly the same as the one there; here, we do not automatically include the edges of the original grid when constructing the graph, but we construct a larger number of edges out of each node.) Simple examples show that one cannot directly formulate a general negative result in this model for the case of exponents γ > 1 [38]. At a higher level, the group-based model is clearly not the only way to generalize the results thus far; in the next section we will discuss one other recent approach, and the development of other general models is a natural direction for further research.

6. Design principles and network data In addition to their formulation as basic questions about search algorithms in graphs, the models we have been discussing thus far have been used as design principles in

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file-sharing systems; and they have been found to capture some of the large-scale structure of human social networks as reflected in on-line data. Peer-to-peer systems and focused web crawling. A recurring theme in recent work on complex networks is the way in which simple probabilistic models can rapidly become design principles for new types of networked systems. In the case of smallworld networks, one observes this phenomenon in the development of protocols for peer-to-peer file sharing. The design of such protocols has become an active topic of research in the area of computer systems, motivated in part by the explosion of popular interest in peer-to-peer applications following the emergence of Napster and music file-sharing in 1999. The goal of such applications was to allow a large collection of users to share the content residing on their personal computers, and in their initial conception, the systems supporting these applications were based on a centralized index that simply stored, in a single place, the files that all users possessed. This way, queries for a particular piece of content could be checked against this index, and routed to the computer containing the appropriate file. The music-sharing application of these systems, of course, ran into significant legal difficulties; but independent of the economic and intellectual property issues raised by this particular application, it is clear that systems allowing large user communities to share content have a much broader range of potential, less controversial uses, provided they can be structured in a robust and efficient way. This has stimulated much subsequent study in the research community, focusing on decentralized approaches in which one seeks file-sharing solutions that do not rely on a single centralized index of all the content. In this decentralized version of the problem, the crux of the challenge is clear: each user has certain files on his or her own computer, but there is no single place that contains a global list of all these files; if someone poses a query looking for a specific piece of content, how can we efficiently determine which user (if any) possesses a copy of it? Without a central index, we are in a setting very much like that of the Milgram experiment: users must pose the query to a subset of their immediate network neighbors, who in turn can forward the query to some of their neighbors, and so forth. And this is where small-world models have played a role: a number of approaches to this problem have tried to explicitly set up the network on which the protocol operates so that its structure makes efficient decentralized search possible. We refer the reader to the surveys by Aspnes and Shah [6] and Lua et al. [44] for general reviews of this body of work, and the work of Clarke et al. (as described in [32]), Zhang et al. [67], Malkhi et al. [45], and Manku et al. [46] for more specific discussions of the relationship to small-world networks. A related set of issues comes up in the design of focused Web crawlers. Whereas standard Web search engines first compile an enormous index of Web pages, and then answer queries by referring to this index, a focused crawler attempts to locate pages on a specific topic by following hyperlinks from one page to another, without first compiling an index. Again, the underlying issue here is the design of decentralized


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search algorithms, in this case for the setting of the Web: when searching for relevant pages without global knowledge of the network, what are the most effective rules for deciding which links to follow? Motivated by these issues, Menczer [49] studied the extent to which the hierarchical model described in the previous section captures the patterns of linkage in large-scale Web data, using the hierarchical organization of topics provided by the Open Directory. Social network data. The previous two applications – peer-to-peer systems and focused Web crawling – are both concerned with the structure of computer and information networks, although in both cases there are obvious social forces underlying their construction. Recent work has also investigated the extent to which the models described in the previous sections are actually reflected in data on human social networks. In other words, these small-world models make very concrete claims about the ways in which networks should be organized to support efficient search, but it is not a priori clear whether or not naturally occurring networks are organized in such ways. Two recent studies of this flavor have both focused on social networks that exist in on-line environments – as with the previous applications, we again see an intertwining of social and technological networks, but in these cases the emphasis is on the social component, with the on-line aspect mainly providing an opportune means of performing fine-grained analysis on a large scale. In one study of this flavor, Adamic and Adar [1] considered the e-mail network of a corporate research lab: they collected data over a period of time, and defined an edge between any two people who exchanged at least a certain number of messages during this period. They overlaid the resulting network on a set system representing the organizational structure, with a set for each subgroup of the lab’s organizational hierarchy. Among other findings, they showed that the probability of a link between individuals v and w scaled approximately proportional to g(v, w)−3/4 , compared with the value g(v, w)−1 for efficient search from Theorem 5.2(a). (As above, g(v, w) denotes the size of the smallest group containing both v and w.) Thus, interactions in their data spanned large groups at a slightly higher frequency than the optimum for decentralized search. Of course, the e-mail network was not explicitly designed to support decentralized search, although one can speculate about whether there were implicit factors shaping the network into a structure that was easy to search; in any case, it is interesting that the behavior of the links with respect to the collection of groups is approximately aligned with the form predicted by the earlier theorems. An even closer correlation with the structure predicted for efficient search was found in a large-scale study by Liben-Nowell et al. [43]. They considered LiveJournal, a highly active on-line community with several million participants, in which members communicate with one another, update personal on-line diaries, and post messages to community discussions. LiveJournal is a particularly appealing domain for studying the geographic distribution of links, because members provide explicit links to their friends in the system, and a large subset (roughly half a million at the time of the study in [43]) also provide a hometown in the continental U.S. As a result, one has

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the opportunity to investigate, over a very large population, how the density of social network links decays with distance. A non-trivial technical challenge that must be overcome in order to relate this data to the earlier models is that the population density of the U.S. is extremely nonuniform, and this makes it difficult to interpret predictions based on a model in which nodes are distributed uniformly over a grid. The generalization to group structures in the previous section is one way to handle non-uniformity; Liben-Nowell et al. propose an alternative generalization, rank-based friendships, that they argue may be more suitable to the geographic data here [43]. In the rank-based friendship model, one has a set of n people assigned to locations on a two-dimensional grid, where each grid node may have an arbitrary positive number of people assigned to it. By analogy with the grid-based model from Section 4, each person v chooses a local contact arbitrarily in each of the four neighboring grid nodes, and then chooses an additional long-range contact as follows. First, v ranks all other people in order of their distance to herself (breaking ties in some canonical way); we let rank v (w) denote the position of w in v’s ordered list, and say that w is at rank r with respect to v. v then chooses w as her long-range contact with probability proportional to 1/rank v (w). Note that this model generalizes the grid-based model of Section 4, in the sense that the grid-based model with the inverse-square distribution corresponds to rankbased friendship in which there is one person resident at each grid node. However, the rank-based friendship construction is well-defined for any population density, and Liben-Nowell et al. prove that it supports efficient decentralized search in general. They analyze a decentralized greedy algorithm that always forwards the message to a grid node as close as possible to the target’s; and they define the delivery time in this case to be the expected number of steps needed to reach the grid node containing the target. (So we can imagine that the task here is to route the message to the hometown of the target, rather than the target himself; this is also consistent with the data available from LiveJournal, which only localizes people to the level of towns.) Theorem 6.1 ([43]). For an arbitrary population density on a grid, the expected delivery time of the decentralized greedy algorithm in the rank-based friendship model is O(log3 n). On the LiveJournal data, Liben-Nowell et al. examine the fraction of friendships (v, w) where w is at rank r with respect to v. They find that this fraction is very close to inverse linear in r, in close alignment with the predictions of the rank-based friendship model. This finding is notable for several reasons. First, as with the e-mail network considered by Adamic and Adar, there is no a priori reason to believe that a large, apparently amorphous social network should correspond so closely to a distribution predicted by a simple model for efficient decentralized search. Second, geography is playing a strong role here despite the fact that LiveJournal is an on-line system in which there are no explicit limitations on forming links with people arbitrarily far away; as a result, one might have (incorrectly) conjectured that it would be difficult


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to detect the traces of geographic proximity in such data. And more generally, the analytical results of this section and the previous ones have been based on highly stylized models that nonetheless make very specific predictions about the theoretical “optimum” for search; to see these concrete predictions approximately borne out on real social network data is striking, and it suggests that there may be deeper phenomena yet to be discovered here.

7. Further results on small-world networks and decentralized search Long-range percolation. The grid-based models we have been considering are closely related to the problem of long-range percolation. In the basic version of long-range percolation, one takes the infinite d-dimensional integer lattice Zd , and for each pair of nodes (v, w) one includes an undirected edge between them independently with probability ρ(v, w)−α , where ρ(v, w) is the grid distance between v and w and α ≥ 0 is a parameter of the model. Note that there are some small differences from the grid-based model described in Section 4: the graph is infinite, it is undirected, its nodes do not all have the same degree, and it does not automatically include edges between nearest neighbors on the lattice. In addition to these, a broader difference is in the nature of the questions investigated, with the initial work on longrange percolation focusing on the range of parameters for which an infinite connected component is likely to exist [3], [51], [57]. Motivated in part by the interest in small-world networks, work on long-range percolation began to investigate diameter issues – the maximum D for which every node is connected by a path of at most D steps. Benjamini and Berger [11] studied this problem in one dimension, modifying the model so that the graph is finite (restricted to the integers {1, 2, . . . , n}), and so that edges are guaranteed to exist between adjacent integers. (They also studied the case in which the distance ρ(·, ·) is defined by assuming that the integers are “wrapped” into a cycle, so that ρ(i, j ) is not |j − i| but min(|j − i|, n − |j − i|).) Their work was followed by results of Coppersmith et al. [20] and Biskup [13], who obtained sharper bounds in some cases and considered higher-dimensional lattices as well, in which the node set is {1, 2, . . . , n}d . As a result of this work, we know that the diameter of the graph changes qualitatively at the “critical values” α = d and α = 2d. In particular, with high probability, the diameter is constant when α < d (due in essence to a result of [12]), is proportional to log n/ log log n when α = d [20], is polylogarithmic in n when d < α < 2d (with an essentially tight bound provided in [13]), and is lower-bounded by a polynomial in n when α > 2d [11], [20]. The case α = 2d is largely open, and conjectured to have diameter polynomial in n with high probability [11], [13]. It is also open whether the diameter for α > 2d is in fact linear in n; this has been proved for the one-dimensional case [11] and conjectured to hold for higher dimensions as well [11], [13], [20]. This pair of transitions at α = d and α = 2d was observed in a somewhat different setting by Kempe et al. [34], resolving a conjecture of Demers et al. [21] on the behavior of gossip algorithms. In this model, there are nodes located on the finite

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d-dimensional lattice {1, 2, . . . , n}d , and in each time step each node v picks a single other node and tells everything it currently knows to w; node w is selected as the recipient of this information with probability proportional to ρ(v, w)−α . Information originating at one node thus spreads to other nodes, relayed in an epidemic fashion over time. Now, if a single node v initially possesses a new piece of information at time 0, how long will it take before knowledge of this information has spread to a given node w? The main result of [34] is that the time required for this is polylogarithmic in n for α ≤ d, is polylogarithmic in ρ(v, w) but independent of n for d < α < 2d, and is polynomial in ρ(v, w) for α > 2d. Here too the case α = 2d is not well understood, which is interesting because this transitional value has particular importance in applications of gossip algorithms to distributed computing systems [54]. (See [34] for partial results concerning α = 2d.) For the specific grid-based model described in Section 4, Martel and Nguyen showed that with high probability the diameter is proportional to log n for α ≤ d, in the d-dimensional case [48]. They also identified transitions at α = d and α = 2d analogous to the case of long-range percolation [53]. In particular, their results show that while decentralized search can construct a path of length O(log2 n) when α = d, there in fact exist paths that are shorter by a logarithmic factor. (Note also the contrast with the corresponding results for the long-range percolation model when α ≤ d; in the grid-based model, the out-degree of each node is bounded by a constant, so a diameter proportional to log n is the smallest one could hope for; in the case of longrange percolation, on the other hand, the node degrees will be unbounded, allowing for smaller diameters.) Decentralized search with additional information. A number of papers have studied the power of decentralized search algorithms that are provided with small amounts of additional information [28], [42], [47], [48], [66]. Whereas the model of decentralized algorithms in Section 4 charged unit cost to the algorithm for each node visited, the models in these subsequent papers make the following distinction: a node may “consult” a small number of nearby nodes, and then based on what it learns from this consultation, it chooses a node to forward the messages to. In bounding the number of steps taken by the algorithm, only the message-forwarding operations are counted, not the consultation. In particular, Lebhar and Schabanel [42] consider an algorithm in which the node currently holding the message consults a set S of up to O(log n) nodes within a small number of steps of it; after this, it forwards the message along a path to the node w in S that is closest to the target in grid distance. They show that, in total, the expected number of nodes consulted by this process is O(log2 n) (as in the decentralized algorithm from Section 4), and that the actual path constructed to the target has only O(log n(log log n)2 ) steps. Manku, Naor, and Wieder [47] consider a simpler algorithm in the long-range percolation model on the d-dimensional lattice {1, 2, . . . , n}d with α = d. Note that nodes here will have unbounded degrees – proportional to log n in expectation, rather


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than constant as in the grid-based model. Manku et al. analyze a neighbor-of-neighbor search algorithm in which the current message-holder v consults each of its neighbors to learn the set S of all of their neighbors; v then forwards the message along the twostep path to the node in S that lies closest to the target. They show that with high probability, this algorithm produces a path to the target of at most O(log n/ log log n) steps, matching the bound of Coppersmith et al. [20] on the diameter of this network. Moreover, they show that the basic greedy algorithm, which simply forwards the message to the neighbor closest to the target, requires an expected number of steps proportional to log n to reach the target. Thus, one step of lookahead provides an asymptotic improvement in delivery time; and since one step of lookahead yields path lengths matching the diameter, additional lookahead does not offer any further asymptotic improvements. Thus, the results of Manku et al. provide a rather sharp characterization of the power of lookahead in the long-range percolation model at the exponent α = d that allows for efficient decentralized search; determining a similarly precise delineation on the power of lookahead in the grid-based model (extending the aforementioned results of Lebhar and Schabanel) is an interesting open question. Small-world networks built on arbitrary underlying graphs. The results in Section 5 describe various methods for constructing searchable networks based on underlying structures other than d-dimensional grids. In several recent papers, a number of further structures have been proposed as “scaffolds” for small-world networks [9], [27], [31], [53], [59]. In principle, one can consider adding long-range edges to any underlying graph G; Fraigniaud [27] asks whether any G can be converted through such a process into a network that is efficiently searchable by a greedy algorithm. Specifically, suppose we choose a distribution over long-range contacts for each node of G, and we use this to generate a random graph G by adding a single long-range edge out of each node of G. We then consider the natural greedy algorithm for forwarding the message to a target t: the current message-holder passes the message to a neighbor that has the shortest path to the target as measured in G (not in G ). Is it the case that for every graph G, there is a distribution over long-range contacts such that this algorithm has a delivery time that is polylogarithmic in n? This question is open in general; note that the challenge in resolving it comes from the fact that a single choice of distribution per node must work (in expectation) over any possible destination, and that even if the graph G has nicely-structured short paths, the search algorithm is constrained to behave “greedily” in the original graph G. Fraigniaud answers the question in the affirmative for graphs of bounded tree-width as well as graphs in which there is no induced cycle of greater than a fixed length [27]; he also discusses some respects in which such underlying graphs are qualitatively consistent with observed properties of social networks. Duchon et al. answer the question in the affirmative for graphs satisfying a certain “bounded growth rate” property [24].

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Slivkins [59] considers a different setting, in which nodes are embedded in an underlying metric space. He shows that if the metric is doubling, in the sense that every ball can be covered by a constant number of balls of half the radius (see e.g. [7], [30]), then there is a model such that each node generates a polylogarithmic number of long-range contacts from specified distributions, and a decentralized algorithm is then able to achieve a polylogarithmic delivery time. (Some of the logarithmic dependence here is on the aspect ratio of the metric – the ratio of the largest to the smallest distance – but it is possible to avoid this dependence in the bound on the delivery time. See [59] for further details on this issue.) Finally, other work has studied search algorithms that exploit differences in node degrees. There are indications that people navigating social structures, in settings such as small-world experiments, take into account the fact that certain of their acquaintances simply know a large number of people [22]. Similarly, in peer-to-peer networks, it is also the case that certain nodes have an unusually large number of neighbors, and may thus be more useful in helping to forward queries. Adamic et al. [2] formalize these considerations by studying a random graph model in which high-degree nodes are relatively abundant, and decentralized search algorithms only have access to information about degrees of neighboring nodes, not to any embedding of the graph (spatial or otherwise). Through simulation, they find that for certain models, knowledge of degrees provides an improvement in search performance. Simsek and Jensen [58] consider a model which combines spatial embedding with variable node degrees. Specifically, they study a variant of the grid-based model from Section 4 in which nodes have widely varying degrees, and a decentralized algorithm has access both to the locations of its neighbors and to their degrees. Through simulation, they find that a heuristic taking both these factors into account can perform more efficiently than decentralized algorithms using only one of these sources of information. Finding the optimal way to combine location and degree information in decentralized search, and understanding the range of networks that are searchable under such optimal strategies, is an interesting direction for further research.

8. Conclusion We have followed a particular strand of research running through the topic of complex networks, concerned with short paths and the ability of decentralized algorithms to find them. As suggested initially, the sequence of ideas here is characteristic of the flavor of research in this area: an experiment in the social sciences that highlights a fundamental and non-obvious property of networks (efficient searchability, in this case); a sequence of random graph models and accompanying analysis that seeks to capture this notion in a simple and stylized form; a set of measurements on large-scale network data that parallels the properties of the models, in some cases to a surprising extent; and a range of connections to further results and questions in algorithms, graph theory, and discrete probability.


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To indicate some of the further directions in which research on this topic could proceed, we conclude with a list of open questions and issues related to small-world networks and decentralized search. Some of these questions have already come up implicitly in the discussion thus far, so one goal of this list is to collect a number of these questions in a single place. Other questions here, however, bring in issues that reach beyond the context of the earlier sections. And as with any list of open questions, we must mention a few caveats: the questions here take different forms, since some are concretely specified while other are more designed to suggest problems in need of a precise formulation; the questions are not independent, in that the answer to one might well suggest ways of approaching others; and several of the questions may well become more interesting if the underlying model or formulation is slightly varied or tweaked. 1. Variation in node degrees. As we discussed at the end of the previous section, decentralized search in models that combine wide variation in node degrees with some kind of spatial embedding is an interesting issue that is not well understood. Simsek and Jensen’s study [58] of this issue left open the question of proving bounds on the efficiency of decentralized algorithms. For example, consider the d-dimensional grid-based model with exponent α, and suppose that rather than constructing a fixed number of long-range contacts for each node, we draw the number of long-range contacts for each node v independently from a given probability distribution. To be concrete, we could consider a distribution in which one selects k long-range contacts with probability proportional to k −δ for a constant δ. We now have a family of grid-based models parameterized by α and δ, and we can study the performance of decentralized search algorithms that know not only the longrange contacts out of the current node, but also the degrees of the neighboring nodes. Decentralized selection of a neighbor for forwarding the message has a stochastic optimization aspect here, balancing the goal of forwarding to a node close to the target with the goal of forwarding to a high-degree node. We can now ask the general question of how the delivery time of decentralized algorithms varies in both α and δ. Note that it is quite possible this question becomes more interesting if we vary the model so that long-range links are undirected; this way, a node with a large degree is both easy to find and also very useful once it is found. (In a directed version, a node with large out-degree may be relatively useless simply because it has low in-degree and so is unlikely to be found.) 2. The case of α = 2d. In both the grid-based model and the related long-range percolation models, very little is known about the diameter of the graph when α is equal to twice the dimension. (It appears that a similar question arises in other versions of the group-based models from Section 5, when nodes form links with probability inversely proportional to the square of the size of the smallest group containing both of them.) Resolving the behavior of the diameter would shed light on this transitional point, which lies at the juncture between “small worlds” and “large worlds.” This open

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question also manifests itself in the gossip problem discussed in Section 7, where we noted that the transitional value α = 2d arises in distributed computing applications (see the discussion in [34], [54]). 3. Paths of logarithmic length. It would be interesting to know whether there is a decentralized algorithm in the d-dimensional grid-based model, at the “searchable exponent” α = d, that could construct paths of length O(log n) while visiting only a polylogarithmic number of nodes. This would improve the result of Lebhar and Schabanel [42] to an asymptotically tight bound on path length. 4. Small-world networks with an arbitrary base graph. It would also be interesting to resolve the open problem of Fraigniaud [27] described in Section 7, formalizing the question of whether any graph can be turned into an efficiently searchable small world by appropriately adding long-range links1 . 5. Extending the group-based model. Theorem 5.2 on the group-based model contained a positive result generalizing the ones for grids and hierarchies, and it contained a general negative result for the case when long-range connection were “too long-range” (i.e. with exponent γ < 1). However, it does not fully generalize the results for grids and hierarchies, because there are set systems satisfying conditions (i), (ii), and (iii) of the theorem for which efficient decentralized search is possible even for exponents γ > 1. It would be interesting to find a variation on these three properties that still generalizes grids and hierarchies in a natural way, and for which γ = 1 is the unique exponent at which efficient decentralized search is possible. 6. Multiple hierarchies. Obtaining provable bounds for decentralized search in the “multiple hierarchies” model of Watts, Dodds, and Newman [63] is also an open question. Such results could form an interesting parallel with the findings they discovered through simulation. With some small modifications to the model of Watts et al., one can cast it in the group-based model of Section 5, and so it is entirely possible that progress on this question and the previous could be closely connected. 7. The evolution of searchable networks. The remaining questions have a more general flavor, where much of the challenge is the formalization of the underlying issue. To begin with, the current models supporting efficient decentralized search are essentially static, in that they describe how the underlying network is organized without suggesting how it might have evolved into this state. What kinds of growth processes or selective pressures might exist to cause networks to become more efficiently searchable? Interesting network evolution models addressing this question have been proposed by Clauset and Moore [19] and by Sandberg [56], both based on feedback mechanisms by which nodes repeatedly perform decentralized searches and 1 Note added in proof: Fraigniaud, Lebhar, and Lotker have very recently announced a negative resolution of this question, constructing a family of graphs that cannot be turned into efficiently searchable small worlds by this process.


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in the process partially “rewire” the network. Obtaining provable guarantees for these models, or variations on them, is an open question. A number of peer-to-peer filesharing systems include similar feedback mechanisms, achieving good performance in practice. Freenet [18] is a good example of such a system, and the relationship of its feedback mechanism to the evolution of small-world networks is studied by Zhang et al. [67]. Game theory may provide another promising set of techniques for studying the evolution of small-world networks. A growing body of recent work has considered game-theoretic models of network formation, in which agents controlling nodes and edges interact strategically to construct a graph – the basic question is to understand what types of structures emerge when each agent is motivated by self-interest. For surveys of this area, see [5], [33], [65]. In the present case, it would be interesting to understand whether there are ways to define incentives such that the collective outcome of self-interested behavior would be a searchable small-world network. 8. Decentralized search in the presence of incentives. Game-theoretic notions can provide insight not just into the growth of a network, but also into the processes that operate on it. A topic of interest in peer-to-peer systems, as well as in the design of on-line communities, is the way in which the incentives offered to the members of the system influence the extent to which they are willing to forward queries and information. In the case of decentralized search, suppose that there is some utility associated with routing the message from the starting node to the target, and intermediate nodes behave strategically, demanding compensation for their participation in the construction of the path. How do results on decentralized path formation change when such behavior is incorporated into the model? In [40], this question is made precise in a setting where the underlying network is a random tree, constructed via a branching process. It would be interesting to consider analogous issues in richer classes of networks. 9. Reconstruction. The networks we have considered here have all been embedded in some underlying “reference frame” – grids, hierarchies, or set systems – and most of our analysis has been predicated on a model in which the network is presented together with this embedding. This makes sense in many contexts; recall, for example, the discussion from Section 6 of network data explicitly embedded in Web topic directories [49], corporate hierarchies [1], or the geography of the U.S. [43]. In some cases, however, we may be presented with just the network itself, and the goal is to determine whether it has a natural embedding into a spatial or hierarchical structure, and to recover this embedding if it exists. For example, we may have data on communication within an organization, and the goal is to reconstruct the hierarchical structure under the assumption that the frequency of communication decreases according to a hierarchical model – or to reconstruct the positions of the nodes under the assumption that the frequency of communication decreases with distance according to a grid-based or rank-based model.

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One can formulate many specific questions of this flavor. For example, given a network known to be generated by the grid-based model with a given exponent α, can we approximately reconstruct the positions of the nodes on the grid? What if we are not told the exponent? Can we determine whether a given network was more likely to have been generated from a grid-based model with exponent α or α ? Or what if there are multiple long-range contacts per node, and we are only shown the long-range edges, not the local edges? A parallel set of questions can be asked for the hierarchical model. Questions of this type have been considered by Sandberg [55], who reports on the results of computational experiments but leaves open the problem of obtaining provable guarantees. Benjamini and Berger [11] pose related questions, including the problem of reconstructing the dimension d of the underlying lattice when presented with a graph generated by long-range percolation on a finite piece of Zd . 10. Comparing network datasets. As we saw earlier, the models proposed in Sections 4 and 5 suggest a general perspective from which to analyze network datasets, by studying the way in which the density of links decays with increasing distance or increasing group size (e.g. [1], [43]). One could naturally use this style of analysis to compare related network datasets – for example taking the patterns of communication within k different organizations (as Adamic and Adar did for the corporate lab they studied), and determining exponents γ1 , γ2 , . . . , γk for each such that the probability of a link between individuals v and w in a group of size g scales approximately as g −γi in the i th organization. Differences among these exponents would suggest structural differences between the organizations at a global level – communication in some is more long-range, while in others it is more clustered at the low levels of the hierarchy. It would be interesting to understand whether these differences in turn were naturally reflected in other aspects of the organizations’ behavior and performance. More generally, large-scale social, technological, and information networks are sufficiently complex objects that the guiding principles provided by simple models seem crucial for our understanding of them. The perspective suggested here has offered one such collection of principles, highlighting in particular the ways in which these networks are intertwined with the spatial and organizational structures that they inhabit. One can hope that as we gather an increasing range of different perspectives, our understanding of complex networks will continue to deepen into a rich and informative theory.

References [1] Adamic, L., Adar, E., How to search a social network. Social Networks 27 (3) (2005), 187–203. [2] Adamic, L. A., Lukose, R. M., Puniyani, A. R., Huberman, B. A., Search in Power-Law Networks. Phys. Rev. E 64 (2001), 46135–46143.


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[3] Aizenman, M., Newman, C. M., Discontinuity of the Percolation Density in OneDimensional 1/|x − y|2 Percolation Models. Comm. Math. Phys. 107 (1986), 611–647. [4] Albert, R., Barabási, A.-L., Statistical mechanics of complex networks. Rev. Modern Phys. 74 (2002), 47–97. [5] Anshelevich, E., Network Design and Management with Strategic Agents. Ph.D. thesis, Cornell University, 2005. [6] Aspnes, J., Shah, G., Distributed data structures for P2P systems. In Theoretical and Algorithmic Aspects of Sensor, Ad Hoc Wireless and Peer-to-Peer Networks (ed. by Jie Wu, ed.), CRC Press, Boca Raton, FL, 2005, 685–700. [7] Assouad, P., Plongements lipschitziens dans Rn . Bull. Soc. Math. France 111 (1983), 429–448. [8] Barabási, A.-L., Linked: the new science of networks. Perseus, Cambridge, Mass., 2002. [9] Barrière, L., Fraigniaud, P., Kranakis, E., Krizanc, D., Efficient Routing in Networks with Long Range Contacts. In Distributed Computing, Lecture Notes in Comput. Sci. 2180, Springer-Verlag, Berlin 2001, 270–284. [10] Ben-Naim, Eli, Frauenfelder, Hans, Toroczkai, Zoltan (eds.), Complex Networks. Lecture Notes in Phys. 650, Springer-Verlag, Berlin 2004. [11] Benjamini, I., Berger, N., The diameter of long-range percolation clusters on finite cycles. Random Structures Algorithms 19 (2001), 102–111. [12] Benjamini, I., Kesten, H., Peres,Y., Schramm, O., Geometry of the uniform spanning forest: transitions in dimensions 4, 8, 12, . . . . Ann. of Math. (2) 160 (2004), 465–491. [13] Biskup, M., On the scaling of the chemical distance in long range percolation models. Ann. Probab. 32 (2004), 2938–2977. [14] Bollobás, B., Random Graphs. 2nd edition, Cambridge Stud. Adv. Math. 73, Cambridge University Press, Cambridge 2001. [15] Bollobás, B., Mathematical results on scale-free random graphs. In Handbook of Graphs and Networks (ed. by S. Bornholdt, H. G. Schuster), Wiley/VCH, Weinheim 2003, 1–34. [16] Bollobás, B., Chung, F. R. K., The diameter of a cycle plus a random matching. SIAM J. Discrete Math. 1 (1988), 328–333. [17] Bollobás, B., de la Vega, W. F., The diameter of random regular graphs. Combinatorica 2 (1982), 125–134. [18] Clarke, I., Sandberg, O., Wiley, B., Hong, T. W., Freenet: A Distributed Anonymous Information Storage and Retrieval System. In Designing Privacy Enhancing Technologies, Lecture Notes in Comput. Sci. 2009, Springer-Verlag, Berlin 2001, 46–66 [19] Clauset, A., Moore, C., How Do Networks Become Navigable? Preprint, 2003; arxiv.org, cond-mat/0309415. [20] Coppersmith, D., Gamarnik, D., Sviridenko, M., The diameter of a long-range percolation graph. Random Structures Algorithms 21 (2002), 1–13. [21] Demers, A., Greene, D., Hauser, C., Irish, W., Larson, J., Shenker, S., Sturgis, H., Swinehart, D., and Terry, D., Epidemic algorithms for replicated database maintenance. ACM SIGOPS Operating Systems Review 22 (1) (1988), 8–32. [22] Dodds, P., Muhamad, R., Watts, D. J., An Experimental Study of Search in Global Social Networks. Science 301 (2003), 827.

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[23] Dorogovtsev, S. N., Mendes, J. F. F., Evolution of Networks: from biological networks to the Internet and WWW. Oxford University Press, New York 2003. [24] Duchon, P., Hanusse, N., Lebhar, E., Schabanel, N., Could any graph be turned into a small world? Theoret. Comput. Sci. 355 (1) (200), 96–103. [25] Durrett, R., Random Graph Dynamics. Cambridge University Press, Cambridge 2006. [26] Erdös, P., and Rényi, A., On the Evolution of Random Graphs. Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 17–61. [27] Fraigniaud, P., A New Perspective on the Small-World Phenomenon: Greedy Routing in Tree-Decomposed Graphs. In Proceedings of the 13th Annual European Symposium on Algorithms (ESA), 2005. [28] Fraigniaud, P., Gavoille, C., and Paul, C., Eclecticism shrinks even small worlds. In Proceedings of the 23rd Annual Symposium on Principles of Distributed Computing, ACM Press, New York 2004, 169–178. [29] Guare, J., Six Degrees of Separation: A Play. Vintage Books, New York 1990. [30] Gupta,A., Krauthgamer, R., and Lee, J. R., Bounded geometries, fractals, and low-distortion embeddings. In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 2003, 534–543. [31] Higham, D., Greedy Pathlengths and Small World Graphs. University of Strathclyde Mathematics Research Report 08, 2002. [32] Hong, T., Performance. In Peer-to-Peer: Harnessing the Power of Disruptive Technologies (ed. by A. Oram), O’Reilly and Associates, Sebastopol, CA, 2001, 203–241. [33] Jackson, M., A Survey of Models of Network Formation: Stability and Efficiency. In Group Formation in Economics; Networks, Clubs and Coalitions (ed. by G. Demange and M. Wooders), Cambridge University Press, Cambridge 2004, 11–57. [34] Kempe, D., Kleinberg, J., Demers, A., Spatial Gossip and Resource Location Protocols. In Proceedings of the 33rd Annual Symposium on Theory of Computing, ACM Press, New York 2001, 163–172. [35] Killworth, P., and Bernard, H., Reverse small world experiment. Social Networks 1 (1978), 159–192. [36] Kleinberg, J., Navigation in a Small World. Nature 406 (2000), 845. [37] Kleinberg, J., The small-world phenomenon: An algorithmic perspective. In Proceedings of the 32nd Annual Symposium on Theory of Computing, ACM Press, New York 2000, 163–170. [38] Kleinberg, J., Small-World Phenomena and the Dynamics of Information. In Advances in Neural Information Processing Systems (NIPS) 14 (2001), 431–438. [39] Kleinberg, J., Lawrence, S., The Structure of the Web. Science 294 (2001), 1849–1850. [40] Kleinberg, J., Raghavan, P., Query Incentive Networks. In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 2005. [41] Kleinfeld, J., Could it be a Big World After All? The ‘Six Degrees of Separation’ Myth. Society, April 2002. [42] Lebhar, E., Schabanel, N., Almost optimal decentralized routing in long-range contact networks. In Proceedings of the International Colloquium on Automata, Languages, and Programming (ICALP), 2004.


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[43] Liben-Nowell, D., Novak, J., Kumar, R., Raghavan, P., Tomkins, A., Geographic routing in social networks. Proc. Natl. Acad. Sci. USA 102 (2005), 11623–11628. [44] Lua, E.-K., Crowcroft, J., Pias, M., Sharma, R., and Lim, S., A Survey and Comparison of Peer-to-Peer Overlay Network Schemes. IEEE Communications Surveys and Tutorials 7 (2005), 72–93. [45] Malkhi, D., Naor, M., Ratajczak, D., Viceroy: a scalable and dynamic emulation of the butterfly. In Proceedings of the 21st Annual Symposium on Principles of Distributed Computing, ACM Press, New York 2002, 183–192. [46] Manku, G. S., Bawa, M., Raghavan, P., Symphony: Distributed hashing in a small world. In Proceedings of the 4th USENIX Symposium on Internet Technologies and Systems, USENIX, 2003, 127–140. [47] Manku, G., Naor, M., and Wieder, U., Know Thy Neighbor’s Neighbor: The Power of Lookahead in Randomized P2P Networks. In Proceedings of the 36th Annual Symposium on Theory of Computing, ACM Press, New York 2004, 54–63. [48] Martel, C., Nguyen, V., Analyzing Kleinberg’s (and other) small-world models. In Proceedings of the 23rd Annual Symposium on Principles of Distributed Computing, ACM Press, New York 2004, 179–188. [49] Menczer, F., Growing and Navigating the Small World Web by Local Content. Proc. Natl. Acad. Sci. USA 99 (22) (2002), 14014–14019. [50] Milgram, S., The small world problem. Psychology Today 1 (1967), 60–67. [51] Newman, C. M., Schulman, L. S., One Dimensional 1/|j − i|s Percolation Models: The Existence of a Transition for s ≤ 2. Comm. Math. Phys. 104 (1986), 547–571. [52] Newman, M. E. J., The structure and function of complex networks. SIAM Review 45 (2003), 167–256. [53] Nguyen, V., and Martel, C., Analyzing and characterizing small-world graphs. In Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, PA, 2005, 311–320. [54] van Renesse, R., Birman, K. P., Vogels, W., Astrolabe: A robust and scalable technology for distributed system monitoring, management, and data mining. ACM Trans. Computer Sys. 21 (2003), 164–206. [55] Sandberg, O., Distributed Routing in Small-World Networks. In Proceedings of the 8th Workshop on Algorithm Engineering and Experiments, 2006. [56] Sandberg, O., Searching a Small World. Licentiate thesis, Chalmers University, 2005. [57] Schulman, L. S., Long-range percolation in one dimension. J. Phys. A 16 (17) (1986), L639–L641. [58] Simsek, O., and Jensen, D., Decentralized search in networks using homophily and degree disparity. Proceedings of the 19th International Joint Conference on Artificial Intelligence, 2005. [59] Slivkins, A., Distance Estimation and Object Location via Rings of Neighbors. In Proceedings of the 24rd Annual Symposium on Principles of Distributed Computing, ACM Press, New York 2005, 41–50. [60] Strogatz, S., Exploring complex networks. Nature 410 (2001), 268. [61] Travers, J., and Milgram, S., An experimental study of the small world problem. Sociometry 32 (1969), 425–443.

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[62] Watts, Duncan J., Six Degrees: The Science of a Connected Age. W. W. Norton, Scranton, PA, 2003. [63] Watts, D. J., Dodds, P. S., Newman, M. E. J., Identity and Search in Social Networks. Science 296 (2002), 1302–1305. [64] Watts, D. J. and Strogatz, S. H., Collective dynamics of ’small-world’ networks. Nature 393 (1998), 440–442. [65] Wexler, T., Pricing Games with Selfish Users. Ph.D. thesis, Cornell University, 2005. [66] Zeng, J., Hsu, W.-J., Wang, J., Near Optimal Routing in a Small-World Network with Augmented Local Awareness. In Proceedings of the 3rd International Symposium on Parallel and Distributed Processing and Applications, 2005. [67] Zhang, H., Goel, A., Govindan, R., Using the Small-World Model to Improve Freenet Performance. In Proc. IEEE Infocom, IEEE Comput. Soc. Press, Los Alamitos, CA, 2002.

Department of Computer Science, Cornell University, Ithaca, NY 14853, U.S.A.

On expander graphs and connectivity in small space Omer Reingold∗

Abstract. This presentation is aimed to communicate a recently found deterministic algorithm for determining connectivity in undirected graphs [40]. This algorithm uses the minimal amount of memory possible, up to a constant factor. Specifically, the algorithm’s memory is comparable to that needed to store the name of a single vertex of the graph (i.e., it is logarithmic in the size of the graph). Our algorithm also implies a deterministic, short (i.e. of polynomial length), universal sequence of steps which explores all the edges of every regular undirected graph. Such a sequence will get one out of every maze, and through the streets of every city. More formally we give universal exploration sequences for arbitrary graphs and universal traversal sequences for graphs with some natural restriction on their labelling. Both sequences are constructible with logarithmic memory and are thus only polynomially long. To obtain this algorithm, we give a method to transform (using small memory), an arbitrary connected undirected graph into an expander graph (which is a sparse but highly connected graph). Mathematics Subject Classification (2000). Primary 05C40; Secondary 68Q15. Keywords. Graph connectivity, expander graphs, randomness in computation, symmetric logspace (SL), pseudorandomness.

1. Introduction An undirected graph is a pair of finite sets G = (V , E) where V is the set of vertices and E is the set of edges. An edge is simply a pair of vertices (we say that the edge is adjacent to these two vertices and the two vertices are adjacent to it). Special cases of graphs that may be useful to visualize are mazes and road maps where a vertex corresponds to an intersection and an edge directly connects the two vertices adjacent to it.1 Given as input an undirected graph G and two vertices s and t, the undirected st-connectivity (denoted USTCON) problem is to decide whether or not the two vertices are connected by a path (i.e. a sequence of edges) in G . This fundamental combinatorial problem has received a lot of attention in the last few decades and was studied in a large variety of computational models. It is a basic building block for more complex graph algorithms and is complete2 for an interesting class ∗ Incumbent of the Walter and Elise Haas Career Development Chair, Research supported by US-Israel Bina-

tional Science Foundation Grant 2002246. 1 In a directed graph an edge is an ordered pair of vertices and we can therefore think of an edge as a unidirectional road going from the first vertex to the second vertex. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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of computational problems known as SL (these are problems solvable by symmetric, non-deterministic, log-space computations [25]). A few interesting examples of problems in SL are deciding if a graph is bipartite (equivalently if it is 2-colorable), if a bounded degree graph is planar and identifying chordal graphs, interval graphs, split graphs and permutation graphs (see [8] for a recent study of SL and quite a few of its complete problems). The time complexity of USTCON is well understood as basic search algorithms, particularly breadth-first search (BFS) and depth-first search (DFS), are capable of solving USTCON in linear time. In fact, these algorithms apply to the more complex problem of st-connectivity in directed graphs, denoted STCON (which is complete for the class NL of non-deterministic log-space computations). Unfortunately, the space required to run these algorithms is linear as well. In a recent paper [40] we resolve the space (memory) complexity of USTCON, up to a constant factor, by presenting a log-space (polynomial-time) algorithm for solving it (our algorithm also solves the corresponding search problem, of finding a path from s to t if such a path exists). In this presentation (which is in large part borrowed from [40]) we will discuss this algorithm. We will also discuss the connection of this question to the possible tradeoff between two resources of computation: randomness and memory. Finally, we will discuss explicit (and space efficient) constructions of fascinating combinatorial objects known as universal traversal and universal exploration sequences. Loosely, these are short sequences of simple directions that lead a walk through all of the edges of any graph of an appropriate size. Our main technical tool is borrowed from a combinatorial construction of expander graphs due to Reingold, Vadhan and Wigderson [43]. We will further elaborate on the connection to expander graphs. Background. Given the inefficiency in terms of memory of BFS and DFS, Savitch’s [47] introduced an algorithm which solves STCON in space log2 ( · ) (and super-polynomial time). Major progress in understanding the space complexity of USTCON was made by Aleliunas, Karp, Lipton, Lovász, and Rackoff [2], who gave a randomized log-space algorithm for the problem. Specifically, they showed that a random walk (a path that selects a uniform edge at each step) starting from an arbitrary vertex of any connected undirected graph will visit all the vertices of the graph in polynomial number of steps. Therefore, the algorithm can perform a random walk starting from s and verify that it reaches t within the specified polynomial number of steps. Essentially all that the algorithm needs to remember is the name of the current vertex and a counter for the number of steps already taken. With this result we get the following view of space complexity classes: L ⊆ SL ⊆ RL ⊆ NL ⊆ L2 (where RL is the class of problems that can be decided by randomized log-space algorithms with one-sided error and Lc is the class of problems that can be decided deterministically in space logc ( · )). 2A complete problem for a class is such that any other problem “efficiently reduces” to it. Therefore, an algorithm for this problem implies an algorithm (which is almost as efficient) for any other problem in the class.

On expander graphs and connectivity in small space

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The existence of a randomized log-space algorithm for USTCON puts this problem in the context of derandomization. Can this randomized algorithm be derandomized without substantial increase in space? Furthermore, the study of the space complexity of USTCON has gained additional motivation as an important test case for understanding the tradeoff between two central resources of computations, namely between memory space and randomness. Particularly, a natural goal on the way to proving RL = L is to prove that USTCON ∈ L, as USTCON is undoubtedly one of the most interesting problems in RL. Following [2], most of the progress on the space complexity of USTCON indeed relied on the tools of derandomization. In particular, this line of work greatly benefited from the development of pseudorandom generators that fool space-bounded algorithms [1], [10], [33], [19] and it progressed concurrently with the study of the L vs. RL problem. Another very influential notion, introduced by Stephen Cook in the late 70s, is that of a universal-traversal sequence. Loosely, this is a fixed sequence of directions that guides a deterministic walk through all of the vertices of any connected graph of the appropriate size (see further discussion below). While Nisan’s space-bounded generator [33], did not directly imply a more space efficient USTCON algorithm it did imply quasi-polynomially-long, universal-traversal sequences, constructible in space log2 ( · ). These were extremely instrumental in the work of Nisan, Szemeredi and Wigderson [34] who showed that USTCON ∈ L3/2 – The first improvement over Savitch’s algorithm in terms of space (limited of course to the case of undirected graphs). Using different methods, but still heavily relying on [33], Saks and Zhou [46] showed that every RL problem is also in L3/2 (their result in fact generalizes to randomized algorithms with two-sided error). Relying on the techniques of both [34] and [46], Armoni, et. al. [9] showed that USTCON ∈ L4/3 . Their USTCON algorithm was the most space-efficient one previous to this work. We note that the most space-efficient polynomial-time algorithm for USTCON previously known was Nisan’s [32], which still required space log2 ( · ). Independent of our work (and using different techniques), Trifonov [49] has presented an O(log n log log n)space, deterministic algorithm for USTCON. Our approach. The essence of our algorithm is in the following very natural approach: If you want to solve a connectivity problem on your input graph, first improve its connectivity. In other words, transform your input graph (or rather, each one of its connected components), into an expander.3 We will also insist on the final graph being constant degree (i.e., every vertex is adjacent to a constant number of edges). Once the connected component of s is a constant-degree expander, then it is trivial to decide if s and t are connected: Since expander graphs have logarithmic diameter, 3 The exact definition of expander graphs is less important for now, and the following description could be understood by viewing expanders as graphs with very strong connectivity properties. Still, for the knowledgable reader, the particular measure that seems the most convenient to work with is the second eigenvalue (in absolute value) of the adjacency matrix of the graph (we will only need to work with regular graphs). It may however be that other, more combinatorial, measures will also do (see [41] for a more detailed discussion).

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it is enough to enumerate all logarithmically long paths starting with s and to see if one of these paths visits t. Since the degree is constant, the number of such paths is polynomial and they can easily be enumerated in log space. How can we turn an arbitrary graph into an expander? First, we note that every connected, non-bipartite, graph can be thought of as an expander with very small (but non-negligible) expansion. Consider for example an arbitrary connected graph with self-loops added to each one of its vertices. The number of neighbors of every strict subset of the vertices is larger than its size by at least one. In this respect, the graph can be thought of as expanding by a factor 1 + 1/N (where N is the total number of vertices in the graph). Now, a very natural operation that improves the expansion of the graph is powering. The k th power of G contains an edge between two vertices v and w for every path of length k in G. Formally, it can be shown that by taking some polynomial power of any connected non-bipartite graph (equivalently, by repeatedly squaring the graph logarithmic number of times), it will indeed turn into an expander. The down side of powering is of course that it increases the degree of the graph. Taking a polynomial or any non-constant power is prohibited if we want to maintain constant degree. Fortunately, there exist operations that can counter this problem. Consider for example the replacement product of a D-regular graph G with a d-regular graph H on D vertices (with d D). This can be loosely defined as follows: Each vertex v of G is replaced with a “copy” Hv of H . Each of the D vertices of Hv is connected to its neighbors in Hv but also to one vertex in Hw , where (v, w) is one of the D edges going out of v in G. The degree in the product graph is d + 1 (which is smaller than D). Therefore, this operation can transform a graph G into a new graph (the product of G and H ) of smaller degree. It turns out that if H is a “good enough” expander, the expansion of the resulting graph is “not worse by much” than the expansion of G. Formal statements to this affect were proven by Reingold, Vadhan and Wigderson [43] for both the replacement product and the zig-zag product, introduced there. Independently, Martin and Randall [30], building on previous work of Madras and Randall [27], proved a decomposition theorem for Markov chains that also implies that the replacement product preserves expansion. Given the discussion above, we are ready to informally describe our USTCON algorithm. First, turn the input graph into a constant-degree, regular graph with each connected component being non-bipartite (this step is very easy). Then, the main transformation turns each connected component of the graph, in logarithmic number of phases, into an expander. Each phase starts by raising the current graph to some constant power and then reducing the degree back via a replacement or a zigzag product with a constant-size expander. We argue that each phase enhances the expansion at least as well as squaring the graph would, and without the disadvantage of increasing the degree. Finally, all that is left is to solve USTCON on the resulting graph (which is easy as the diameter of each connected component is only logarithmic). To conclude that USTCON ∈ L, we need to argue that all of the above can be done in logarithmic space, which easily reduces to showing that the main transformation can be carried out in logarithmic space. For that, consider the graph Gi obtained after i


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phases of the transformation. We note that a step on Gi (i.e., evaluating the j ’th neighbor of some vertex v in Gi ) is composed of a constant number of operations that are either a step on the graph Gi−1 from the previous phase or an operation that only requires a constant amount of memory. As the memory for each of these operations can be freed after it is performed, the memory for carrying out a step on Gi is only larger by an additive constant than the memory for carrying out a step on Gi−1 . This implies that the entire transformation is indeed log space. Universal traversal sequences While universal-traversal sequences were introduced as a way for proving USTCON ∈ L, these are interesting combinatorial objects in their own right. A universal-traversal sequence for D-regular graphs on N-vertices, is a sequence of edge labels in {1, . . . , D} such that for every such graph, for every labelling of its edges, and for every start vertex, the deterministic walk defined by these labels (where in the i’th step we take the edge labeled by the i’th element of the sequence), visits all of the vertices of the graph. Aleliunas et. al. [2] showed that polynomial-length universal-traversal sequence exists, and in fact almost every sequence of the appropriate length will do. We are interested in obtaining a polynomially-long, universal-traversal sequence that is constructible in logarithmic space (even less explicit sequences may still be very interesting). This is again a derandomization problem. Namely, can we derandomize the probabilistic construction of universal-traversal sequences? Explicit constructions of polynomially-long universal-traversal sequences are only known for extremely limited classes of graphs. Even for expander graphs, such sequences are only known when the edges are “consistently labelled” [18] (this means that the labels of all edges that lead to any particular vertex are distinct). It is therefore not very surprising that our algorithm on its own does not imply full fledged universaltraversal sequences. Still, our algorithm can be shown to imply a very local, and quite oblivious, deterministic procedure for exploring a graph. We can think of our algorithm as maintaining a single pebble, that is placed on the edges of the graph. The pebble is moved either from one side of the edge to another, or between different edges that are adjacent to the same vertex (say to the next or to the previous edge). As with universal-traversal sequences, the fixed sequence of instructions is good for every graph, for every labelling of its edges, and for any starting point on the graph. The only difference from universal-traversal sequences is that the pebble here is placed on the edges rather than on the vertices of the graph. In particular, we get polynomially-long, universal-exploration sequences for all undirected graphs. In universal-exploration sequences, introduced by Koucky [23], the elements of the sequence are not interpreted as absolute edge-labels but rather as offsets from the previous edge that was traversed. In terms of traversal sequences, our algorithm implies a polynomially-long, universaltraversal sequence that is constructible in logarithmic space under some restrictions on the labelling. These restrictions were relaxed in a subsequent work [41] to be identical to those of [18]. For more details see Section 5.

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More on previous work Graph connectivity problems and space-bounded derandomization are the focus of a vast and diverse body of research. The scope of this paper only allows for an extremely partial discussion of this area. Some very beautiful and influential research (as many of the papers mentioned above) is only briefly touched upon, other areas will not be discussed at all (examples include, time-space tradeoffs for deterministic and randomized connectivity algorithms, restricted constructions of universal traversal sequences, and analysis of connectivity in many other computational models). Insightful, though somewhat outdated, surveys on these topics were given by Wigderson [50] and by Saks [45]. Useful discussion and pointers were also given by Koucky [24]. We continue here by mentioning a few of the most related previous results (most of which are subsumed by the results of this paper). A more technical comparison with some previous work appears in Section 6. Following Aleliunas et. al. [2], Borodin et. al. [12] gave a zero-error, randomized, log-space algorithm for USTCON. An upper bound of different nature on SL was given by Karchmer and Wigderson [21], who showed SL ⊆ ⊕L. Nisan and Ta-Shma [35] showed that SL is closed under complement, thus collapsing the “symmetric log-space hierarchies” of both Reif [39] and Ben Asher et. al. [11], and putting some very interesting problems into SL. To give just one example, the planarity of bounded-degree undirected graphs was placed in SL as a corollary (we refer again to [8] for a list of SL-complete problems). A research direction initiated by Ajtai et. al. [1], and continued with Nisan and Zuckerman [36] is to fully derandomize (i.e., to put in L) log n-space computations that use fewer than n random bits (poly log n√ bits in the case of [36]). Raz and Reingold [38] showed how to derandomize 2 log n bits for subclasses of RL. One √ of their main applications can be viewed as derandomizing 2 log n bits for SL. It is interesting to note (and personally gratifying to the author) that the techniques of [38] played a major roll in the definition of the zig-zag product and with this work found their way back to the study of space-bounded derandomization. Goldreich and Wigderson [17] gave an algorithm that on all but a tiny fraction of the graphs, evaluates USTCON correctly (and on the rest of the graphs outputs an error message). Based on rather relaxed computational hardness assumptions, Klivans and van Melkebeek [22] proved both that RL = L and that efficiently constructible, polynomial length, universal traversal sequences exist.

2. Preliminaries This section discusses various aspects of graphs: their representation, eigenvalue expansion, graph powering, and two graph products (the replacement product and the zig-zag product). The definitions and notation used here are borrowed directly from [43].


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2.1. Graphs representations. There are several standard representations of graphs. Fortunately, there exist log-space transformations between natural representations. Thus, the space complexity of USTCON is to a large extent independent of the representation of the input graph. When discussing the eigenvalue expansion of a graph, we will consider its adjacency matrix. That is, the matrix whose (nonnegative, integral) entry (u, v) equals to the number of edges that go from vertex u to vertex v. Note that this representation allows graphs with self loops and parallel edges (and indeed such graphs may be generated by our algorithm). A graph is undirected iff its adjacency matrix is symmetric (implying that for every edge from u to v there is an edge from v to u). It is D-regular if the sum of entries in each row (and column) is D (so exactly D edges are incident to every vertex). Let G be a D-regular undirected graph on N vertices. When considering a walk on G, we would like to assume that the edges leaving each vertex of G are labeled from 1 to D in some arbitrary, but fixed, way. We can then talk about the i’th edge incident to a vertex v, and similarly about the i’th neighbor of v. A central insight of [43] is that when taking a step on a graph from vertex v to vertex w, it may be useful to keep track of the edge traversed to get to w (rather than just remembering that we are now at w). This gave rise to a new representation of graphs through the following permutation on pairs of vertex name and edge label: Definition 2.1. For a D-regular undirected graph G, the rotation map Rot G : [N] × [D] → [N] × [D] is defined as follows: RotG (v, i) = (w, j ) if the i’th edge incident to v leads to w, and this edge is the j ’th edge incident to w. Rotation maps will indeed be the representation of choice for this work. Specifically, the first step of our algorithm will be to transform the input graph into a regular one specified by its rotation map (in particular, this step will give labels to the edges of the graph). 2.2. Eigenvalue expansion and st-connectivity for expanders. Expanders are sparse graphs which are nevertheless highly connected. The strong connectivity properties of expanders make them very desirable in our context. Specifically, since the diameter of expander graphs is only logarithmically long, there is a trivial log-space algorithm for finding paths between vertices in constant-degree expanders. The particular formalization of expanders used in this paper is the (algebraic) characterization based on the spectral gap of their adjacency matrix. Namely, the gap between the first and second eigenvalues of the (normalized) adjacency matrix. The normalized adjacency matrix M of a D-regular undirected graph G, is the adjacency matrix of G divided by D. In terms of the rotation map, we have: 1 Mu,v = · {(i, j ) ∈ [D]2 : RotG (u, i) = (v, j )} . D M is simply the transition probability matrix of a random walk on G. By the D-regularity of G, the all-1’s vector 1N = (1, 1, . . . , 1) ∈ RN is an eigenvector of M of

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eigenvalue 1. It turns out that all the other eigenvalues of M have absolute value at most 1. We denote by λ(G), the second largest eigenvalue (in absolute value) of G’s normalized adjacency matrix. We refer to a D-regular undirected graph G on N vertices such that λ(G) ≤ λ as an (N, D, λ)-graph. It is well-known that the second largest eigenvalue of G is a good measure of G’s expansion properties. In particular, it was shown by Tanner [48] and Alon and Milman [5] that second-eigenvalue expansion implies (and is in fact equivalent [3]) to the standard notion of vertex expansion. In particular, for every λ < 1 there exists ε > 0 such that for every (N, D, λ)-graph G and for any set S of at most half the vertices in G, at least (1 + ε) · |S| vertices of G are connected by an edge to some vertex in S. This immediately implies that G has a logarithmic diameter: Proposition 2.2. Let λ < 1 be some constant. Then for every (N, D, λ)-graph G and any two vertices s and t in G, there exists a path of length O(log N) that connects s to t. Proof. By the vertex expansion of G, for some = O(log N) both s and t have more than N/2 vertices of distance at most from them in G. Therefore, there exists a 2 vertex v that is of distance at most from both s and t. We can therefore conclude that st-connectivity in constant-degree expanders can be solved in log-space: Proposition 2.3. Let λ < 1 be some constant. Then there exists a space O(log D · log N) algorithm A such that when a D-regular undirected graph G on N vertices is given to A as input, the following hold: 1. If s and t are in the same connected component and this component is an (N , D, λ)-graph then A outputs ‘connected’. 2. If A outputs ‘connected’ then s and t are indeed in the same connected component. Proof. The algorithm A simply enumerates all D paths of length = O(log N) from s. (Where the leading constant in the big-O notation depends on λ as in Proposition 2.2.) The algorithm A outputs ‘connected’ if and only if at least one of these paths encounters t. Following any particular path from s of length requires space O(log N), (when given as input the sequence of edge labels in [D] = {1, 2, . . . D} traversed by this path). Enumerating all these D paths requires space O(log D · log N). By Proposition 2.2, in case (1), s and t are of distance at most of each other and A will indeed find a path from s to t and will output ‘connected’. On the other hand, A never outputs ‘connected’ unless it finds a path from s to t, implying (2). 2 Using the probabilistic method, Pinsker [37] showed that most 3-regular graphs are expanders (in the sense of vertex expansion), and this result was extended to eigenvalue


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bounds in [3], [13], [15], [14]. Various explicit families of constant-degree expanders, some with optimal tradeoff between degree and expansion, were given in literature (cf. [28], [16], [20], [5], [4], [26], [29], [31], [43]). Our algorithm will employ a single constant size expander with rather weak parameters. This expander can be obtained by exhaustive search or by any of the explicit constructions mentioned above. In fact, one can use simpler explicit constructions than the ones given above, as we can afford a rather large degree (with respect to the number of vertices), rather than a constant degree. An example of a simpler construction that would suffice is the one given by Alon and Roichman [6], (see also related discussions in [43] regarding their “base graph”). Proposition 2.4. There exists some constant De and a ((De )16 , De , 1/2)-graph. Finally, a key fact for our algorithm is that every connected, non-bipartite graph has a spectral gap which is at least inverse polynomial in the size of the graph (recall that a graph is non-bipartite if there is no partition of the vertices such that all the edges go between the two sides of the partition). Lemma 2.5 ([7]). For every D-regular, connected, non-bipartite graph G on [N] it holds that λ(G) ≤ 1 − 1/DN 2 . 2.3. Powering. Our main transformation will take a graph and transform each one of its connected components (that in itself will be a connected, non-bipartite graph), into a constant degree expander. If we ignore the requirement that the graph remains constant degree, a simple way of amplifying the (inverse polynomial) spectral gap of a graph is by powering. Definition 2.6. Let G be a D-regular multigraph on [N] given by rotation map RotG . The t’th power of G is the D t -regular graph Gt whose rotation map is given by Rot Gt (v0 , (a1 , a2 , . . . , at )) = (vt , (bt , bt−1 , . . . , b1 )), where these values are computed via the rule (vi , bi ) = Rot G (vi−1 , ai ). Proposition 2.7. If G is an (N, D, λ)-graph, then Gt is an (N, D t , λt )-graph. Proof. The normalized adjacency matrix of Gt is the t’th power of the normalized adjacency matrix of G, so all the eigenvalues also get raised to the t’th power. 2 2.4. Two graph products. While taking a power of a graph reduces its second eigenvalue, it also increases its degree. As we are interested in producing constantdegree graphs, we need a complementing operation that reduces the degree of a graph without harming its expansion by too much. We now discuss two graph products that are capable of doing exactly that. The first is the very natural product, known as the replacement product. Assume that G is a D-regular graph on [N] and H is a d-regular graph on [D] (where d is significantly smaller than D). Very intuitively, the replacement product of the two

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graphs is defined as follows: Each vertex v of G is replaced with a “copy” Hv of H . Each of the D vertices of Hv is connected to its neighbors in Hv but also to one vertex in Hw , where (v, w) is one of the D edges going out of v in G. The degree in the product graph is d + 1 (which is smaller than D).4 A second, slightly more evolved, product introduced by Reingold, Vadhan and Wigderson [43], is the zig-zag graph product. Here too we replace each vertex v of G with a “copy” Hv of H . However, the edges of the zig-zag product of G and H correspond to a subset of the paths of length three in the replacement product of these graphs5 (see formal definition below). The degree of the product graph here is d 2 (which should still be thought of as significantly smaller that D). It is immediate from their definition, that both products can transform a graph G to a new graph (the product of G and H ) of smaller degree. As discussed in the introduction, it was previously shown [43], [30] that if H is a “good enough” expander, then the expansion of the resulting graph is “not worse by much” than the expansion of G (see formal statement below for the zig-zag product). Either one of these products can be used in our USTCON algorithm (with some variation in the parameters). We find it more convenient to work with the zig-zag product (even though it is a bit more involved), hence we proceed by formally defining it. Definition 2.8 ([43]). If G is a D-regular graph on [N] with rotation map Rot G and H zH is a d-regular graph on [D] with rotation map RotH , then their zig-zag product G 2 is defined to be the d -regular graph on [N ] × [D] whose rotation map RotGz H is as follows (see Figure 1 for an illustration): ' s

$ '

(v, a )

(v, a)

i

i

&

Hv

' sj (w, b )

&

$

s

% $

j s Hw (w, b)

s a

b s

w &

%

v

G

%

Figure 1. On the left – an edge of the zig-zag product is composed of three steps: a “short step” (in Hv ), a “big step” (between Hv and Hw which corresponds to an edge of G between v and w), and a final “small step” (in Hw ). The values i ,i , j and j are labels of edges of H (going out of the H vertices a, a , b and b respectively). On the right – the projection of these steps on the graph G (which corresponds to the middle step specified by (w, b ) = Rot G (v, a )). 4 Sometimes it is better to consider the balanced replacement product, where every edge in G is taken d times

in parallel. The degree of the product graph in this case is 2d instead of d + 1. 5 Those length three paths that are composed of a “short edge” (an edge inside one of the copies H ), a “long v edge” (one that corresponds to an edge of G), and finally one additional “short edge”.


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RotGz H ((v, a), (i, j )): 1. Let (a , i ) = Rot H (a, i). 2. Let (w, b ) = Rot G (v, a ). 3. Let (b, j ) = Rot H (b , j ). 4. Output ((w, b), (j , i )). z H ) was bounded as a function of λ(G) and λ(H ). The interesting In [43], λ(G case there was when both λ(G) and λ(H ) were small constants (and in fact, λ(G) is significantly smaller than λ(H )). In our context, λ(H ) will indeed be a small constant but G may have an extremely small spectral gap (recall that the spectral gap of G is z H to be roughly the same 1 − λ(G)). In this case, we want the spectral gap of G as that of G (i.e., smaller by at most a constant factor). It turns out that the stronger z H ), given in [43] implies a useful bound also in this case. We note bound on λ(G that a simpler proof for the sort of bound on the zig-zag product we need is given in [41] (in a more general setting than the one considered in [43]). Theorem 2.9 ([43]). If G is an (N, D, λ)-graph and H is a (D, d, α)-graph, then z H is a (N · D, d 2 , f (λ, α))-graph, where G f (λ, α) =

1 1 (1 − α 2 )λ + (1 − α 2 )2 λ2 + 4α 2 . 2 2

z H is smaller than that As a simple corollary, we have that the spectral gap of G of G by a factor that only depends on λ(H ). Corollary 2.10. If G is an (N, D, λ)-graph and H is a (D, d, α)-graph, then z H) ≥ 1 − λ(G

1 (1 − α 2 ) · (1 − λ). 2

Proof. Since λ ≤ 1 we have that 1 1 1 1 (1 − α 2 )2 λ2 + 4α 2 ≤ (1 − α 2 )2 + 4α 2 = (1 + α 2 ) = 1 − (1 − α 2 ). 2 2 2 2 Therefore, f (λ, α) from Theorem 2.9 satisfies f (λ, α) ≤ 1 − 21 (1 − α 2 )(1 − λ).

2

3. Transforming graphs into expanders This section gives a log-space transformation that essentially turns each one of the connected components of a graph into an expander. This is the main part of our USTCON algorithm.

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Definition 3.1 (Main transformation). On input G and H , where G is a D 16 -regular graph on [N ] and H is a D-regular graph on [D 16 ], both given by their rotation maps, the transformation T outputs the rotation map of a graph G defined as follows:

• Set to be the smallest integer such that (1 − 1/DN 2 )2 < 1/2. • Set G0 to equal G, and for i > 0 define Gi recursively by the rule: z H )8 . Gi = (Gi−1 Denote by Ti (G, H ) the graph Gi , and T (G, H ) = G Note that by the basic properties of powering and the zig-zag product, it follows inductively that each Gi is a D 16 -regular graph over [N] × ([D 16 ])i . In particular, the zig-zag product of Gi and H is well defined. In addition, if D is a constant, then = O(log N) and G has poly(N) vertices. Our first lemma shows that T is capable of turning an input graph G into an expander G (as long as H is in itself an expander). Lemma 3.2. Let G and H be the inputs of T as in Definition 3.1. If λ(H ) ≤ 1/2 and G is connected and non-bipartite then λ(T (G, H )) ≤ 1/2. Proof. Since G = G0 is connected and non-bipartite we have by Lemma 2.5 that λ(G0 ) ≤ 1 − 1/DN 2 . By the choice of it is therefore enough to prove that for every i > 0, it holds that λ(Gi ) ≤ max{λ(Gi−1 )2 , 1/2}. Denote λ = λ(Gi−1 ). Since z H ) ≤ 1 − 3/8(1 − λ) < λ(H ) ≤ 1/2, we have by Corollary 2.10 that λ(Gi−1 1 − 1/3(1 − λ). By the definition of Gi and by Proposition 2.7 we have that λ(Gi ) < [1 − 1/3(1 − λ)]8 . We now consider two cases. First, if λ < 1/2 then λ(Gi ) < (5/6)8 < 1/2. Otherwise, elementary calculation shows that [1 − 1/3(1 − λ)]4 ≤ λ and therefore λ(Gi ) < λ2 . The lemma follows. 2 As we are working our way to solving st-connectivity, rather than solving connectivity (the problem of deciding if the input graph is connected or not), our transformation should be meaningful even for graphs that are not connected (as even in this case the two input vertices s and t may still be in the same connected component). For that, we will argue that T operates separately on each connected component of G. The reason is that T is composed of two operations (the zig-zag product and powering), that also operate separately on each connected component. We will need some additional notation: For any graph G and subset of its vertices S, denote by G|S the subgraph of G induced by S (i.e., the graph on S which contains all of the edges in G between vertices in S). A set S is a connected component of G if G|S is connected and the set S is disconnected from the rest of G (i.e., there are no edges in G between vertices in S and vertices outside of S). Lemma 3.3. Let G and H be the inputs of T as in Definition 3.1. If S ⊆ [N] is a connected component of G then T (G|S , H ) = T (G, H )|S×([D 16 ]) .


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Proof. We will only rely on S being disconnected from the rest of G, and will prove inductively that Ti (G|S , H ) = Ti (G, H )|S×([D 16 ])i . Note that for i > 0 this directly implies that S × ([D 16 ])i is disconnected from the rest of Ti (G, H ) (since both Ti (G|S , H ) and Ti (G, H ) are D 16 -regular, and thus all of the D 16 edges incident to a vertex in S × ([D 16 ])i reside inside Ti (G, H )|S×([D 16 ])i ). The base case i = 0 is trivial, and here too S × ([D 16 ])i = S is disconnected from the rest of Ti (G, H ) = G, by assumption. Assume by induction that Ti (G|S , H ) = Ti (G, H )|S×([D 16 ])i . Set Gi = Ti (G, H ) and Si = S × ([D 16 ])i (and recall that Si is disconnected from the rest of Gi ). Then, by the definition of the zig-zag product, Si × [D 16 ] is disconnected from the rest of z H and the edges incident to Si ×[D 16 ] in Gi z H are exactly as in Gi |Si ×[D 16 ] z H. Gi 16 By the definition of powering we now have that Si × [D ] is disconnected from the z H )8 and the edges incident to Si × [D 16 ] in (Gi z H )8 are exactly as rest of (Gi 8 z H ) . This proves the induction hypothesis for i + 1 and completes in (Gi |Si ×[D 16 ] the proof. 2 Finally, we need to argue that T is a log-space transformation (when D is a constant). The reason is that the evaluation of the rotation map RotGi+1 of each graph Gi+1 in the definition of T requires just a constant additional amount of memory over the evaluation of RotGi . Simply, the evaluation of RotGi+1 is composed of a constant number of operations, where each operation is either an evaluation of RotGi or it requires constant amount of memory (and the same memory can be used for each one of these operations). So the additional memory needed for evaluating RotGi+1 is essentially a constant size counter (keeping track of which operation we are currently performing). Lemma 3.4. For every constant D the transformation T of Definition 3.1 can be computed in space O(log N) on inputs G and H , where G is a D 16 -regular graph on [N ] and H is a D-regular graph on [D 16 ]. Proof. We describe an algorithm A that on inputs G and H computes the rotation map RotG of G = T (G, H ). Namely, given G and H (written on the readonly input tape), it enumerates all values (v, a) in the domain of RotG and outputs [(v, a), RotG (v, a)]. Recall that a value (v, a) in the domain of RotG consists of v ∈ [N] × ([D 16 ]) which is the name of a G vertex, and a ∈ [D 16 ], which is the label of a G edge. Since = O(log N) and D is a constant, the length of each value (v, a) is O(log N) and therefore enumerating all of these values can be done in space O(log N). It remains to show that for any particular value (v, a), evaluating Rot G (v, a) can also be done in the required space. The algorithm A will first allocate the following variables: v which will take value in [N] (specifying a vertex of G), and + 1 variables a0 , a1 , . . . , a each taking value in [D 16 ] (and each specifying a vertex name of H ; In addition, a0 may specify an edge label of G). It is sometimes convenient to view each one of a1 , . . . , a as specifying a sequence of 16 edge labels of H . In this case we denote ai = ki,1 . . . ki,16 . Now, A

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will copy the value (v, a) into the above mentioned variables: v into v, a0 , . . . , a−1 and a into a . Throughout the execution of A, the values of these variables will slowly evolve such that when A finishes (for this particular (v, a)), the same variables will contain the desired output RotG (v, a) (which is of the same range as the input (v, a)). We describe the operation of A in a recursive manner that closely follows the definition of T . Particularly, at each level of the recursion, A will evaluate RotGi for some i on the appropriate prefix v, a0 , . . . , ai of the variables defined above. For the base case i = 0, Rot G0 = Rot G is written on the input tape, and can therefore be evaluated in space O(log N) by simply searching the input tape for the desired entry. For larger i, the evaluation of RotGi is as follows: For j = 1 to 16: • Set ai−1 , ki,j ← Rot H (ai−1 , ki,j ). • If j is odd, recursively set v, a0 . . . ai−1 ← Rot Gi−1 ((v, a0 . . . ai−2 ), ai−1 ). • If j = 16, reverse the order of the individual labels in ai : Set ki,1 , . . . , ki,16 ← ki,16 , . . . , ki,1 . The correctness of A immediately follows from the definition of T and from the operations of which it consists (powering and the zig-zag product). Essentially, going over the operations (in the first two bullets) for any two consecutive values z H ). Repeating eight times implies a path of j corresponds to one step on (Gi−1 z H ), or alternatively one step on (Gi−1 z H )8 . The third of length eight on (Gi−1 bullet reverses the order of labels to fit the definition of zig-zag and powering. We therefore concentrate on the space complexity of A. Note that each node of the recursion tree performs a constant number of operations and makes a constant number of recursive calls. In addition the depth of the recursion is + 1 = O(log N). Therefore, maintaining the recursion can be done in space O(log N). Furthermore, each one of the basic operations (evaluating RotG , evaluating RotH , and reversing the order of labels in the last step) can be performed in space O(log N). Finally, the only memory that needs to be kept after a basic operation is performed, is the memory holding the variables v, a0 , . . . , a (that are shared by all of these operations), and the memory for maintaining the recursion. We therefore conclude that the space complexity of A is O(log N) which completes the proof. 2

4. A log-space algorithm for USTCON This section puts together the tools developed above into a deterministic log-space algorithm that decides undirected st-connectivity. As will be discussed in Section 5, the algorithm can also output a path from s to t if such a path exists. Theorem 4.1. USTCON ∈ L.


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As undirected USTCON is complete for SL [25], Theorem 4.1 can be rephrased as follows. Theorem 4.2. SL = L. Proof of Theorem 4.1. We give an algorithm A that gets as input a graph G over the set of vertices [N], and two vertices s and t in [N ]. For concreteness, we assume that the graph is given via the adjacency matrix representation. A will answer ‘connected’ if and only if there exists a path in G between s and t (i.e., s and t are in the same connected component). Furthermore, G will use space which is logarithmic in its input size. The algorithm A will need to evaluate the rotation map of a ((De )16 , De , 1/2)graph H , where De is some constant. By Proposition 2.4, there exists such a graph and therefore A can obtain it by exhaustive search using constant amount of memory (a more efficient alternative is of course to obtain H by any of the explicit constructions of expanders mentioned in Section 2.2). Let T be the transformation given by Definition 3.1. We would like to apply T to G and H in order to obtain a graph where each connected component is an expander. For such graphs, st-connectivity can be solved in logarithmic space by Proposition 2.3. However, we will first need to preprocess G in order to get a new graph Greg such that (Greg , H ) is a correct input to T . In particular, we need Greg to be a De16 -regular graph given by its rotation map. There are various ways of transforming G to Greg . The one given here was selected for its simplicity even though it is not the most efficient one possible (in terms of the size of Greg ). Essentially, we replace every vertex of G with a cycle of length N and each of the vertices (v, w), where there is an edge between v and w in G, is also connected to (w, v) (the rest of the edges are self loops). The rotation map RotGreg : ([N] × [N]) × [De16 ] → ([N] × [N]) × [De16 ] of Greg is formally defined as follows: • RotGreg ((v, w), 1) = ((v, w ), 2), where w = w + 1 if w < N and w = 1 otherwise. • Rot Greg ((v, w), 2) = ((v, w ), 1), where w = w − 1 if w > 1 and w = N otherwise. • In case there is an edge between v and w in G then RotGreg ((v, w), 3) = ((w, v), 3). Otherwise, RotGreg ((v, w), 3) = ((v, w), 3). • For i > 3, RotGreg ((v, w), i) = ((v, w), i). The transformation from G (given by its adjacency matrix) to Greg (given by its rotation map) is clearly computable in logarithmic space. Furthermore, Greg is De16 regular by definition and all its connected components are non-bipartite (as every vertex in Greg has self loops). Finally, for every connected component S ⊆ [N] of G we have that S × [N] is a connected component in Greg . To see that, we first note that for every vertex v ∈ [N] the set of vertices v × [N] is in the same connected

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component of Greg (as this set is connected by a cycle). Furthermore, there is an edge in Greg between some vertex in v × [N] and some vertex in w × [N] if and only if v and w are connected by an edge in G (the only possible edge that can connect these subsets is an edge between (v, w) and (w, v) which only exists in Greg if there is an edge between v and w in G). Now define Gexp = T (Greg , H ), and = O(log N) is the corresponding value as in Definition 3.1. Let S be the connected component of G, such that s ∈ S. By the arguments above, S × [N] is a connected component of Greg , and Greg |S×[N] is non-bipartite. By Lemma 3.3, S × [N] × ([D 16 ]) is a connected component of Gexp (as both Gexp and Gexp |S×[N ]×([D 16 ]) are De16 -regular). By Lemma 3.2 and Lemma 3.3, we have that λ(Gexp |S×[N ]×([D 16 ]) ) ≤ 1/2. Let A be the algorithm guaranteed by Proposition 2.3 (which decides undirected st-connectivity correctly in graphs where the connected component of the starting vertex is an expanders). The algorithm A will now invoke A , on the graph Gexp and the vertices s = (s, 1+1 ) and t = (t, 1+1 ). If A outputs that s and t are connected in Gexp then A will output that s and t are connected in G. Otherwise, A will output that s and t are not connected. The algorithm A is log-space since it is composed of a constant number of logspace procedures: (1) The transformation from G to Greg . (2) The transformation from Greg to Gexp , which is log-space by Lemma 3.4. (3) The algorithm A which is log-space by Proposition 2.3. Correctness of A is argued as follows. First, s and t are connected in Gexp if and only if s and t are connected in G (since S × [N] × ([D 16 ]) is a connected component of Gexp , where S is the connected component of G that contains s). The correctness of A now follows since Proposition 2.3 implies that A

will output ‘connected’ if and only if s and t are indeed connected in Gexp (as λ(Gexp |S×[N]×([D 16 ]) ) ≤ 1/2). 2

5. Universal traversal and exploration sequences In this section, we look closer into our USTCON algorithm and conclude that it also solves the corresponding search problem (i.e., finding the path from s to t if such a path exist). In addition, it implies efficiently-constructible universal-traversal sequences for graphs with restricted labelling, and universal exploration sequences for general graphs. The sort of restriction we pose on the labelling of graphs is a strengthening of the “consistent labelling” used in [18]. In a subsequent work [41], our restriction is relaxed to that of [18]. We start by analyzing T , the main transformation of the algorithm, given by Definition 3.1. We show that every edge in T (G, H ) translates to a path in G between the appropriate vertices, and that this path is log-space constructible (as this path is indeed computed during the log-space evaluation of T ). Looking ahead to the universal-traversal sequences, we note that if we restrict the labelling of G, then the labels of edges, traversed along this path, are independent of G.


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Definition 5.1. Let π be a permutation over [D] and RotG the rotation map of a D-regular graph G. Then Rot G is π-consistent if for every v, i, w and j such that Rot G (v, i) = (w, j ), it holds that j = π(i). In such a case we may also say that the labelling of G is π -consistent. An example of a π-consistent labelling is symmetric labelling where π is simply the identity. Namely, every edge is labelled in the same way from both its end points. However, other kinds of π-consistent labellings come up naturally. An example for that is the labelling of Greg in the proof of Theorem 4.1. We can now state the appropriate technical lemma regarding the transformation T . Lemma 5.2. Let D be some constant. Let G be a D 16 -regular graph on [N] and let H be a D-regular graph on [D 16 ], both given by their rotation maps. Let G = T (G, H ), where T and are given by Definition 3.1. There exists a log-space algorithm such that given RotG , Rot H and (v, a) in the domain of Rot G , it outputs a sequence of labels in [D 16 ] with the following property: If the first element of v is a vertex u ∈ [N] and the first element of Rot G (v, a) is a vertex w ∈ [N], then the walk on G from u using the labels that the algorithm outputs leads to w. Furthermore, for every fixed permutation π on [D 16 ], if the labelling of G is π-consistent, the log-space algorithm can evaluate the sequence of labels without access to RotG . Proof. Consider the log-space algorithm A in the proof of Theorem 3.4, as it evaluates Rot G (v, a). We enhance it a bit, to define an algorithm A as claimed by the lemma. Consider in particular the two variables v and a0 used by A. To begin with, v will be initialized to the value u (the first element of v). At the end, v will contain the value w. Throughout the run of A, the variable v is only updated by the rule v, a0 ← RotG (v, a0 ) (used at the bottom of the recursion). Therefore, all that A

needs to do is to output the value of a0 just before each time A updates v. Regarding the second part of the lemma. We note that the value of a0 is only influenced by RotG , through the evaluations v, a0 ← RotG (v, a0 ). If G is π -consistent, then A can completely ignore the variable v and the rotation map of G. To simulate A, it is sufficient that whenever A evaluates v, a0 ← Rot G (v, a0 ), then A will evaluate a0 ← π(a0 ). 2 Using Lemma 5.2, it is not hard to obtain the algorithm that finds paths in undirected graphs. Theorem 5.3. There exists a log-space algorithm that gets as input a graph G over the set of vertices [N], and two vertices s and t in [N], and outputs a path from s to t if such a path exists (otherwise it outputs ‘not connected’). Proof. Consider the algorithm A from the proof of Theorem 4.1. We revise it to an algorithm A as required by the theorem. First, we note that it is enough for A to

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output a path from (s, 1) to (t, 1) in Greg if such a path exists, as it is easy to transform (in log-space) such a path to a path from s to t in G (and the existence of the two paths is equivalent). Next we note that A enumerates all logarithmically-long paths from s = (s, 1+1 ) in Gexp . If it does not find a path that visits t = (t, 1+1 ), it concludes that s and t are not connected in G. Therefore, in such a case, A can output ‘not connected’. Otherwise A found a short path from s to t . Apply the algorithm guaranteed by Lemma 5.2 on each edge on the path from s to t . Each time the algorithm outputs a sequence of edge-labels in Greg . Let a be the concatenation of these sequences. It follows from Lemma 5.2 that the path in Greg starting from (s, 1) and following the edges according to the labels in a leads to (t, 1). The theorem now follows. 2 To give our result regarding universal-traversal sequences, we need some notation. Let a = {a1 , . . . , am } be a sequence of values in [D] (these are interpreted as edge labels). a is an (N, D)-universal traversal sequence, if for every connected D-regular, labelled graph G on N vertices, and every start vertex s ∈ [N], the walk that starts at s and follows the edges labelled a1 , . . . , am , visits every vertex in the graph. For a permutation π over [D], we say that a is an (N, D) π-universal traversal sequence, if the above property holds for every connected D-regular graph on N vertices, that has a π-consistent labelling, (rather than for all such graphs). Theorem 5.4. There exists a log-space algorithm that takes as input 1N and a permutation π over [D] and outputs an (N, D) π-universal traversal sequence. Proof. First we argue that it is enough to construct an (N · D, De16 ) π -universal sequence for the following simple permutation: π (1) = 2, π (2) = 1 and for every i > 2 π (i) = i. Furthermore, all we need is that the sequence will traverse nonbipartite graphs. Consider a (connected) D-regular graph G on N vertices that has a π-consistent labelling. This graph can be transformed into a De16 -regular (connected and non-bipartite) graph G on N · D vertices that has a π -consistent labelling. Each vertex v ∈ N is transformed into a cycle over D vertices (v, 1), . . . , (v, D), the edges of the cycle are labelled 1 and 2 (just as in the definition of Greg in the proof of Theorem 4.1). The edge labelled 3 going out of (v, i) will lead to RotG (v, i) (and will be labelled 3 from that end as well). All other edges are self loops. Assume that a sequence of labels a1 , . . . , am , visits every vertex of G starting from every vertex (v, 1) (this is even less general than what we obtain). We can translate this (in log space) into a sequence of labels b1 , . . . , bm that traverses G from every vertex v. To do that, we simulate the walk on G from an arbitrary vertex (v, 1). As v is unknown and our simulation does not rely on G, it will only know at each point the value b such that the walk at this point visits some vertex (w, b) of G (where w is unknown). First b is set to 1. Then, during the simulation, labels ai > 3 can be ignored (as they are self loops). Given labels 1 and 2, b can easily be updated (these are edges on the cycle). Finally, when encountering ai = 3 the walk moves from a vertex (w, b) to a vertex (w , π(b)) (as the labelling of G is π -consistent), and so it is


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easy to update the value of b (given access to π ). The projection of the walk on G is exactly the edges labelled 3 that are taken by the walk on G . Therefore, to transform the sequence of ai ’s to the sequence of bi ’s we can simply output (throughout the simulation) the current value of b, whenever we encounter a label ai = 3. Now we consider a De16 -regular (connected and non-bipartite) graph G on N · D vertices that has a π -consistent labelling. Let H be a ((De )16 , De , 1/2)-graph. Finally let G = T (G, H ), where T and are given by Definition 3.1. By Lemma 3.2, λ(G ) ≤ 1/2 and therefore its diameter is logarithmic. Therefore, for every two vertices v and u of G one of the polynomially many sequences of labels (of the appropriate logarithmic length) will visit (u, 1 ), starting at (v, 1 ). Let B be the set of all these sequences of labels. Lemma 5.2 gives a way to translate in log-space each one of the sequences in B into a corresponding sequence of edge-labels of G . Let B be the set of translated sequences. By Lemma 5.2 and the above argument, for every two vertices v and u of G one of the sequences in B will lead a walk in G that starts in v through the vertex u. We should also note that given a sequence a = a1 , . . . , am that leads from a vertex v to a vertex u, we have that the sequence π −1 (am ), . . . , π −1 (a1 ) leads from u to v (this operation simply reverses the walk). We refer to this latter sequence as the reverse of a . Finally, we can define a sequence that traverses all of the vertices of G regardless of the starting vertex. Simply, we concatenate for each sequences in B its reversed sequence and concatenate all of these sequences one after the other. By the arguments above, for every vertex v, the sequence we obtain will visit v after every pair of a sequence and its reversed sequence. Furthermore, for every vertex u, one of these sequences will lead to u. As the log-space construction of this sequence ignores the graph G (and only relies on π ), we obtained the desired (N · D, De16 ) π -universal sequence for non-bipartite graphs. The theorem follows. 2 In an (N, D)-universal exploration sequence, the sequence of labels is interpreted as offsets rather than absolute labels. This means that if we entered a vertex v on an edge labelled a (from v’s view point), and we are reading the label b, then we will leave v on the edge labelled a + b (or a + b − D if a + b > D). In fact this notion can apply to graphs that are not-regular (it then makes sense to allow negative elements in the sequence). Universal-exploration sequences have more flexibility than universal-traversal sequences. For example, it is not clear how to transform a universal-traversal sequence for degree-3 graphs to one for higher-degree graphs. This is easy for universal-exploration sequences (and seems desirable as USTCON can easily be reduced to USTCON for regular-graphs of any degree larger than 2). Koucky [24] showed how to transform a universal-traversal sequence to a universalexploration sequence. His transformation (which is essentially the same as the one from G to G in the proof of Theorem 5.4), only needs the universal-sequence to work for graphs with π -consistent labelling for some simple permutation π. We can therefore conclude from Theorem 5.4 a log-space construction for general universalexploration sequences.

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Corollary 5.5. There exists a log-space algorithm that takes as input (1N , 1D ) and produces an (N, D)-universal exploration sequence.

6. Discussion and further research We start by comparing the techniques of this paper with some previous ones, with the goal of shading some light on the source of our improvements. We continue by discussing some open problems and the results of a subsequent work. Comparison with previous techniques The USTCON algorithms of [47], [34], [9] also operate by transforming, in phases, the input graph into a more accommodating one. In each one of these algorithms, each phase “charges” logarithmic amount to the space complexity of the algorithm. The improvement in the space complexity is directly correlated to reducing the number of phases needed for the transformation. With this approach, the only way to obtain a log-space algorithm is to reduce the number of phases to a constant. We deviate from this direction, as we use a logarithmic number of phases (just as in Savitch’s algorithm), to gradually improve the connectivity of the input graph. The space efficiency of our algorithm stems from each transformation being significantly less costly in space. The parameter being improved by [34], [9], is the size of the graph (each transformation shrinks the graph by collapsing it to a “representative” subset of the vertices). In contrast, our transformation will in fact expand the graph by a polynomial factor (as each phase, enlarges our graph by a constant factor). The parameter Savitch’s transformation improves is the diameter of the graph, which is much closer to the parameter we improve (the expansion). In fact, each phase of Savitch’s algorithm can be described very similarly to our algorithm. Each one of these phases consists of squaring the graph and then removing parallel edges (which may reduce the degree). Although all that is needed is indeed that the diameter of the resulting graph will be small, our analysis relies on bounding the expansion of intermediate graphs – a stronger notion of connectivity than the diameter. This allows our transformation to preserve constant degree of the graph (rather than linear degree in Savitch’s algorithm), which is crucial for our analysis of the space complexity. It also seems instructive to compare with the combinatorial construction of expander graphs of [43]. There, an arbitrarily large expander graphs was constructed, starting with a constant size expander. This small expander is made larger and larger, while its degree is kept constant via the zig-zag or the replacement product. Our main transformation shows how to turn any connected graph (which is already large) into an expander. This means that the above mentioned products need to be applied when one of the graphs is an extremely weak expander (whereas in [43] both graphs were fairly good expanders). Very fortunately, both products work quite well in this unusual setting of parameters.


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Further Research There are many open problems and new research directions brought up by this work, we discuss just a few of those. A very natural question is whether the techniques of this paper can be used towards a proof of RL = L. While progress in the context of RL does not seem immediate (as the case of symmetric computations does seem easier), we feel that it is still quite plausible. We also feel that this paper should give an opportunity to reevaluate the common conjecture that Savitch’s algorithm is optimal for STCON. While this conjecture may very well be correct, we feel that there is not enough evidence supporting it. Another open problem is to come up with full-fledged, efficiently-constructible, universal-traversal sequences. Interestingly, it seems that this problem shares some of the obstacles that one encounters when trying to generalize the USTCON algorithm to solving RL (this is formalized to some extent in the results of [41] mentioned below). In a subsequent work, Reingold, Trevisan and Vadhan [41], make some progress on extending our techniques to dealing with the general RL case, obtaining the following results: 1. They exhibit a new complete problem for RL: STCON restricted to directed graphs for which the random walk is promised to have polynomial mixing time. 2. Generalizing our techniques, they present a deterministic, log-space algorithm that given a regular directed graph G (i.e., a directed graph where all in-degrees and out-degrees are equal) and two vertices s and t, finds a path between s and t if one exists. 3. Using the same techniques as in Item 2, they give a “pseudorandom generator” for random walks on “consistently labelled” regular directed graphs. Roughly speaking, given a random seed of logarithmic length, the generator constructs, in log-space, a “short” pseudorandom walk that ends at an almost-uniformly distributed vertex when taken in any consistently labelled regular directed graph. 4. They prove that if their pseudorandom generator from Item 3 could be generalized to all regular directed graphs (instead of just consistently labelled ones), then their complete problem from Item 1 can be solved in log-space and hence RL = L. Finally, we have made no attempt to optimize our algorithm in terms of running time (or the constant in the space complexity). Major improvements in efficiency can come about by better analysis of the zig-zag and replacement products. These may also determine which one of these products yields a more efficient algorithm. In a subsequent work Rozenman and Vadhan [44] give a log-space algorithm for USTCON. Their algorithm makes substantial progress in terms of reducing the running time of the algorithm (compared to ours). Their key technical tool is a new operation they introduce and name “derandomized squaring”. This operation reduces the second eigenvalue of a graph “similarly” to standard squaring but increases the degree much more moderately. Very loosely, this operation can replace in our algorithm the combination of (standard) powering and zig-zag product. Their analysis for

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the new operation is tight (unlike the analysis we currently know for the zig-zag and replacement products) and it is simple and very appealing.6 Acknowledgments. This work came about during a delightful visit to UC Berkeley. I am most grateful to Irit Dinur and Luca Trevisan for many hours of stimulating discussions on closely related topics and for creating the most conducive research environment possible for me. I would like to thank Moni Naor, Ran Raz, Salil Vadhan and Avi Wigderson for many discussions that helped me form my intuitions on the derandomization of space bounded computations. Among other contributions, I want to thank Moni for steering me towards this topic early on during my PhD studies, and to thank Ran, Salil and Avi for intuitions formed during our joint work on [38], [42].

References [1] Ajtai, Miklós, Komlós, János, and Szemerédi, E., Deterministic simulation in LOGSPACE. In Proceedings of the 19th Annual ACM Symposium on Theory of Computing, ACM Press, New York 1987, 132–140. [2] Aleliunas, Romas, Karp, Richard M., Lipton, Richard J., Lovász, László, and Rackoff, Charles, Random walks, universal traversal sequences, and the complexity of maze problems. In Proceedings of the 20th Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 1979, 218–223. [3] Alon, Noga, Eigenvalues and expanders. Combinatorica 6 (2) (1986), 83–96. [4] Alon, Noga, Galil, Zvi, and Milman, Vitali D., Better expanders and superconcentrators. J. Algorithms 8 (3) (1987), 337–347. [5] Alon, Noga, and Milman, Vitali D., λ1 , isoperimetric inequalities for graphs, and superconcentrators. J. Combin. Theory Ser. B 38 (1) (1985), 73–88. [6] Alon, Noga, and Roichman, Yuval, Random Cayley graphs and expanders. Random Structures Algorithms 5 (2) (1994), 271–284. [7] Alon, Noga, and Sudakov, Benny, Bipartite subgraphs and the smallest eigenvalue. Combin. Probab. Comput. 9 (1) (2000), 1–12. [8] Àlvarez, Carme, and Greenlaw, Raymond, A compendium of problems complete for symmetric logarithmic space. Comput. Complexity 9 (2000), 123–145. [9] Armoni, Roy, Ta-Shma, Amnon, Wigderson, Avi, and Zhou, Shiyu, An o(log(n)4/3 ) space algorithm for (s, t) connectivity in undirected graphs. J. ACM 47 (2), (2000), 294–311. [10] Babai, László, Nisan, Noam, and Szegedy, Márió, Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. Comput. System Sci. 45 (1992), 204–232. [11] Ben-Asher, Y., Lange, K., Peleg, D., and Schuster, A., The complexity of reconfiguring network models. Inform. and Comput. 21 (1) (1995), 41–58. 6 In essence, their analysis implies that an expander can be viewed as a convex combination of a complete graph (the ultimate expander) and an error term. This insightful observation can be exploited to simplify the analysis of the zig-zag and replacement products as well (but the obtained analysis is still not tight).


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[12] Borodin, Allan, Cook, Stephen A., Dymond, Patrick W., Ruzzo, Walter L., and Tompa, Martin, Two applications of inductive counting for complementation problems. SIAM J. Comput. 18 (3) (1989), 559–578. [13] Broder, Andrei, and Shamir, Eli, On the second eigenvalue of random regular graphs. In Proceedings of the 28th Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 1987, 286–294. [14] Friedman, Joel, On the second eigenvalue and random walks in random d-regular graphs. Combinatorica 11 (4) (1991), 331–362. [15] Friedman, Joel, Kahn, Jeff, and Szemerédi, Endre, On the second eigenvalue in random regular graphs. In Proceedings of the 21th Annual ACM Symposium on Theory of Computing, ACM Press, New York 1989, 587–598. [16] Gabber, Ofer, and Galil, Zvi, Explicit constructions of linear-sized superconcentrators. J. Comput. System Sci. 22 (3) (1981), 407–420. [17] Goldreich, Oded, and Wigderson, Avi, Derandomization that is rarely wrong from short advice that is typically good. In RANDOM 2002, Lecture Notes in Comput. Sci. 2483, Springer-Verlag, Berlin 2002, 209–223. [18] Hoory, Shlomo, and Wigderson, Avi, Universal traversal sequences for expander graphs. Inform. Process. Lett. 46 (2) (1993), 67–69. [19] Impagliazzo, Russell, Nisan, Noam, and Wigderson, Avi, Pseudorandomness for network algorithms. In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, ACM Press, New York 1994, 356–364. [20] Jimbo, Shuji, and Maruoka, Akira, Expanders obtained from affine transformations. Combinatorica 7 (4) (1987), 343–355. [21] Karchmer, Mauricio, and Wigderson, Avi, On span programs. In Proceedings of the 8th Structures in Complexity Conference, IEEE Comput. Soc. Press, Los Alamitos, CA, 1993, 102–111. [22] Klivans, Adam, and van Melkebeek, Dieter, Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput. 31 (5) (2002), 1501–1526. [23] Koucky, Michal, Universal traversal sequences with backtracking. J. Comput. System Sci. 65 (2002), 717–726. [24] Koucky, Michal, On traversal sequences, exploration sequences and completeness of Kolmogorov random strings. PhD thesis, Rutgers University, 2003. [25] Lewis, Harry R., and Papadimitriou, Christos H., Symmetric space-bounded computation. Theoret. Comput. Sci. 19 (1982), 161–187. [26] Lubotzky, Alex, Phillips, Ralph, and Sarnak, Peter, Ramanujan graphs. Combinatorica 8 (3) (1988), 261–277. [27] Madras, Neal, and Randall, Dana, Factoring graphs to bound mixing rates. In Proceedings of the 37th Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 1996, 194–203. [28] Margulis, Gregory A., Explicit constructions of expanders. Problemy Peredaˇci Informacii 9 (4) (1973), 71–80. [29] Margulis, Gregory A., Explicit group-theoretic constructions of combinatorial schemes and their applications in the construction of expanders and concentrators. Problemy Peredachi Informatsii 24 (1) (1988), 51–60.

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[30] Martin, Russell A., and Randall, Dana, Sampling adsorbing staircase walks using a new markov chain decomposition method. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, 492–502. [31] Morgenstern, Moshe, Existence and explicit constructions of q + 1 regular Ramanujan graphs for every prime power q. J. Combin. Theory. Series B 62 (1) (1994), 44–62. [32] Nisan, Noam, RL ⊆ SC. In Proceedings of the 24th Annual ACM Symposium on Theory of Computing, ACM Press, New York 1992, 619–623. [33] Nisan, Noam, Pseudorandom generators for space-bounded computation. Combinatorica 12 (4) (1992), 449–461. [34] Nisan, Noam, Szemeredi, Endre, and Wigderson, Avi, Undirected connectivity in o(log1.5 n) space. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 1989, 24–29. [35] Nisan, Noam, and Ta-Shma, Amnon, Symmetric logspace is closed under complement. Chicago J. Theor. Comput. Sci., 1995 (electronic). [36] Nisan, Noam, and Zuckerman, David, Randomness is linear in space. J. Comput. System Sci. 52 (1) (1996), 43–52. [37] Pinsker, Mark S., On the complexity of a concentrator. In 7th Annual Teletraffic Conference, Stockholm, 1973, 318/1–318/4. [38] Raz, Ran, and Reingold, Omer, On recycling the randomness of the states in space bounded computation. In Proceedings of the 31st Annual ACM Symposium on the Theory of Computing, ACM Press, New York 1999, 159–168. [39] Reif, John H., Symmetric complementation. J. ACM 31 (2) (1984), 401–421. [40] Reingold, Omer, Undirected st-connectivity in log-space. In Proceedings of the 37th ACM Symposium on Theory of Computing, ACM Press, New York 2005, 376–385. [41] Reingold, Omer, Trevisan, Luca, and Vadhan, Salil, Pseudorandom walks in biregular graphs and the RL vs. L problem. Electronic Colloquium on Computational Complexity Technical Report TR05-022, 2005. [42] Reingold, Omer, Vadhan, Salil, and Wigderson, Avi, Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, 3–13. [43] Reingold, Omer, Vadhan, Salil, and Wigderson, Avi, Entropy waves, the zig-zag graph product, and new constant-degree expanders. Ann. of Math. 155 (1) (2002), 157–187. [44] Rozenman, Eyal, and Vadhan, Salil, Derandomized squaring of graphs. In Approximation, randomization and combinatorial optimization (Berkeley, CA, 2005), ed. by Chandra Chekuri et al., Lecture Notes in Comput. Sci. 3624, Springer-Verlag, Berlin 2005, 436–447. [45] Saks, Michael, Randomization and derandomization in space-bounded computation. In Proceedings of the 11th Annual Conference on Structure in Complexity Theory, IEEE Comput. Soc. Press, Los Alamitos, CA, 1996, 128–149. [46] Saks, Michael, and Zhou, Shiyu, bph space(S) ⊆ dspace(S 3/2 ). J. Comput. System Sci. 58 (2) (1999), 376–403. [47] Savitch, J., Relationship between nondeterministic and deterministic tape complexities. J. Comput. System Sci. 4 (2) (1970), 177–192.


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[48] Tanner, Michael R., Explicit concentrators from generalized n-gons. SIAM J. Algebraic Discrete Methods 5 (3) (1984), 287–293. [49] Trifonov, Vladimir, An o(log n log log n) space algorithm for undirected s,t-connectivity. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, ACM Press, New York 2005, 626–633. [50] Wigderson, Avi, The complexity of graph connectivity. In Mathematical Foundations of Computer Science 1992 (ed. by I. M. Havel and V. Koubek), Lecture Notes in Comput. Sci. 629, Springer-Verlag, Berlin 1992, 112–132. Department of Computer Science, Weizmann Institute of Science, Rehovot 76100, Israel E-mail: [email protected]

Potential functions and the inefficiency of equilibria Tim Roughgarden∗

Abstract. We survey one area of the emerging field of algorithmic game theory: the use of approximation measures to quantify the inefficiency of game-theoretic equilibria. Potential functions, which enable the application of optimization theory to the study of equilibria, have been a versatile and powerful tool in this area. We use potential functions to bound the inefficiency of equilibria in three diverse, natural classes of games: selfish routing networks, resource allocation games, and Shapley network design games. Mathematics Subject Classification (2000). 68Q25, 68W25, 90B35, 91A65. Keywords. Game theory, inefficiency of equilibria, Nash equilibrium, network design, potential functions, price of anarchy, price of stability, resource allocation, selfish routing.

1. Introduction The interface between theoretical computer science and microeconomics, often called algorithmic game theory, has been an extremely active research area over the past few years. Recent points of contact between the two fields are diverse and include, for example, increased attention to computational complexity and approximation in combinatorial auctions (e.g. [9]); a new focus on worst-case analysis in optimal auction design (e.g. [17]); and a renewed emphasis on the computability and learnability of equilibrium concepts (e.g. [14], [18], [26]). This survey touches on just one connection between theoretical computer science and game theory: the use of approximation measures to quantify the inefficiency of game-theoretic equilibria. 1.1. Quantifying the inefficiency of equilibria. Even in very simple settings, selfish behavior can lead to highly inefficient outcomes [11]. A canonical example of this phenomenon is provided by the “Prisoner’s Dilemma” [28], in which strategic behavior by two captured and separated prisoners inexorably draws them into the worst-possible outcome. We will see several concrete examples of the inefficiency of selfish behavior in networks later in the survey. Must more recently, researchers have sought to quantify the inefficiency of selfish behavior. Koutsoupias and Papadimitriou [23] proposed a framework to systematically study this issue. The framework presupposes a strategic environment (a game), a definition for the outcome of selfish behavior (an equilibrium concept), and a real∗ Supported in part by ONR grant N00014-04-1-0725, DARPA grant W911NF-04-9-0001, and an NSF CAREER Award.


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valued, nonnegative objective function defined on the possible outcomes of the game. The price of anarchy [23], [26] is then defined as the ratio between the objective function value of an equilibrium and that of an optimal solution. (For the moment, we ignore the question of whether or not equilibria exist and are unique.) If the price of anarchy of a game is 1, then its equilibria are fully efficient. More generally, bounding the price of anarchy in a class of games provides a guarantee on the worst-possible inefficiency of equilibria in these games. The price of anarchy is directly inspired by other popular notions of approximation in theoretical computer science [23]. One example is the approximation ratio of a heuristic for a (typically NP-hard) optimization problem, defined as the largest ratio between the objective function value of the solution produced by the heuristic and that of an optimal solution. While the approximation ratio measures the worst-case loss in solution quality due to insufficient computational effort, the price of anarchy measures the worst-case loss arising from insufficient ability (or willingness) to control and coordinate the actions of selfish individuals. Much recent research on the price of anarchy is motivated by optimization problems that naturally occur in the design and management of large networks (like the Internet), in which users act selfishly, but implementing an optimal solution is not practical. 1.2. Potential functions. The price of anarchy has been successfully analyzed in a diverse array of game-theoretic models (see e.g. [32], [33] and the references therein). This survey discusses three of these models, with the goal of illustrating a single mathematical tool for bounding the price of anarchy: potential functions. The potential function technique is by no means the only one known for bounding the inefficiency of equilibria, but (so far) it has been the most versatile and powerful. Potential functions enable the application of optimization theory to the study of equilibria. More precisely, a potential function for a game is a real-valued function, defined on the set of possible outcomes of the game, such that the equilibria of the game are precisely the local optima of the potential function. This idea was first used to analyze selfish behavior in networks by Beckmann, McGuire, and Winsten [4], though similar ideas were used earlier in other contexts. When a game admits a potential function, there are typically consequences for the existence, uniqueness, and inefficiency of equilibria. For example, suppose a game admits a potential function and either: (1) there are a finite number of distinct outcomes; or (2) the set of outcomes is compact and the potential function is continuous. In either case, the potential function achieves a global optimum, which is also a local optimum, and hence the game has at least one equilibrium. This is a much more elementary approach to establishing the existence of equilibria than traditional fixed-point proofs (e.g. [25]). Moreover, if the potential function has a unique local optimum, then the game has a unique equilibrium. Finally, if the potential function is “close to” the true objective function, then the equilibria that are global optima of the potential function have nearly-optimal objective function value, and are thus approximately efficient.

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The power of the potential function approach might suggest that its applicability is limited. Fortunately, many important and natural classes of games admit wellbehaved potential functions. To suggest what such functions look like, we briefly interpret some classical results about electric networks in terms of potential functions. Consider electrical current in a two-terminal network of resistors. By Kirchhoff’s equations and Ohm’s law, we can interpret this current as an “equilibrium”, in the sense that it equalizes the voltage drop along all paths in the network between the two terminals. (View current as a large population of “selfish particles”, each seeking out a path with minimum voltage drop.) On the other hand, Thomson’s principle states that electrical current also minimizes the dissipated energy over all flow patterns that achieve the same total current. In other words, energy dissipation serves as a potential function for current in an electrical network. For further details and discussion, see Kelly [21] and Doyle and Snell [10]. 1.3. Survey overview. Each of the next three sections introduces a model of selfish behavior in networks, and uses a potential function to bound the inefficiency of equilibria in the model. We focus on these three examples because they are simple, natural, and diverse enough to illustrate different aspects of potential function proof techniques. In order to emphasize the most important concepts and provide a number of self-contained proofs, we often discuss only special cases of more general models and results. Section 2 discusses selfish routing networks, a model that generalizes the electrical networks of Subsection 1.2 and has been extensively studied by the transportation, networking, and theoretical computer science communities. Section 3 analyzes the performance of a well-studied distributed protocol for allocating resources to heterogeneous users. Section 4 bounds the inefficiency of equilibria in a model of selfish network design. Section 5 concludes.

2. Selfish routing and the price of anarchy 2.1. The model. In this section, we study the inefficiency of equilibria in the following model of noncooperative network routing. A multicommodity flow network, or simply a network, is a finite directed graph G = (V , E), with vertex set V and (directed) edge set E, together with a set (s1 , t1 ), . . . , (sk , tk ) of source-sink vertex pairs. We also call such pairs commodities. We denote the set of simple si -ti paths by Pi , and always assume that this set is non-empty for each i. We allow the graph G to contain parallel edges, and a vertex can participate in multiple source-sink pairs. A flow in a network G is a nonnegative vector indexed by the set P = ki=1 Pi . For a flow f and a path P ∈ Pi , we interpret fP as the amount of traffic of commodity i that chooses the path P to travel from si to ti . We use r to denote a nonnegative vector of traffic rates, indexed by the commodities of G. A flow f is feasible for r if it routes all of the prescribed traffic: for each i ∈ {1, 2, . . . , k}, P ∈Pi fP = ri .

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We model the negative consequences of network congestion in the following simple way. For a flow f in a network G and an edge e of G, let fe = P ∈P : e∈P fP denote the total amount of traffic employing edge e. Each edge e then has a nonnegative, continuous, and nondecreasing cost function ce , which describes the cost incurred by traffic using the edge e as a function of fe . We call a triple of the form (G, r, c) a selfish routing network or simply an instance. Next we describe a notion of equilibrium in selfish routing networks – the expected outcome of “selfish routing”. Define the cost of a path P with respect to a flow f as the sum of the costs of the constituent edges: cP (f ) = e∈P ce (fe ). Assuming that selfish traffic attempts to minimize its incurred cost, we obtain the following definition of a Wardrop equilibrium [38]. Definition 2.1 ([38]). Let f be a feasible flow for the instance (G, r, c). The flow f is a Wardrop equilibrium if, for every commodity i ∈ {1, 2, . . . , k} and every pair P , P˜ ∈ Pi of si -ti paths with fP > 0, cP (f ) ≤ cP˜ (f ). In Definition 2.1, we are implicitly assuming that every network user controls a negligible portion of the overall traffic, so that the actions of an individual user have essentially no effect on the network congestion. In the game theory literature, games with this property are called nonatomic [35]. Atomic variants of selfish routing have also been extensively studied (see e.g. [32]). We will study other types of atomic games in Sections 3 and 4. Example 2.2 (Pigou’s example [27]). Consider the two-vertex, two-edge network shown in Figure 1. There is one commodity and the traffic rate is 1. Note that the lower edge is cheaper than the upper edge if and only if less than one unit of traffic uses it. There is thus a unique Wardrop equilibrium, with all traffic routed on the lower edge. In this flow, all traffic incurs one unit of cost. c(x) = 1

t

s

c(x) = x Figure 1. Pigou’s example (Example 2.2).

Pigou’s example already illustrates that equilibria can be inefficient. More specifically, note that routing half of the traffic on each of the two edges would produce a “better” flow: all of the traffic would incur at most one unit of cost, while half of the traffic would incur only 1/2 units of cost.

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The inefficiency of the Wardrop equilibrium in Example 2.2 arises from what is often called a congestion externality – a selfish network user accounts only for its own cost, and not for the costs that its decision imposes on others. The “better” routing of traffic in Example 2.2 is not a Wardrop equilibrium because a selfish network user routed on the upper edge would switch to the lower edge, indifferent to the fact that this switch (slightly) increases the cost incurred by a large portion of the population. In Example 2.2, there is a unique Wardrop equilibrium. In Subsection 2.2 we will use a potential function to prove the following theorem, which states that Wardrop equilibria exist and are “essentially unique” in all selfish routing networks. Theorem 2.3 ([4]). Let (G, r, c) be an instance. (a) The instance (G, r, c) admits at least one Wardrop equilibrium. (b) If f and f˜ are Wardrop equilibria for (G, r, c), then ce (fe ) = ce (f˜e ) for every edge e. The Wardrop equilibrium in Example 2.2 is intuitively inefficient; we next quantify this inefficiency. We define our objective function, the cost of a flow, as the sum of the path costs incurred by traffic: cP (f )fP = ce (fe )fe . (1) C(f ) = P ∈P

e∈E

The first equality in (1) is a definition; the second follows easily from the definitions. An optimal flow for an instance (G, r, c) is feasible and minimizes the cost. Since cost functions are continuous and the set of feasible flows is compact, every instance admits an optimal flow. In Pigou’s example (Example 2.2), the Wardrop equilibrium has cost 1, while routing half of the traffic on each edge yields an optimal flow with cost 3/4. Definition 2.4 ([23], [26]). The price of anarchy ρ(G, r, c) of an instance (G, r, c) is ρ(G, r, c) =

C(f ) , C(f ∗ )

where f is a Wardrop equilibrium and f ∗ is an optimal flow. The price of anarchy ρ( ) of a non-empty set of instances is sup(G,r,c)∈ ρ(G, r, c). Definition 2.1 and Theorem 2.3(b) easily imply that all Wardrop equilibria have equal cost, and thus the price of anarchy of an instance is well defined unless there is a flow with zero cost. In this case, all Wardrop equilibria also have zero cost, and we define the price of anarchy of the instance to be 1. Example 2.5 (Nonlinear Pigou’s example [34]). The inefficiency of the Wardrop equilibrium in Example 2.2 can be amplified with a seemingly minor modification to the network. Suppose we replace the previously linear cost function c(x) = x on

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the lower edge with the highly nonlinear one c(x) = x p for p large (Figure 2). As in Example 2.2, the cost of the unique Wardrop equilibrium is 1. The optimal flow routes a small ε fraction of the traffic on the upper edge and has cost ε + (1 − ε)p+1 . Since this approaches 0 as ε tends to 0 and p tends to infinity, the price of anarchy of this selfish routing network grows without bound as p grows large. c(x) = 1

t

s

c(x) = x p Figure 2. A nonlinear variant of Pigou’s example (Example 2.5).

Example 2.5 demonstrates that the price of anarchy can be large in (very simple) networks with nonlinear cost functions. In Subsection 2.2 we use a potential function to show the converse: the price of anarchy is large only in networks with “highly nonlinear” cost functions. 2.2. A potential function for wardrop equilibria. We now show that Wardrop equilibria can be characterized as the minima of a potential function, and use this characterization to prove both Theorem 2.3 and upper bounds on the price of anarchy of selfish routing. To motivate this potential function, we first characterize the optimal flows of an instance. Optimal flows for an instance (G, r, c) minimize the cost (1) subject to linear flow feasibility constraints. Assume for the moment that for every edge e, the function x · ce (x) is convex. The cost (1) is then a convex (separable) function, and we can apply the Karush–Kuhn–Tucker conditions (see e.g. [5]) to characterize its global minima. To state this characterization cleanly, assume further that all cost functions are differentiable, and let ce∗ (x) = (x · ce (x)) = ce (x) + x · ce (x) denote the marginal cost function for the edge e. The KKT conditions then give the following. Proposition 2.6 ([4]). Let (G, r, c) be an instance such that, for every edge e, the function x ·ce (x) is convex and differentiable. Let ce∗ denote the marginal cost function of the edge e. Then f ∗ is an optimal flow for (G, r, c) if and only if, for every commodity i ∈ {1, 2, . . . , k} and every pair P , P˜ ∈ Pi of si -ti paths with fP > 0, cP∗ (f ) ≤ c∗˜ (f ). P

Comparing Definition 2.1 and Proposition 2.6, we discover that Wardrop equilibria and optimal flows are essentially the same thing, just with different sets of cost functions.


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Corollary 2.7. Let (G, r, c) be an instance such that, for every edge e, the function x · ce (x) is convex and differentiable. Let ce∗ denote the marginal cost function of the edge e. Then f ∗ is an optimal flow for (G, r, c) if and only if it is a Wardrop equilibrium for (G, r, c∗ ). To construct a potential function for Wardrop equilibria, we need to “invert” Corollary 2.7: of what function do Wardrop equilibria arise as global minima? The answer is simple: to recover Definition 2.1 as an optimality condition, we seek a function he (x) for each edge x e – playing the previous role of x ·ce (x) – such that he (x) = ce (x). Setting he (x) = 0 ce (y) dy for each edge e thus yields the desired potential function. Precisely, call fe (f ) = ce (x) dx (2) e∈E 0

the potential function for an instance (G, r, c). Analogously to Corollary 2.7, the following proposition holds. Proposition 2.8 ([4]). Let (G, r, c) be an instance. A flow feasible for (G, r, c) is a Wardrop equilibrium if and only if it is a global minimum of the corresponding potential function given in (2). Remark 2.9. Thomson’s principle for electrical networks (Subsection 1.2) can be viewed as the special case of Proposition 2.8 for single-commodity flow networks with linear cost functions (of the form ce (x) = ae x). Theorem 2.3 now follows easily. Proof of Theorem 2.3 (Sketch). Since cost functions are continuous and the set of feasible flows is compact, part (a) of the theorem follows immediately from Proposition 2.8 and Weierstrass’s Theorem. Since cost functions are nondecreasing, the potential function in (2) is convex; moreover, the set of feasible flows is convex. Part (b) of the theorem now follows from routine convexity arguments. 2 Much more recently, the potential function (2) has been used to upper bound the price of anarchy of selfish routing. The intuition behind this connection is simple: if Wardrop equilibria exactly optimize a potential function (2) that is a good approximation of the objective function (1), then they should also be approximately optimal. Formally, we have the following. Theorem 2.10 ([34]). Let (G, r, c) be an instance, and suppose that x · ce (x) ≤ x γ · 0 ce (y) dy for all e ∈ E and x ≥ 0. Then the price of anarchy ρ(G, r, c) is at most γ . Proof. Let f and f ∗ be a Wardrop equilibrium and an optimal flow for (G, r, c), respectively. Since cost functions are nondecreasing, the cost of a flow (1) is always

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at least its potential function value (2). The hypothesis ensures that the cost of a flow is at most γ times its potential function value. The theorem follows by writing C(f ) ≤ γ · (f ) ≤ γ · (f ∗ ) ≤ γ · C(f ∗ ), with the second inequality following from Proposition 2.8.

2

Theorem 2.10 implies that the price of anarchy of selfish routing is large only in networks with “highly nonlinear” cost functions. For example, if ce is a polynomial function x with degree at most p and nonnegative coefficients, then x · ce (x) ≤ (p + 1) 0 ce (y) dy for all x ≥ 0. Applying Theorem 2.10, we find that the price of anarchy in networks with cost functions that are polynomials with nonnegative coefficients grows at most linearly with the degree bound p. Corollary 2.11 ([34]). If (G, r, c) is an instance with cost functions that polynomials with nonnegative coefficients and degree at most p, then ρ(G, r, c) ≤ p + 1. This upper bound is nearly matched by Example 2.5. (The upper and lower bounds differ by roughly a ln p multiplicative factor.) Qualitatively, Example 2.5 and Corollary 2.11 imply that a large price of anarchy can be caused by highly nonlinear cost functions, but not by complex network topologies or by a large number of commodities. 2.3. An optimal bound on the price of anarchy. We have established that the price of anarchy of selfish routing depends on the “degree of nonlinearity” of the network cost functions. However, even in the simple case of polynomial cost functions, there is gap between the lower bound on the price of anarchy provided by Example 2.5 and the upper bound of Theorem 2.10. We conclude this section by showing how a different analysis, which can be regarded as a more “global” application of potential function ideas, provides a tight bound on the price of anarchy for essentially every set of allowable cost functions. We first formalize a natural lower bound on the price of anarchy based on “Pigoulike examples”. Definition 2.12 ([7], [31]). Let C be a nonempty set of cost functions. The Pigou bound α(C) for C is r · c(r) , c∈C x,r≥0 x · c(x) + (r − x)c(r)

α(C) = sup sup

(3)

with the understanding that 0/0 = 1. The point of the Pigou bound is that it lower bounds the price of anarchy in instances with cost functions in C. Proposition 2.13. Let C be a set of cost functions that contains all of the constant cost functions. Then ρ(C) ≥ α(C).

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Proof. Fix a choice of c ∈ C and x, r ≥ 0. We can complete the proof by exhibiting a selfish routing network with cost functions in C and price of anarchy at least c(r)r/[c(x)x + (r − x)c(r)]. Since c is nondecreasing, this expression is at least 1 if x ≥ r; we can therefore assume that x < r. Let G be a two-vertex, two-edge network as in Figure 1. Give the lower edge the cost function c1 (y) = c(y) and the upper edge the constant cost function c2 (y) = c(r). By assumption, both of these cost functions lie in C. Set the traffic rate to r. Routing all of the traffic on the lower edge yields a Wardrop equilibrium with cost c(r)r. Routing x units of traffic on the lower edge and r − x units of traffic on the upper edge gives a feasible flow with cost [c(x)x + (r − x)c(r)]. The price of anarchy in 2 this instance is thus at least c(r)r/[c(x)x + (r − x)c(r)], as desired Proposition 2.13 holds more generally for every set C of cost functions that is inhomogeneous in the sense that c(0) > 0 for some c ∈ C [31]. We next show that, even though the Pigou bound is based only on Pigou-like examples, it is also an upper bound on the price of anarchy in general multicommodity flow networks. The proof requires the following variational inequality characterization of Wardrop equilibria, first noted by Smith [36]. Proposition 2.14 ([36]). A flow f feasible for (G, r, c) is a Wardrop equilibrium if and only if ce (fe )fe ≤ ce (fe )fe∗ e∈E

for every flow f ∗ feasible for (G, r, c).

e∈E

Proposition 2.14 can be derived as an optimality condition for minimizers of the potential function (2), or can be proved directly using Definition 2.1. We now show that the Pigou bound is tight. Theorem 2.15 ([7], [31]). Let C be a set of cost functions and α(C) the Pigou bound for C. If (G, r, c) is an instance with cost functions in C, then ρ(G, r, c) ≤ α(C). Proof. Let f ∗ and f be an optimal flow and a Wardrop equilibrium, respectively, for an instance (G, r, c) with cost functions in the set C. The theorem follows by writing C(f ∗ ) = ce (fe∗ )fe∗ e∈E

≥

1 ce (fe )fe + (fe∗ − fe )ce (fe ) α(C) e∈E

e∈E

C(f ) , ≥ α(C) where the first inequality follows from Definition 2.12 and the second from Proposition 2.14. 2

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Different, more recent proofs of Theorem 2.15 can be found in [8], [37]. Proposition 2.13 and Theorem 2.15 establish the qualitative statement that, for essentially every fixed restriction on the allowable network cost functions, the price of anarchy is maximized by Pigou-like examples. Determining the largest-possible price of anarchy in Pigou-like examples (i.e., the Pigou bound) is a tractable problem in many cases. For example, it is precisely 4/3 when C is the affine functions [34], and more generally is [1 − p · (p + 1)−(p+1)/p ]−1 ≈ p/ ln p when C is the set of polynomials with degree at most p and nonnegative coefficients [31]. In these cases, the maximum price of anarchy (among all multicommodity flow networks) is achieved by the instances in Examples 2.2 and 2.5. For further examples, see [7], [31]. For much more on topics related to the price of anarchy of selfish routing, including many extensions and generalizations of the results described in this section, see [32], [33] and the references therein.

3. Efficiency loss in resource allocation protocols We next study the performance of a protocol for allocating resources to heterogeneous users. While there are a number of conceptual differences between this model and the selfish routing networks of Section 2, the inefficiency of equilibria in these models can be analyzed in a similar way. 3.1. The model. We consider a single divisible resource – the capacity of a single network link, say – to be allocated to a finite number n > 1 of competing users. These users are heterogeneous in the sense that different users can have different values for capacity. We model this by giving each user i a nonnegative real-valued utility function Ui that expresses this user’s value for a given amount of capacity. We assume that each Ui is concave, strictly increasing, and continuously differentiable. A resource allocation game is defined by the n utility functions U1 , . . . , Un and the link capacity C > 0. Anallocation for a resource allocation game is a nonnegative vector (x1 , . . . , xn ) with ni=1 xi = C. We study the following protocol for allocating capacity. Each user i submits a nonnegative bid bi for capacity. The protocol allocates capacity in proportion to bids, with bi xi = n ·C (4) j =1 bj units of capacity allocated to user i. The payoff Qi to a user i is defined as its value for the capacity it receives, minus the bid that it made (and presumably now has to pay): bi Qi (b1 , . . . , bn ) = Ui (xi ) − bi = Ui n · C − bi . j =1 bj Assume that if all users bid zero, then all users receive zero payoff.

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An equilibrium is then a bid vector in which each user bids optimally, given the bids of the other users. To state this precisely, we use the notation b−i = (b1 , b2 , . . . , bi−1 , bi+1 , . . . , bn ) to denote the bids of the users other than i, and sometimes write (bi , b−i ) for a bid vector (b1 , . . . , bn ). Definition 3.1. A bid vector (b1 , . . . , bn ) is an equilibrium of the resource allocation game (U1 , . . . , Un , C) if for every user i ∈ {1, 2, . . . , n}, Qi (bi , b−i ) = sup Qi (b˜i , b−i ).

(5)

b˜i ≥0

One easily checks that in every equilibrium, at least two users submit strictly positive bids. While equilibria are most naturally defined for bid vectors, we will be interested in the quality of the corresponding allocations. An equilibrium allocation is an allocation (x1 , . . . , xn ) induced by an equilibrium bid vector – i.e., there is an equilibrium (b1 , . . . , bn ) such that (4) holds for each user i. We next give a characterization of equilibrium allocations that will be crucial for designing a potential function for resource allocation games. First, a simple calculation shows that concavity of the utility function Ui (in xi ) implies strict concavity of the payoff function Qi (in bi ) for every fixed vector b−i with at least one strictly positive component. Similarly, the latter function is continuously differentiable for each such fixed b−i . We can therefore characterize solutions to (5) via standard first-order optimality conditions, which yields the following. Proposition 3.2 ([16], [20]). Let (U1 , . . . , Un , C) be a resource allocation game and (b1 , . . . , bn ) a bid vector with at least two strictly positive bids. Let B = jn=1 bj denote the sum of the bids. This bid vector is an equilibrium if and only if

bi bi B ≤ ·C 1− B B C for every user i ∈ {1, 2, . . . , n}, with equality holding whenever bi > 0. Ui

Reformulating Proposition 3.2 in terms of allocations gives the following corollary (cf., Definition 2.1). Corollary 3.3 ([16], [20]). Let (U1 , . . . , Un , C) be a resource allocation game. An allocation (x1 , . . . , xn ) is an equilibrium if and only if for every pair i, j ∈ {1, 2, . . . , n} of users with xi > 0, xj

xi

≥ Uj (xj ) 1 − . Ui (xi ) 1 − C C Proof. The “only if” direction follows easily from Proposition 3.2 and equation (4). For the “if” direction, suppose (x1 , . . . , xn ) satisfies the stated condition. There is then a scalar λ ≥ 0 such that Ui (xi )[1 − (xi /C)] ≤ λ for all users i, with equality holding whenever xi > 0. Setting bi = λxi for each i yields a bid vector that meets the equilibrium condition in Proposition 3.2. 2

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Example 3.4 ([20]). Consider a resource allocation game in which the capacity C is 1, one user has the utility function U1 (x1 ) = 2x1 , and the other n − 1 users have the utility function Ui (xi ) = xi . Corollary 3.3 implies that in the unique equilibrium allocation, the first user receives 21 + ε units of capacity, while each of the other n − 1 users receive δ units of capacity (with ε, δ → 0 as n → ∞). In this allocation, Ui (xi )(1−xi ) is the same for each user i, and is slightly less than 1. The corresponding equilibrium bid vector is roughly the same as the equilibrium allocation vector. In the next subsection, we use a potential function to show that every resource allocation game has a unique equilibrium allocation. We claim that the equilibrium allocation in Example 3.4 is suboptimal. As in the previous section, we formalize this claim by introducing an objective function and studying the price of anarchy. We define the efficiency of an allocation (x1 , . . . , xn ) of a resource allocation game to be the sum of the users’ utilities: E (x1 , . . . , xn ) =

n

Ui (xi ).

(6)

i=1

An optimal allocation has the maximum-possible efficiency. The price of anarchy of a resource allocation game is the ratio E (x)/E (x ∗ ), where x is the equilibrium allocation and x ∗ is an optimal allocation. Note that the price of anarchy of such a game is at most 1. In Example 3.4, the optimal allocation gives all of the capacity to the first user and has efficiency 2. The equilibrium allocation has efficiency approaching 3/2 as n → ∞; the price of anarchy can therefore be arbitrarily close to 3/4 in this family of examples. Why does inefficiency arise in Example 3.4? First, note that if the first user is the only one bidding a strictly positive amount (leading to the optimal allocation), then the bid vector cannot be an equilibrium: the first user can bid a smaller positive amount and continue to receive all of the capacity. A similar argument holds whenever the first user’s bid comprises a sufficiently large fraction of the sum of the users’bids: if the first user lowers its bid, its allocation diminishes, but the price it pays per unit of bandwidth decreases by a large enough amount to increase its overall payoff. This intuition is mathematically reflected in Corollary 3.3 in the term Ui (xi )(1 − xi ) – the marginal benefit of increased capacity to a user becomes increasing tempered as its allocation grows. Inefficiency thus arises in Example 3.4 because of “market power” – the fact that the actions of a single user have significant influence over the effective price of capacity. Indeed, resource allocation games were initially studied by Kelly [22] under the assumption that no users have nontrivial market power. Under this assumption, equilibria are fully efficient – i.e., the price of anarchy is always 1 [22]. See [19, §1.3–1.4] for further discussion. Remark 3.5. Selfish routing networks and resource allocation games differ in a number of ways. In the former, there is a continuum of selfish network users that each have a finite set of strategies (paths); in the latter, there is a finite set of users, each with a

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continuum of strategies (bids). In selfish routing, the objective is cost minimization; in resource allocation, it is efficiency maximization. Finally, and perhaps most fundamentally, inefficiency appears to arise for different reasons in the two models. Recall that in selfish routing networks, inefficiency stems from congestion externalities (see the discussion following Example 2.2). Example 3.4 shows that market power is the culprit behind inefficient equilibria in resource allocation games. Despite all of these conceptual differences, the next two subsections show that the inefficiency of equilibria can be quantified in the two models via remarkably similar analyses. 3.2. A potential function for equilibria. As in Subsection 2.2, our first step toward constructing a potential function for equilibrium allocations is to characterize optimal allocations. Since efficiency (6) is a separable concave function, a straightforward application of first-order optimality conditions yields the following. Proposition 3.6. Let (U1 , . . . , Un , C) be a resource allocation game. An allocation (x1 , . . . , xn ) is optimal if and only if for every pair i, j ∈ {1, 2, . . . , n} of users with xi > 0, Ui (xi ) ≥ Uj (xj ). Given the near-identical characterizations of equilibrium and optimal allocations in Corollary 3.3 and Proposition 3.6, respectively, we again ask: of what function does an equilibrium allocation arise as the global maximum? To recover Corollary 3.3 as an optimality condition, we seek a function Hi for each user i such that H i ) = Ui (x i )[1 − (xi /C)] for all xi ≥ 0. Setting Hi (xi ) = Ui (xi )[1 − (xi /C)] + i (x xi 0 Ui (y) dy /C thus yields the desired potential function. Precisely, for the resource allocation game (U1 , . . . , Un , C), define RA (x1 , . . . , xn ) =

n

Uˆ i (xi ),

(7)

i=1

where

xi 1 xi

xi ˆ Ui (xi ) = 1 − · Ui (xi ) + · Ui (y) dy . C C xi 0

(8)

A simple calculation shows that each function Uˆ i is strictly concave, increasing, and continuously differentiable. Regarding (Uˆ 1 , . . . , Uˆ n , C) as a resource allocation game, applying Proposition 2.6 to it, and appealing to Corollary 3.3 formalizes the fact that RA is a potential function. Proposition 3.7 ([16], [20]). An allocation of the game (U1 , . . . , Un , C) is an equilibrium allocation if and only if it is a global maximum of the corresponding potential function RA . Existence and uniqueness of equilibrium allocations follow immediately. Proposition 3.8 ([16], [20]). In every resource allocation game, there is a unique equilibrium allocation.

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Proof. Existence follows from Proposition 3.7 and the facts that the potential function (7) is continuous and the set of all allocations is compact. Uniqueness follows from Proposition 3.7 and the fact that the potential function (7) is strictly concave. 2 Proposition 3.7 also has consequences for the price of anarchy in resource allocation games. To see why, note that the value of Uˆ i (xi ) in (8) can be viewed as a average xi weighted of two quantities – the “true utility” Ui (xi ) and the “average utility” U (y) dy /xi . Since Ui is increasing, the latter quantity can only underestimate i 0 the utility Ui (xi ), and hence Uˆ i (xi ) ≤ Ui (xi ) for all i and xi ≥ 0. On the other hand, since Ui is nonnegative and concave, the average utility between 0 and xi is at least half of the utility Ui (xi ) at xi . Thus Uˆ i (xi ) ≥ Ui (xi )/2 for all i and xi ≥ 0. It follows that E (x1 , . . . , xn ) ≥ RA (x1 , . . . , xn ) ≥ E (x1 , . . . , xn )/2 for every allocation (x1 , . . . , xn ). Following the proof of Theorem 2.10 now gives a lower bound of 1/2 on the price of anarchy in resource allocation games. Theorem 3.9 ([20]). In every resource allocation game, the price of anarchy is at least 1/2. 3.3. An optimal bound on the price of anarchy. There is a gap between the lower bound of 1/2 on the price of anarchy given in Theorem 3.9 and the upper bound of 3/4 that is achieved (in the limit) in Example 3.4. As in Subsection 3.3, an optimal (lower) bound can be obtained by leveraging the potential function characterization of equilibria (Proposition 3.7) in a less crude way. Our argument will again be based on a “variational inequality”, which can be derived directly from Corollary 3.3 or viewed as a first-order optimality condition for the potential function (7). Proposition 3.10. Let (U1 , . . . , Un , C) be a resource allocation game. For each user i, define the modified utility function Uˆ i as in (8). An allocation xˆ is an equilibrium for (U1 , . . . , Un , C) if and only if n i=1

Uˆ i (xî )xî ≥

n

Uˆ i (xî )xi

i=1

for every feasible allocation x. Next is the analogue of the Pigou bound (Definition 2.12) for resource allocation games. This definition is primarily motivated by the upper bound on the price of anarchy provided by Example 3.4; we state it in a form that also permits easy application of Proposition 3.10 in the proof of Lemma 3.13 below. Definition 3.11. Let U denote the set of real-valued, nonnegative, strictly increasing, continuously differentiable, and concave (utility) functions. Define the JT bound β by ˆ ∗ − x) ˆ U (x) ˆ + Uˆ (x)(x β = inf inf inf , (9) ∗ U ∈U C>0 0≤x,x U (x ) ˆ ∗ ≤C


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where Uˆ is defined as in (8), as a function of U and C. In the rest of this section, we show that the JT bound is exactly the worst price of anarchy occurring in resource allocation games, and explicitly compute the bound. Lemma 3.12. For every ε > 0, there is a resource allocation game with price of anarchy at most β + ε, where β is the JT bound. Lemma 3.13. In every resource allocation game, the price of anarchy is at least the JT bound β. Lemma 3.14. The JT bound β is exactly 3/4. Lemmas 3.12–3.14 give an explicit optimal bound on the price of anarchy in resource allocation games. Theorem 3.15 ([20]). In every resource allocation game, the price of anarchy is at least 3/4. Moreover, this bound is tight. We now prove Lemmas 3.12–3.14 in turn. Proof of Lemma 3.12. Fix a choice of a utility function U , a capacity C > 0, and values for x, ˆ x ∗ ∈ [0, C]. We aim to exhibit a resource allocation game with price of anarchy (arbitrarily close to) U (x) ˆ + Uˆ (x)(x ˆ ∗ − x) ˆ . ∗ U (x )

(10)

Recall from (8) that Uˆ (x) ˆ = U (x) ˆ · [1 − (x/C)]. ˆ A calculation shows that (10) is at ∗ least 1 if xˆ ≥ x , so we can assume that xˆ < x ∗ . Since (10) is decreasing in C, we can assume that C = x ∗ . Define a resource allocation game in which the capacity is C, the first user has the utility function U1 (x1 ) = U (x1 ), and the other n − 1 users each have the linear utility function Ui (xi ) = Uˆ (x) ˆ · xi . Giving all of the capacity to the first user is a feasible allocation with efficiency U1 (C) = U (x ∗ ). Arguing as in Example 3.4, the equilibrium allocation has efficiency approaching U1 (x) ˆ + (C − x) ˆ · Uˆ (x) ˆ = ∗ U (x) ˆ + Uˆ (x)(x ˆ − x) ˆ as the number n of users tends to infinity. The price of anarchy in this family of instances thus tends to (at most) the expression in (10) as n → ∞, completing the proof. 2 Proof of Lemma 3.13. Let (U1 , . . . , Un , C) be a resource allocation game. Let x ∗ and xˆ denote optimal and equilibrium allocations, respectively. Define the modified utility function Uˆ i as in (8). The lemma follows by writing n n 1 ∗ ∗ ˆ Ui (xi ) ≤ Ui (xî ) + Ui (xî )(xi − xî ) β i=1

i=1

≤

n 1 Ui (xî ), β i=1

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where the first inequality follows from Definition 3.11 and the second from Proposition 3.10. 2 Proof of Lemma 3.14. Setting U to the identity function, xˆ = 1/2, and C = x ∗ = 1 shows that the JT bound is at most 3/4. Now fix arbitrary choices of U , C, and x, ˆ x ∗ ∈ [0, C]. We need to show that (10) is at least 3/4. As in the proof of Lemma 3.12, we can assume that xˆ < x ∗ = C. We can then write

xˆ U (x) ˆ + Uˆ (x)(x ˆ ∗ − x) ˆ = U (x) ˆ + 1 − ∗ U (x)(x ˆ ∗ − x) ˆ x xˆ ˆ ≥ U (x) ˆ + 1 − ∗ U (x ∗ ) − U (x) x xˆ xˆ · U (x) ˆ + 1 − ∗ · U (x ∗ ) = x∗ x 2 xˆ xˆ ∗ ≥ · U (x ) + 1 − · U (x ∗ ) x∗ x∗ 3 ≥ · U (x ∗ ), 4 where the first equality follows from the definition of Uˆ in (8), the first and second inequalities follow from the concavity and nonnegativity of U , and the final inequality follows from the fact that the function y 2 −y +1 is uniquely minimized when y = 1/2. The proof is complete. 2 Remark 3.16. The original proof of Theorem 3.15 is fairly different than the one given here. Specifically, Johari and Tsitsiklis [20] first show that the price of anarchy is minimized in games in which all users have linear utility functions, and then explicitly determine a worst-case example (the same as Example 3.4) by analyzing a linear program. We instead presented the proof above to further highlight the connections between resource allocation games and selfish routing networks. Despite the numerous common features in our analyses of the price of anarchy in selfish routing networks and in resource allocation games, the precise relationship between the two models is not completely understood. In particular, we lack a unifying analysis of the price of anarchy in the two models. Open Question 1. Find a compelling generalization of selfish routing networks and resource allocation games in which the price of anarchy can be analyzed in a uniform way. Ideally, such a generalization would unify Theorems 2.15 and 3.15, and would also apply to several of the more general classes of games described in [19], [32]. As with selfish routing networks, we have only scratched the surface of the literature on the price of anarchy in resource allocation games. For much more on the subject, including generalizations of these games to general networks, see Johari and Tsitsiklis [20] and Johari [19].

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4. The price of stability in network design games Our final class of games is a model of network design with selfish users. These games share some features with selfish routing networks, but also differ in a few fundamental respects. 4.1. The model. In this section we study Shapley network design games, first proposed by Anshelevich et al. [1]. The game occurs in a directed graph G = (V , E), in which each edge e ∈ E has a fixed nonnegative cost ce . There is a finite set of k selfish players, and each player i ∈ {1, 2, . . . , k} is identified with a source-sink vertex pair (si , ti ). Let Pi denote the set of simple si -ti paths. Each player i chooses apath Pi ∈ Pi from its source to its destination. This creates a network V , i Pi , and we define the cost of this outcome as ce . (11) c(P1 , . . . , Pk ) =

e∈

i

Pi

We assume that this cost is shared among the players in the following way. First, if edge e lies in fe of the chosen paths, then each player choosing such a path pays a proportional shareπe = ce /fe of the cost. The overall cost ci (P1 , . . . , Pk ) to player i is then the sum e∈Pi πe of these proportional shares. Selfish players naturally attempt to minimize their incurred cost. We next define our notion of equilibria for Shapley network design games. In contrast to selfish routing networks and resource allocation games, these network design games are finite games – there is a finite set of players, each with a finite set of strategies. This is the classical setting for Nash equilibria [25]. As in Definition 3.1, we use P−i to denote the vector of strategies chosen by the players other than i. Definition 4.1. An outcome (P1 , . . . , Pk ) of a Shapley network design game is a (pure-strategy) Nash equilibrium if for every player i, ci (Pi , P−i ) = min ci (P˜i , P−i ). P˜i ∈Pi

In a pure-strategy Nash equilibrium, every player chooses a single strategy. In a mixed-strategy Nash equilibrium, a player can randomize over several strategies. We will not discuss mixed-strategy Nash equilibria in this survey, though the price of anarchy of such equilibria has been studied in different models (see e.g. [3], [23]). Example 4.2 ([2]). Consider the network shown in Figure 3. There are k players, each with the same source s and sink t. The edge costs are k and 1 + ε, where ε > 0 is arbitrarily small. In the minimum-cost outcome, all players choose the lower edge. This outcome is also a Nash equilibrium. On the other hand, suppose all of the players choose the upper edge. Each player i then incurs cost 1, and if player i deviates to the lower edge it pays the full cost of 1 + ε. This outcome is thus a second Nash equilibrium, and it has cost k.

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t

s

1+ε Figure 3. Multiple Nash equilibria in Shapley network design games (Example 4.2).

Example 4.2 shows that Shapley network design games are more ill-behaved than selfish routing networks and resource allocation games in a fundamental respect: there can be multiple equilibria, and different equilibria can have very different objective function values. (Cf., Theorem 2.3 and Proposition 3.8.) The definition of the price of anarchy is ambiguous in games with multiple equilibria – we would like to quantify the inefficiency of an equilibrium, but of which one? The price of anarchy is historically defined as the ratio between the objective function value of the worst equilibrium and that of an optimal solution [23], [26]. This definition is natural from the perspective of worst-case analysis. In Example 4.2, the price of anarchy is (arbitrarily close to) k. It is also easy to show that the price of anarchy in every Shapley network design game is at most k. In this section, we instead focus on the ratio between the cost of the best Nash equilibrium of a Shapley network design game and that of an optimal solution. This measure is called the price of stability [1]. Our motivation is twofold. First, as Example 4.2 shows, the price of anarchy is large and trivial to determine. Second, the price of stability has a reasonably natural interpretation in network design games – if we envision the network as being designed by a central authority for subsequent use by selfish players, then the best Nash equilibrium is an obvious solution to propose. In this sense, the price of stability measures the necessary degradation in solution quality caused by imposing the game-theoretic constraint of stability. See [1], [2], [6], [7] for further discussion and examples of the price of stability. The price of stability in Example 4.2 is 1. We conclude this subsection with an example showing that this is not always the case. Example 4.3 ([1]). Consider the network shown in Figure 4. There are k players, all with the same sink t, and ε > 0 is arbitrarily small. For each i ∈ {1, 2, . . . , k}, the edge (si , t) has cost 1/i. In the minimum-cost outcome, each player i chooses the path si → v → t and the cost is 1 + ε. This is not a Nash equilibrium, as player k can decrease its cost from (1 + ε)/k to 1/k by switching to the direct path sk → t. More generally, this direct path is a dominant strategy for the kth player – it is the minimumcost strategy, independent of the paths chosen by the other players. It follows that in every Nash equilibrium, the kth player selects its direct path. Arguing inductively

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about the players k − 1, k − 2, . . . , 1, we find that the unique Nash equilibrium is t to the sink. The the outcome in which each player i chooses its direct path si → cost of this outcome is exactly the kth harmonic number Hk = ki=1 (1/i), which is roughly ln k. The price of stability can therefore be (arbitrarily close to) Hk in Shapley network design games.

t

... 1

1/2 1/3

s1

s2 0

...

s3 0

1/(k − 1) 1/k

0

...

sk−1 0

sk

0

1+ε v

Figure 4. The price of stability in Shapley network design games can be at least Hk (Example 4.3).

4.2. A potential function for Nash equilibria. In this subsection we use a potential function to prove the existence of pure-strategy Nash equilibria and upper bound the price of stability in Shapley network design games. Recall that for both selfish routing networks and resource allocation games, we designed potential functions using a characterization of optimal solutions as a guide (see Propositions 2.6 and 3.6). In Shapley network design games, computing an optimal solution is an NP-hard network design problem [15], and we cannot expect to find an analogous characterization. There are two ways that Shapley network design games differ from selfish routing networks that prevent the characterization of optimal solutions (Proposition 2.6) from carrying over. First, there are a finite number of players in the former model, and a continuum of players in the latter model. Second, cost functions in selfish routing networks are nondecreasing, whereas Shapley network design games effectively have cost functions that are decreasing in the “congestion” – if x ≥ 1 players use an edge e with fixed cost ce , then the per-player cost on that edge is ce /x. On the bright side, the potential function (2) for selfish routing networks is easily modified to account for these two differences. First, note that this function remains well-defined for decreasing cost functions. Second, passing from an infinite player set to a finite one merely involves changing the integrals in (2) to sums. This motivates

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the following proposal for a potential function for a Shapley network design game: N D (P1 , . . . , Pk ) =

fe ce e∈E i=1

i

,

(12)

where fe denotes the number of paths Pi that include edge e. While equilibria in selfish routing networks and resource allocation games can be characterized as the global optima of their respective potential functions (2) and (7), we will see that the Nash equilibria of a Shapley network design game are characterized as the local optima of the potential function (12). This idea is originally due to Rosenthal [29], [30], who also considered the broader context of “atomic congestion games”. The next lemma, which is crucial for the rest of this section, states that the potential function “tracks” the change in cost experienced by a deviating player. Lemma 4.4 ([1], [30]). Let (G, c) denote a Shapley network design game with k players and N D the corresponding potential function (12). Let i ∈ {1, 2, . . . , k} be a player, and let (Pi , P−i ) and (P˜i , P−i ) denote two outcomes that differ only in the strategy chosen by the ith player. Then ci (P˜i , P−i ) − ci (Pi , P−i ) = N D (P˜i , P−i ) − ND (Pi , P−i ).

(13)

Proof. Let fe denote the number of players that choose a path containing the edge e in the outcome (Pi , P−i ). Then both sides of (13) are equal to e∈P˜i \Pi

ce ce . − fe + 1 fe e∈Pi \P˜i

2

In the game theory literature, equation (13) is often taken as the definition of a potential function in the context of finite games. See Monderer and Shapley [24] for a fairly general treatment of potential functions for finite games. While simple, Lemma 4.4 has a number of non-trivial consequences. First, Nash equilibria of a Shapley network design game are the local minima of the corresponding potential function. Formally, two outcomes of a Shapley network design game are neighbors if they differ in at most one component, and an outcome is a local minimum of ND if it has no neighbor with strictly smaller potential function value. Corollary 4.5 ([1], [30]). An outcome of a Shapley network design game is a Nash equilibrium if and only if it is a local minimum of the corresponding potential function ND . Proof. Immediate from the definitions and Lemma 4.4.

2

Since every Shapley network design game has a finite number of outcomes, its corresponding potential function has a global (and hence local) minimum.


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Corollary 4.6 ([1], [30]). In every Shapley network design game, there is at least one (pure-strategy) Nash equilibrium. We note in passing that several related classes of network games do not always have pure-strategy Nash equilibria [2], [6], [13], [30]. A stronger version of Corollary 4.6 also holds. In a finite game, better-response dynamics refers to the following process: start with an arbitrary initial outcome; if the current outcome is not a Nash equilibrium, pick an arbitrary player that can decrease its cost by switching strategies, update its strategy to an arbitrary superior one, and repeat. Better-response dynamics terminate if and only if a Nash equilibrium is reached. Even in extremely simple two-player games, better-response dynamics need not terminate (e.g., in “rock-paper-scissors”). On the other hand, the potential function (12) ensures that such dynamics always converge in Shapley network design games. Corollary 4.7 ([1], [30]). In every Shapley network design game, better-response dynamics always converges to a Nash equilibrium in a finite number of iterations. Proof. By Lemma 4.4, every iteration of better-response dynamics strictly decreases the value of the potential function N D . Better-response dynamics therefore cannot visit an outcome more than once and eventually terminates, necessarily at a Nash equilibrium. 2 Corollary 4.7 does not address the number of iterations required to reach a Nash equilibrium; see [1], [12] for further study of this issue. Finally, the potential function (12) has direct consequences for the price of stability in Shapley network design games. Comparing the definitions of cost (11) and potential function value (12) of such a game, we have c(P1 , . . . , Pk ) ≤ N D (P1 , . . . , Pk ) ≤ Hk · c(P1 , . . . , Pk )

(14)

for every outcome (P1 , . . . , Pk ). As a result, a global minimum of the potential function N D of a Shapley network design game is both a Nash equilibrium (by Corollary 4.6) and has cost at most Hk times that of optimal (by the argument in the proof of Theorem 2.10). This gives the following theorem Theorem 4.8 ([1]). In every k-player Shapley network design game, the price of stability is at most Hk . A similar argument shows that the bound of Hk in Theorem 4.8 applies to every Nash equilibrium reachable from an optimal solution via better-response dynamics. The bound also carries over to numerous extensions of Shapley network design games; see [1] for details. Example 4.3 shows that the bound in Theorem 4.8 is tight for every k ≥ 1. Thus, unlike for selfish routing networks and resource allocation games, a direct application

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of a potential function argument yields an optimal upper bound on the inefficiency of equilibria. The upper bound in Theorem 4.8 is not optimal for some important special cases of Shapley network design games, however. For example, suppose we insist that the underlying network G is undirected. There is no known analogue of Example 4.3 for undirected Shapley network design games – the best lower bound known on the price of stability in such games is 2. On the other hand, it is not clear how to significantly improve the Hk bound in Theorem 4.8 for undirected networks. Open Question 2. Determine the largest-possible price of stability in undirected Shapley network design games.

5. Conclusion This survey has discussed three natural types of games: selfish routing networks, resource allocation games, and Shapley network design games. These classes of games differ from each other, both conceptually and technically, in a number of ways. Despite this, the worst-case inefficiency of selfish behavior is fairly well understood in all of these models, and in each case can be determined using a potential function characterization of equilibria. While the entire field of algorithmic game theory is still in a relatively nascent stage, several broad research agendas are emerging. For the problem of quantifying the inefficiency of noncooperative equilibria, a central research issue is to understand characteristics of games that guarantee approximately optimal equilibria, and to develop flexible mathematical techniques for proving such guarantees. While many research accomplishments from the past few years have improved our understanding of these intertwined goals, there is clearly much left to be done. Perhaps the current state of the art in bounding the inefficiency of equilibria can be compared to the field of approximation algorithms circa twenty-five years ago, when the most fundamental problems and the most powerful algorithmic techniques (such as linear programming) were only beginning to crystallize. Motivated by this analogy, we conclude with the following question: will potential functions be as ubiquitous in bounds on the inefficiency of equilibria as linear programming is in bounds on the performance of approximation algorithms? Open Question 3. We have seen that a potential function characterization of equilibria leads a bound on the inefficiency of equilibria. Under what conditions and to what extent does a converse hold? When does a bound on the inefficiency of the equilibria of a game imply the existence of some form of a potential function for the game?


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References [1] Anshelevich, E., Dasgupta, A., Kleinberg, J., Tardos, É., Wexler, T., and Roughgarden, T., The price of stability for network design with fair cost allocation. In Proceedings of the 45th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 2004, 295–304. [2] Anshelevich, E., Dasgupta, A., Tardos, É., and Wexler, T., Near-optimal network design with selfish agents. In Proceedings of the 35th Annual ACM Symposium on Theory of Computing, ACM Press, New York 2003, 511–520. [3] Awerbuch, B., Azar, Y., and Epstein, E., The price of routing unsplittable flow. In Proceedings of the 37th Annual ACM Symposium on Theory of Computing, ACM Press, New York 2005, 57–66. [4] Beckmann, M. J., McGuire, C. B., and Winsten, C. B. Studies in the Economics of Transportation. Yale University Press, 1956. [5] Bertsekas, D. P., Nedic, A., and Ozdaglar, A. E., Convex Analysis and Optimization. Athena Scientific, 2003. [6] Chen, H., and Roughgarden, T., Network design with weighted players. Submitted, 2005. [7] Correa, J. R., Schulz, A. S., and Stier-Moses, N. E., Selfish routing in capacitated networks. Math. Oper. Res. 29 (4) (2004), 961–976. [8] Correa, J. R., Schulz, A. S., and Stier-Moses, N. E., On the inefficiency of equilibria in congestion games. In Integer Programming and Combinatorial Optimization, Lecture Notes in Comput. Sci. 3509, Springer-Verlag, Berlin 2005, 167–181. [9] Cramton, P., Shoham, Y., and Steinberg, R., Combinatorial Auctions. MIT Press, 2006. [10] Doyle, P. G., and Snell, J. L., Random Walks and Electrical Networks. Mathematical Association of America, 1984. [11] Dubey, P., Inefficiency of Nash equilibria. Math. Oper. Res. 11 (1) (1986), 1–8. [12] Fabrikant, A., Papadimitriou, C. H., and Talwar, K., The complexity of pure Nash equilibria. In Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM Press, New York 2004, 604–612. [13] Fotakis, D., Kontogiannis, S. C., and Spirakis, P. G., Selfish unsplittable flows. Theoret. Comput. Sci. 348 (2–3) (2005), 226–239. [14] Friedman, E. J., and Shenker, S., Learning and implementation on the Internet. Working paper, 1997. [15] Garey, M. R., and Johnson, D. S., Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, 1979. [16] Hajek, B., and Gopalakrishnan, G., Do greedy autonomous systems make for a sensible Internet? Presentation at the Conference on Stochastic Networks, Stanford University, June 2002 (Cited in [20]). [17] Hartline, J. D., Optimization in the Private Value Model: Competitive Analysis Applied to Auction Design. PhD thesis, University of Washington, 2003. [18] Jain, K., A polynomial time algorithm for computing the Arrow-Debreu market equilibrium for linear utilities. In Proceedings of the 45th Annual Symposium on Foundations of Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 2004, 286–294.

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[19] Johari, R., Efficiency Loss in Market Mechanisms for Resource Allocation. PhD thesis, MIT, 2004. [20] Johari, R., and Tsitsiklis, J. N., Efficiency loss in a network resource allocation game. Math. Oper. Res. 29 (3) (2004), 407–435. [21] Kelly, F. P., Network routing. Philos. Trans. Roy. Soc. London Ser. A 337 (3) (1991), 343–367. [22] Kelly, F. P., Charging and rate control for elastic traffic. European Transactions on Telecommunications 8 (1) (1997), 33–37. [23] Koutsoupias, E., and Papadimitriou, C. H., Worst-case equilibria. In STACS 99, Lecture Notes in Comput. Sci. 1563, Springer-Verlag, Berlin 1999, 404–413. [24] Monderer, D., and Shapley, L. S., Potential games. Games Econom. Behav. 14 (1) (1996), 124–143. [25] Nash, J. F., Equilibrium points in N-person games. Proc. National Academy of Science 36 (1) (1959), 48–49. [26] Papadimitriou, C. H., Algorithms, games, and the Internet. In Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, ACM Press, New York 2001, 749–753. [27] Pigou, A. C., The Economics of Welfare. Macmillan, 1920. [28] Rapoport, A., and Chammah, A. M., Prisoner’s Dilemma. University of Michigan Press, 1965. [29] Rosenthal, R. W., A class of games possessing pure-strategy Nash equilibria. Internat. J. Game Theory 2 (1) (1973), 65–67. [30] Rosenthal, R. W., The network equilibrium problem in integers. Networks 3 (1) (1973), 53–59. [31] Roughgarden, T., The price of anarchy is independent of the network topology. J. Comput. System Sci. 67 (2) (2003), 341–364. [32] Roughgarden, T., Selfish Routing and the Price of Anarchy. MIT Press, 2005. [33] Roughgarden, T., Selfish routing and the price of anarchy. OPTIMA 71 (2006), to appear. [34] Roughgarden, T., and Tardos, É., How bad is selfish routing? J. ACM 49 (2) (2002), 236–259. [35] Schmeidler, D., Equilibrium points of nonatomic games. J. Statist. Phys. 7 (4) (1973), 295–300. [36] Smith, M. J., The existence, uniqueness and stability of traffic equilibria. Transportation Res. Part B 13 (4) (1979), 295–304. [37] Tardos, É., CS684 course notes. Cornell University, 2004. [38] Wardrop, J. G., Some theoretical aspects of road traffic research. In Proceedings of the Institute of Civil Engineers, Pt. II, volume 1, 1952, 325–378. Stanford University, Department of Computer Science, 462 Gates Building, 353 Serra Mall, Stanford, CA 94305, U.S.A. E-mail: [email protected]

Sublinear time algorithms Ronitt Rubinfeld ∗

Abstract. Sublinear time algorithms represent a new paradigm in computing, where an algorithm must give some sort of an answer after inspecting only a very small portion of the input. We discuss the sorts of answers that one might be able to achieve in this new setting. Mathematics Subject Classification (2000). Primary 68Q25; Secondary 68W20, 68W25. Keywords. Sublinear time algorithms, property testing.

1. Introduction The goal of algorithmic research is to design efficient algorithms, where efficiency is typically measured as a function of the length of the input. For instance, the elementary school algorithm for multiplying two n digit integers takes roughly n2 steps, while more sophisticated algorithms have been devised which run in less than n log2 n steps. It is still not known whether a linear time algorithm is achievable for integer multiplication. Obviously any algorithm for this task, as for any other nontrivial task, would need to take at least linear time in n, since this is what it would take to read the entire input and write the output. Thus, showing the existence of a linear time algorithm for a problem was traditionally considered to be the gold standard of achievement. Nevertheless, due to the recent tremendous increase in computational power that is inundating us with a multitude of data, we are now encountering a paradigm shift from traditional computational models. The scale of these data sets, coupled with the typical situation in which there is very little time to perform our computations, raises the issue of whether there is time to consider any more than a miniscule fraction of the data in our computations? Analogous to the reasoning that we used for multiplication, for most natural problems, an algorithm which runs in sublinear time must necessarily use randomization and must give an answer which is in some sense imprecise. Nevertheless, there are many situations in which a fast approximate solution is more useful than a slower exact solution. A first example of sublinear time computation that comes to mind is the classical result from the theory of sampling that one can, in time independent of the size of the data, determine a good estimate of the average value of a list of numbers of bounded ∗ supported by NSF grant 012702-001


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magnitude. But what about more interesting algorithmic questions? For example, given access to all transcripts of trades in the stock exchange, can we determine whether there is a trend change? This is easily detectable after a careful scan of the entire transcript, but by the time the scan is performed, it might be too late to make use of the information. However, it might be feasible to construct much faster algorithms based on random sampling. The recently emerging theory of sublinear time algorithms addresses questions of precisely this nature for problems in various domains. This paper will describe a number of problems that can be solved in sublinear time, using different types of approximations. Outline of the paper. We begin by giving a motivating example of a sublinear time algorithm in Section 2. In Section 3 we formalize the definitions of approximations that sublinear time algorithms are able to achieve. We then describe various examples of sublinear time algorithms.

2. A simple example: monotonicity of a list Let us begin with a simple example, which will motivate our subsequent definitions of the types of approximations that we will be interested in achieving. A list of integers x = x1 , . . . , xn , is monotone (increasing) if xi ≤ xj for all 1 ≤ i < j ≤ n. Given input x, the task is to determine whether or not x is monotone. In order to construct an algorithm that runs in sublinear time for determining whether x is monotone, we first need to make our model of computation precise. For example, if our algorithm must scan x1 , . . . , xi−1 in order to reach xi , then there is no hope for the existence of a sublinear time algorithm. However, it is often natural to assume that our algorithms have query (also called oracle) access to the input. That is, they can access xi in one step for any 1 ≤ i ≤ n. Even with this model of computation, it is clear that finding a sublinear time algorithm for the above task is impossible, since any algorithm that does not look at some xj could be fooled by an input for which all the xi ’s are in monotone order for i = j . Thus, we can only hope to solve an approximate version of this problem, but what is a meaningful notion of an approximation? One natural approximate version is defined as follows: Say that x1 , . . . , xn is ε-close to monotone if by changing at most εn of the values of the xi ’s one can transform x1 , . . . , xn into a monotone list. Then, a property tester for monotonicity is a randomized algorithm that on input x, ε, must output “pass” if x1 , . . . , xn is monotone, and “fail” if x1 , . . . , xn is not ε-close to monotone. The algorithm is allowed to err with probability at most 1/3. However, once an algorithm with error probability at most 1/3 is achieved, it is easy to see that for any β, of a probability error of at most β can be achieved by repeating the algorithm O log β1 times and taking the majority answer. Note that if x1 , . . . , xn is, say, ε/2-close to monotone,

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the property testing algorithm is allowed to output “pass” or “fail”. Indeed, in this case, since the list is close to monotone, it may not be too harmful to pass it. On the other hand, since it is not actually monotone, it is also not a big injustice to fail it. How do we construct a property tester for monotonicity? On first thought, one might try picking random indices i, j and performing tests of the form “is xi ≤ xj ?” or “is xi ≤ xi+1 ?”. However, these tests do not work very well.1 It is easy to construct √ examples showing that there are lists that are not even 1 − n -close to monotone, 1 . This means that at least n1/4 yet pass such tests with probability at least 1 − n1/4 such tests must be performed if one is to find a reason to output “fail”. Though this does not rule out the possibility of a sublinear time property tester, we will see that one can do much better. In the following, we describe an O(log n) time algorithm from the work of Ergün et. al. [17] which tests if x has a long monotone increasing subsequence. Note that the problem is known to require (log n) queries [17], [20]. Let c be a constant that is set appropriately. For simplicity, let us assume that the elements in x are distinct. The last assumption is without loss of generality, since one can append the index of an item to the least significant bits of its value in order to break ties.

1. Let = c/ε. Choose indices i1 , . . . , i uniformly from [n]. 2. For each such chosen index ij , assume the list is monotone and perform a binary search in x as if to determine whether xij is present in x or not. 3. Output “fail” if the binary search fails to find any xij in location ij or finds a pair of out-of-order elements along the search path. Output “pass” if all the binary searches succeed. The running time of the algorithm is O((1/ε) log n). Moreover, if x is monotone, then the algorithm will always output “pass” as each of the binary searches will succeed. To show that if x is not ε-close to monotone, then the algorithm will output “fail” with probability at least 2/3, we show the contrapositive. Namely, assume that the input is such that the algorithm outputs “pass” with probability at least 1/3. To see that x has a long increasing subsequence, let G ⊆ [n] denote the set of indices for which the binary search would succeed, i.e., i ∈ G if and only if xi can be found by a binary search on x that sees no pair of out-of-order elements along the search path. The constant c can be chosen such that if |G| < (1 − ε)n, then the algorithm would pick some ij ∈ G with probability at least 1/3, causing it to output “fail”. Thus, since the algorithm outputs “pass” with probability at least 1/3, we know that |G| ≥ (1 − ε)n. We now argue that the restriction of x to the indices in G is an increasing subsequence: Let i, j ∈ G and i < j . Let k be the least common ancestor index where the binary searches for xi and xj diverge. Then xi < xk and xk < xj , which implies xi < xj . Finally, if x has an increasing subsequence of size at least (1 − ε)n then it is easy to see that x is ε-close to monotone. Thus we have the following theorem: Theorem 2.1 ([17]). There is an algorithm that, given a sequence x = x1 , . . . , xn and an ε > 0, runs in O((1/ε) log n) time and outputs (1) “pass”, if x is monotone

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and (2) “fail”, with probability 2/3, if x does not have an increasing subsequence of length at least (1 − ε)n (in particular, if x is ε-far from monotone).

3. What do we mean by an “approximation”? Now that we have developed some intuition, we present our model and definitions in more detail: We are interested in computing some function f on input x without reading all of x. This is an impossible task in general, since a change to a single bit of x could alter the value of f . When f is the characteristic function of a property, i.e., f (x) = 1 if x has the property and f (x) = 0 otherwise, the following notion of approximation has emerged: Given an input, a property tester tries to distinguish whether the input has the property from the case where the input is not even close to having the property. We first formalize what it means to be close. Definition 3.1. An input x, represented as a function x : D → R, is ε-close to satisfying property P if there is some y satisfying P such that x and y differ on at most ε|D| places in their representation. Otherwise, x is said to be ε-far from satisfying P . In the monotonicity example of the previous section, D = [n] and x(i) returns the i th element of the list. We now formalize what it means for an algorithm to test a property. As in the previous section, we assume in our model of computation that algorithms have query access to the input. Definition 3.2. Let P be a property. On input x of size n = |D| and ε, a property tester for P must satisfy the following: • If x satisfies property P , the tester must output “pass” with probability at least 2/3. • If x is ε-far from satisfying P , the tester must output “fail” with probability at least 2/3. The probability of error may depend only on the coin tosses of the algorithm and not on any assumptions of the input distribution. The number of queries made by the property tester q = q(ε, n) is referred to as the query complexity of the property tester. We say that a property tester for P has 1-sided error if it outputs “pass” with probability 1 when x satisfies P . If the query complexity is independent of n, then we say that the property is easily testable. Note that if x does not satisfy P but x is also not ε-far from satisfying P , then the output of the property tester can be either “pass” or “fail”. We have already seen that it is this gap which allows property testers to be so efficient.


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The probability that the property tester errs is arbitrarily set to 1/3 and may alternatively be defined to be any constant less than 1/2. It is then easy to see that for any of error of at most β can be achieved by repeating the algorithm β, a probability O log β1 times and taking the majority answer. Property testing was first defined by Rubinfeld and Sudan [37] in the context of program testing. Goldreich, Goldwasser, Ron [23] refined and generalized the definition. Various more general definitions are given in several works, including [17], [25], [31], which mostly differ in terms of the generality of the distance function and natural generalizations as to when the tester should accept and reject.

4. Algebraic problems In this section, we consider property testing algorithms for problems that are algebraic in nature. We begin with the problem of testing whether a function is a homomorphism. We then show that the ideas used to construct property testers for homomorphisms extend to other properties with similar underlying structure. 4.1. Homomorphism testing. We begin with an example that was originally motivated by applications in program testing [14] and was later used in the construction of Probabilistically Checkable Proof systems [6]. Suppose you are given oracle access to a function f : D → R, that is, you may query the oracle on any input x ∈ D and it will reply with f (x). Is f a homomorphism? In order to determine the answer exactly, it is clear that you need to query f on the entire domain D. However, consider the property testing version of the problem, for which on input ε, the property tester should output “pass” with probability at least 2/3 if f is a homomorphism and “fail” with probability at least 2/3 if f is ε-far from a homomorphism (that is, there is no homomorphism g such that f and g agree on at least (1 − ε)|D| inputs). In order to construct such a property tester, a natural idea would be to test that the function satisfies certain relationships that all homomorphisms satisfy. We next describe two such relationships and discuss their usefulness in constructing property testers. Two characterizations of homomorphisms over Zq . Consider the case when f is over the domain and range D = R = Zq for large integer q. The set of homomorphisms over Zp can be characterized as the set of functions which satisfy f (x + 1) − f (x) = f (1) for all x. This suggests that a property tester might test that f (x + 1) − f (x) = f (1) for most x. However, it is easy to see that there are functions f which are very far from any homomorphism, but would pass such a test √ with overwhelmingly high probability. For example, g(x) = x mod q satisfies g(x + 1) − g(x) = g(1) for 1 − √1q fraction of the x ∈ Zq but g(x) is 1 − √1q -far from a homomorphism.

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The set of homomorphisms over D can alternatively be characterized as the set of functions which satisfy f (x) + f (y) = f (x + y) for all x, y. This suggests that one might test that f (x) + f (y) = f (x + y) for most x, y. It might be worrisome to note that when q = 3n, the function h(x) defined by h(x) = 0 if x ≡ 0 mod 3, h(x) = 1 if x ≡ 1 mod 3 and h(x) = 3n − 1 if x ≡ −1 mod 3 passes the above test for 7/9 fraction of the choices of pairs x, y ∈ D and that h(x) is 2/3-far from a homomorphism [16]. However, here the situation is much different: one can show that for any δ < 2/9, if f (x) + f (y) = f (x + y) for at least 1 − δ fraction of the choices of x, y ∈ D, then there is some homomorphism g, such that f (x) = g(x) on at least 1 − δ/2 fraction of the x ∈ D [13]. Once one has established such a theorem, then one can construct a property tester based on this characterization by sampling O(1/ε) pairs x, y and ensuring that each pair in the sample satisfies f (x) + f (y) = f (x + y). This property tester clearly passes all homomorphisms. On the other hand, if f is ε-far from a homomorphism then the above statement guarantees that at least 2ε fraction of the choices of x, y pairs do not satisfy f (x) + f (y) = f (x + y), and the property tester is likely to fail. In both cases, homomorphisms are characterized by a collection of local constraints, where by local, we mean that few function values are related within each constraint. What is the difference between the first and the second characterization of a homomorphism that makes the former lead to a bad test and the latter to a much better test? In [37] (see also [36]), the notion of a robust characterization was introduced to allow one to quantify the usefulness of a characterization in constructing a property test. Loosely, a robust characterization is one in which the “for all” quantifier can be replaced by a “for most” quantifier while still characterizing essentially the same functions. That is, for a given ε, δ, a characterization is (ε, δ)-robust if for any function f that satisfies at least 1 − δ fraction of the constraints, f must be ε-close to some function g that satisfies all of the constraints and is thus a solution of the “for all” characterization. As we saw above, once we have an (ε, δ)-robust characterization for a property, it is a trivial matter to construct a property tester for the property. We are interested in the relationship between ε and δ as well as the range of δ for which the property is (ε, δ)-robust, since the value of δ directly influences the running time of the property tester. Homomorphism testing, a history. Let G, H be two finite groups. For an arbitrary map f : G → H , define δ, the probability of group law failure, by 1 − δ = Pr [f (x) + f (y) = f (x + y)] . x,y

Define ε such that ε is the minimum τ for which f is τ -close to a homomorphism. We will be interested in the relationship between ε and δ. Blum, Luby and Rubinfeld [14], considered this question and showed that over cyclic groups, there is a constant δ0 , such that if δ ≤ δ0 , then the one can upper bound ε in terms of a function of δ that is independent of |G|. This yields a homomorphism

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tester with query complexity that depends (polynomially) on 1/ε, but is independent of |G|, and therefore shows that the property of being a homomorphism is easily testable. The final version of [14] contains an improved argument due to Coppersmith [16], which applies to all Abelian groups, shows that δ0 < 2/9 suffices, and that ε is upper bounded by the smaller root of x(1 − x) = δ (yielding a homomorphism tester with query complexity linear in 1/ε). Furthermore, the bound on δ0 was shown to be tight for general groups [16]. In [13], it was shown that for general (nonAbelian) √ groups, for δ0 < 2/9, then f is ε-close to a homomorphism where ε = (3 − 9 − 24δ)/12 ≤ δ/2 is the smaller root of 3x − 6x 2 = δ. The condition on δ, and the bound on ε as a function of δ, are shown to be tight, and the latter improves that of [14], [16]. Though δ0 < 2/9 is optimal over general Abelian groups, using Fourier techniques, Bellare et. al. [12] have shown that for groups of the form (Z/2)n , δ0 ≤ 45/128 suffices. A proof of a homomorphism test. We describe the following proof, essentially due to Coppersmith [14], [16] of the robustness of the homomorphism characterization over Abelian groups. Though this is not the strongest known result, we include this proof to give a flavor of the types of arguments used to show robustness of algebraic properties. Theorem 4.1. Let G be a finite Abelian group and f : G → G. Let δ be such that 1 − δ = Pr [f (x) + f (y) = f (x + y)] . x,y

Then if δ < 2/9, f is 2δ-close to a homomorphism. Proof. Define φ(x) = majy∈G (f (x + y) − f (y)), that is, let φ(x) be the value that occurs with the highest probability when evaluating f (x + y) − f (y) over random y (breaking ties arbitrarily). The theorem follows immediately from the following two claims showing that φ is a homomorphism and that f and φ are 2δ-close. Claim 4.2. |{y|f (y) = φ(y)}| ≥ (1 − 2δ)|G|. Proof of Claim 4.2. Let B = {x ∈ G : Pr y [f (x) = f (x + y) − f (y)] > 1/2]}. |B| If x ∈ / B, then φ(x) = f (x). Thus, it suffices to bound |G| . If x ∈ B, then Pr y [f (x) + f (y) = f (x + y)] > 1/2. Thus δ = Pr x,y [f (x) = f (x + y) − f (y)] ≥ |B| 1 |B| 2 |G| · 2 or equivalently |G| ≤ 2δ. Claim 4.3. If δ < 2/9, then φ(x) + φ(z) = φ(x + z) for all x, z. Proof of Claim 4.3. Fix x, we first show that most pairs y1 , y2 agree to vote for the same value of φ(x). Pick random y1 , y2 ∈ G, and we have: Pr y1 ,y2 [f (x + y1 ) − f (y1 ) = f (x + y2 ) − f (y2 )] = Pr y1 ,y2 [f (x + y1 ) + f (y2 ) = f (x + y2 ) + f (y1 )].

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x + y1 and y2 are both uniformly distributed elements of G. Therefore we have Pr y1 ,y2 [f (x + y1 ) + f (y2 ) = f (x + y1 + y2 )] = δ < 2/9. Similarly, we have Pr y1 ,y2 [f (x + y2 ) + f (y1 ) = f (x + y1 + y2 )] = δ < 2/9. If neither of the above events happens, then f (x + y1 ) − f (y1 ) = f (x + y2 ) − f (y2 ). Via a union bound we have that Pr y1 ,y2 [f (x + y1 ) − f (y1 ) = f (x + y2 ) − f (y2 )] > 1 − 2δ ≥ 5/9. It is straightforward to show that for any distribution in which the collision probability is at least 5/9, the maximum probability element must have probability at least 2/3. Thus, (1) Pr [φ(x) = f (x + y) − f (y)] < 1/3 for all x ∈ G. y

To show that for all x, z ∈ G, φ(x) + φ(z) = φ(x + z), fix x and z. Then apply Equation (1) to x, z and x + z to get Pr y [φ(x) = f (x + (y − x)) − f (y − x)] < 1/3, Pr y [φ(z) = f (z + y) − f (y)] < 1/3, Pr y [φ(x + z) = f ((x + z) + (y − x)) − f (y − x)] < 1/3.

(2) (3) (4)

Thus Pr y [φ(x) = f (y) − f (y − x) and φ(z) = f (z + y) − f (y) and φ(x + z) = f (z + y) − f (y − x)] > 0, and so there exists a y for which φ(x)+φ(z) = (f (y)−f (y−x))+(f (z+y)−f (y)) = f (z+y)−f (y−x) = φ(x+z). The above equality holds for every x, z ∈ G, showing that φ is a homomorphism and completing the proof of Claim 4.3. 2 2 A word about self-correcting. In the proof, we note that φ is defined so that it is the “self-correction” of f . Observe that there is a simple randomized algorithm that computes φ(x) given oracle access to f : pick c log 1/β values y, compute f (x + y) − f (y) and output the value that you see most often. If f is 18 -close to a homomorphism φ, then since both y and x + y are uniformly distributed, we have that for at least 3/4 of the choices of y, φ(x + y) = f (x + y) and φ(y) = f (y), in which case f (x + y) − f (y) = φ(x). Thus it is easy to show that there is a constant c such that if f is 18 -close to a homomorphism φ, then the above algorithm will output φ(x) with probability at least 1 − β. 4.2. Other algebraic functions. It is natural to wonder what other classes of functions have robust characterizations as in the case of homomorphisms? There are many other classes of functions that are defined via characterizations that are local. The field of functional equations is concerned with the prototypical problem of characterizing the set of functions that satisfy a given set of properties (or functional equations). For

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example, the class of functions of the form f (x) = tan Ax are characterized by the functional equation f (x + y) =

f (x) + f (y) 1 − f (x)f (y)

for all x, y.

D’Alembert’s equation f (x + y) + f (x − y) = 2f (x)f (y) for all x, y characterizes the functions 0, cos Ax, cosh Ax. Multivariate polynomials of total degree d over Zp for p > md can be characterized by the equation d+1

ˆ =0 αi f (xˆ + i h)

for all x, ˆ hˆ ∈ Zpm ,

i=0

where αi = (−1)i+1 d+1 i . All of the above characterizations are known to be (ε, δ)robust for ε and δ independent of the domain size (though for the case of polynomials, there is a polynomial dependence on the total degree d) thus showing that the corresponding properties are easily testable [36], [37]. A long series of works have given increasingly robust characterizations of functions that are low total degree polynomials (cf. [6], [32], [7], [34], [3], [29], [27]). We note that all of these results can be extended to apply over domains that are subsets of infinite cyclic groups. They can further be extended to the case of computation with finite precision, which requires that one address the stability of functional equations [18], [30]. Convolutions of distributions. We now turn to a seemingly unrelated question about distributions that are close to their self-convolutions: Let A = {ag | g ∈ G} be a distribution on group G. The convolution of distributions A, B is C = A ∗ B, cx = ay bz . y,z∈G yz=x

Let A be the self-convolution of A, A ∗ A, i.e. ax = y,z∈G;yz=x ay az . It is known that A = A exactly when A is the uniform distribution over a subgroup of G. Suppose we know that A is close to A , can we say anything about A in this case? Suppose 1 dist(A, A ) = 2 x∈G |ax − ax | ≤ ε for small enough ε. Then [13] show that A must be close to the uniform distribution over a subgroup of G. More precisely, in [13] it is shown that for a distribution A over a group G, if dist(A, A ) = 21 x∈G |ax − ax | ≤ ε ≤ 0.0273, then there is a subgroup H of G such that dist(A, UH ) ≤ 5ε, where UH is the uniform distribution over H [13]. On the other hand, in [13] there is an example of a distribution A such that dist(A, A ∗ A) ≈ .1504, but A is not close to uniform on any subgroup of the domain.

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A weaker version of this result, was used to prove a preliminary version of the homomorphism testing result in [14]. To give a hint of why one might consider the question on convolutions of distributions when investigating homomorphism testing, consider the distribution Af achieved by picking x uniformly from G and outputting f (x). It is easy to see that the error probability δ in the homomorphism test is at least dist(Af , Af ∗ Af ). The other, more useful, direction is less obvious. In [13] it is shown that this question on distributions is “equivalent” in difficulty to homomorphism testing: Theorem 4.4. Let G, H be finite groups. Assume that there is a parameter β0 and function φ such that the following holds: For all distributions A over group G, if dist(A ∗ A, A) ≤ β ≤ β0 then A is φ(β)-close to uniform over a subgroup of G. Then, for any f : G → H and δ < β0 such that 1 − δ = Pr[f (x) ∗ f (y) = f (x ∗ y)], and φ(δ) ≤ 1/2, we have that f is φ(δ)-close to a homomorphism.

5. Combinatorial objects In 1996, Goldreich, Goldwasser and Ron [23] focused attention on the problem of testing various properties of graphs and other combinatorial objects. Their work introduced what is now referred to as the dense graph model of property testing. In this model, a graph on n nodes is represented via an n × n adjacency matrix, where the (i, j )th entry of the matrix contains a 1 if the edge (i, j ) is present in the graph and a 0 otherwise. Two graphs G and H are ε-close if at most εn2 edges need to be modified (inserted or deleted) to turn G into H . In [23], several graph properties were shown to be easily testable. In fact, as we shall soon see, the question of which graph properties are easily testable has led to a series of intriguing results. One property that [23] consider is that of k-colorability – is it possible to assign one of k colors to each of the nodes so that no pair of nodes that have an edge between them are assigned the same color? The property of k-colorability is NP-complete to determine – meaning, that though we know how to verify that a certain coloring is a valid k-coloring, we have no idea how to determine whether a graph has a k coloring in time polynomial in the size of the graph. Somewhat surprisingly, k-colorability is easily testable, so we can distinguish k-colorable graphs from those that are ε-far from k-colorable in constant time. Thus we see that the efficiency of a property tester is not directly related to the complexity of deciding the property exactly. Though the proof of correctness of the property tester for k-colorability is involved, the algorithm used to conduct the property test is easy to describe: It simply picks a constant sized random sample of the vertices, queries all the edges among this random sample and then outputs “pass” or “fail”, according to whether the sample is


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k-colorable. Since the sample is of constant size, the determination of whether the sample is k-colorable can be made in constant time. Such algorithms that (1) pick a constant sized random sample of the vertices, (2) query all the edges among this random sample, and then (3) output “pass” or “fail” based on whether the subgraph has the property or not, are referred to as “natural algorithms”. Modulo a technicality about how the final output decision is made, Goldreich and Trevisan [26] essentially show that for any graph property that is easily testable, the natural algorithm gives a property tester. Thus, all easily testable graph properties provably have easy-to-describe algorithms. The work of [23] sparked a flurry of results in the dense graph model. A very interesting line of work was initiated in the work of Alon, Fischer, Krivelevich and Szegedy [2], in which they use the Szemerédi Regularity Lemma to show that the property of a graph being H -free (that is, the graph does not contain any copy of H as a subgraph) is easily testable for any constant sized graph H . Very recently, the above line of work culminated in the following amazing result: Alon and Shapira [5] have shown that one can completely characterize the classes of graph properties that are easily testable with 1-sided error in the dense graph model. Before describing their result, we make two definitions. A graph property P is hereditary if it is closed under the removal of vertices (but not necessarily under the removal of edges). A graph property P is semi-hereditary if there is a hereditary graph property H such that (1) any graph satisfying P also satisfies H and (2) for any ε > 0, there is an M(ε), such that any graph G of size at least M(ε) which is ε-far from satisfying P does not satisfy H . The result of Alon and Shapira is then the following: Theorem 5.1. A graph property P is easily testable with one-sided error if and only if P is semi-hereditary. Hereditary graph properties include all monotone graph properties (including kcolorability and H -freeness), as well as other interesting non-monotone graph properties such as being a perfect, chordal, or interval graph. The techniques used by Alon and Shapira are quite involved, and are based on developing new variants of the Szemerédi Regularity Lemma. Previously in the literature, the “Regularity Lemma type” arguments were used to develop testers for graph properties that were characterized by a finite set of forbidden subgraphs. Here the set of forbidden subgraphs may be infinite, and they are forbidden as induced subgraphs. Several interesting questions regarding easily testable graph properties remain. For example, because of the use of the Szemerédi Regularity Lemma, the upper bounds given by the previously mentioned results have a dependence on 1/ε that is enormous. It would be interesting to characterize which problems have property testers whose dependence on 1/ε is polynomial (cf. [1]). There are many interesting properties that are not easily testable, but do have sublinear time property testers. For example, the graph isomorphism problem asks whether two graphs are identical under relabeling of the nodes. In [21], it is shown

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that the property testing problem requires (n) queries and that there is a property tester for this problem which uses O(n5/4 polylog n) queries, which is sublinear in the input size n2 . The area of property testing has been very active, with a number of property testers devised for other models of graphs as well as other combinatorial objects. The testability of a problem is very sensitive to the model in which it is being tested. In contrast to the dense graph model, where k-colorability is easily testable, it is known that there are no sublinear time property testers for the k-colorability problem in models suitable for testing sparse graphs [15]. Property testers have also been studied in models of general graphs, and threshold-like behaviors have been found for the complexity of the testing problems in terms of the average degree of the graph [28], [4]. Property testers for combinatorial properties of matrices, strings, metric spaces and geometric objects have been devised. We refer the reader to the excellent surveys of Goldreich [22], Ron [35] and Fischer [19].

6. Testing properties of distributions In a wide variety of settings, data is most naturally viewed as coming from a probability distribution. In order to effectively make use of such data, one must understand various properties of the underlying probability distribution. Some of these properties are “local” in nature, for example focusing on whether or not a specific domain element appears with large probability. Other properties have a rather “global” feel in the sense that they are a property of the distribution as a whole and not of a small subset of the domain elements. Unlike the case for local properties, it makes sense to characterize a distribution in terms of some meaningful distance measure to the closest distribution that has the global property. This yields a somewhat different model than the property testing model in terms of the assumption on how the data is presented for here we do not assume that an explicit description of the distribution is given. In the following, we assume that there is an underlying distribution from which the testing algorithm receives independent identically distributed (iid) samples. The complexity of the algorithm is measured in terms of the number of samples required in order to produce a meaningful answer (the sample complexity). As mentioned in the introduction, it is a classical result from the theory of sampling that one can, in time independent of the size of the data, determine a good estimate of the average value of a list of numbers of bounded magnitude. However, more recently, properties such as closeness between two distributions, closeness to an explicitly given distribution, independence, and high entropy, have been studied in this model [24], [10], [9]. For many properties, well-known statistical techniques, such as the χ 2 -test or the straightforward use of Chernoff bounds, have sample complexities that are at least linear in the size of the support of the underlying probability distribution. In contrast, there are algorithms whose sample complexity is sublinear in the size of the support

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for various properties of distributions. We mention one illustrative example: Given samples of a distribution X on [n], for example all the previous winners of the lottery, how can one tell whether X is close to uniform? We will measure closeness in terms of the L2 norm, i.e., letting X(i) denote the probability that X assigns to i, X2 = (X(i) − 1/n)2 . i∈[n]

Goldreich and Ron [24] note that since (X(i) − 1/n)2 = X(i)2 − 1/n, i∈[n]

i∈[n]

it is enough to estimate the √ collision probability. They then show that this can be done by considering only O( n) samples and counting the number of pairs that are the same. By bounding the variance of their estimator, they obtain the following: Theorem 6.1 ([24]). There is an algorithm that, given a distribution X on [n] via a generation √ oracle, approximates X2 to within a multiplicative factor of (1 ± ε) using O( n/ε2 ) samples, with constant probability. Such techniques are very useful for achieving sublinear time algorithms for testing whether distributions satisfy several other global properties. For example, for the properties of closeness of two arbitrary distributions [10], independence of a joint distribution [9], high entropy [8], and monotonicity of the probability density function (when the distribution is over a totally ordered domain) [11], the testing problem can be reduced to the problem of testing the near-uniformity of the distribution on various subdomains.

7. Some final comments We have seen several contexts in which one can test properties in sublinear time. The study of sublinear time algorithms has led to a new understanding of many problems that had already been well-studied. Though we have mentioned only property testing problems in this survey, other, more traditional, types of approximations are achievable in sublinear time. Such algorithms have been used to design very fast approximation algorithms for graph problems and for string compressibility problems (cf. [23], [10], [33]). Some of these algorithms have even resulted in better linear time approximation algorithms than what was previously known. Probabilistically Checkable Proof Systems (PCPs) can be thought of as a way to write down a proof so that another person can verify it by viewing only a constant number of locations (cf. [6]). PCPs can thus be viewed as a type of robust characterization and their verification is a sublinear algorithm. More interestingly, property

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testers for homomorphisms and low degree polynomials are used as key ingredients in the construction of Probabilistically Checkable Proof Systems. As we have seen, the study of sublinear algorithms gives a new perspective that has yielded insights to other areas of theoretical computer science. Much still remains to be understood about the scope of sublinear time algorithms, and we expect that this understanding will lead to further insights. Acknowledgements. We thank Ran Canetti, Tali Kaufman, and Madhu Sudan for their extensive and helpful comments on this manuscript.

References [1] Alon, N., Testing subgraphs in large graphs. In Proceedings of the 46th Symposium on Foundations of Computer Science, IEEE Computer Soc. Press, Los Alamitos, CA, 2001, 434–441. [2] Alon, N., Fischer, E., Krivelevich, M., and Szegedy, M., Efficient testing of large graphs. Combinatorica 20 (2000), 451–476. [3] Alon, N., Kaufman, T., Krivelevich, M., Litsyn, S., and Ron, D., Testing low-degree polynomials over GF(2). In Approximation, randomization, and combinatorial optimization, Lecture Notes in Comput. Sci. 2764, Springer-Verlag, Berlin 2003, 188–199. [4] Alon, N., Kaufman, T., Krivelevich, M., Ron, D., Testing triangle-freeness in general graphs. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM Press, New York 2006, 279–288. [5] Alon, N., and Shapira, A., A characterization of the (natural) graph properties testable with one-sided error. In Proceedings of the 46th Annual Symposium on Foundations of Computer Science, IEEE Computer Soc. Press, Los Alamitos, CA, 2005, 429–438. [6] Arora, S., Lund, C., Motwani, R., Sudan, M., and Szegedy, M., Proof verification and the hardness of approximation problems. J. ACM 45 (3) (1998), 501–555. [7] Arora, S., and Sudan, M., Improved low degree testing and its applications. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computing, ACM Press, New York 1997, 485–495. [8] Batu, T., Dasgupta, S., Kumar, R., and Rubinfeld, R., The complexity of approximating the entropy. In Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, ACM Press, New York 2002, 678–687. [9] Batu, T., Fischer, E., Fortnow, L., Kumar, R., Rubinfeld, R., and White, P., Testing random variables for independence and identity. In Proceedings of the 42nd Conference on Foundations of Computer Science, IEEE Computer Soc. Press, Los Alamitos, CA, 2001, 442–451. [10] Batu, T., Fortnow, L., Rubinfeld, R., Smith, W., and White, P., Testing that distributions are close. In Proceedings of the 41st Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 2000, 259–269. [11] Batu, T., Kumar, R., and Rubinfeld, R., Sublinear algorithms for testing monotone and unimodal distributions. In Proceedings of the 36th Annual ACM Symposium on the Theory of Computing, ACM Press, New York 2004, 381–390.


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[12] Bellare, M., Coppersmith, D., Håstad, J., Kiwi, M., and M. Sudan, M., Linearity testing over characteristic two. IEEE Trans. Inform. Theory 42 (6) (1996), 1781–1795. [13] Ben-Or, M., Coppersmith, D., Luby, M., and Rubinfeld, R., Non-abelian homomorphism testing, and distributions close to their self-convolutions. In Approximation, randomization, and combinatorial optimization. Algorithms and techniques, Lecture Notes in Comput. Sci. 3122, Springer-Verlag, Berlin 2004, 273–285. [14] Blum, M., Luby, M., and Rubinfeld, R., Self-testing/correcting with applications to numerical problems. J. Comput. System Sci. 47 (1993), 549–595. [15] Bogdanov, A., Obata, K., and Trevisan, L., A lower bound for testing 3-colorability in bounded-degree graphs. In Proceedings of the 43rd Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 2002, 93–102. [16] Coppersmith, D., Manuscript, 1989. [17] Ergün, F., Kannan, S., Kumar, S. R., Rubinfeld, R., and Viswanathan, M., Spot-checkers. J. Comput. System Sci. 60 (3) (2000), 717–751. [18] Ergün, F., Kumar, R., and Rubinfeld, R., Checking approximate computations of polynomials and functional equations. SIAM J. Comput 31 (2) (2001), 550–576. [19] Fischer, E., The art of uninformed decisions: A primer to property testing. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 75 (2001), 97–126. [20] Fischer, E., On the strength of comparisons in property testing. In Electronic Colloquium on Computational Complexity 8 (20), 2001. [21] Fischer, E., and Matsliah,A., Testing graph isomorphism. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, ACM Press, New York 2006, 299–308. [22] Goldreich, O., Combinatorial property testing - a survey. In Randomization Methods in Algorithm Design, DIMACS Ser. Discrete Math. Theoret. Comput. Sci. 43, Amer. Math. Soc., Providence, RI, 1999, 45–60. [23] Goldreich, O., Goldwasser, S., and Ron, D., Property testing and its connection to learning and approximation. J. ACM 45 (4) (1998), 653–750. [24] Goldreich, O., and Ron, D., On testing expansion in bounded-degree graphs. Electronic Colloquium on Computational Complexity 7 (20), 2000. [25] Goldreich, O., and Ron, D., Property testing in bounded degree graphs. Algorithmica 32 (2002), 302–343. [26] Goldreich, O., and Trevisan, L., Three theorems regarding testing graph properties. In Proceedings of the 42nd Conference on Foundations of Computer Science, IEEE Computer Soc. Press, Los Alamitos, CA, 2001, 460–469. [27] Jutla, C. S., Patthak, A. C., Rudra, A., and Zuckerman, D., Testing low-degree polynomials over prime fields. In Proceedings of the 45th Annual Symposium on Foundations of Computer Science, IEEE Computer Soc. Press, Los Alamitos, CA, 2004, 423–432. [28] Kaufman, T., Krivelevich, M., Ron, D., Tight bounds for testing bipartiteness in general graphs. SIAM J. Comput. 33 (2004), 1441–1483. [29] Kaufman, T., and Ron, D., Testing polynomials over general fields. In Proceedings of the 45th Annual Symposium on Foundations of Computer Science, IEEE Computer Soc. Press, Los Alamitos, CA, 2004, 413–422. [30] Kiwi, M., Magniez, F., and Santha, M., Approximate testing with error relative to input size. J. Comput. System Sci. 66 (2) (2003), 371–392.

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[31] Parnas, M., and Ron, D., Testing the diameter of graphs. Random Structures Algorithms 20 (2) (2002), 165–183. [32] Polischuk, A., and Spielman, D., Nearly linear-size holographic proofs. In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, ACM Press, New York 1994, 194–203. [33] Raskhodnikova, S., Ron, D., Rubinfeld, R., Shpilka, A., and Smith, A., Sublinear algorithms for string compressibility and the distribution support size. Electronic Colloquium on Computational Complexity 5 (125), 2005. [34] Raz, R., and Safra, S., A sub-constant error-probability low-degree test, and a sub-constant error-probability PCP characterization of NP. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computing, ACM Press, New York 1997, 475–484. [35] Ron, D., Property testing. In Handbook of randomized computing, Vol. II, Comb. Optim., Kluwer Acad. Publ., Dordrecht 2001, 597–649. [36] Rubinfeld, R., On the robustness of functional equations. SIAM J. Comput 28 (6) (1999), 1972–1997. [37] Rubinfeld, R., and Sudan, M., Robust characterization of polynomials with applications to program testing. SIAM J. Comput. 25 (2) (1996), 252–271. Computer Science and Artificial Intelligence Laboratory, MIT, Cambridge, MA 02139, U.S.A. E-mail: [email protected]

Pseudorandomness and combinatorial constructions Luca Trevisan∗

Abstract. In combinatorics, the probabilistic method is a very powerful tool to prove the existence of combinatorial objects with interesting and useful properties. Explicit constructions of objects with such properties are often very difficult, or unknown. In computer science, probabilistic algorithms are sometimes simpler and more efficient than the best known deterministic algorithms for the same problem. Despite this evidence for the power of random choices, the computational theory of pseudorandomness shows that, under certain complexity-theoretic assumptions, every probabilistic algorithm has an efficient deterministic simulation and a large class of applications of the probabilistic method can be converted into explicit constructions. In this survey paper we describe connections between the conditional “derandomization” results of the computational theory of pseudorandomness and unconditional explicit constructions of certain combinatorial objects such as error-correcting codes and “randomness extractors.” Mathematics Subject Classification (2000). Primary 68Q10; Secondary 05D40. Keywords. Computational complexity, pseudorandomness, derandomization, randomness extraction.

1. Introduction 1.1. The probabilistic method in combinatorics. In extremal combinatorics, the probabilistic method is the following approach to proving existence of objects with certain properties: prove that a random object has the property with positive probability. This simple idea has been amazingly successful, and it gives the best known bounds for most problems in extremal combinatorics. The idea was introduced (and, later, greatly developed) by Paul Erd˝os [18], who originally applied it to the following question: define R(k, k) to be the minimum value n such that every graph on n vertices has either an independent set of size at least k or a clique of size at least k.1 It was known that R(k, k) is finite and that it is at most 4k , and the question was to prove a lower bound. Erd˝os proved that a random graph with 2k/2 vertices has a positive probability of having no clique and no independent set larger than k, and so ∗ The author is supported in part by NSF grant CCF 0515231. 1 Here by “graph” we mean an undirected graph, that is, a pair G = (V , E) where V is a finite set of vertices and E is a set of pairs of elements of E, called edges. A clique in a graph G = (V , E) is a set C ⊆ V of vertices such that {u, v} ∈ E for every two vertices u, v ∈ C. An independent set is a set I ⊆ V of vertices such that {u, v} ∈ E for every two vertices u, v ∈ I .


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R(k, k) ≥ 2k/2 . The method, of course, gives no indication of how to actually construct a large graph with no small clique and no small independent set. Remarkably, in the past 60 years, there has been no asymptotic improvement to Erd˝os’s lower bound and, perhaps more significantly, the best explicit construction of a graph without a clique of size k and without an independent set of size k has only about k log k vertices [20], a bound that has not been improved in 25 years. Shannon [59] independently applied the same idea to prove the existence of encoding schemes that can optimally correct from errors in a noisy channel and optimally compress data. The entire field of information theory arose from the challenge of turning Shannon’s non-constructive results into algorithmic encoding and decoding schemes. We will return to the problem of encodings for noisy channels in Section 2. Around the same time, Shannon [60] applied the probabilistic method to prove the existence of boolean functions of exponential circuit complexity (see Section 4). Proving that certain specific boolean functions (for example, satisfiability of boolean formulae or 3-colorability of graphs) require exponential size circuits is a fundamental open problem in computational complexity theory, and little progress has been made so far. The probabilistic method has found countless applications in the past 60 years, and many of them are surveyed in the famous book by Alon and Spencer [4]. For most problems in extremal combinatorics, as well as in information theory and complexity theory, probabilistic methods give the best known bound, and explicit constructions either give much worse bounds, or they give comparable bounds but at the cost of technical tours de force. 1.2. Probabilistic methods in computer science. In computer science, an important discovery of the late 1970s was the power of probabilistic algorithms. The most famous (and useful) of such algorithms are probably the polynomial time probabilistic algorithms for testing primality of integers [65], [47]. In these algorithms one looks for a “certificate” that a given number n is composite; such a certificate could be for example an integer a such that a n ≡ a (mod n), or four distinct square roots (mod n) of the same integer. Rabin, Solovay and Strassen [65], [47] proved that there is a good chance of finding such certificates just by picking them at random, even though no efficient method to deterministically construct them was known.2 Two other important and influential algorithms were discovered around the same time: an algorithm to test if two implicitly represented multivariate polynomials are identical [85], [56] (evaluate them at a random point chosen from a domain larger than the degree) and an algorithm to check if two vertices in a graph are connected by a path [2] (start a random walk at the first vertex, and see if the second vertex is reached after a bounded number of steps).3 2 Note the similarity with the probabilistic method. 3 The algorithm of Aleliunas et al. [2] broke new grounds in terms of memory use, not running time. It was already known that the Depth-First-Search algorithm could be used to solve the problem using linear time and a linear amount of memory. The random walk algorithm, however, needs only O(log |V |) bits of memory, and

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A different type of probabilistic algorithms was developed starting in the late 1980s with the work of Sinclair and Jerrum [61]. These algorithms solve approximate “counting” problems, where one wants to know the number of solutions that satisfy a given set of combinatorial constraints. For example, given a bipartite graph, one would like to know, at least approximately, how many perfect matchings there are.4 Sinclair and Jerrum introduced an approach based on a reduction to the problem of approximately sampling from the uniform distribution of all possible solutions. Since the latter problem involves randomness in its very definition, this approach inevitably leads to probabilistic algorithms. 1.3. The computational theory of pseudorandomness. In light of such algorithmic results, it was initially conjectured that probabilistic algorithms are strictly more powerful than deterministic ones and that, for example, there exist problems that can be solved probabilistically in polynomial time but that cannot be solved in polynomial time using deterministic algorithms. This belief has been overturned by developments in the computational theory of pseudorandomness. The theory was initiated by Blum [9], Goldwasser and Micali [22], and Yao [84], with the motivation of providing sound foundations for cryptography. From the very beginning, Yao [84] realized that the theory also provides conditional derandomization results, that is, theorems of the form “if assumption X is true, then every problem that can be solved by a probabilistic polynomial time algorithm can also be solved by a deterministic algorithm of running time Y.” Yao showed that we can take X to be “there is no polynomial time algorithm that on ε input a random integer finds its prime factorization”5 and Y to be “time 2n for every ε > 0.” An important project in complexity theory in the 1980s and 1990s was to strengthen Y to be “polynomial time” with a plausible X. The goal was achieved 1997 in a landmark paper by Impagliazzo and Wigderson [31], building on a considerable body of previous work. At a very high level, the Impagliazzo–Wigderson result is proved in two steps. The first step (which is the new contribution of [31]) is the proof that an assumption about the worst-case complexity of certain problems implies a seemingly stronger assumption about the average-case complexity of those problems. A result of this kind is called an amplification of hardness result, because it “amplifies” a worst-case hardness assumption to an average-case one. The second step, already established exponential improvement. 4A bipartite graph is a triple G = (U, V , E) where U, V are disjoint sets of vertices and E ⊆ U × V is a set of edges. A perfect matching is a subset M ⊆ E such that for every u ∈ U there is precisely one v ∈ V such that (u, v) ∈ M, and vice versa. 5 More generally, Yao showed that X can be “one-way permutations exist,” see 5 for more details. The assumption about integer factorization implies the existence of one-way permutations, provided that we restrict ourselves to “Blum integers.”

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ten years earlier by Nisan and Wigderson [44], is the proof that the average-case assumption suffices to construct a certain very strong pseudorandom generator, and that the pseudorandom generator suffices to simulate deterministically in polynomial time every polynomial time probabilistic algorithm. In conclusion, assuming the truth of a plausible complexity-theoretic assumption, every polynomial time probabilistic algorithm can be “derandomized,” including the approximation algorithms based on the method of Sinclair and Jerrum. Furthermore, under the same assumption, a large class of applications of the probabilistic method in combinatorics can be turned into explicit constructions. We give some more details about the Impagliazzo–Wigderson Theorem in Section 6. The reader is also referred to the excellent survey paper of Impagliazzo [28] in the proceedings of the last ICM. It is remarkable that many of the “predictions” coming from this theory have been recently validated unconditionally: Agrawal et al. [1] have developed a deterministic polynomial time algorithm for testing primality and Reingold [53] has developed a deterministic O(log n) memory algorithm for undirected graph connectivity. One can read about such developments elsewhere in these proceedings. Here is an example of a question that is still open and that has a positive answer under the complexity-theoretic assumption used in the Impagliazzo–Wigderson work: • Is there a deterministic algorithm that, on input an integer n, runs in time polynomial in log n and return a prime between n and 2n? 1.4. When randomness is necessary. Suppose that, in a distant future, someone proves the assumption used in the Impagliazzo–Wigderson work, so that we finally have an unconditional polynomial time derandomization of all probabilistic algorithms. Would this be the end of the use of randomness in computer science? The answer is no, for at least two reasons. One reason is that such derandomization would probably not be practical. At a broad qualitative level, we consider polynomial-time algorithms as “efficient” and super-polynomial-time algorithms as “inefficient,” and then such a result would establish the deep fact that “efficient” probabilistic algorithms and “efficient” deterministic algorithms have the same power. If the derandomization, however, causes a considerable (albeit polynomial) slow-down, and if it turns a practical probabilistic algorithm into an impractical deterministic one, then the probabilistic algorithm will remain the best choice in applications. A more fundamental reason is that there are several applications in computer science where the use of randomness is unavoidable. For example, consider the task of designing a secure cryptographic protocol in a setting where all parties behave deterministically. These observations lead us to consider the problem of generating randomness to be used in probabilistic algorithms, cryptographic protocols, and so on. Such generation begins by measuring a physical phenomenon that is assumed to be unpredictable (such as a sequence of physical coin flips) and that will be called a random source in


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the following. Typically, one has access to random sources of very poor quality, and converting such measurements into a sequence of independent and unbiased random bits is a difficult problem. In Section 8 we discuss various approaches and impossibility results about this problem, leading to the definition of seeded randomness extractor due to Nisan and Zuckerman [45], [86]. Seeded randomness extractors have an amazing number of applications in computer science, often completely unrelated to the original motivation of extracting random bits from physical sources. They are related to hash functions, to pseudorandom graphs, to error-correcting codes and they are useful in complexity theory to prove, among other things, negative results for the approximability of optimization problems. Ironically, the problem of generating high-quality random bits for cryptographic application is not satisfactorily solved by seeded randomness extractors (even though it was the original motivation for the research program that led to their definition). Seedless randomness extractors are needed for such application, and their theory is still being developed. 1.5. Connections. So far, we have discussed (i) the power of probabilistic methods, (ii) the conditional results proving that all probabilistic algorithms have a polynomialtime derandomization under complexity assumption, and (iii) the use of seeded randomness extractors to unconditionally run probabilistic algorithms in a setting in which only a weak source of randomness is available. In Section 8 we describe a recently discovered connection between (ii) and (iii) and, more generally, between conditional results proved in the computational theory of pseudorandomness and unconditional explicit constructions of combinatorial objects. One connection is between error-correcting codes and “hardness amplification” results. This connection has led to the application of coding-theoretic techniques in the study of average-case complexity. It is also possible to use complexity-theoretic techniques to build error-correcting codes, but so far this approach has not been competitive with previously known coding-theoretic techniques. The second connection is between pseudorandom generators and seeded randomness extractors. This connection has led to improvements in both settings. Various impossibility results are known for error-correcting codes and randomness extractors. Via these connections, they imply impossibility results for hardness amplification and conditional derandomization. In Section 7 we discuss approaches to sidestep these negative results.

2. Pseudorandom objects: codes and graphs In this section we introduce two examples of very useful combinatorial objects whose existence easily follows from the probabilistic method: error-correcting codes and expander graphs. Explicit constructions of such objects are also known.

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2.1. Error-correcting codes. Consider the process of picking a random set S ⊆ {0, 1}n of size 2k , k < n. If, say, k = n/2, then it is easy to show that there is an absolute constant δ > 0 such that, with high probability, every two elements u, v ∈ S differ in at least δn coordinates. By a more careful estimate, we can also see that there is an absolute constant c such that, for every ε > 0, it is likely that every two elements of S differ in at least (1/2 − ε)n coordinates with high probability, provided k ≤ cε2 n. For reasons that will be clear shortly, let us change our perspective slightly, and consider the (equivalent) process of picking a random injective function C : {0, 1}k → {0, 1}n : clearly the same bounds apply. For two strings u, v ∈ {0, 1}n , the Hamming distance between u and v (denoted dH (u, v)) is the number of coordinates where u and v differ, that is dH (u, v) := |{i : ui = vi }|.

(1)

Definition 2.1 (Error-correcting code). We say that C : {0, 1}k → {0, 1}n is an (n, k, d)-code if dH (C(x), C(y)) ≥ d for every two distinct x, y ∈ C. This concept is due to Hamming [25]. Error-correcting codes are motivated by the following scenario. Suppose we, the sender, have a k-bit message M ∈ {0, 1}k that we want to transmit to a receiver using an unreliable channel that introduces errors, and suppose we have an (n, k, d)-code C. Then we can compute c = C(M) and transmit c over the channel. The receiver gets a string c , which is a corrupted version of c, and looks for the message M that minimizes dH (C(M ), c ). If the channel introduces fewer than d/2 errors, then the receiver correctly reconstructs M.6 Keeping this application in mind, for every given k, we would like to construct (n, k, d)-codes where d is as large as possible (because then the receiver can tolerate more errors) and n is as small as possible (so that we do not have to communicate a very long encoding). Furthermore, we would like C and the decoding procedure run by the receiver to be computable by efficient algorithms. One trade-off between the parameters is that dn ≤ 21 + ok (1). Keeping in mind that the number of errors that can be corrected is at most d/2, this means that the receiver can correctly reconstruct the message only if the number of errors is at most 1 4 + ok (1) n. It is possible to do better if we are willing to settle for the notion of “listdecodability,” introduced by Elias [17]. Definition 2.2 (List-decodable code). We say that C : {0, 1}k → {0, 1}n is (L, δ)-list decodable if for every u ∈ {0, 1}n , |{x ∈ {0, 1}k : dH (C(x), u) ≤ δn}| ≤ L. 6 It is also easy to see that this analysis is tight. If there are two messages M, M such that d (C(M), C(M )) = H d and we send M, then it is possible that even a channel that introduces only d/2 errors can fool the receiver into thinking that we sent M .


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Here the idea is that we send, as before, the encoding C(M) of a message M. The receiver gets a string u and computes the list of all possible messages M such that dH (C(x), u) ≤ δn. If C is an (L, δ)-code, then the list is guaranteed to be of length at most L, and if the channel introduces at most δn errors then our message is guaranteed to be in the list. Using the probabilistic method, it is easy to show the existence of (L, 1/2 − ε)-list decodable codes C : {0, 1}k → {0, 1}n for every k and ε, where n = O(kε−2 ) and L = O(ε −2 ). It was also known how to define efficiently encodable codes with good (but not optimal) parameters. It took, however, 40 years until Sudan [66] defined the first efficient list-decoding algorithm for such codes. Sudan’s algorithm suffices to define (ε−O(1) , 1/2 − ε)-list decodable codes C : {0, 1}k → {0, 1}n with n = (k/ε)O(1) for every k, ε, and the codes are encodable and list-decodable in time polynomial in n. This means that even if the channel introduces close to n/2 errors, it is still possible for the receiver to gain considerable information about the message. (Namely, the fact that the message is one out of a small list of possibilities.) Other list-decoding algorithms are now known, but they are beyond the scope of this survey. See Sudan’s survey [67], Guruswami’s thesis [23] and two recent breakthrough papers [24], [46]. 2.2. Expander graphs. Consider the process of picking at random a graph according to the Gn, 1 distribution.(The Gn, 1 distribution is the uniform distribution over the 2 2 n shows that for every two disset of 2(2) graphs over n vertices.) A simple calculation 1 joint sets of vertices A, B there are 2 ± on (1) |A||B| edges with one endpoint in A and one endpoint in B. Chung, Graham and Wilson [15] call a family of graphs satisfying the above properties a family of quasi-random graphs, and prove that six alternative definitions of quasi-randomness are all equivalent. Explicit constructions of quasi-random graphs are known, and the notion has several applications in combinatorics. (See the recent survey paper by Krivelevich and Sudakov [37].) Consider now a process where we randomly generate an n-vertex graph where every vertex has degree at most d (think of d as a fixed constant and n as a parameter). For example, consider the process of picking d perfect matchings and then taking their union. Then it is possible to show that for every two disjoint sets of vertices A, B there are edges with one endpoint in A and one endpoint in B. (Families (1 ± on,d (1))d |A||B| n of) graphs with this property are called expanders, and they have several applications in computer science. To gain a sense of their usefulness, imagine that an expander models a communication network and note that if o(dn) edges are deleted, the graph still has a connected component with (1 − o(1))n vertices. Furthermore, expander graphs have several other interesting properties: they have small diameter, it is possible to find several short edge-disjoint paths between any two vertices, and so on. There are other possible definitions of expanders, which are related but not equivalent. In one possible (and very useful) definition, expansion is measured in terms of the eigenvalue gap of the adjacency matrix of the graph (see e.g. the discussion in [37]). For this definition, Lubotzky, Phillips and Sarnak [41] provide an optimal explicit construction. Another possible measure is the edge expansion of the graph. Optimal

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explicit constructions for this measure are not known, but considerable progress is made in [13], [3].

3. Randomness extractor Randomness extractors are procedures originally designed to solve the problem of generating truly random bits. As we will see, randomness extractors can be seen as a sort of pseudorandom graphs, they can be constructed using techniques from the field of pseudorandomness, and they are tightly related to constructions error-correcting codes, expanders and other random-like combinatorial objects. 3.1. Generating random bits. In order to generate random bits in practice, one starts by measuring a physical phenomenon that is assumed to contain randomness.7 For example, in many computer systems one starts by collecting statistics on the user’s keystrokes or mouse movement, or on the latency time of disk access, and so on. This raw data, which is assumed to contain some amount of entropy, is then passed to a “hash function,” and the output of the function is assumed to be a sequence of truly random bits. Such systems, widely used in practice, are typically not validated by any rigorous analysis. In a mathematical modeling of this situation, we have a random variable X representing our physical measurement, ranging, say, over {0, 1}n . We would like to construct a function Ext : {0, 1}n → {0, 1}m such that, by making as little assumptions on X as possible, we can prove that Ext(X) is distributed uniformly over {0, 1}m , or at least it is approximately so. Von Neumann [82] studied a version of this problem where X is a sequence of independent and identically distributed biased coin tosses. The independence assumption is crucially used. The general problem was extensively studied in computer science in the 1980s [55], [80], [79], [78], [14], [16]. Notably, the goal was to define a single function Ext that would work for as large as possible a class of distributions X. An early conclusion was that the extraction problem is impossible [55], as defined above, even if just very weak forms of dependencies between different bits are allowed in the distribution of X. Two approaches have been considered to circumvent this impossibility. 1. One approach is to consider a model with a small number of mutually independent random variables X1 , . . . , Xk , each satisfying weak randomness requirements. This line of work, initiated in [55], [80], [14], saw no progress for a long time, until recent work by Barak et al. [7] made such progress possible by a breakthrough in additive combinatorics [12], [36]. This is now a very active area of research [8], [53], [11], [87], [48] with connections to other areas of combinatorics. 7 We will not get into the physical and philosophical problems raised by such assumption.

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2. The other approach, initiated in [79], is to stick to the model of a single sample X and to consider the following question: suppose we have a randomized algorithm A (that is correct with high probability given the ability to make truly random choices) and suppose we have an input x. Can we efficiently find what is the most probable output of A(x)? 3.2. The definition of randomness extractors. To formalize approach (2) it is convenient to think of a probabilistic algorithm A( · , · ) as having two inputs: a “random” input r and a “regular” input I . We say that “A computes a function f with high probability” if, for every I , P[A(r, I ) = f (I )] ≥ .9 where the probability is taken with respect to the uniform distribution over bit strings r of the proper length.8 Let Un denote a random variable uniformly distributed over {0, 1}n . Suppose that our algorithm A requires m truly random bits to process a given input x. Furthermore, suppose that we can define a function Ext : {0, 1}n × {0, 1}d → {0, 1}m such that if X is our physical source and Ud is uniformly distributed over {0, 1}d then Ext(X, Ud ) is uniformly distributed over {0, 1}m . Here is a way to simulate A( · ) using X: (i) get a sample x ∼ X, (ii) for every s ∈ {0, 1}d , compute as := A(Ext(x, s), I ), (iii) output the most common value among the as . It is now easy to show that the above algorithm computes f (I ) with probability at least .8, over the choice of X. This is because P[A(Ext(Ud , X), I ) = f (I )] ≥ .9 and so

P P [A(Ext(Ud , X), I ) = f (I )] >

X Ud

1 2

≥ .8.

(2)

The running time of our simulation of A is 2d times the running time of A, which is polynomial in the running time of A provided that d is logarithmic. For this reasoning to work it is not necessary that Ext(X, Ud ) be distributed exactly uniformly, but it is enough if it approximates the uniform distribution in an appropriate technical sense. If X and Y are two random variables taking values in , then we define their variational distance (also called statistical distance) as X − Y SD := max | P[X ∈ T ] − P[Y ∈ T ]|. T ⊆

(3)

We will sometimes call sets T ⊆ statistical tests. If X − Y SD ≤ ε then we say that X is ε-close to Y . We say that Ext : {0, 1}n → {0, 1}d → {0, 1}m is a seeded extractor for a distribution X with error parameter ε if Ext(X, Ud ) is ε-close to Um . 8 The reader may find .9 to be a poor formalization of the notion of “with high probability,” but it is easy to reduce the error probability at the cost of a moderate increase of the running time.

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Vazirani and Vazirani [79] provided extractors for a certain class of distributions. (Their terminology was different.) Zuckerman [86] was the first to show that extractors exist for a very general class of distributions. Define the min-entropy of X as 1 . If H∞ (X) ≥ k, then we say that X is a k-source. H∞ (X) := mina log2 P[X=a] Definition 3.1. A function Ext : {0, 1}n → {0, 1}d → {0, 1}m is a (k, ε) seeded extractor if Ext(X, Ud ) is ε-close to Um for every k-source X. The definition is implicit in [86]. The term extractor was coined in [45]. The term “seeded” refer to the truly random input of length d, which is called a seed. From now, we will refer to seeded extractors as simply “extractors.” Let Ext : {0, 1}n × {0, 1}d → {0, 1}m be a (k, ε)-extractor. Construct a bipartite graph G = ([N], [M], E) with N = 2n vertices on the left, M = 2m vertices on the right. Connect two vertices u, v if there is an s that v = Ext(u, s). Then if we pick any subset S ⊆ [N] on the left and any subset T ⊆ [M] on the right, the number of edges is |S| · 2d · |T |/2m plus or minus ε|S|2d , provided |S| ≥ 2k . This is similar to one of the definitions of expander. Zuckerman and Wigderson [83] prove that one can derive expanders with very strong “edge expansion” from extractors. Radakrishnan and Ta-Shma show that, in every extractor, d ≥ log(n − k) + 2 log(1/ε) − O(1) and that m ≤ k + d − O(1). Non-constructively, one can show that such bounds are achievable up to the additive constant factor, but explicit constructions are difficult. We will discuss explicit constructions later. 3.3. Applications. Randomness extractors have several applications, some of which are described below. See the tutorial by Salil Vadhan [76] and the survey by Ronen Shaltiel [57] for more examples and a broader discussion. Simulation of randomized algorithms. Suppose we have a randomized algorithm A that on input I computes f (I ) with probability, say, .9, and suppose that Ext is a (k , 1/4)-extractor and that X is a k-source. As before, let us sample x ∼ X and compute A(Ext(x, s), I ) for every s and output the majority value. Let B be the

set of x such that the algorithm fails. If |B| ≥ 2k , then consider a random variable Y uniformly distributed over B. It has entropy k, so Ext(Y, Ud ) should be 1/4-close to uniform. Consider the statistical test T defined as T := {r : A(r, I )} = f (I )}.

(4)

Then P[Un ∈ T ] ≥ .9 by assumption and P[Ext(Y, Ud ) ∈ T ] ≤ 1/2 by construction.

This would contradict Ext being an extractor. We then conclude that |B| ≤ 2k , and

so the probability that our algorithm fails is at most P[X ∈ B] ≤ |B|/2k ≤ 2k −k . This is very useful even in a setting in which we assume access to a perfect random source. In such a case, by using n truly random bits we achieve an error probability

that is only 2k −n . Note that, in order to achieve the same error probability by running the algorithm several times independently we would have used O((n−k )·m) random bits instead of n.


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Other applications. Randomness extractors are also very useful in settings where we assume a fully random distribution, say, over n bits, that is unknown to us, except for some partial information of entropy at most n − k bits. Then the distribution of the unknown string conditioned on our knowledge still has entropy at least k. If an extractor is applied to the unknown string, then the output of the extractor will be uniformly distributed even conditioned on our knowledge. In other words, our knowledge in useless in gaining any information about the output of the extractor. This approach is used in the cryptographic settings of privacy amplification and everlasting security and in the design of pseudorandom generators for space-bounded algorithms. See [39], [77] and the references therein for the application to everlasting security and [45], [30], [51] for the application to pseudorandom generators.

4. Circuit complexity In order to discuss the computational approach to pseudorandomness we need to define a measure of efficiency for algorithms. We will informally talk about the “running time” of an algorithm on a given input without giving a specific definition. The reader can think of it as the number of elementary operations performed by an implementation of the algorithm on a computer. A more formal definition would be the number of steps in a Turing machine implementation of the algorithm. (See e.g. [64] for a definition of Turing machine.) We say that a set L ⊆ {0, 1}∗ is decidable in time t (n) if there is an algorithm that on input x ∈ {0, 1}n decides in time ≤ t (n) whether x ∈ L. We are also interested in a more “concrete” measure of complexity, called circuit complexity. For integers n and i ≤ n, define the set Pi,n := {(a1 , . . . , an ) ∈ {0, 1}n : ai = 1}. We say that a set S ⊆ {0, 1}n has a circuit of size K if there is a sequence of sets S1 , . . . , SK such that: (i) SK = S and (ii) each Sj is either a set Pi,n , or it is the complement of a set Sh , h < j , or it is the union Sh ∪ S of two sets, with h, < j or it is the intersection Sh ∩ S of two sets, with h, k < j . We say that a function f : {0, 1}n → {0, 1} has a circuit of size K if it is the characteristic function of a set that has a circuit of size K. The circuit complexity of a set S is the minimum K such that S has a circuit of size K. (Similarly for boolean functions.) It is easy to see that there are subsets of {0, 1}n whose circuit complexity is at least 2n c n for some constant c > 0: if a set has circuit complexity at most K, then it can be described by using only O(K log K) bits, and so there are 2O(K log K) sets of circuit n complexity at most K. If this number is less than 22 then there exists a set of circuit complexity larger than K. Indeed, by the same argument, a random set has circuit n complexity at least c 2n with very high probability. If L ⊆ {0, 1}∗ is a set decidable in time t (n), then for every n there is a circuit

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of size O((t (n))2 ) for L ∩ {0, 1}n . This implies that in order to prove lower bounds on the running time of algorithms for a given decision problem it is enough to prove lower bounds for the circuit complexity of finite fragments of it.9 So far there has been very little success in proving circuit complexity lower bounds for “explicit sets,” such as sets in NP. The strongest known lower bound is 5n [38], [32], and even an n log n lower bound is considered hopelessly out of reach of current techniques. This is perhaps surprising given the simplicity of the definition of circuit complexity. The definition looks like a finite version of the definition of complexity for Borel sets, and one may hope that one could transfer techniques from topology to this setting. Sipser describes this idea in [62], [63], but, unfortunately, so far it has not led to any lower bound for general circuits. Complexity theorists’ failure to prove strong circuit lower bounds is partly explained by a famous paper by Razborov and Rudich [52]. They describe a general class of approaches to lower bounds that they call “natural proofs.” Razborov and Rudich show that all known methods to prove lower bounds for restricted classes of circuits yield natural proofs, but that (under certain complexity-theoretic assumptions) natural proofs cannot prove lower bounds for general circuits. The complexity theoretic assumption is itself about circuit lower bounds, and it is used to construct certain pseudorandom generators. The pseudorandom generators, in turn, imply the impossibility result. Somewhat inaccurately, the Razborov–Rudich result can be summarized as: Circuit lower bounds are difficult to prove because they are true.

5. Pseudorandom generators and their application to derandomization Informally, a pseudorandom generator is an efficiently computable map G : {0, 1}t → {0, 1}m , where m is much bigger than t, such that, for a uniformly selected x ∈ {0, 1}t , the distribution G(x) is pseudorandom, that is, it “looks like” the uniform distribution over {0, 1}m . We begin by describing how to formalize the notion of a distribution “looking like” the uniform distribution, and, more generally, the notion of two distributions “looking like” one other. Recall that we use Un to denote a random variable that is uniformly distributed in {0, 1}n . Ideally, we would like to say that G( · ) is a good pseudorandom generator if G(Ut ) and Um are close in statistical distance. Then, as we already discussed in Section 3, every application in which m truly random bits are needed could be realized using the output of the generator (with a small increase in the probability of error). Unfortunately, this is too strong a definition: consider the statistical test T defined to be the set of all possible outputs of G. Then P[G(Ut ) ∈ T ] = 1 but P[Um ∈ T ] ≤ 2t−m . 9 The converse is not true: one can have undecidable sets of bounded circuit complexity.

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The great idea that came from the work of Blum, Goldwasser, Micali and Yao in 1982 ([9], [22], [84]) was to modify the notion of statistical distance by considering only efficiently computable statistical tests. Definition 5.1 (Computational indistinguishability). Two distributions μX and μY over {0, 1}m are (K, ε)-indistinguishable if for every set T ⊆ {0, 1}m of circuit complexity at most K, P [x ∈ T ] − P [y ∈ T ] ≤ ε. x∼μX

y∼μY

Definition 5.2 (Pseudorandomness). A distribution μX over {0, 1}m is (K, ε)-pseudorandom if it is (K, ε)-indistinguishable from the uniform distribution. That is, for every T ⊆ {0, 1}m of circuit complexity ≤ K, P [x ∈ T ] − |Tm| ≤ ε. 2 x∼μX

The following definition is due to Nisan and Wigderson [44]. Definition 5.3 (Quick pseudorandom generator). Suppose that for every n there is a Gn : {0, 1}t (n) → {0, 1}n that is (n2 , 1/n)-pseudorandom, and that there is an algorithm G that, given n, s, computes Gn (s) in time 2O(t (n) . Then G is called a t (n)-quick pseudorandom generator. Suppose that an O(log n)-quick pseudorandom generator (abbreviated logQPRG) exists, and suppose that f is a function and A is a polynomial time randomized algorithm that computes f with probability at least 3/4. We now describe a derandomization of the algorithm A. Let I be an input, and let m be the number of random bits used by A on input I . Let K be an efficiently computable upper bound for the circuit complexity pf T := {r : A(r, I ) = f (I )}. Choose n to be large enough so that: (i) n2 ≥ K, (ii) n ≥ m, and (iii) n ≥ 5. Because of our assumption that A runs in polynomial time, n is polynomial in the length of I .10 Now compute A(Gn (s), I ) for each s, and output the value that is returned most often. This completes the description of a polynomial time deterministic algorithm. Regarding correctness, we assumed P[A(Um , I ) = f (I )] ≥ 43 , and so P[A(Gn (Ut (n)), I ) = f (I )] ≥

3 4

−

1 n

> 21 .

(5)

Otherwise, the set T = {r : A(r, I ) = f (I )} contradicts the pseudorandomness of Gn . Something similar can be done if A is only guaranteed to approximate f with high probability, for example if f (I ) is the number of perfect matchings in the graph 10 It should be noted that we may not know how to construct a circuit for T , because it seems that to construct such a circuit we need to know f (I ). In order to compute a polynomially bounded upper bound for the circuit complexity of T , however, we just need to find out how large the circuit for T is that we would be able to build if we knew f (I ).

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represented by I and A is the Jerrum–Sinclair–Vigoda probabilistic approximation algorithm for this problem [33]. The only difference is that we take the median of the outputs instead of the most common one. The applications of logQPRGs to the probabilistic method is as follows. Suppose that: • For every n, we have a set n of “objects of size n” (for example, graphs with n vertices and maximum degree d, where d is a fixed constant). It is convenient to assume the sets n to be disjoint. • We define P ⊆ n n to be the set of interesting objects that we would like to construct. (For example, expander graphs.) • Property P is computable in polynomial time. That is, there is an algorithm that, given n and x ∈ n , runs in time polynomial in n and determines whether x ∈ P. • The probabilistic method proves that such graphs exist and are “abundant.” That is, for every n, we define a probability distribution μn over n and we prove that Px∼n [x ∈ P ] ≥ 1/2. (The constant 1/2 is not important.) • The distributions μn are polynomial time samplable. That is, there is a probabilistic algorithm A that, given n, generates in time polynomial in n a sample from μn . This formalization captures the way the probabilistic method is typically used in practice, with the exception of the efficient computability of P , which sometimes is not true. (For example, in the problem of finding lower bounds for R(k, k).) Finally, suppose that a logQPRG exists. Given n here is how we construct an element in P ∩n . Let m be the number of random bits used by A to sample an element of μn , and let K be an upper bound for the size of a circuit for the set T := {r : A(n, r) ∈ P }. As before, we can use the assumption that A is computable in polynomial time and P is decidable in polynomial time to conclude that m and K are upper bounded by polynomials in n. Let N be large enough so that (i) N ≥ 3, (ii) N 2 ≥ K and (iii) N ≥ m. Then compute A(n, GN (N, s)) for every s, and let s0 be such that A(n, GN (N, s0 )) ∈ P . Such an s0 must exist, otherwise T contradicts the pseudorandomness of GN . Output A(n, GN (N, s0 )).

6. Conditional constructions of pseudorandom generators Blum and Micali [9] construct no(1) QPRGs, according to a slightly different definition, assuming a specific number-theoretic assumption. Yao [84] proves that the Blum–Micali definition is equivalent to a definition based on indistinguishability and


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constructs no(1) QPRGs under the more general assumption that one-way permutao(1) tions exist. Yao [84] also recognizes that no(1) QPRGs imply a 2n derandomization of every probabilistic algorithm. Blum, Micali and Yao do not use the parametrization that we adopted in the definition of quick pseudorandom generators. In the cryptographic applications that motivate their work, it is important that the generator be computable in time polynomial in the length of the output (rather than exponential in the length of the input), and, if m is the length of the output, one desires (S(m), ε(m))-pseudorandomness where S(m) and 1/ε(m) are super-polynomial in m. Their constructions satisfy these stronger requirements. Håstad et al. [26] show that the weaker assumption that one-way functions exist suffices to construct no(1) QPRGs. Their construction satisfies the stronger requirements of [9], [84]. We do not define one-way permutations and one-way functions here and we refer the interested reader to Goldreich’s monograph [21], the definitive treatment of these results. Nisan and Wigderson [44] introduced the definition of quick pseudorandom generator that we gave in the previous section and presented a new construction that works under considerably weaker assumptions than the existence of one-way functions.11 The Nisan–Wigderson construction also “scales” very well, and it gives more efficient QPRGs if one is willing to start from stronger assumptions. A sufficiently strong assumption implies optimal logQPRGs, and this is the only version of the Nisan–Wigderson results that we will discuss. We first need to define the notion of average-case circuit complexity. We say that a set S ⊆ {0, 1}n is (K, ε)-hard on average if for every set T computable by a circuit of size ≤ K we have P[1S (x) = 1T (x)] ≤ 21 + ε, where we use the notation 1S for the characteristic function of the set S. We say that a set L ⊆ {0, 1}∗ is (K(n), ε(n))-hard on average if, for every n, L ∩ {0, 1}n is (K(n), ε(n))-hard on average. Theorem 6.1 (Nisan and Wigderson [44]). Suppose there is a set L such that: (i) L can be decided in time 2O(n) and (ii) there is a constant δ > 0 such that L is (2δn , 2−δn )hard on average. Then a logQPRG exists. When Theorem 6.1 was announced in 1988, average-case complexity was much less understood than worst-case complexity and it was not even clear if the assumption used in the theorem was plausible. This motivated a long-term research program on average-case complexity. Building on work by Babai, Fortnow, Impagliazzo, Nisan and Wigderson [6], [27], Impagliazzo and Wigderson finally proved in 1997 that the assumption of Theorem 6.1 is equivalent to a seemingly weaker worst-case assumption. 11 On the other hand, the Nisan–Wigderson generator does not satisfy the stronger properties of the pseudorandom generators of Blum, Micali, Yao, Håstad et al. [9], [84], [26]. This is unavoidable because the existence of such stronger pseudorandom generators is equivalent to the existence of one-way functions.

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Theorem 6.2 (Impagliazzo and Wigderson [31]). Suppose there is a set L such that: (i) L can be decided in time 2O(n) and (ii) there is a constant δ > 0 such that the circuit complexity of L is at least 2δn . Then there is a set L such that: (i) L can be decided in time 2O(n) and (ii) there

is a constant δ > 0 such that L is (2δ n , 2−δ n )-hard on average. In conclusion, we have optimal logQPRG and polynomial time derandomization of probabilistic algorithms under the assumptions that there are problems of exponential circuit complexity that are computable in exponential time. Such an assumption is considered very plausible. There are other applications of these techniques that we will not have space to discuss, including extensions to the case of pseudorandomness against “nondeterministic statistical tests,” which imply surprising results for the Graph Isomorphism problem [35], [43].

7. Average-case complexity and codes We now come to a connection between the Impagliazzo–Wigderson Theorem and error-correcting codes. Due to space limitations we will only give a short discussion. The interested reader is referred to our survey paper [72] for more details. Impagliazzo and Wigderson derive Theorem 6.2 from the following “hardness amplification” reduction. Theorem 7.1 (Impagliazzo and Wigderson [31]). For every δ > 0 there are constants δ > 0, c > 1, and an algorithm with the following property. If S ⊆ {0, 1}n is a set of circuit complexity at least 2δn , then, on input S, the

algorithm outputs a set S ⊆ {0, 1}cn that is (2δ n , 2−δ n )-hard on average. Like most results in complexity theory, the proof is by contradiction: suppose

we have a set T computable by a circuit of size 2δ n such that Px∼{0,1}n [1S (x) =

1T (x)] ≥ 1/2 + 2−δ n ; then Impagliazzo and Wigderson show how to use such a circuit for T to construct a circuit for S of size 2δn . Phrased this way, the result has a strong coding-theoretic flavor: we can think of S as a “message,” of S as the “encoding” of S, of T as the “corrupted transmission” that the receiver gets, and of the process of reconstructing (a circuit for) S from (a circuit for) T as a “decoding” process. Given this perspective, introduced in [68], it is natural to try and apply coding-theoretic algorithms to hardness amplification. In doing so, we encounter the following difficulty: viewed as a message, a set S ⊆ {0, 1}n is (or can be represented as) a bit-string of length N = 2n , and so a polynomial time coding-theoretic algorithm that reconstructs S from a corrupted encoding of S takes time N O(1) = 2O(n) . In Theorem 7.1 however we need to produce a circuit of size 2δn = N δ , and so the circuit cannot simply be an implementation of the decoding algorithm.


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It seems that what we need is the following type of error-correcting code (we use the notation P (A) to denote the set of all subsets of a set A): a map C : P ({0, 1}n ) →

P ({0, 1}n ) with n = O(n) such that there is an algorithm that – given a set T ∈

P ({0, 1}n ) close to the encoding C(S) of a message S ∈ P ({0, 1}n ) and an element a ∈ {0, 1}n – determines in time at most 2δn whether a ∈ S or not. If we think of a set S ∈ P ({0, 1}n ) as simply a bit-string in {0, 1}N , N = 2n , then we are looking for

an error correcting code C : {0, 1}N → {0, 1}N , N = N O(1) , such that there is an

algorithm that, given a string u ∈ {0, 1}N close to an encoding C(x) and given an index i ∈ {1, . . . , N }, computes in time at most N δ the bit xi . It remains to specify how to “give in input” a string u of length N > N to an algorithm of running time, say, N .001 : the algorithm does not even have enough time to read the input. This can be handled by modeling the input as an “oracle” for the algorithm, which is a standard notion. The existence of error-correcting codes with this kind of “sub-linear time decoding algorithms” was well known, but the problem is that this notion is still not sufficient for the application to Theorem 7.1. The reason is that we have described a decoding algorithm that gives a unique answer and, as discussed in Section 2, such algorithms cannot recover from more than a 1/4 + o(1) fraction of errors. Theorem 7.1, however, requires us to correct from close to 1/2 fraction of errors. In Section 2 we remarked that it is possible to do list-decoding even after almost a 1/2 fraction of errors occur. So we need a definition of sub-linear time list decoding algorithm. The definition is too technical to be given here. It was formulated, for a different application, in [5]. A reasonably simple sub-linear time list-decoding algorithm giving a new proof of Theorem 7.1 is presented in [68]. The codingtheoretic proof is considerably simpler than the original one. The connection between error-correcting and hardness amplification also goes in the other direction: it is possible to view the techniques of [6], [27], [31] as defining list-decodable codes with sub-linear time decoding algorithm. This reverse connection has been used to transfer known coding theoretic impossibility results to the setting of amplification of hardness. Recall that if we want to correct from 1/2 − ε errors, then unique decoding is impossible. Codes that are (L, 1/2 − ε)-list decodable exist, but it is possible to prove that for such codes we need L = (ε−2 ). In our proof [68] of Theorem 6.2, this is not a problem because when we realize the decoding algorithm as a circuit we can “hard-wire” into the circuit the correct choice from the list. Suppose, however, that we want to prove a version of Theorem 6.2 where “algorithm of running time K” replaces “circuits of size K.” Then such a theorem would not follow from [68]: if we

try to follow the proof we see that from a good-on-average algorithm for L ∩ {0, 1}n we can only construct a list of algorithms such that one of them computes L ∩ {0, 1}n correctly, and it is not clear how to choose one algorithm out of this list.12 This problem is solved in [74], where we do prove a version of Theorem 6.2 with “probabilistic 12 This difficulty is discussed in [74].

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algorithm” in place of “circuit.” Viola [81] proves that error-correcting codes cannot be computed in certain very low complexity classes, and this means that the exponentially big error-correcting code computations occurring in [68] must add a very strong complexity overhead. This means that coding-theoretic techniques cannot be used to prove a version of Theorem 6.2 where “computable in time 2O(n) ” is replaced by “computable in NP.” Indeed, it remains a fundamental open question whether a theorem showing equivalence of worst-case complexity and average-case complexity in NP can be proved. Results of [19], [10] show that this is unlikely. Impagliazzo [28] wonders about a positive use of the fact that amplification of hardness results imply error-correcting codes, and whether the techniques of [6], [27], [31] would lead to practical error-correcting codes. We explore this question in [71], focusing on an optimization of the techniques of [27], but our results are far from being competitive with known constructions and algorithms of list-decodable codes. On the other hand, our work in refining the techniques of [27], while not successful in deriving good coding-theoretic applications, has led to interesting applications within complexity theory [71], [73].

8. Extractors and pseudorandom generators We now come to what is perhaps the most surprising result of this survey, the fact that (the proofs of) Theorems 6.1 and 6.2 directly lead to unconditional constructions of extractors. First, let us give a very high-level description of the pseudorandom generator construction that follows from Theorems 6.1 and 6.2. Let L be the set of exponential circuit complexity as in the assumption of Theorem 6.2, and let m be a parameter such that we want to construct a generator Gm : {0, 1}O(log m) → {0, 1}m whose output is (m2 , 1/m)-pseudorandom. First, we define = O(log m) such that L ∩ {0, 1} has circuit complexity at least mc , for a certain absolute constant c. Then we define our generator as Gm (z) = I Wm (L ∩ {0, 1} , z), where I Wm (S, z) is a procedure that takes as input a set S ⊆ {0, 1} and a string z ∈ {0, 1}O(log m) , outputs a string in {0, 1}m , and is such that if S has circuit complexity at least mc then I Wm (S, UO(log m) ) is (m2 , 1/m)-pseudorandom. Proving that I Wm ( ·, ·) has this property is of course quite complicated, but the general outline is as follows. As usual we proceed by contradiction and start from a statistical test T of circuit complexity at most m2 such that, supposedly, | P[Um ∈ T ] − P[I Wm (S, UO(log m) ) ∈ T ]| >

1 m.

Then we modify the circuit for T and build a new circuit for S of size < mc , thus contradicting the hypothesis. The analysis, indeed, proves a more general result. We will need some additional definitions before stating this more general result. For sets T ⊆ {0, 1}m and S ⊆

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{0, 1} , we say that S has a circuit with T -gates of size K if there is a sequence of sets S1 , . . . , Sm such that Sm = S, and each Sj is either a set of the form Pi,n , or it is the complement of a set Sh h < j , or it is the union or the intersection of two sets Sh , Sh

with h, h < j , or it is defined as Sj := {a ∈ {0, 1} : (1Sh1 (a), . . . , 1Shm (a)) ∈ T } for some h1 , . . . , hm < j . It is not hard to show that if S has a circuit with T -gates of size K1 , and T has a regular circuit of size K2 , then S has a regular circuit of size at most K1 · K2 . With these definitions in place we can be more specific about the analysis in [44], [31]: the analysis shows that if S ⊆ {0, 1} and T ⊆ {0, 1}m are two arbitrary sets such that | P[Um ∈ T ] − P[I Wm (S, UO(log m) ) ∈ T ]| >

1 m

then there is a circuit with T -gates for S of size < mc−2 . (Note that this implies our previous statement.) Here is the main idea in [70]: suppose that we have access to a weak random source, that is, a random variable X taking values in {0, 1}n and having min-entropy at least k. Suppose that n = 2 . Then we can, equivalently, regard X as being distributed over P ({0, 1} ), the set of all subsets of {0, 1} . What can we say about the distribution of I Wm (X, UO(log m) )? We claim that, if k is large enough, the distribution I Wm (X, UO(log m) ) is close in statistical distance to the uniform distribution; in other words, I Wm ( ·, ·) is an extractor. Let us see how to prove this by contradiction. Let T be a statistical test such that | P[Um ∈ T ] − P[I Wm (X, UO(log m) ) ∈ T ]| >

1 m

and call a set S ∈ P ({0, 1} ) bad if | P[Um ∈ T ] − P[I Wm (S, UO(log m) ) ∈ T ]| > m2 . Let B be the set of all bad sets. Then, by Markov’s inequality, P[X ∈ B] ≥ m2 , and since X has min-entropy k we have |B| ≥ 2k−log m−1 . On the other hand, if S is bad, then there is a circuit with T -gates of size at most mc−2 that computes S. The number c−1 c−1 of such circuits is at most 2O(m , and so |B| ≤ 2O(m . So if k ≥ c mc−1 , where c, c are absolute constants, we reach a contradiction. Thus, I Wm (X, UO(log m) ) − Um SD ≤ m1 . If we look more closely at how I Wm (S, z) is defined, we see that (especially if we use the proof of Theorem 6.2 in [68]) it can be seen as I Wm (S, z) := NWm (C(S), z), where C is an error-correcting code and NWm is the relatively simple pseudorandom generator construction of Nisan and Wigderson. For the application to derandomization, it is important that C be a “sub-linear time list-decodable” error-correcting code. However, in order for our argument about randomness extraction to work, it is sufficient that C be an arbitrary list-decodable code, and not even a polynomial time list-decoding algorithm is needed. This means that one can get extractors by using standard error-correcting codes and the simple Nisan–Wigderson generator. The

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resulting construction is described and analysed in [70] in about two pages and, at the time, it was the best known extractor construction, improving over very technical previous work. What makes these calculations work is the intuition that the proofs of Theorems 6.1 and 6.2 prove more than the intended statement. In particular, the proof works if we replace “circuit complexity” with “description complexity” which we exploited in the previous argument. See [70] for further discussion of this point. The connection with pseudorandomness and the general idea of analysing an extractor by finding short descriptions of the output of the source based on a hypothetical statistical test (the so-called “reconstruction method” to analyse extractors) has led to remarkable advances in extractor constructions in the past five years, together with other ideas. The best distillation of the reconstruction method is in [58], providing a near-optimal and simple construction of extractors.13 The extractor motivation has also led to improvements in pseudorandom generator constructions, see [58], [75]. Currently, the best known extractor construction [40] uses the notion of “condenser” introduced in [69], [54] and a combination of several components, one of which is analysed with the reconstruction method. The extractors of Lu et al. [40] is almost best possible.

9. Conclusions We have discussed how, starting from worst-case complexity assumptions, it is possible to construct very strong pseudorandom generators and derive conditional derandomization results for all probabilistic algorithms. What about proving circuit lower bounds and deriving unconditional derandomization results? The results of Razborov and Rudich [52] show that a significant departure from current techniques will be required to prove such lower bounds. What about deriving derandomization results without proving lower bounds? Impagliazzo, Kabanets and Wigderson [29] prove that any general derandomization result implies a circuit lower bound.14 Short of proving such elusive circuit lower bounds, we should test the prediction of the theory and look for polynomial time deterministic versions of known probabilistic polynomial time algorithms. The four most important probabilistic algorithms (or collections of algorithms) are: primality testing, graph connectivity using random walks, polynomial identity testing, and algorithms for approximate counting. Primality testing and graph connectivity using random walks have been derandomized [1], [53]. Kabanets and Impagliazzo [34] prove that any derandomized polynomial identity testing algorithms implies circuit lower bounds.15 13 The construction is simple but the analysis is quite non-trivial. 14 Here is what we mean by “general derandomization”: if f is a function and A is randomized algorithm that with high probability achieves a good approximation of f , then there is a deterministic algorithm that achieves a good approximation of f and whose running time is polynomial in the running time of A.


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The possibility of derandomizing approximate counting algorithms with current techniques is quite open. Here is perhaps the simplest question: given an n-variable boolean formula in disjunctive normal form and ε > 0, compute in time polynomial in the size of the formula and in 1/ε an approximation to the number of satisfying assignments up to an additive error ≤ 2n ε. See [42] for a nearly polynomial time deterministic algorithm for this problem. The construction of an optimal (seeded) extractor with parameters matching the known lower bounds remains an elusive open question. It would also be interesting to match the parameters of [40] with a simpler construction. There has been very exciting recent progress towards constructing good seedless extractors for independent sources, and for the related problem of constructing bipartite Ramsey graphs [8], [11]. The broader area of seedless extractor constructions for general classes of distributions has seen much recent progress. In the long run, we would expect this research to define simple and powerful seedless extractors working for a wide and natural class of distributions. Such extractors would be very useful in practice, giving a principled approach to the production of random bits for cryptographic applications.

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[83] Wigderson, Avi, and Zuckerman, David, Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica 19 (1) (1999), 125–138. [84] Yao, Andrew C., Theory and applications of trapdoor functions. In Proceedings of the 23th IEEE Symposium on Foundations of Computer Science, IEEE, New York 1982, 80–91. [85] Zippel, Richard, Probabilistic algorithms for sparse polynomials. In Symbolic and algebraic computation (ed. by Edward W. Ng), Lecture Notes in Comput. Sci. 72, Springer-Verlag, Berlin 1979, 216–226. [86] Zuckerman, David, General weak random sources. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, IEEE, New York 1990, 534–543. [87] Zuckerman, David, Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the 38th ACM Symposium on Theory of Computing, ACM Press, New York 2006, 681–690. Computer Science Division, U.C. Berkeley, 679 Soda Hall, Berkeley, CA 94720-1776, U.S.A. E-mail: [email protected]

Least-squares finite element methods Pavel Bochev∗ and Max Gunzburger†

Abstract. Least-squares finite element methods are an attractive class of methods for the numerical solution of partial differential equations. They are motivated by the desire to recover, in general settings, the advantageous features of Rayleigh–Ritz methods such as the avoidance of discrete compatibility conditions and the production of symmetric and positive definite discrete systems. The methods are based on the minimization of convex functionals that are constructed from equation residuals. This paper focuses on theoretical and practical aspects of least-square finite element methods and includes discussions of what issues enter into their construction, analysis, and performance. It also includes a discussion of some open problems. Mathematics Subject Classification (2000). 65N30, 65N99, 65N15. Keywords. Least squares, finite element methods, compatible discretizations.

1. Introduction Finite element methods (FEMs) for the approximate numerical solution of partial differential equations (PDEs) were first developed and analyzed for problems in linear elasticity and other settings for which solutions can be characterized as (unconstrained) minimizers of convex, quadratic functionals on infinite-dimensional Hilbert spaces [46]. A Rayleigh–Ritz approximation of such solutions is defined by minimizing the functional over a family of finite-dimensional subspaces. An FEM results when these spaces consist of piecewise polynomial functions defined with respect to a family of grids. When applied to problems such as linear elasticity or the Poisson equation, the Rayleigh–Ritz setting gives rise to FEMs with several advantageous features that led to their great success and popularity: 1. general regions and boundary conditions are relatively easy to treat in a systematic manner; 2. the conformity1 of the finite element spaces suffices to guarantee the stability ∗ Supported by the Applied Mathematical Sciences program, U.S. Department of Energy, Office of Energy Research, and performed at Sandia National Labs, a multiprogram laboratory operated by Sandia Corporation, a Lockheed-Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under contract DE-AC-94AL85000. † Supported in part by CSRI, Sandia National Laboratories, under contract 18407 and by the National Science Foundation under grant number DMS-0240049. 1An approximating space is referred to as being conforming if it is a subspace of the underlying infinitedimensional Hilbert space.


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Pavel Bochev and Max Gunzburger

and optimal accuracy2 of the approximate solutions; 3. all variables can be approximated using a single type of finite element space, e.g., the same degree piecewise polynomials defined with respect to a same grid; 4. the resulting linear systems are a) sparse; b) symmetric; c) positive definite. The success of FEMs in the Rayleigh–Ritz setting quickly led both engineers and mathematicians to apply and analyze FEMs in other settings, motivated by the fact that properties 1 and 4a are retained for all FEMs.3 For example, mixed FEMs arose from minimization problems constrained by PDEs such as the Stokes problem; the Lagrange multiplier rule was applied to enforce the constraints, resulting in saddle-point problems [19]. In this setting, the only other property retained from the Rayleigh–Ritz setting is 4b. More generally, Galerkin FEMs can, in principle, be defined for any PDE by forcing the residual of the PDE (posed in a weak, variational formulation) to be orthogonal to the finite element subspace [3]. In this general setting, one usually loses all the features of the Rayleigh–Ritz setting other than 1 and 4a. Using the same formalisms, Galerkin FEMs were even applied to nonlinear problems such as the Navier–Stokes equations [34]. It is a testament to the importance of advantage 1 that despite the loss of other advantages, mixed and Galerkin FEMs are in widespread use and have also been extensively analyzed.4 Not surprisingly, despite the success of mixed and Galerkin FEMs in general settings, there has been substantial interest and effort devoted to developing finite element approaches that recover at least some of the advantages of the Rayleigh–Ritz setting. Notable among these efforts have been penalty and stabilized FEMs, e.g., for the Stokes problem, stabilized FEMs [4]–[5], [6], [12], [15], [29], [30], [36], [37], [44] recover advantages 2 and 3 but fail to recover advantage 4c and often lose advantage 4b. Least-squares finite element methods (LSFEMs) can be viewed as another attempt at retaining the advantages of the Rayleigh–Ritz setting even for much more general problems. In fact, they offer the possibility of, in principle, retaining all of the advantages of that setting for practically any PDE problem. In §2, we show how this is possible. However, this is not the whole story. Any FEM, including an LSFEM, must also meet additional practicality criteria: A. bases for conforming subspaces are easily constructed; B. linear systems are easily assembled; C. linear systems are relatively well conditioned. 2An approximate solution is referred to as being optimally accurate if the corresponding error is bounded by a constant times the error of the best approximation. 3 These properties follow from the way finite element spaces are constructed, e.g., based on grids and choosing basis functions of compact support. 4 It should be noted that in the general settings for which FEMs lose many of the advantages they possess in the Rayleigh–Ritz setting, they do not suffer from any disadvantages compared to other discretization methods such as finite difference, finite volume, and spectral methods.

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In judging whether or not an LSFEM meets theses criteria, we will measure them up against Galerkin FEMs for the Poisson equation; in particular, we will ask the questions: can we use standard, piecewise polynomial spaces that are merely continuous and for which bases are easily constructed? can the assembly of the linear systems be accomplished by merely applying quadrature rules to integrals? and, are the condition numbers of the linear systems of5 O(h−2 )? Unfortunately, naively defined LSFEMs often fail to meet one or more of the practicality criteria. LSFEMs possess two additional advantageous features that other FEMs, even in the Rayleigh–Ritz setting, do not possess. First, least-square functionals provide an easily computable residual error indicator that can be used for adapting grids. Second, the treatment of general boundary conditions, including nonhomogeneous ones, is greatly facilitated because boundary condition residuals can be incorporated into the least-square functional.

2. The most straightforward LSFEM Let denote a bounded domain in d , d = 2 or 3, with boundary . Consider the problem Lu = f in and Ru = g on , (1) where L is a linear differential operator and R is a linear boundary operator. We assume that the problem (1) is well posed so that there exists a solution Hilbert space S, data Hilbert spaces H and H , and positive constants α1 and α2 such that α1 u2S ≤ Lu2H + Ru2H ≤ α2 u2S

for all u ∈ S.

(2)

Then consider the least-squares functional6 J (u; f, g) = Lu − f 2H + Ru − g2H

(3)

and the unconstrained minimization problem min J (u; f, g). u∈S

(4)

Note that the functional (3) measures the residuals of the components of the system (1) using the data space norms H and H and the minimization problem (4) seeks a solution in the solution space S for which (2) is satisfied. It is clear that the problems (1) and (4) are equivalent in the sense that u ∈ S is a solution of (4) if and only if it is also a solution, perhaps in a generalized sense, of (1). An LSFEM can be defined by choosing a family of finite element subspaces S h ⊂ S parameterized by h tending to zero and then restricting the minimization problem (4) 5 Usually, h is a measure of the size of the grid used in the construction of the finite element space. 6A least-squares functional may be viewed as an “artificial” energy that plays the same role for LSFEMs as a bona fide physically energy plays for Rayleigh–Ritz FEMs.

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to the subspaces. Thus, the LSFEM approximation uh ∈ S h to the solution u ∈ S of (1) or (4) is the solution of the problem min J (uh ; f, g).

(5)

uh ∈S h

The Euler–Lagrange equations corresponding to the minimization problems (4) and (5) are given by seek u ∈ S seek u ∈ S h

such that B(u, v) = F (v) h

such that B(u , v ) = F (v ) h

h

h

for all v ∈ S,

(6)

for all v h ∈ S h ,

(7)

respectively, where for all u, v ∈ S, B(u, v) = (Lv, Lu)H +(Rv, Ru)H and F (v) = (Lv, f )H +(Rv, g)H . (8) If we choose a basis {Uj }jJ=1 , where J = dim(S h ), then we have that uh = J J j =1 cj Uj for some constants {cj }j =1 and then the discretized problem (7) is equivalent to the linear system Kc = f, (9) where the elements of the matrix K ∈ J ×J and the vectors f ∈ J and c ∈ J are given, for i, j = 1, . . . , J , by cj = cj , Kij = (LUi , LUj )H + (RUi , RUj )H ,

and

f i = (LUi , f )H + (RUi , g)H .

The results of the following theorem follow directly from (2). Theorem 2.1. Assume that (2) holds and that S h ⊂ S. Then, – the bilinear form B( · , · ) defined in (8) is continuous, symmetric, and coercive; – the linear functional F (·) defined in (8) is continuous; – the problem (6) has a unique solution u ∈ S that is also the unique solution of the minimization problem (4); – the problem (7) has a unique solution uh ∈ S h that is also the unique solution of the minimization problem (5); – for some constant C > 0, we have that uS ≤ C(f H + gH ) and uh S ≤ C(f H + gH ) ; – for some constant C > 0, u and uh satisfy the error estimate u − uh S ≤ C inf u − v h S ; v h ∈S h

(10)

– the matrix K of (9) is symmetric and positive definite. Note that it is not assumed that the system (1) is self-adjoint or positive as it would have to be in the Rayleigh–Ritz setting; it is only assumed that it is well posed. Despite the generality of the system (1), the LSFEM based on (5) recovers all desirable


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features of FEMs in the Rayleigh–Ritz setting. Note that (10) shows that least-squares finite element approximations are optimally accurate with respect to solution norm · S for which the system (1) is well posed. In defining the least-squares principle (4), we have not restricted the spaces S and S h to satisfy the boundary conditions. Instead, we have included the residual Ru − g of the boundary condition in the functional J ( · ; ·, ·) defined in (3). Thus, we see that LSFEMs possess a desirable feature that is absent even from standard FEMs in the Rayleigh–Ritz setting: the imposition of boundary conditions can be effected through the functional and need not be imposed on the finite element spaces.7 Notwithstanding this advantage, one can impose essential boundary conditions on the space S in which case all terms in (2)–(8) involving the boundary condition are omitted and we also set H = H . Note also that since J (uh ; f, g) = Luh − f 2H + Ruh − g2H = B(uh , uh ) − 2F (uh ) + (f, f )H + (g, g)H , the least-square functional J (uh ; f, g) provides a computable indicator for the residual error in the LSFEM approximation uh . Such indicators are in widespread used for grid adaption. The problems (6) and (7) display the normal equation form typical of least-squares systems; see (8). It is important to note that since L is a differential operator, (6) involves a higher-order differential operator. We shall see that this observation has a profound effect on how practical LSFEMs are defined. 2.1. The practicality of the straightforward LSFEM. The complete recovery, in general settings, of all desirable features of the Rayleigh–Ritz setting is what makes LSFEMs intriguing and attractive. But, what about the practicality of the method defined by (5)? We explore this issue using examples. 2.1.1. An impractical application of the straightforward LSFEM. Consider the problem −u = f in and u = 0 on , (11) where we assume that is either a convex, Lipschitz domain or that it has a smooth boundary. Of course, this is a problem which fits into the Rayleigh–Ritz framework so that there is no apparent need8 to use any other type of FEM. However, let us proceed and use the LSFEM method anyway, and see what happens. Here we have that (2) holds with9 S = H 2 () ∩ H01 (), H = L2 (), and L = −. We then have 7 This advantage of LSFEM can be useful for imposing inhomogeneous boundary conditions, essential boundary conditions such as Dirichlet boundary conditions for second-order elliptic PDEs, and boundary conditions involving a particular component, e.g., the normal component, of a vector variable. 8 Inhomogeneous Dirichlet boundary conditions provide a situation in which one might want to use LSFEMs even for the Poisson problem. 9 We use standard Sobolev space notation throughout the paper. Also, in this and most of our examples, we will be imposing the boundary condition on the solutions space S.

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that, for all u, v ∈ H 2 () ∩ H01 (), 2 f v d, J (u; f ) = u + f 0 , F (v) =

and

B(u, v) =

v u d.

Note that minimizing the least-squares functional has turned the second-order Poisson problem into a fourth-order problem. An LSFEM is defined by choosing a subspace S h ⊂ S = H 2 () ∩ H01 () and then posing the problem (7). It is well known that in this case, the finite element space S h has to consist of continuously differentiable functions; this requirement greatly complicates the construction of bases and the assembly of the matrix problem. Furthermore, it is also well known that the condition number of the matrix problem is O(h−4 ) which should be contrasted with the O(h−2 ) condition number obtained through a Rayleigh–Ritz discretization of the Poisson equation. Thus, for this problem, the straightforward LSFEM fails all three practicality tests. Since it is also true that (2) holds with S = H01 () and H = H −1 (), one could develop an LSFEM based on the functional J (u; f ) = u + f −1 and the solution space S = H01 (). This approach would allow one to use a finite element space S h consisting of merely continuous functions so that bases may be easily constructed. Moreover, it can be shown that because of the use of the H −1 () inner product, the condition number of the resulting matrix system is O(h−2 ) which is the same as for a Rayleigh–Ritz discretization. However, the H −1 () inner product is computed by inverting the Laplacian operator which leads to the loss of property 4a and also makes the assembly of the matrix problem more difficult. So, as it stands, the straightforward LSFEM remains impractical for the second-order Poisson problem. 2.1.2. A practical application of the straightforward LSFEM. Consider now the problem −∇ · u = f and ∇ × u = g in

and

n · u = 0 on .

(12)

Here u ∈ S = H 1n () = {v ∈ H 1 () | n · v = 0 on } and {f, g} ∈ H = L20 () × L2s (), where L20 () = {f ∈ L2 () | f d = 0}, and L2s () = {g ∈ L2 () | ∇ · g = 0 in }. We then have that (2) holds so that we may define the least-squares functional J (u; f, g) = ∇ · u + f 20 + ∇ × u − g20

for all u ∈ S = H 1n ()

(13)

that results in (∇ · u)(∇ · v) + (∇ × u) · (∇ × v) d for all u, v ∈ S = H 1n () B(u, v) =

and

F (v) =

− f ∇ · v + g · ∇ × v d for all v ∈ S = H 1n ().


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An LSFEM is defined by choosing a subspace S h ⊂ S = H 1n () and then solving the problem (7). The LSFEM based on the functional (13) not only recovers all the good properties of the Rayleigh–Ritz setting for the problem (12), but also satisfies all three practicality criteria. Since we merely require that S h ⊂ H 1n (), we can choose standard finite element spaces for which bases are easily constructed. Furthermore, since the functional (13) only involves L2 () inner products, the assembly of the matrix system is accomplished in a standard manner. Finally, it can be shown that the condition number of the matrix system is O(h−2 ). 2.2. Norm-equivalence vs. practicality. Since (2) and (3) imply that α1 u2S ≤ J (u; 0, 0) ≤ α2 u2S ,

(14)

we refer to the functional J ( · ; ·, ·) as being norm equivalent. This property of the functional causes the LSFEM defined by (5) to recover all the desirable properties of the Rayleigh–Ritz setting. However, the norms that enter the definition of the functional J ( · ; ·, ·) as well as the form of the PDE system (1) can render the resulting LSFEM impractical. Thus, in order to define a practical LSFEM, one may have to define a least-squares functional that is not norm equivalent in the sense of (14). We take up this issue in §3. Here, we examine the examples of §2.1 to see what guidance they give us about what makes an LSFEM practical. 2.2.1. First-order system form of the PDEs. Perhaps the most important observation that can be made from the examples of §2.1 is that the example of §2.1.2 involved a first-order system of PDEs and an LSFEM that allowed for the easy construction of finite element bases (because one could work with merely continuous finite element spaces) and resulted in matrix systems with relative good conditioning. As a result, all modern LSFEMs are based on first-order formulations of PDE systems. Of course, many if not most PDEs of practical interest are not usually posed as first-order systems. Thus, the first step in defining an LSFEM should be recasting a given PDE system into a first-order system. Unfortunately, there is no unique way to do this. For example, the three problems ⎫ ⎧ ⎫ ⎧ ⎫ ⎧ u + ∇φ = 0 in ⎪ ⎪ ⎪ ⎪ u + ∇φ = 0 in ⎬ ⎨ ∇ · u = f in ⎬ ⎬ ⎨ ⎨ ∇ ·u=f in ∇ × u = 0 in ∇ ·u=f in , , ∇ × u = 0 in ⎪ ⎭ ⎩ ⎭ ⎪ ⎩ ⎪ ⎪ n × u = 0 on φ=0 on ⎭ ⎩ φ=0 on are all first-order systems that are equivalent to the Poisson problem (11). Each happens to be norm equivalent, but with respect to different norms. If we assume that in each case the boundary condition is imposed on the solutions space, we have that, for the three problems, the space S in (2) is respectively given by H01 ()×H (, div), H01 () × H 1 (), and H 1τ (), where H (, div) = {v ∈ L2 () | ∇ · v ∈ L2 ()} and H 1τ () = {v ∈ H 1 () | n × v = 0 on }.

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2.2.2. Functionals formed using L2 norms of equation residuals. Another observation that can be gleaned from the examples of §2.1 is that if one wants to be able to assemble the matrix system using standard finite element techniques, then one should use L2 norms of equation residuals in the definition of the least-squares functional. Unfortunately, it is not always the case that the resulting least-squares functional is norm equivalent. Let us explore this issue in more detail. Consider the Stokes problem −u + ∇p = f , ∇ · u = 0 in

and

u = 0 on .

(15)

The most popular LSFEM for this problem is based on the first-order system ∇ × ω + ∇p = f , ω = ∇ × u, ∇ · u = 0 in

and

u = 0 on

(16)

that is known for obvious reasons as the velocity–vorticity–pressure formulation. One would then be tempted to use the functional J0 (u, ω, p; f ) = ∇ × ω + ∇p − f 20 + ∇ × u − ω20 + ∇ · u20

(17)

that involves only L2 () norms of equation residuals. Indeed, this is the most popular approach for defining LSFEM for the Stokes equations. Unfortunately, the functional (17) is not norm equivalent [13]. On the other hand, the functional J−1 (u, ω, p; f ) = ∇ × ω + ∇p − f 2−1 + ∇ × u − ω20 + ∇ · u20 is equivalent to u21 + ω20 + p20 [13]. So, on the one hand, the lack of norm equivalence for the functional J0 ( · , · , · ; ·) results in a loss of accuracy of the LSFEM approximations based on that functional. On the other hand, the appearance of the H −1 () norm in the functional J−1 ( · , · , · ; ·) results in an impractical LSFEM because the matrix systems are not easily assembled.10

3. More sophisticated LSFEMs To define the least-squares principle (4), one had to choose the pair {S, J ( · ; f, g)}, where S denotes a solution Hilbert space and J ( · ; f, g) a functional defined over S that satisfies the norm-equivalence relation (14). We refer to the variational principle (4) as the continuous least-squares principle.11 The straightforward LSFEM was defined by choosing a finite element subspace S h ⊂ S and then minimizing the functional J ( · ; f, g) over S h ; see (5). We refer to the straightforward LSFEM as the conforming LSFEM. For such LSFEMs, we obtain the error estimate (10). 10A similar dilemma arises when one imposes boundary conditions through the least-squares functional. 11 Here, “continuous” refers to the fact that solutions of (4) are also solutions of the PDE system (1). Recall also that (14) follows from the well-posedness relation (2) for the PDE system.

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Conforming LSFEMs can be generalized so that their applicability and practicality are enhanced. Here, we briefly discuss some of these generalizations. We still have in mind approximating solutions of the continuous least-squares principle (4) or what is equivalent, solutions of the PDE system (1). We again choose a finite element space S h and a convex, quadratic functional Jh ( · ; f, g) defined over S h . The pair {S h , Jh ( · ; f, g)} gives rise to the discrete least-squares principle min Jh (uh ; f, g).

(18)

uh ∈S h

Since we only require that the functional Jh ( · ; f, g) be defined for functions in S h , we refer to LSFEMs constructed in this manner as discrete LSFEMs. The functional Jh ( · ; f, g) is required to satisfy the following non-restrictive assumptions. H1. There exists a discrete energy inner product ( · , · )h : S h × S h → and a 1/2 discrete energy norm · h = ( · , · )h such that Jh (uh ; 0, 0) = (uh , uh )h = uh 2h for all uh ∈ S h . H2. There exist bilinear forms E( · , · ) and T ( · , · ) such that for all smooth functions u ∈ S and all uh ∈ S h Jh (uh ; Lu, Ru) = u − uh 2h + E(u, uh ) + T (u, u).

(19)

The two assumptions are sufficient to prove the following results about solutions of (18). Theorem 3.1. Assume that hypotheses H1 and H2 hold for the discrete principle {S h , Jh ( · ; f, g)} and let u denote a sufficiently smooth solution of (1). Then the problem (18) has a unique solution uh ∈ S h . Moreover, uh satisfies u − uh h ≤ inf u − v h h + sup v h ∈S h

v h ∈S h

E(u, v h ) . v h h

(20)

A discrete least-squares functional Jh ( · ; f, g) will be referred to being order rconsistent if there exists a positive number r such that for all sufficiently smooth functions u ∈ S, the second term on the right-hand side of (20) can be bounded from above by C(u)hr , where C(u) is a positive number whose value may depend on u but not on h. If Jh ( · ; f, g) is order r-consistent, then (20) implies that u − uh h ≤ inf u − v h h + C(u)hr . v h ∈S h

(21)

Theorem 3.1 shows that discrete LSFEMs can work under a minimal set of assumptions. It also explains why LSFEMs tend to be much more robust than their mixed FEM counterparts; unlike the inf-sup conditions that are required for the latter type of method, defining pairs {S h , Jh ( · ; f, g)} such that the assumptions H1 and H2 are satisfied is not a difficult task.

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Constructing discrete least-squares functionals. Theorem 3.1 provides estimates for the error with respect to the discrete norm · h . Of greater interest is estimating errors using the (mesh-independent) solution norm · S associated with the PDE problem (1). Since S h ⊂ S, it is certainly true that · S acts as another norm on S h , in addition to · h . Thus, since S h is finite dimensional, these two norms are comparable. However, the comparability constants may depend on h; if they do, then error estimates analogous to (20) and (21) but in terms of the norm · S will surely involve constants that depend on inverse powers of h and, at the least, accuracy may be compromised. We conclude that hypotheses H1 and H2 do not sufficiently connect Jh ( · ; f, g) to the problem (1) for us to determine much about the properties of the error in the discrete LSFEM solution with respect to · S norm. Thus, we now discuss how to construct discrete least-squares functionals so that we can get a handle on these properties. We assume that (2) and (14) hold for the problem (1), the least-squares functional J ( · ; f, g), the solution space S, and the data spaces H and H . Let DS , D , and D denote norm-generating operators that allow us to relate the norms on S, H , and H , respectively, to12 L2 () norms, i.e., such that, for all u ∈ S, f ∈ H , and g ∈ H , uS = DS u0 , f H = D f 0 , and gH = D g0, . We then let h h Jh (uh ; f, g) = D (Lh uh − Q f )20 + Dh (Rh uh − Qh g)20 , h , D h , Lh , and Rh are approximations of the operators D , D , L, and R, where D h : H → L2 () and Qh : H → L2 () are projections. It can respectively, and Q be shown that Jh (uh ; f, g) satisfies (19) with a specific form for E(u, v h ). The operators L and R define the problem (1) that is being solved so that the main objective in choosing Lh and Rh is to make Jh (u; f, g) as small as possible for the exact solutions u. An appropriate choice is to use operators that will lead to truncation errors of order r in (19), i.e., Lh and Rh should be such that (21) holds. On the other hand, D and D define the energy balance of (1), i.e., the proper scaling between h and D h data and solution spaces. As a result, the main objective in the choice of D is to ensure that the scaling induced by Jh ( · ; f, g) is as close as possible to (2), i.e., to “bind” {S h , Jh ( · ; f, g)} to the energy balance of {S, J ( · ; f, g)}. For norm-equivalent discrete least-squares principles, Jh (·, f, g) satisfies

αˆ 1 uh S ≤ Jh ( · ; 0, 0) ≤ αˆ 2 uh S

for all uh ∈ S h .

If the finite element space satisfies standard inverse assumptions, minimizers of this functional satisfy the error estimate

E(u, v h ) u − u S ≤ C inf u − v S + inf u − v h + sup h v h ∈S h v h ∈S h v h ∈S h v h h

h

h

.

12 Recall from §2.2 that the use of L2 () norms in the definition of the least-squares functional is a key factor to making an LSFEM practical.


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For quasi norm-equivalent discrete least-squares principles, Jh ( · ; f, g) satisfies αˆ 1h uh S ≤ Jh ( · ; 0, 0) ≤ αˆ 2h uh S , where αˆ 1h > 0 and αˆ 2h > 0 for all h > 0 but may depend on h. Under additional assumptions, error estimates can also be derived in this case.

4. Compatible LSFEMs Stable mixed finite element methods (MFEMs) for the Poisson equation13 based on first-order formulations involving a scalar variable φ and a vector (or flux) variable u require the use of finite element spaces that satisfy an appropriate inf-sup condition [19], [20]. It is well known that pairs of standard, nodal-based, continuous finite element spaces fail the inf-sup condition and lead to unstable mixed methods. It is also well known that the inf-sup condition is circumvented if one uses such simple element pairs in LSFEMs based on L2 least-squares functionals. Ever since such LSFEMs for first-order formulations of the Poisson equation were first considered in [38], this fact has been deemed as an important advantage of those methods over MFEMs. On the other hand, such LSFEMs suffer from two deficiencies. Computationally-based observations indicate that nodal-based LSFEMs do a poor job, compared to stable MFEMs, of conserving mass, i.e., of locally satisfying ∇ · u = 0. In addition, excepting in one special case, such methods produce suboptimally accurate (with respect to L2 () norms) flux approximations.14 Already in [38], optimal L2 error estimates for LSFEMs were established for the scalar variable; however, there and in all subsequent analyses, optimal L2 error estimates for the flux could not be obtained15 without the addition of a “redundant” curl equation; see, e.g., [23], [24], [26], [39], [43]. Moreover, computational studies in [32] strongly suggested that optimal L2 convergence for flux approximations may in fact be nearly impossible to obtain if one uses pairs of standard, nodal-based, continuous finite element spaces. A notable exception was a case studied in [32] for which optimal L2 error estimates for both the scalar variable and the flux were obtained when these variables were approximated by continuous nodal spaces corresponding to a criss-cross grid. The key to proving these results was the validity of a grid decomposition property (GDP) which was established for the criss-cross grid in [33]. So far, the criss-cross grid remains the only known case of a continuous, nodal-based 13Although we consider only the Poisson problem, much of what we discuss can be easily extended to other systems of elliptic PDEs. 14 The least-squares functionals in question are norm equivalent so that optimally accurate approximations are obtained with respect to the norms for which the equivalences hold. Here, we are interested in error estimates in weaker L2 () norms for which the norm equivalence of the least-square functional does not by itself guarantee optimal accuracy. 15A somewhat different situation exists for negative-norm-based LSFEMs for which it is known that the L2 accuracy of the flux is optimal with respect to the spaces used; however, for such methods, no error bound for the divergence of the flux could be established; see [18].

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finite element space for which the GDP can be verified. More importantly, it was shown in [33] (see also [17]) that the GDP is necessary and sufficient for the stability of MFEMs. The correlation between the stability of MFEMs and the optimal accuracy of LSFEMs, established in [32], opens up the intriguing possibility that optimal L2 accuracy for the flux may be obtainable for an LSFEM, provided that this variable is approximated using finite element spaces that are stable for an appropriate MFEM. Today, the stability of MFEMs is well understood, and many examples of stable finite element pairs are known. We will show that the use of some of these spaces in an LSFEM indeed can help improve the L2 accuracy of flux approximations. What we conclude is that if one gives up the use of nodal-based, continuous finite element spaces for the approximation of the flux, one can obtain optimally accurate approximations of the flux with respect to L2 () norms. While this conclusion may disappoint the adherents of equal-order implementations,16 our results do not void LSFEMs as a viable or even preferable computational alternative to MFEMs. To the contrary, they demonstrate that an LSFEM can be designed that combines the best computational properties of two dual MFEMs and at the same time manages to avoid the inf-sup conditions and indefinite linear systems that make the latter more difficult to solve. Although we reach this conclusion in the specific context of MFEMs and LSFEMs for the Poisson problem, the idea of defining the latter type of method so that it inherits the best characteristics of a pair of mixed methods that are related through duality may have considerably wider application. In the rest of this section, we focus the Poisson equation −φ = f in ,

φ = 0 on d ,

and

∂φ/∂n = 0 on n ,

(22)

where denotes a bounded region in d , d = 2, 3, with a Lipschitz continuous boundary that consists of two disjoint parts denoted by d and n . 4.1. MFEMs for the Poisson problem. So as to provide a background for subsequent discussions concerning LSFEMs, we first consider two17 (dual) MFEMs for the Poisson problem (22) written in the first-order form ∇ · u = f,

u + ∇φ = 0 in ,

φ = 0 on d ,

u · n = 0 on n .

(23)

4.1.1. Stable MFEMs for the Dirichlet principle. Continuous, nodal finite element spaces built from mth degree polynomials, m ≥ 1, and whose elements satisfy 0 . Note that S 0 ⊂ {ψ ∈ the boundary condition φ = 0 on n are denoted by Sm m 1 the space ∇(S 0 ).18 H 1 () | ψ = 0 on d }. We denote by Sm m 16 Recall that the ability to approximate all variables using simple nodal finite element spaces was one or the advantages of the FEMs in the Rayleigh–Ritz setting that we set out to recover using LSFEMs. 17 Because they can be derived from two classical optimization problems, we will refer to the two methods as the discretized Dirichlet and Kelvin principles, respectively. 18 Except for m = 1, S 1 is not a complete (m − 1)st degree polynomial space. However, characterizing S 1 is m m not difficult and turns out to be unnecessary in practice.

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0 Sm

A stable MFEM based on the Dirichlet principle is defined as follows: seek ψh ∈ 1 = ∇(S 0 ) such that and uh ∈ Sm m ⎧ ⎪ 1, ⎪ uh · v h d + ∇φh · v h d = 0 for all v h ∈ Sm ⎨ (24) ⎪ ⎪ 0 ⎩ ∇ψh · uh d = − f ψh d for all ψ ∈ Sm .

1 = ∇(S 0 ), even at the discrete level, we may eliminate the flux Note that since Sm m 0 approximation to obtain the equivalent discrete problem for φh ∈ Sm 0 ∇φh · ∇ψh d = (25) f ψh d for all ψ ∈ Sm

that we recognize as the standard Galerkin discretization of (22). In fact, (24) and (25) are equivalent in that whenever φh is a solution of (25), then φh and uh = ∇φh are a solution pair for (24) and conversely. In this way we see that for (24), i.e., the Dirichlet principle, the required inf-sup condition is completely benign in the sense that it can be avoided by eliminating the flux approximation uh from (24) and solving (25) instead. The required inf-sup condition is implicitly satisfied by the pair 0 and S 1 = ∇(S 0 ). If one insists on solving (24), then one needs to of spaces Sm m m 1 ; this is easily accomplished. explicitly produce a basis for Sm From either (24) or (25) one obtains, for the Dirichlet principle, that if φ ∈ H m+1 () ∩ Hd1 (), then φ −φh 0 ≤ hm+1 φm+1

and

u−uh 0 = ∇(φ −φh )0 ≤ hm φm+1 . (26)

4.1.2. Stable MFEMs for the Kelvin principle. The BDMk and RTk spaces on are built from the individual element spaces defined with respect to a finite element K in a partition Th of BDMk (K) = (Pk (K))n

and

RTk (K) = (Pk (K))n + xPk (K)

in a manner that ensures the continuity of the normal component across element boundaries; see [20] for details and definitions of the corresponding element degrees of freedom. Since BDMk and RTk both contain complete polynomials of degree k, their approximation properties in L2 are the same. Since RTk also contains the higherdegree polynomial component xPk (K), it approximates the divergence of the flux with better accuracy than does BDMk . Note, however, that this additional component does not help to improve the L2 accuracy of RTk spaces because it does not increase to k + 1 the order of the complete polynomials contained in RTk . In what follows, we will denote by Sk2 the RT and BDM spaces having equal approximation orders with respect to the divergence operator, i.e., we set Sk2 = {v ∈ Hn (, div) | v|K ∈ Sk2 (K)}, where Sk2 (K) is one of the finite element spaces

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RTk−1 (K) or BDMk (K) and Hn (, div) = {v ∈ Hn (, div) | v · n = 0 on n }. We denote by Sk3 the space ∇ · (Sk2 ). For characterizations of these spaces, see [20]. A stable MFEM based on the Kelvin principle is defined as follows: we seek uh ∈ Sk2 and φh ∈ Sk3 = ∇ · (Sk2 ) such that ⎧ ⎪ ⎪ φh ∇ · v h d = 0 ⎨ uh · v h d − ⎪ ⎪ ⎩ ψh ∇ · uh d = f ψh d

for all v h ∈ Sk2 , (27) for all ψh ∈

Sk3 .

For (27), the required inf-sup condition is much more onerous than for (24) in the sense that defining a pair of stable finite element spaces for the scalar variable and the flux is not so straightforward a matter. We refer to [20] for a proof that (Sk3 , Sk2 ) is a stable pair for the mixed finite element problem (27). Moreover, one can show [20] that for any sufficiently regular exact solution of (23), one has ⎧ ⎨for 1 ≤ r ≤ k if Sk2 (K) =RTk−1 , u − uh 0 ≤ Chr ur (28) ⎩for 1 ≤ r ≤ k + 1 if S 2 (K) =BDMk , k ∇ · (u − uh )0 ≤ Chr ∇ · ur

for 1 ≤ r ≤ k

(29)

φ − φh 0 ≤ Chr (φr + ur )

for 1 ≤ r ≤ k.

(30)

and

It is important to note that if one uses continuous, nodal based finite element spaces for both the scalar variable and the flux, then (24) and (27) are identical discrete systems. It is well known that this leads to unstable approximations, so that one cannot use such pairs of finite element spaces in the MFEMs (24) or (27). 4.1.3. The grid decomposition property. The following result establishes the GDP for the spaces Sk2 used for the discretized Kelvin principle (27);19 for a proof, see [14]. Theorem 4.1. For every uh ∈ Sk2 , there exist w h , zh in Sk2 such that wh · zh d = 0, uh = w h + zh , ∇ · zh = 0, wh 0 ≤ C ∇ · uh −1 + h∇ · uh 0 .

and

(31)

It was shown in [33] that the GDP, i.e., (31), along with the relation Sk3 = ∇ · (Sk2 ), are necessary and sufficient for the stability of the discretized Kelvin principle (27). 19An analogous GDP can be defined in the context of the finite element spaces S 0 used for the discretized m Dirichlet principle (24) but it is trivially satisfied.

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4.2. LSFEMs for the Poisson problem. An LSFEM for the Poisson problem (22) can be defined based on the quadratic functional J (φ, u; f ) = ∇ · u − f 20 + ∇φ + u20

(32)

and the least-squares principle min

(φ,u)∈Hd1 ()×Hn (,div)

J (φ, u; f ).

(33)

Note that we have used the first-order form (23) of the Poisson problem and that we use L2 () norms to measure the equation residuals. Also, we require the functions in the spaces Hd1 () and Hn (, div) to satisfy the boundary conditions φ = 0 on d and u · n = 0 on n , respectively. The Euler–Lagrange equations corresponding to (33) are given by: seek {φ, u} ∈ Hd1 () × Hn (, div) such that B({φ, u}, {ψ, v}) = F ({ψ, v}) for all {ψ, v} ∈ Hd1 () × Hn (, div), where

B({φ, u}, {ψ, v}) =

(34)

(∇φ + u) · (∇ψ + v) d

(∇ · u)(∇ · v) d +

(35)

and F ({ψ, v}) =

f (∇ · v) d.

(36)

0 ×S 2 ⊂ H 1 ()× To define an LSFEM, we restrict (33) to the conforming subspace Sm k d Hn (, div) or, equivalently, restrict (34) to those subspaces to obtain the discrete 0 × S 2 such that problem: seek {φh , uh } ∈ Sm k

B({φh , uh }, {ψh , v h }) = F ({ψh , v h })

0 × S2. for all {ψh , v h } ∈ Sm k

(37)

The next theorem states that the functional (32) is norm equivalent.20 For a proof, see any of [21], [23], [24], [43]. Theorem 4.2. There exist positive constants α1 and α2 such that for any {φ, u} ∈ Hd1 () × Hn (, div), α1 φ21 + u2H (,div) ≤ J (φ, u; 0) ≤ α2 φ21 + u2H (,div) . (38) Thus, the LSFEM defined through (37) is an example of an LSFEM that recovers all the desirable properties of the Rayleigh–Ritz setting, except that by using the finite 0 and S 2 , we have forced ourselves to not use continuous, nodalelement spaces Sm k based finite element spaces for the flux approximation.21 Because we are using finite element spaces that are compatible for the MFEMs (24) and (27), we refer to the LSFEM defined by (37) as a compatible LSFEM. 2 2 1/2 . H (,div) = (u0 + ∇ · u0 ) 21 We could, of course, use such spaces for the flux approximation, but, as indicated previously, we would then not be able to obtain optimal error estimates with respect to L2 () norms. 20 In the theorem, we have that u

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4.2.1. Error estimates in H 1 () × H (, div). We now review the convergence properties of LSFEMs for the Poisson equation with respect to the H 1 ()×H (, div) norm. For a proof of the following theorem, see [14]. Theorem 4.3. Assume that the solution {φ, u} of (34) satisfies {φ, u} ∈ Hd1 () ∩ H m+1 () × Hn (, div) ∩ H k+1 () for some integers k, m ≥ 1. Let {φh , uh } ∈ 0 × S 2 be the solution of the least-squares finite element problem (37). Then there Sm k exists a constant C > 0 such that (39) φ − φh 1 + u − uh H (,div) ≤ C hk uk+1 + hm φm+1 . The error estimate (39) remains valid if u is approximated in the continuous, nodalbased finite element space (Pk ())n . Theorem 4.3 shows that the errors in uh and φh are equilibrated when k = m and that (Sk0 , Sk2 ) has the same asymptotic accuracy in the norm of H 1 () × H (, div) as the C 0 pair (Sk0 , (Pk )n ). For this reason, in the implementation of the LSFEM, one usually chooses the nodal-based pair (Sk0 , (Pk )n ) because it is easier to implement. Indeed, the ability to use equal-order interpolation has been often cited as a primary reason for choosing to use LSFEMs. Nevertheless, the pair is not flawless because optimal L2 norm errors for the flux approximation have proven impossible to obtain without using the very restrictive criss-cross grid or augmenting (23) with an additional redundant curl constraint equation.22 Also, as we have already mentioned, numerical studies in [32] indicate that the L2 convergence of the flux is indeed suboptimal with such finite element spaces. We will see that if the nodal approximation of the flux is replaced by an approximation in Sk2 , it may be possible to recover optimal L2 convergence rates without adding the curl constraint. As in [32], the key to this is the GDP. 4.2.2. Error estimates in L2 . We assume that the solution of the problem −ψ = η in ,

ψ = 0 on d ,

∂ψ = 0 on n ∂n

satisfies the regularity estimate ψs+2 ≤ Cηs for s = 0, 1 and for all η ∈ H s (). This is needed since L2 error estimates are based on duality arguments. L2 error estimates for the scalar variable. Theorem 4.4. Assume that the regularity assumption is satisfied, and assume that the solution (φ, u) of (34) satisfies (φ, u) ∈ Hd1 ()∩H m+1 ()×Hn (, div)∩H k+1 () 22 The redundant curl constraint ∇ × u = 0, first introduced in the least-squares finite element setting in [26] and subsequently utilized by many others (see, e.g., [21], [23], [24], [39]), renders the least-squares functional norm-equivalent with respect to the H 1 () × H 1 () norm but, in some situations, may unduly restrict the range of the data and should be avoided.


1153

0 × S 2 be the solution of the leastfor some integers k, m ≥ 1. Let (φh , uh ) ∈ Sm k squares finite element problem (37). Then there exists a constant C > 0 such that φ − φh 0 ≤ C(hk+1 uk+1 + hm+1 φm+1 ).

For a proof of this theorem, see [14]. The optimal L2 error bound of Theorem 4.4 for the scalar variable does not require that the finite element space for flux approximations satisfy (31), i.e., the GDP. Thus, it remains valid even when continuous, nodal-based finite element spaces are used for the flux approximations, a result first shown in [38]. On the other hand, we will see that the GDP is needed if one wants to improve the L2 accuracy of the flux. L2 error estimate for the flux. The L2 error estimates for approximations to the flux depend on whether Sk2 represents the RTk−1 or the BDMk family. To this end, we have the following result whose proof may be found in [14]. Theorem 4.5. Assume that the hypotheses of Theorem 4.4 hold with k = m = r. Then there exists a constant C > 0 such that ⎧ ⎨hr (ur+1 + φr+1 ) if Sr2 () = RTr−1 , u − uh 0 ≤ C (40) ⎩hr+1 (ur+1 + φr+1 ) if S 2 () = BDMr . r Consider, for example, the lowest-order case for which r = 1, S10 () = P1 , and S12 () is either RT0 or BDM1 . If the least-squares finite element method is implemented with RT0 elements, (40) specializes to u − uh 0 ≤ h(u2 + φ2 ). If instead we use BDM1 elements, we then obtain the improved error bound u − uh 0 ≤ h2 (u2 + φ2 ). 4.3. Interpretation of results and mass conservation. We have seen that an LSFEM method implemented using equal-order, continuous, nodal-based finite element spaces approximates the scalar variable with the same accuracy (with respect to both H 1 () and L2 () norms) as the Galerkin method (25) (or, equivalently, the mixed method (24) for the Dirichlet principle. However, the approximation properties of the Kelvin principle (27) are only partially inherited in the sense that the accuracy in the approximation to the divergence of the flux is recovered, but the accuracy in the flux approximation itself may be of one order less. This should not be too much of a surprise because continuous, nodal-based finite elements provide stable discretization only for the Dirichlet principle (with the exception of the criss-cross grid; see [32]). While least-squares minimization is stable enough to allow for the approximation of scalar variables and the flux by equal-order, continuous, nodal-based finite element

1154


spaces, it cannot completely recover from the fact that such spaces are unstable for the Kelvin principle. The key observation from §4.2.2 is that an LSFEM can inherit the best properties of both the discretized Dirichlet principle (24) and Kelvin principle (27), provided the scalar variable and the flux are approximated by finite element spaces that are stable with respect to these two principles, respectively. Then least-squares finite element solutions recover the accuracy of the Dirichlet principle for the scalar variable and the accuracy of the Kelvin principle for the flux. In a way, we see that, implemented in this particular manner, the LSFEM represents a balanced mixture of the two principles. In [16], an explanation of this observation using the apparatus of differential form calculus is provided as are the results of several illustrative computational experiments. Unlike LSFEMs based on the use of continuous, nodal-based finite element spaces for all variables, it can be shown that through a simple local post-processing procedure, the compatible LSFEM inherits the local mass conservation properties of the discretized Kelvin principle (27); see [16] for details.

5. Alternative LSFEMs The LSFEMs considered so far follow variants of the template established in §2: first, spaces S, H , and H that verify (2) are determined, then a least-squares functional (3) is defined by measuring equation residuals in the norms of H and H and, finally, an LSFEM is obtained by minimizing (3) over a finite-dimensional subspace S h of S. Here, we provide examples of methods that, while still relying on least-squares notions, deviate in more significant ways from that template. 5.1. Collocation LSFEMs. The least-squares optimization steps (3) and (4) precede the discretization step (5). In the broadest sense, collocation LSFEM (CLSFEM) are methods [25], [31], [41] that reverse the order of these two steps. They are also known as point least-squares or overdetermined collocation methods. Let {Uj (x)}jJ=1 denote a basis for a finite element space. We seek an approximate solution of (1) of the form u(x) ≈ uˆ h (x) = jJ=1 cj Uj (x), where c = (c1 , c2 , . . . , cJ ) 1 is a vector of unknown coefficients. Then collocation points {x i }M i=1 ⊂ and M {x i }i=M1 +1 ⊂ are chosen in such a way that the corresponding point residuals Luˆ h (x i ) − f (x i ) and Ruˆ h (x i ) − g(x i ) are well defined. Then a CLSFEM is defined by minimizing, over c ∈ J , the discrete functional

Jc (c; f, g) =

M1 i=1

M 2 2 αi Luˆ h (x i ) − f (x i ) + βi Ruˆ h (x i ) − g(x i ) . i=M1 +1

The weights αi and βi can be used to adjust the relative importance of the terms in the functional. The necessary condition for the minimization of Jc ( · ; f, g) gives rise


1155

to an M × J linear system Ac = b. If M = J , then the method reduces to a standard collocation method. If M > J , the solution c is obtained in a least-squares sense by solving the normal equations AT Ac = AT b. If the collocation points and weights correspond to a quadrature rule, then the CLSFEM is equivalent to an LSFEM in which integrals are approximated by a quadrature rule. Since only a finite set of collocation points belonging to the domain need be specified, collocation LSFEMs are attractive for problems posed on irregularly shaped domains; see [41]. On the other hand, since the normal equations tend to become ill-conditioned, such methods require additional techniques such as scaling or orthonormalization in order to obtain a reliable solution; see [31]. 5.2. Discontinuous LSFEMs. The LSFEMs of §2, 3, and 4 are defined using a conforming finite element subspace S h of the solution space S. Discontinuous LSFEMs (DLSFEMs) are an alternative approach that use finite element subspaces of L2 () that consist of piecewise polynomial functions that are not constrained by interelement continuity requirements. The degrees of freedom on each element can be chosen independently of each other and the elements can have hanging nodes. These features offer great flexibility in implementing adaptive methods because first, resolution on each element can be adjusted as needed and second, new elements can be added by simple subdivisions of existing elements. In general, the least-squares problem (4) cannot be restricted to a discontinuous space S h because it is not a proper subspace of S. To take advantage of discontinuous spaces, it is necessary to modify (3) so that it is well defined on the “broken” (with respect to a partition Th of the domain ) data space S = {u ∈ L2 () | u ∈ S(K) for all K ∈ Th }. The first DLSFEMs appeared in [2], [22] as least-squares formulations for interface and transmission problems for the Poisson equation. We follow [22], where a DLSFEM is developed for the problem ⎧ ⎨ ∇ · (ai ui ) = fi φi = 0 on i,d ⎩ φ1 = φ2

and ui + ∇φi = 0 in i , i = 1, 2, and ui · ni = 0 on i,n , i = 1, 2, and a1 u1 · n1 + a2 u2 · n2 = 0 on 12

(41)

that is a first-order formulation of a transmission problem for the Poisson equation.23 Here, 1 and 2 are two24 open subsets of such that 1 ∪2 = and 1 ∩2 = ∅. The set 12 = 1 ∪2 is the interface between the two subdomains and i,d = d ∩i and i,n = n ∩ i , i = 1, 2. In the conforming case, an LSFEM for the Poisson equation was defined by using the functional (32) and conforming subspaces of the solution space S = Hd1 () × Hn (, div). For the problem (41), we instead use the “broken” (with respect to the 23 The functions a and a denote a “media property” that is discontinuous across . 1 2 12 24 The generalization to more than two subdomains is straightforward.

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partition {1 ,2 }) solution space S = H 1d () × H n (, div), where = {φ1 , φ2 } | φi ∈ H 1 (i ), i = 1, 2} H 1d () = {φ for the scalar variable, d H n (, div) = { u = {u1 , u2 } | ui ∈ Hn (i , div), i = 1, 2} for the flux. To define a DLSFEM, we also need to replace (32) by a least-squares functional that can be minimized over S. Of course, we also want a functional whose minimizer is a solution of (41). A functional with the desired properties is given by (see [22]) u; f 1 , f 2 ) = J12 (φ,

2 ∇ · (ai ui ) − fi 20,h + ui + ∇φi 20,h i=1

+ |φ1 − φ2 21/2,12

i

i

(42)

+ a1 u1 · n1 + a2 u2 · n2 2−1/2,12 .

Interface terms in (42) are treated in exactly the same way as one would impose weak Dirichlet and Neumann conditions, respectively. To obtain a practical method, they are replaced by weighted L2 norms on 12 . Choosing S h ⊂ S completes the formulation of the DLSFEM; see [22] for further details. The Treffetz element least-squares method [42], [45] can be viewed as a variant of the DLSFEM. The term “Treffetz elements” usually refers to methods that use approximation spaces consisting of piecewise analytic solutions of the PDE. Such spaces provide highly accurate approximations of the broken solution space S so that they also require functionals that are well-posed with respect to that space. Given a Treffetz element space, it is a trivial matter to use (42) to define a DLSFEM; see [42], [45] for further details.

6. Open problems in LSFEM We close with a brief discussion of some of the open problems that exist in the theory and application of LSFEMs. 6.1. Hyperbolic PDEs. Recovery of the Rayleigh–Ritz properties by LSFEMs relies on the existence of Hilbert spaces that validate the bounds (2) for (1). Such bounds are natural for elliptic PDEs and can be derived for any such PDE by using the Agmon– Douglis–Nirenberg theory [1]. On the other hand, for hyperbolic PDEs such bounds are not so natural, partly because they admit data in Lp spaces and their solutions may have contact discontinuities and shock waves. Recall that (7) can be viewed as a Galerkin method applied to a higher-order PDE. As a result, LSFEMs for hyperbolic equations designed using a Hilbert space setting are equivalent to a Galerkin discretization of a degenerate elliptic PDE. The result is an LSFEM that will have excellent stability properties but which will smear shocks and discontinuities; see [11] for numerical examples.

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To illustrate some of the pitfalls that can be encountered with hyperbolic PDEs, it suffices to consider the simple linear convection-reaction problem ∇ · (bu) + cu = f in

and

u = g on − ,

(43)

where b is a given convection vector, c(x) is a bounded measurable function on , and − = {x ∈ | n(x) · b(x) < 0} is the inflow part of the boundary . A straightforward L2 () norm-based least-squares principle for (43) is defined by minimizing the functional J (v; f, g) = ∇ · (bv) + cv − f 20 + v − g20,−

(44)

over the Hilbert space S = {u ∈ L2 () | Lu = ∇ · (bv) + cv ∈ L2 ()}. Then the following theorem can be obtained [10]. Theorem 6.1. Assume that − is non-characteristic and c + 21 ∇ · b ≥ σ > 0 for some constant σ . Then J (v; 0, 0) = ∇ · (bv) + cv20 + v20,− is equivalent to the graph norm v2S = v20 + Lv20 . For every f ∈ L2 () and g ∈ L2 (− ), (44) has a unique minimizer u ∈ S and for that u we have that J (u; 0, 0) ≤ f 0 + g20,− . This theorem shows that if the data belongs to L2 , all the prerequisites needed to define an LSFEM are fulfilled. We can proceed as in §2 and define a method in the most straightforward way by restricting the Euler–Lagrange equation corresponding to the minimization of (44) to a finite dimensional subspace S h ⊂ S. However, the convection-reaction problem (43) is meaningful even if the data25 f belongs only to the Banach space L1 (). In this case, proper solution and data spaces for (43) are given by S = {v ∈ L1 () | ∇ · (bv) ∈ L1 ()} and H = L1 (), respectively. One can show [35] that L is an isomorphism S → H and so, instead of (2), we have a similar bound but in Banach spaces: α1 uS ≤ LuH for all u ∈ S. Now, consider the unconstrained minimization problem associated with the spaces S and H : min J1 (u; f ), where J1 (u; f ) = Lu − f L1 () = |Lu − f | d. (45) u∈S

For our model equation (43), this is the “correct” minimization problem that, restricted to S h ⊂ S, will have solutions that do not smear discontinuities. This fact has been recognized independently in [40] and more recently in [35]. In [35], it is also shown that under some reasonable assumptions on S h , the discrete problem min J1 (uh ; f )

uh ∈S h

(46)

has at least one global minimizer, no local minimizers, and a solution that satisfies the stability bound uh S ≤ Cf H . 25 We assume now that g = 0.

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We can view (45) as yet another example of the conflict between practicality and optimality. In this case, however, the practicality issue is much more severe because (45) is not differentiable, we cannot write a first-order optimality condition, and the discrete problem (46) does not give rise to a matrix problem. This is the chief reason that so far there are only two examples [35], [40] of FEMs for (43) based on the L1 optimization problem (45). In [35], the minimizer of (46) is approximated by solving a sequence of regularized L1 optimization problems that are differentiable. The method of [40] uses a sequence of more conventional L2 least-squares approaches, but defined using an adaptively weighted L2 inner product. The weights are used to weaken contributions to the least-squares functional from elements that contain solution discontinuities. At this point, there is very limited experience with solving hyperbolic PDEs by minimizing functionals over Banach spaces. For problems with non-smooth data, computational experiments with the methods of [35] and [40] show that they are superior to LSFEMs defined through the minimization of (44); most notable is their ability to provide sharp discontinuity profiles without over- and under-shooting. A series of experiments in [35] also points strongly towards a possibility that the numerical solutions actually obey a maximum principle on general unstructured grids and that the L1 -based algorithm seems to be able to select viscosity solutions. However, at present, there are no mathematical confirmations of these facts, nor is it known whether such algorithms for hyperbolic conservation laws are able to provide accurate shock positions and speeds. Despite the promise of L1 optimization techniques, the state of LSFEMs for hyperbolic problems is far from satisfactory. Straightforward L2 norm-based LSFEMs are clearly not the most appropriate as they are based on the “wrong” stability estimate for the problem. L1 norm-based techniques give far better results but are more complex and, in the case of [35], require the solution of nonlinear optimization problems. Thus, the jury is still out on whether or not it is possible to define a simple, robust, and efficient LSFEMs for hyperbolic problems that will be competitive with specially designed, upwind schemes employing flux limiters. 6.2. Mass conservation. In §4 it was shown that LSFEMs for the Poisson equation can be implemented in a way that allows them to inherit the best computational properties of MFEMs for the same problem. In particular, it is possible to define an LSFEM for (23) so that the approximation locally conserves mass. Currently, the methods in §4 are the only such example. Achieving local mass conservation in LSFEMs for incompressible, viscous flows remains an important open problem. All existing LSFEMs for incompressible, viscous flows conserve mass only approximately so that ∇ · uh 0 = O(hr ), where r is the approximation order of the finite element space. For low-order elements, which are among the most popular and easy to use elements, LSFEMs have experienced severe problems with mass conservation. For LSFEMs based on the velocity–vorticity–pressure system (16), these problems were first identified in [27] where also a solution was proposed that

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combines least-squares principles and Lagrange multipliers to achieve element-wise mass conservation. Then the resulting restricted LSFEM treats the continuity equation ∇ · u = 0 as an additional constraint that is enforced on each element by a Lagrange multiplier. The method achieves remarkable local conservation but compromises the main motivation underlying LSFEMs: to recover a Rayleigh–Ritz setting for the PDE. In particular, property 4c does not hold. An alternative to exact local conservation is an LSFEM with enhanced total mass conservation. This can be effected by increasing the importance of the continuity residual by using weights. A weighted LSFEM for (16) using the functional h2K W ∇ · u20,K + ∇ × u − ω20,K JW (ω, p, u) = ∇ × ω + ∇p − f 20 + K∈Th

was studied in [28] where numerical studies showed that fairly a small weight, e.g., W = 10, helps to significantly improve total mass conservation. Thus, for the Stokes problem, at present there are methods that either recover local mass conservation but forfeit some important advantages of the Rayleigh–Ritz settings or retain all those advantages but can at best provide improved global conservation. It is of interest to explore whether or not the ideas of §4 can be extended to develop compatible LSFEMs for viscous flows that retain all the Rayleigh–Ritz advantages and at the same time locally conserve mass. 6.3. LSFEMs for nonlinear problems. Consider the nonlinear version of (1) Lu + G(u) = f in

and

Ru = g on ,

(47)

where G(u) is a nonlinear term. Formally, a least-squares principle for (1) can be easily extended to handle (47) by modifying (4) and (3) to min JG (u; f, g), u∈S

where JG (u; f, g) = Lu + G(u) − f 2H + Ru − g2H (48)

and then define an LSFEM by restricting (48) to a family S h ⊂ S. While the extension of LSFEMs to (47) is trivial, its analysis is not and remains one of the open problems in LSFEMs. Compared with the well-developed mathematical theory for linear elliptic problems [2], [13], [18], [21], [23], [24], [26], [32], [38], analyses of LSFEMs for nonlinear problems are mostly confined to the Navier–Stokes equations [7]–[9]. It can be shown that the Euler–Lagrange equation associated with the least-squares principle (48) for the Navier–Stokes equations has the abstract form F (λ, U ) ≡ U + T · G(λ, U ) = 0,

(49)

where λ is the Reynolds number, T is a least-squares solution operator for the associated Stokes problem, and G is a nonlinear operator. As a result, the corresponding discrete nonlinear problem has the same abstract form F h (λ, U h ) ≡ U h + T h · G(λ, U h ) = 0,

(50)

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where T h is an approximation of T . The importance of (50) is signified by the fact that discretization in (50) is introduced solely by means of an approximation to the linear operator T in (49). As a result, under some assumptions, one can show that the error in the nonlinear approximation defined by (50) is of the same order as the error in the least-squares solution of the linear Stokes problem. One of the obstacles in extending this approach to a broader class of nonlinear problems is that after the application of a least-squares principle, the (differentiation) order of the nonlinear term may change.

References [1] Agmon, S., Douglis, A., Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17 (1964), 35–92. [2] Aziz, A., Kellogg, R., Stephens, A., Least-squares methods for elliptic systems. Math. Comp. 44 (1985), 53–70. [3] Babuška, I., Aziz, K., Survey lectures on the mathematical foundations of the finite element method. In The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations (ed. by K. Aziz and I. Babuška), Academic Press, New York 1972. [4] Barth, T., Bochev, P., Gunzburger, M., Shadid, J., A Taxonomy of consistently stabilized finite element methods for the Stokes problem. SIAM J. Sci. Comput. 25 (2004), 1585–1607. [5] Becker, R., Braack, M., A finite element pressure gradient stabilization for the Stokes equations based on local projections. Calcolo 38 (2001), 173–199. [6] Blasco, J., Codina, R., Stabilized finite element method for the transient Navier-Stokes equations based on a pressure gradient projection. Comput. Methods Appl. Mech. Engrg. 182 (2000), 277–300. [7] Bochev, P., Analysis of least-squares finite element methods for the Navier-Stokes equations. SIAM J. Numer. Anal. 34 (1997), 1817–1844. [8] Bochev, P., Cai, Z., Manteuffel, T., McCormick, S., Analysis of velocity-flux least squares methods for the Navier-Stokes equations, Part-I. SIAM. J. Numer. Anal. 35 (1998), 990–1009. [9] Bochev, P., Manteuffel, T., McCormick, S., Analysis of velocity-flux least-squares methods for the Navier-Stokes equations, Part-II. SIAM. J. Numer. Anal. 36 (1999), 1125–1144. [10] Bochev, P., Choi, J., Improved least-squares error estimates for scalar hyperbolic problems. Comput. Meth. Appl. Math. 1 (2001), 115–124. [11] Bochev, P., Choi, J., A comparative numerical study of least-squares, SUPG and Galerkin methods for convection problems. Int. J. Comput. Fluid Dyn. 15 (2001), 127–146. [12] Bochev, P., Dohrmann, C., Gunzburger, M., Stabilization of low-order mixed finite elements for the Stokes equations. SIAM J. Numer. Anal. 44 (1) (2006), 82–101. [13] Bochev, P., Gunzburger, M., Analysis of least-squares finite element methods for the Stokes equations. Math. Comp. 63 (1994), 479–506.


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[14] Bochev, P., Gunzburger, M., On least-squares finite element methods for the Poisson equation and their connection to the Dirichlet and Kelvin principles. SIAM J. Numer. Anal. 43 (2005), 340–362. [15] Bochev, P., Gunzburger, M., An absolutely stable pressure-Poisson stabilized method for the Stokes equations. SIAM J. Numer. Anal. 42 (2005), 1189–1207. [16] Bochev, P., Gunzburger, M., Compatible least-squares finite element methods. SIAM J. Numer. Anal., to appear. [17] Boffi, D., Brezzi, F., Gastaldi, L., On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comp. 69 (2000), 121–140. [18] Bramble, J., Lazarov, R., Pasciak, J., A least squares approach based on a discrete minus one inner product for first order systems. Math. Comp. 66 (1997), 935–955. [19] Brezzi, F., On existence, uniqueness and approximation of saddle-point problems arising from Lagrange multipliers. RAIRO Model. Math. Anal. Numer. 21 (1974) 129–151. [20] Brezzi, F., Fortin, M., Mixed and Hybrid Finite Element Methods. Springer Ser. Comput. Math. 15, Springer-Verlag, New York 1991. [21] Cai, Z., Lazarov, R., Manteuffel, T., McCormick, S., First-order system least squares for second-order partial differential equations: Part I. SIAM J. Numer. Anal. 31 (1994), 1785–1799. [22] Cao, Y., Gunzburger, M., Least-squares finite element approximations to solutions of interface problems. SIAM J. Numer. Anal. 35 (1998), 393-405. [23] Carey, G., Pehlivanov, A., Error estimates for least-squares mixed finite elements. Math. Model Numer. Anal. 28 (1994), 499–516. [24] Chang, C.-L., Finite element approximation for grad-div type systems in the plane. SIAM J. Numer. Anal. 29 (1992), 452–461. [25] Chang, C.-L., Gunzburger, M., A subdomain Galerkin/least squares method for first order elliptic systems in the plane. SIAM J. Numer. Anal. 27 (1990), 1197–1211. [26] Chang, C.-L., Gunzburger, M., A finite element method for first order elliptic systems in three dimensions. Appl. Math. Comp. 23 (1987), 171–184. [27] Chang, C.-L., Nelson, J., Least squares finite element method for the Stokes problem with zero residual of mass conservation. SIAM J. Numer. Anal. 34 (1997), 480–489. [28] Deang, J., Gunzburger, M., Issues related to least-squares finite element methods for the Stokes equations. SIAM J. Sci. Comput. 20 (1998), 878–906. [29] Dohrmann, C., Bochev, P., A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Internat. J. Numer. Methods Fluids 46 (2004), 183–201. [30] Douglas, J., Wang, J.,An absolutely stabilized finite element method for the Stokes problem. Math. Comp. 52 (1989) 495–508. [31] Eason, E., A review of least-squares methods for solving partial differential equations. Internat. J. Numer. Methods Engrg. 10 (1976), 1021–1046 [32] Fix, G., Gunzburger, M., Nicolaides, R., On finite element methods of the least-squares type. Comput. Math. Appl. 5 (1979), 87–98. [33] Fix, G., Gunzburger, M., Nicolaides, R., On mixed finite element methods for first-order elliptic systems. Numer. Math. 37 (1981), 29–48.

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[34] Girault, V., Raviart, P.-A., Finite Element Methods for Navier-Stokes Equations. Springer Ser. Comput. Math. 5, Springer-Verlag, Berlin 1986. [35] Guermond, J.-L., A finite element technique for solving first-order PDEs in Lp . SIAM J. Numer. Anal. 42 (2004), 714–737. [36] Hughes, T., Franca, L., A new finite element formulation for computational fluid dynamics: VII. The Stokes problem with various well-posed boundary conditions: symmetric formulations that converge for all velocity pressure spaces. Comput. Methods Appl. Mech. Engrg. 65 (1987), 85–96. [37] Hughes, T., Franca, L., Balestra, M., A new finite element formulation for computational fluid dynamics: Circumventing the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation of the Stokes problem accommodating equal-order interpolations. Comput. Methods Appl. Mech. Engrg. 59 (1986), 85–99. [38] Jesperson, D., A least-squares decomposition method for solving elliptic equations. Math. Comp. 31 (1977), 873–880. [39] Jiang, B.-N., Povinelli, L., Optimal least-squares finite element methods for elliptic problems. Comput. Methods Appl. Mech. Engrg. 102 (1993), 199–212. [40] Jiang, B.-N., Non-oscillatory and non-diffusive solution of convection problems by the iteratively reweighted least-squares finite element method. J. Comput. Phys. 105 (1993), 108–121. [41] Liable, J., Pinder, G., Least-squares collocation solution of differential equations on irregularly shaped domains using orthogonal meshes. Numer. Methods Partial Differential Equations 5 (1989), 347–361. [42] Monk, P., Wang, D.-Q., A least-squares method for the Helmholtz equation. Comput. Meth. Appl. Mech. Engrg. 175 (1999), 121–136. [43] Pehlivanov, A., Carey, G., Lazarov, R., Least-squares mixed finite elements for secondorder elliptic problems. SIAM J. Numer. Anal. 31 (1994), 1368–1377. [44] Silvester, D., Optimal low order finite element methods for incompressible flow. Comput. Meth. Appl. Mech. Engrg. 111 (1994), 357–368. [45] Stojek, M., Least-squares Treffetz-type elements for the Helmholtz equation. Internat. J. Numer. Methods Engrg. 41 (1998), 831–849. [46] Strang, G., Fix, G., An Analysis of the Finite Element Method. Prentice Hall, Englewood Cliffs, NJ, 1973. Computational Mathematics and Algorithms Department, Sandia National Laboratories, Albuquerque, NM 87185-1110, U.S.A. E-mail: [email protected] School of Computational Science, Florida State University, Tallahassee, FL 32306-4120, U.S.A. E-mail: [email protected]

A posteriori error analysis and adaptive methods for partial differential equations Zhiming Chen∗

Abstract. The adaptive finite element method based on a posteriori error estimates provides a systematic way to refine or coarsen the meshes according to the local a posteriori error estimator on the elements. One of the remarkable properties of the method is that for appropriately designed adaptive finite element procedures, the meshes and the associated numerical complexity are quasi-optimal in the sense that in two space dimensions, the finite element discretization error is proportional to N −1/2 in terms of the energy norm, where N is the number of elements of the underlying mesh. The purpose of this paper is to report some of the recent advances in the a posteriori error analysis and adaptive finite element methods for partial differential equations. Emphases will be paid on an adaptive perfectly matched layer technique for scattering problems and a sharp L1 a posteriori error analysis for nonlinear convection-diffusion problems. Mathematics Subject Classification (2000). Primary 65N15; Secondary 65N30. Keywords. A posteriori error estimates, adaptivity, quasi-optimality.

1. Introduction The aim of the adaptive finite element method (AFEM) for solving partial differential equations is to find the finite element solution and the corresponding mesh with least possible number of elements in terms of discrete errors. The task to find the mesh with the desired property is highly nontrivial because the solution is a priori unknown. The basic idea of the seminal work [3] is to find the desired mesh under the principle of error equidistribution, that is, the discretization errors should be approximately equal on each element. The errors on the elements which are also unknown can, however, be estimated by a posteriori error estimates. Today AFEM based on a posteriori error estimates attracts increasing interests and becomes one of the central themes of scientific computation. The purpose of this paper is to report some of the recent advances in the a posteriori error analysis and AFEM for partial differential equations. A posteriori error estimates are computable quantities in terms of the discrete solution and data, which provide information for adaptive mesh refinement (and coarsening), error control, and equidistribution of the computational effort. We describe here briefly the basic idea of AFEM using the example of solving the Possion equation ∗ The author is grateful to the support of China National Basic Research Program under the grant 2005CB321701 and the China NSF under the grant 10025102 and 10428105.


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Zhiming Chen

on a polygonal domain in R2 −u = f in ,

u = 0 on ∂.

(1.1)

Here the source function f is assumed to be in L2 (). It is well known that the solution of the problem (1.1) may be singular due to the reentrant corners of the domain in which case the standard finite element methods with uniform meshes are not efficient. Let Mh be a regular triangulation of the domain and Bh be the collection of all inter-element sides of Mh . Denote by uh the piecewise linear conforming finite element solution over Mh . For any inter-element side e ∈ Bh , let e be the collection of two elements sharing e and define the local error indicator ηe as 1/2 ηe2 := hK f 2L2 (K) + he Je 2L2 (e) , K∈e

where hK := diam(K), he := diam(e), and Je := [[ ∇uh ]]e · ν stands for the jump of flux across side e which is independent of the orientation of the unit normal ν to e. The following a posteriori error estimate is well known [2]: u − uh 2H 1 () ≤ C ηe2 . e∈Bh

That ηe really indicates the error locally is explained by the following local lower bound [39]: ηe2 ≤ C u − uh 2L2 (K) + C hK (f − fK ) 2L2 (K) , K∈e

K∈e

1 where fK = |K| K f dx. Based on the local error indicator, the usual adaptive algorithm solving the elliptic problem (1.1) reads as follows:

Solve → Estimate → Refine. The important convergence property, which guarantees the iterative loop terminates in finite steps starting from any initial coarse mesh, is proved in [23], [30]. It is also widely observed that for appropriately designed adaptive finite element procedures, the meshes and the associated numerical complexity are quasi-optimal in the sense that ∇(u − uh ) L2 () ≈ CN −1/2

(1.2)

is valid asymptotically, where N is the number of elements of the underlying finite element mesh. Since the nonlinear approximation theory [5] indicates that N −1/2 is the highest attainable convergence order for approximating functions in H 1 () in

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Adaptive finite element methods

two space dimensions over a mesh with N elements, one concludes that AFEM is an optimal discretization method for solving the elliptic problem (1.1). In Section 2 we consider to use AFEM to solve the Helmholtz-type scattering problems with perfectly conducting boundary u + k 2 u = 0 ∂u = −g ∂n √ ∂u − iku → 0 r ∂r

¯ in R2 \D,

(1.3a)

on D ,

(1.3b)

as r = |x| → ∞.

(1.3c)

Here D ⊂ R2 is a bounded domain with Lipschitz boundary D , g ∈ H −1/2 (D ) is determined by the incoming wave, and n is the unit outer normal to D . We assume the wave number k ∈ R is a constant. We study an adaptive perfectly matched layer (APML) technique to deal with the Sommerfeld radiation condition (1.3c) in which the PML parameters such as the thickness of the layer and the fictitious medium property are determined through sharp a posteriori error estimates. The APML technique combined with AFEM provides a complete numerical strategy for solving the scattering problem in the framework of finite element which has the nice property that the total computational costs are insensitive to the thickness of the PML absorbing layers. The quasi-optimality of underlying FEM meshes is also observed. Things become much more complicated when applying AFEM to solve timedependent partial differential equations. One important question is if one should use the adaptive method of lines (AML) in which variable timestep sizes (but constant at each time step) and variable space meshes at different time steps are assumed, or one should consider the space-time adaptive method in which space-time domain is considered as a whole and AFEM is used without distinguishing the difference of time and space variables. Our recent studies in [9], [10], [11] reveal that with sharp a posteriori error analysis and carefully designed adaptive algorithms, the AML method produces the very desirable quasi-optimal decay of the error with respect to the computational complexity |||u − U |||×(0,T ) ≤ CM −1/3

(1.4)

for a large class of convection-diffusion parabolic problems in two space dimensions using backward Euler scheme in time and conforming piecewise linear finite elements in space. Here |||u − U |||×(0,T ) is the energy norm of the error between the exact solution u and the discrete solution U , and M is the sum of the number of elements of the space meshes over all time steps. Thus if one takes the quasi-optimality of the computational complexity as the criterion to assess the adaptive methods, then the space-time adaptive method which is less studied in the literature will not have much advantage over the AML method. A posteriori error analysis for parabolic problems in the framework of AML has been studied intensively in the literature. The main tool in deriving a posteriori error

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Zhiming Chen

estimates in [25], [26], [14], [31], [7] is the analysis of linear dual problems of the corresponding error equations. The derived a posteriori error estimates, however, depend on the H 2 regularity assumption on the underlying elliptic operator. Without using this regularity assumption, energy method is used in [34], [9] to derive an a posteriori error estimate for the total energy error of the approximate solution for linear heat equations. A lower bound for the local error is also derived for the associated a posteriori error indicator in [34], [9]. In [9] an adaptive algorithm is constructed which at each time step, is able to reduce the error indicators (and thus the error) below any given tolerance within finite number of iteration steps. Moreover, the adaptive algorithm is quasi-optimal in terms of energy norm. In [10] an quasi-optimal AML method in terms of the energy norm is constructed for the linear convection-dominated diffusion problems based on L1 a posteriori error estimates. In Section 3 we study the AML method for the initial boundary value problems of nonlinear convection-diffusion equations of the form ∂u + divf (u) − A(u) = g. ∂t We derive sharp L∞ (L1 ) a posteriori error estimates under the non-degeneracy assumption A (s) > 0 for any s ∈ R. The problem displays both parabolic and hyperbolic behavior in a way that depends on the solution itself. It is discretized implicitly in time via the method of characteristic and in space via continuous piecewise linear finite elements. The analysis is based on the Kružkov “doubling of variables” device and the recently introduced “boundary layer sequence” technique to derive the entropy error inequality on bounded domains. The derived a posteriori error estimate leads to a quasi-optimal adaptive method in terms of the L∞ (L1 ) norm of the error.

2. The APML technique for scattering problems In this section we consider the APML technique for the scattering problem (1.3a)–(1.3c). Since [4] proposed a PML technique for solving the time dependent Maxwell equations, various constructions of PML absorbing layers have been proposed and studied in the literature [38], [37]. Here we introduce the PML technique for (1.3a)–(1.3c) following the method in [19]. Let D be contained in the interior of the circle BR = {x ∈ R2 : |x| < R}. In the domain R2 \B¯ R , the solution u of (1.3a)–(1.3c) can be written under the polar coordinates as follows: 2π Hn(1) (kr) 1 inθ u(r, θ) = u(R, θ)e−inθ dθ. (2.1) uˆ n e , uˆ n = 2π 0 Hn(1) (kR) n∈Z

where Hn(1) is the Hankel function of the first kind and order n. The series in (2.1) converges uniformly for r > R [20].

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The basic idea of PML technique is to surround the fixed domain R = BR \D¯ with a PML layer of thickness ρ −R and choose the fictitious medium property so that either the wave never reaches its external boundary or the amplitude of the reflected wave is so small that it does not essentially contaminate the solution in R . Let α = 1 + iσ be the model medium property satisfying σ ∈ C(R), σ ≥ 0, and σ = 0 for r ≤ R. The most widely used model medium property σ in the literature is the power function, that is, r −R m σ = σ0 , m ≥ 1, σ0 > 0 constant. (2.2) ρ−R Denote by r˜ the complex radius defined by r r˜ = r˜ (r) = r 0 α(t)dt = rβ(r) Since Hn(1) (z) ∼

if r ≤ R, if r ≥ R.

2 i (z− π2 n− π4 ) πz e

as |z| → ∞, [19] obtained the PML equation by considering the following extension of u in the exterior domain R2 \B¯ R : Hn(1) (k r˜ ) w(r, θ) = uˆ n einθ , (1) H (kR) n n∈Z

1 uˆ n = 2π

2π

u(R, θ)e−inθ dθ.

(2.3)

0

It is easy to check that w satisfies ∇ · (A∇w) + αβk 2 w = 0

in R2 \B¯ R ,

where A = A(x) is a matrix which satisfies, in polar coordinates, 1 ∂ βr ∂ α 1 ∂2 ∇ · (A∇) = . + r ∂r α ∂r β r 2 ∂θ 2 The PML problem then becomes ¯ ∇ · (A∇ u) ˆ + αβk 2 uˆ = 0 in Bρ \D, ∂ uˆ = −g on D , uˆ = 0 on ρ . ∂n

(2.4a) (2.4b)

It is proved in [22], [21] that the resultant PML solution converges exponentially to the solution of the original scattering problem as the thickness of the PML layer tends to infinity. We remark that in practical applications involving PML techniques, one cannot afford to use a very thick PML layer if uniform finite element meshes are used because it requires excessive grid points and hence more computer time and more storage. On the other hand, a thin PML layer requires a rapid variation of the artificial material property which deteriorates the accuracy if too coarse mesh is used in the PML layer.

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The APML technique was first proposed in [16] for solving scattering by periodic structures (the grating problem) which uses a posteriori error estimates to determine the PML parameters such as the thickness and the medium property σ0 in the (2.2). For the scattering problem (1.3a)–(1.3c), the main difficulty of the analysis is that in contrast to the grating problems in which there are only finite number of outgoing modes, now there are infinite number of outgoing modes expressed in terms of Hankel functions. We overcome this difficulty by the by exploiting the following uniform estimate for the Hankel functions Hν1 , ν ∈ R. Lemma 2.1. For any ν ∈ R, z ∈ C++ = {z ∈ C : (z) ≥ 0, (z) ≥ 0}, and ∈ R such that 0 < ≤ |z|, we have |Hν (z)| ≤ e (1)

2 1/2 −(z) 1− 2 |z|

|Hν(1) ()|.

The proof of the lemma which depends on the Macdonald formula for the modified Bessel functions can be found in [12]. Lemma 2.1 allows us to prove the exponentially decaying property of the PML solution without resorting to the integral equation technique in [22] or the representation formula in [21]. As a corollary of Lemma 2.1, we know that the function w in (2.3) satisfies w H 1/2 (ρ ) ≤ e

−k(ρ) ˜ 1−

R2 |ρ| ˜2

1/2

u H 1/2 (R ) .

We remark that in [22], [21], it is required that the fictitious absorbing coefficient must be linear after certain distance away from the boundary where the PML layer is placed. We also remark that since (2.5) is valid for all real order ν, the results of [12] can be extended directly to study three dimensional Helmholtz-type scattering problems. Let Mh be a regular triangulation of Bρ \D¯ and uh be the finite element solution of the PML problem (2.4a)–(2.4b). Let Bh denote the set of all sides that do not lie on D and ρh . For any K ∈ Mh , we introduce the residual Rh := ∇ · (A∇uh |K ) + αβk 2 uh |K . For any interior side e ∈ Bh which is the common side of K1 and K2 ∈ Mh , we define the jump residual across e: Je := (A∇uh |K1 − A∇uh |K2 ) · νe , using the convention that the unit normal vector νe to e points from K2 to K1 . If e = D ∩ ∂K for some element K ∈ Mh , then we define the jump residual Je := 2(∇uh |K · n + g) For any K ∈ Mh , denote by ηK the local error estimator which is defined by

1/2 1 he Je 2L2 (e) , ηK = max ω(x) · hK Rh 2L2 (K) + 2 x∈K˜ e⊂∂K

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where K˜ is the union of all elements having nonempty intersection with K, and ⎧ ¯ ⎨1 if x ∈ B¯ R \D,

1/2 2 ω(x) = ⎩|α α|e−k(˜r ) 1− |˜rr |2 if x ∈ B¯ ρ \BR . 0 Theorem 2.2. There exists a constant C depending only on the minimum angle of the mesh Mh such that the following a posterior error estimate is valid:

1/2 2 u − uh H 1 (R ) ≤ C(kR)1/2 (1 + kR) ηK K∈Mh

2 1/2 ˜ 1− R 2 2 −k(ρ) |ρ| ˜

+ C(1 + kR) |α0 | e (1) |H0 (kR)| . Here (kR) = max 1, (1) 2

uh H 1/2 (R ) .

|H0 (kR)|

From Theorem 2.2 we know that the a posteriori error estimate consists of two parts: the PML error and the finite element discretization error. An adaptive algorithm is developed in [12] which uses the a posteriori error estimate to determine the PML parameters. We first choose ρ and σ0 such that the exponentially decaying factor ωˆ = e

−k(ρ)(1− ˜

R 2 1/2 ) |ρ| ˜2

≤ 10−8 ,

which makes the PML error negligible compared with the finite element discretization errors. Once the PML region and the medium property are fixed, we use the standard finite element adaptive strategy to modify the mesh according to the a posteriori error estimate. The extensive numerical experiments reported in [12] show the competitive behavior of the proposed adaptive method. In particular, the quasi-optimality of meshes is observed and the adaptive algorithm is robust with respect to the choice of the thickness of PML layer: the far fields of the scattering solutions are insensitive to the choices of the PML parameters.

3. The AML method for nonlinear convection diffusion problems Let is a bounded domain in Rd (d = 1, 2, 3) with Lipschitz boundary and T > 0. In this section we consider the following nonlinear convection-diffusion equation: ∂u + divf (u) − A(u) = g ∂t

in Q

(3.1)

with the initial and boundary conditions u|t=0 = u0 ,

u|∂×(0,T ) = 0.

(3.2)

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Here u = u(x, t) ∈ R, with (x, t) ∈ Q = × (0, T ). We assume that the function f : R → Rd is locally Lipschitz continuous, the function A : R → R is nondecreasing and locally Lipschitz continuous, g ∈ L∞ (Q) and u0 ∈ L∞ (). Problems of the type (3.1) model a wide variety of physical phenomena including porous media flow, flow of glaciers and sedimentation processes, or flow transport through unsaturated porous media which is governed by the so-called Richards equation. For the Richards equation, the existence of weak solutions is considered in [1] and the uniqueness of weak solutions is proved in [33] based on the Kružkov “doubling of variables” technique. Entropy solutions for (3.1) are studied in [6], [29]. The discretization of (3.1) is based on combining continuous piecewise linear finite elements in space with the characteristic finite difference in time. The method of characteristic originally proposed in [24], [35] is widely used to solve convectiondiffusion problems in finite element community (cf. e.g. [26], [14]). Given Uhn−1 as the finite element approximation of the solution at time t n−1 , let τn and V0n ⊂ H01 () be the time step and the conforming linear finite element space at the nth time step, then our discrete scheme reads as following: find Uhn ∈ V0n such that Uhn − U¯ hn−1 , v + ∇A(Uhn ), ∇v = g¯ n , v for all v ∈ V0n , (3.3) τn tn ˜ n−1 )), and the approximate where g¯ n = τn−1 t n−1 g(x, t)dt, U¯ hn−1 (x) = Uhn−1 (X(t ˜ characteristic X(t) is defined by ˜ ˜ d X/dt = f (Uhn−1 (X(t))),

˜ n ) = x. X(t

The well-known Kružkov “doubling of variables” technique originally appeared in [28] plays a decisive role in the error estimation (both a posteriori and a priori) for numerical schemes solving the Cauchy problems of nonlinear conservation laws (see e.g. [17], [18], [27] and the reference therein). It is also used recently in [32] for the implicit vortex centered finite volume discretization of the Cauchy problems of (3.1) for general non-negative A (s) ≥ 0 for all s ∈ R. The √ common feature of these studies is that the derived error indicators are of the order h in the region where the solution is smooth, where h is the local mesh size. We remark that in the region where the diffusion is dominant, the error indicators developed for the parabolic equations (cf. e.g. [34], [9]) are of order h. Thus the degeneration of the order of the error indicators used in [32] may cause over-refinements for the solution of (3.1) in the region where the diffusion is dominant. The basic assumption in this paper is that the diffusion is positive A (s) > 0,

for all s ∈ R.

This assumption includes the Richards equation and the viscosity regularization of degenerate parabolic equations, for example, the regularized continuous casting problem which is considered in [14]. The novelty of our analysis with respect to the analysis

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for nonlinear conservation laws in [17], [18], [27] or nonlinear degenerate parabolic equations in [32] lies in the following aspects. Firstly, only Cauchy problems are considered in [17], [18], [27], [32]. The difficulty to include boundary condition is essential. Here we use the recently introduced technique of “boundary layer sequence” in [29] to overcome the difficulty. The technique of “boundary layer sequence” allows us to truncate the standard Kružkov test function (see Definition 3.4 below) to obtain the admissible test function in the entropy error identity. Secondly, the nature of the estimators are different: our estimators emphasize the diffusion effect of the problem which requires the assumption A (s) > 0 for any s ∈ R; the estimates in [32] are valid for any nonlinear function A such that A (s) ≥ 0. The nice consequence of the analysis is that our a posteriori error estimates are able to recover the standard sharp a posteriori error estimators in the literature derived for parabolic problem with diffusion coefficients bounded uniformly away from zero. Now we elaborate the main steps to derive sharp L1 a posteriori error estimate for the discrete scheme (3.3) based on the Kružkov “doubling of variables” device. By testing (3.1) with any function ϕ ∈ L2 (0, T ; H01 ()) such that φ( ·, 0) = φ( ·, T ) = 0, we have T ∂t u, ϕ dt + (−f (u) + ∇A(u)) · ∇ϕ dxdt = gϕ dxdt. (3.4) 0

Q

Q

For any ε > 0, let Hε (z) = sgn(z) min(1, |z|/ε) be the regularization of the sign function sgn(z). For any k ∈ R, define the entropy pair (Uε , Fε ) by z z Uε (z, k) = Hε (A(r) − A(k)) dr, Fε (z, k) = Hε (A(r) − A(k))f (r) dr. k

k

The following result is well known (cf. e.g. [6], [29]) by taking ϕ = Hε (A(u)−A(k))φ in (3.4). Lemma 3.1. For any φ ∈ L2 (0, T ; H01 ()) such that φ( ·, 0) = φ( ·, T ) = 0, and any k ∈ R, we have − Uε (u, k)∂t φ − Fε (u, k) · ∇φ + Hε (A(u) − A(k))∇A(u) · ∇φ Q Q Q Hε (A(u) − A(k))|∇A(u)|2 φ + (3.5) Q = gHε (A(u) − A(k))φ. Q

Let (H 1 ()) be the dual space of H 1 (), we define the discrete residual R ∈ L2 (0, T ; (H 1 ()) ) through the following relation, for any ϕ ∈ H 1 (), ∂t Uh , ϕ − f (Uh ), ∇ϕ + ∇A(Uh ), ∇ϕ = g, ϕ − R, ϕ.

(3.6)

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For any k ∈ R, by taking ϕ = Hε (A(Uh )−A(k ))φ in (3.6), we have the following result. Lemma 3.2. For any φ ∈ L2 (0, T ; H01 ()) such that φ( ·, 0) = φ( ·, T ) = 0, and any k ∈ R, we have − Uε (Uh , k )∂t φ − Fε (Uh , k ) · ∇φ Q Q Hε (A(Uh ) − A(k ))∇A(u) · ∇φ + Q (3.7) Hε (A(Uh ) − A(k ))|∇A(Uh )|2 φ + Q

=

T

gHε (A(Uh ) − A(k ))φ −

R, Hε (A(Uh ) − A(k ))φ.

0

Q

Now we are going to apply the Kružkov “doubling of variables” technique and will always write u = u(y, s), Uh = Uh (x, t), unless otherwise stated. By taking k = Uh (x, t) in (3.5) and k = u(y, s) in (3.7), we have the following entropy error identity. Lemma 3.3. Let φ = φ(x, t; y, s) be non-negative function such that (x, t) → φ(x, t; y, s) ∈ Cc∞ (Q) for every (y, s) ∈ Q, (y, s) → φ(x, t; y, s) ∈ Cc∞ (Q) for every (x, t) ∈ Q. Then we have −

Uε (u, Uh )(∂t φ + ∂s φ) − Fε (u, Uh )(∇x φ + ∇y φ) Q×Q + Hε (A(u) − A(Uh ))∇y A(u) · (∇x φ + ∇y φ) Q×Q Hε (A(Uh ) − A(u))∇x A(Uh ) · (∇x φ + ∇y φ) + Q×Q Hε (A(u) − A(Uh ))|∇x A(Uh ) − ∇y A(u)|2 φ + Q×Q =− ∂t [Uε (Uh , u) − Uε (u, Uh ))]φ Q×Q − ∇x [Fε (Uh , u) − Fε (u, Uh ))]φ

Q×Q

Q×Q

T

− Q(y,s)

0

R, Hε (A(Uh ) − A(u))φ dt.

(3.8)

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The next objective is to remove the restriction that the test functions in the entropy error identity (3.8) must have vanishing trace. This is achieved by using the technique of boundary layer sequence introduced in [29]. For any δ > 0, the boundary layer sequence ζδ is defined as the solution of the elliptic problem −δ 2 ζδ + ζδ = 1 in ,

ζδ = 0 on ∂.

We specify now the choice of the test function φ in the entropy error identity (3.8), which is similar to that used in [29]. Definition 3.4. Let φ(x, t, y, s) = ζδ (x)ζη (y)ξ(x, t, y, s)θ(t), where θ ∈ Cc∞ (0, T ) such that θ ≥ 0, and ξ is defined as follows. Let {ϕj }0≤j ≤J be ¯ ⊂ ∪J Bj , a partition of unity subordinate to open sets B0 , B1 , . . . , BJ such that j =0 B0 ⊂⊂ and ∂ ⊂ ∪jJ=1 Bj . Let ϕˆj ∈ Cc∞ (Rd ), 0 ≤ ϕˆj ≤ 1, such that supp(ϕˆj ) ⊂ Bj and ϕˆj (x) = 1 on the support of ϕj so that ϕj (x)ϕˆj (x) = ϕj (x). We use ϕj as a function of y and ϕˆj as a function of x, and denote ϕˆj (x)ϕj (y) = ψj (x, y). Define ξ(x, t, y, s) =

J

ωl (t − s)ωm (x − y )ωn (xd − yd )ψj (x, y),

j =0

where ωl , ωn are sequences of symmetric mollifiers in R, ωm is a sequence of symmetric mollifier in Rd−1 , and for j = 1, 2, . . . , J , x = (x , xd ), y = (y , yd ) are local coordinates induced by ψj (x, y) in Bj , that is, Bj ∩ ∂ = {x ∈ Bj : xd = ρj (x )}, B ∩ = {x ∈ Bj : xd < ρj (x )} for some Lipschitz continuous function ρj : Rd−1 → R. By taking limit δ, η → 0 in the entropy error identity (3.8), we obtain the following entropy error inequality. Theorem 3.5. Let θ and ξ be defined in Definition 3.4. Then we have the following entropy error inequality: − Uε (u, Uh )ξ θt − Kε (u, Uh ) · (∇x ξ + ∇y ξ )θ Q×Q Q×Q Hε (A(u) − A(Uh ))|∇x A(Uh ) − ∇y A(u)|2 ξ θ + Q×Q ≤− ∂t [Uε (Uh , u) − Uε (u, Uh ))]ξ θ (3.9) Q×Q − ∇x [Fε (u, Uh ) − Fε (u, Uh ))]ξ θ Q×Q

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Zhiming Chen

Fε (u, Uh ) − Hε (A(u) − A(Uh ))∇y A(u) · νx ξ θ

− Q(y,s)

(x,t)

Fε (u, Uh ) − Hε (A(Uh ) − A(u))∇x A(Uh ) · νy ξ θ

− Q(x,t)

(y,s)

T

− Q(y,s)

R, Hε (A(Uh ) − A(u))ξ θ dt,

0

where Kε (u, Uh ) = Fε (u, Uh ) − Hε (A(u) − A(Uh ))(∇y A(u) − ∇x A(Uh )), = ∂ × (0, T ), and (x,t) or (y,s) are the domain of integration of with respect to (x, t) or (y, s) respectively. For any ε > 0 and z ∈ R, define ν(ε, z) = min{A (s) : |A(s) − A(z)| ≤ ε}. Assume A A−1 is Lipschitz, then we have the following elementary estimate which extends the result in [18, Corollary 6.4]: ε K1 , ν(ε, z) ε K2 , |∂z [Fε (z, k) − Fε (k, z)]| ≤ ν(ε, z)

|∂z [Uε (z, k) − Uε (k, z)]| ≤

(3.10)

where k, z ∈ R, K1 = L(A A−1 ), K2 = K1 f L∞ (R) + L(f ) with L(A A−1 ) and L(f ) being the Lipschitz constant of A A−1 and f respectively. To complete the Kružkov “doubling of variables” technique, we let first l, m → ∞ then n → ∞ in the entropy error inequality (3.9). The first two terms on the righthand side of (3.9) can be treated by using (3.10) and the third and fourth terms can be shown to tend to zero. Thus we have − Uε (u, Uh )θt + Hε (A(u) − A(Uh ))|∇(A(Uh ) − A(u))|2 θ Q

Q

≤ Kε Q

1 (|∂t Uh | + |∇x Uh |)θ − ν(ε, Uh )

T

R, Hε (A(Uh ) − A(u))θ dt.

0

where K = max(K1 , K2 ). To proceed, we introduce the interior residual R n := g¯ n −

Uhn − U¯ hn−1 + A(Uhn ) τn

where we recall that g¯ n = τn−1

tn t n−1

g(x, t) dt.

on any K ∈ Mn ,

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Theorem 3.6. Let ε0 = 3i=1 Ei , where E1 , E2 , E3 are the error indicators defined below. For any m ≥ 1, let Qm = × (0, t m ), and define 1 1 (|∂t Uh | + |∇Uh |) + , (3.11) m = max 1, m Qm ν(ε0 , Uh ) ν(ε0 , Uh ) where for any z ∈ R, ν(ε0 , z) = min{A (s) : |A(s) − A(z)| ≤ ε0 }. Then there exists a constant C depending only on the minimum angles of the meshes Mn , n = 1, . . . , m, such that the following a posteriori error estimate is valid: u

m

− Uhm L1 ()

≤

1/2 E0 + E4 + E5 + Cm

3

Ei ,

i=1

where the error indicators Ei , i = 0, . . . , 5, are defined by E0 = u0 − Uh0 L1 () E1 =

m

initial error

1/2

τn |||hn [[∇A(Uhn )]]|||2L2 ()

1/2

jump residual

n=1

E2 =

m

τn hn R n 2L2 ()

1/2

interior residual

n=1

E3 =

m

τn ∇(A(Uhn ) − A(Uhn−1 )) 2L2 ()

1/2

time residual

n=1

E4 =

m n=1

E5 =

m n=1

U n − U¯ n−1 h h − (∂ U + divf (U )) t h h n−1 τ n t tn

tn

t n−1

g − g¯ n L1 () dt

dt L1 ()

characteristic and coarsening source.

In the case of strong diffusion A (s) ≥ β > 0 for any s ∈ R and A is uniformly Lipschitz continuous, then n is bounded by β −1 Uh BV (Qn ) which is expected to be bounded in practical computations. The a posteriori error estimator in Theorem 3.6 then recovers the standard a posteriori error estimator derived in the literature for parabolic problems [34], [9]. In particular, the space error indicators E1n , E2n , which control the adaptation of finite element meshes at each time step, are sharp in the sense that a local lower bound for the error can be established by extending the argument in [9, Theorem 2.2] for linear parabolic equations. We also remark that the method of the a posteriori error analysis here is different from those for nonlinear conservation laws in [17], [18], [27] or nonlinear degenerate parabolic equations in [32]. Recall that there are several parameters introduced in the analysis:

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Zhiming Chen

• The regularizing parameter ε in Hε (z). • The boundary layer sequence parameters δ and η, and the mollifier parameters l, m and n. The analysis for Cauchy problems in [17], [18], [27] is based on letting ε → 0 and taking finite mollifier parameters l, m, n. The analysis in [32] takes both finite ε and finite mollifier parameters l, m, n. Note that there are no boundary layer sequence parameters δ, η for the analysis for Cauchy problems. The analysis in this paper is based on letting δ, η → 0 and l, m, n → ∞ but taking a finite ε. We are not able to use the same technique as that in [17], [18], [27], [32] by choosing finite mollifier parameters l, m, n to treat the problem with boundary conditions. Based on the a posteriori error estimate in Theorem 3.6, an adaptive algorithm is proposed and implemented in [11]. In particular, the numerical experiments in [11] indicate that the total estimated error is roughly proportional to M −1/3 , i.e. η ≈ CM −1/3 for some constant C > 0. This implies the quasi-optimal decay of the error Hε (A(u) − A(Uh ))|∇(A(Uh ) − A(u))|2 ≤ CM −1/3 u − Uh L∞ (0,T ;L1 ()) + Q

is valid asymptotically. Here M is the sum of the number of elements of the space meshes over all time steps. Figure 3.1 shows the meshes and the surface plots of the solutions at time t = 0.251278 and t = 0.500878 for the Burger’s equation with small viscosity ∂u + u∂x u − εu = 0 ∂t

in Q,

where = (0, 1)2 , T = 1.0, ε = 10−3 , and the initial condition and boundary condition u(x, y, t)|∂ = u0 (x, y) = 0.5 sin(πx) + sin(2π x). The adaptive algorithm is based on the a posteriori error estimate in Theorem 3.6 and is described in [11]. We observe from Figure 3.1 that the method captures the internal and boundary layers of the solution. Acknowledgment. The author would like to thank Shibin Dai, Guanghua Ji, Feng Jia, Xuezhe Liu, Ricardo H. Nochetto, Alfred Schmidt, and Haijun Wu for the joint work through the years.

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Adaptive finite element methods 1

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Figure 3.1. The meshes (top) and the surface plots (bottom) of the solutions t = 0.400317 (left) and t = 1.0 (right) with 35286 and 5020 nodes.

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References [1] Alt, H. W., and Luckhaus, S., Quasilinear elliptic-parabolic differential equations. Math. Z. 183 (1983), 311–341. [2] Babuška, I., and Miller, A., A feedback finite element method with a posteriori error estimation: Part I. The finite element method and some basic properties of the a posteriori error estimator. Comput. Meth. Appl. Mech. Engrg. 61 (1987), 1–40. [3] Babuška, I., and Rheinboldt, C., Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15 (1978), 736–754. [4] Berenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Physics 114 (1994), 185–200. [5] Binev, P., Dahmen, W., and DeVore, R., Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004), 219–268. [6] Carrillo, J., Entropy solutions for nonlinear degenerate problems. Arch. Rational Mech. Anal. 147 (1999), 269–361. [7] Chen, Z., and Dai, S., Adaptive Galerkin methods with error control for a dynamical Ginzburg-Landau model in superconductivity. SIAM J. Numer. Anal. 38 (2001), 1961–1985. [8] Chen, Z., and Dai, S., On the efficiency of adaptive finite element methods for elliptic problems with discontinuous coefficients. SIAM J. Sci. Comput. 24 (2002), 443–462. [9] Chen, Z., and Jia, F., An adaptive finite element method with reliable and efficient error control for linear parabolic problems. Math. Comp. 73 (2004), 1163–1197. [10] Chen, Z., and Ji, G., Adaptive computation for convection dominated diffusion problems. Sci. China Ser. A 47 (Supplement) (2004), 22–31. [11] Chen, Z., and Ji, G., Sharp L1 a posteriori error analysis for nonlinear convection-diffusion problems. Math. Comp. 75 (2006), 43–71. [12] Chen, Z., and Liu, X., An Adaptive Perfectly Matched Layer Technique for Time-harmonic Scattering Problems. SIAM J. Numer. Anal. 43 (2005), 645–671. [13] Chen, Z., and Nochetto, R. H., Residual type a posteriori error estimates for elliptic obstacke problems. Numer. Math. 84 (2000), 527–548. [14] Chen, Z., Nochetto, R. H., and Schmidt, A., A characteristic Galerkin method with adaptive error control for continuous casting problem. Comput. Methods Appl. Mech. Engrg. 189 (2000), 249–276. [15] Chen, Z., Nochetto, R. H., and Schmidt, A., Error control and adaptivity for a phase relaxation model. Math. Model. Numer. Anal. 34 (2000), 775–797. [16] Chen, Z., and Wu, H., An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures. SIAM J. Numer. Anal. 41 (2003), 799–826. [17] Cockburn, B., Coquel, B. F., and Lefloch, P. G., An error estimate for finite volume methods for multidimensional conservation laws. Math. Comp. 63 (1994), 77–103. [18] Cockburn, B., and Gremaud, P.-A., Error estimates for finite element methods for scalar conservation laws. SIAM J. Numer. Anal. 33 (1996), 522–554. [19] Collino, F., and Monk, P. B., The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19 (1998), 2061–2090.


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[20] Colton D., and Kress R., Integral Equation Methods in Scattering Theory. John Wiley & Sons, New York 1983. [21] Hohage, T., Schmidt, F., and Zschiedrich, L., Solving time-harmonic scattering problems based on the pole condition. II: Convergence of the PML method. SIAM J. Math. Anal., to appear. [22] Lassas, M., and Somersalo, E., On the existence and convergence of the solution of PML equations. Computing 60 (1998), 229–241. [23] Dörfler, W., A convergent adaptive algorithm for Possion’s equations. SIAM J. Numer. Anal. 33 (1996), 1106–1124. [24] Douglas Jr., J., and Russell, T. F., Numerical methods for convection-dominated diffusion problem based on combining the method of characteristic with finite element or finite difference procedures. SIAM J. Numer. Anal. 19 (1982), 871–885. [25] Eriksson, K., and Johnson, C., Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal. 28 (1991), 43–77. [26] Houston, P., and Süli, E., Adaptive Lagrange-Galerkin methods for unsteady convectiondiffusion problems. Math. Comp. 70 (2000), 77–106. [27] Kröner, D., and Ohlberger, M., A posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi-dimensions. Math. Comp. 69 (2000), 25–39. [28] Kružkov, N. N., First order quasi-linear equations in several independent variables. Math. USSR Sb. 10 (1970), 217–243. [29] Mascia, C., Porretta, A. and Terracina, A., Nonhomogeneous Dirichlet problems for degenerate parabolic-hyperbolic equations. Arch. Rational Mech. Anal. 163 (2002), 87–124. [30] Morin, P., Nochetto. R. H., and Siebert, K. G., Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38 (2000), 466–488. [31] Nochetto, R. H., Schmidt, A., and Verdi, C., A posteriori error estimation and adaptivity for degenerate parabolic problems. Math. Comp. 69 (2000), 1–24. [32] Ohlberger, M., A posteriori error estimates for vertex centered finite volume approximations of convection-diffusion-reaction equations. Math. Model. Numer. Anal. 35 (2001), 355–387. [33] Otto, F., L1 -contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Diff. Equations 131 (1996), 20–38. [34] Picasso, M., Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998), 223–237. [35] Pironneau, O., On the transport-diffusion algorithm and its application to the Navier-Stokes equations. Numer. Math. 38 (1982), 309–332. [36] Schmidt, A., and Siebert, K. G., ALBERT: An adaptive hierarchical finite element toolbox. IAM, University of Freiburg, 2000; http://www.mathematik.unifreiburg.de/IAM/Research/projectsdz/albert. [37] Teixeira, F. L., and Chew, W. C., Advances in the theory of perfectly matched layers. In Fast and Efficient Algorithms in Computational Electromagnetics (ed. by W. C. Chew), Artech House, Boston 2001, 283–346. [38] Turkel, E., and Yefet, A., Absorbing PML boundary layers for wave-like equations. Appl. Numer. Math. 27 (1998), 533–557.

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[39] Verfürth, R., A Review of A Posteriori Error Estimation and adaptive Mesh Refinement Techniques. Advances in Numerical Mathematics, John Wiley & Sons, B. G. Teubner, Chichester, Stuttgart 1996. LSEC, Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China E-mail: [email protected]

Error estimates for anisotropic finite elements and applications Ricardo G. Durán ∗

Abstract. The finite element method is one of the most frequently used techniques to approximate the solution of partial differential equations. It consists in approximating the unknown solution by functions which are polynomials on each element of a given partition of the domain, made of triangles or quadrilaterals (or their generalizations to higher dimensions). A fundamental problem is to estimate the error between the exact solution u and its computable finite element approximation. In many situations this error can be bounded in terms of the best approximation of u by functions in the finite element space of piecewise polynomial functions. A natural way to estimate this best approximation is by means of the Lagrange interpolation or other similar procedures. Many works have considered the problem of interpolation error estimates. The classical error analysis for interpolations is based on the so-called regularity assumption, which excludes elements with different sizes in each direction (called anisotropic). The goal of this paper is to present a different approach which has been developed by many authors and can be applied to obtain error estimates for several interpolations under more general hypotheses. An important case in which anisotropic elements arise naturally is in the approximation of convection-diffusion problems which present boundary layers. We present some applications to these problems. Finally we consider the finite element approximation of the Stokes equations and present some results for non-conforming methods. Mathematics Subject Classification (2000). Primary 65N30; Secondary 65N15. Keywords. Finite elements, Mixed methods, anisotropic elements, Stokes equations, convectiondiffusion.

1. Introduction The finite element method in its different variants is one of the most frequently used techniques to approximate the solution of partial differential equations. The general idea is to use weak or variational formulations in an infinite dimensional space and to replace that space by a finite dimensional one made of piecewise polynomial functions. In this way, the original differential equation is transformed into an algebraic problem which can be solved by computational methods. Although the main idea goes back ∗ Supported by ANPCyT under grant PICT 03-05009, by Universidad de Buenos Aires under grant X052 and by Fundación Antorchas. The author is a member of CONICET, Argentina.


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to the works of Galerkin and Ritz in the early twentieth-century (or even to previous works, see for example [9] for a discussion of the history of these ideas), the finite element method became more popular since the middle of the twentieth century mainly because of its application by engineers to structural mechanics. On the other hand, the general mathematical analysis started only around forty years ago. The theory of finite elements can be divided into a priori and a posteriori error analysis. The main goals of the a priori analysis are to prove convergence of the methods, to know the order of convergence (in terms of parameters associated with the finite dimensional problem, such as degree of approximation, mesh-size, size of the discrete problem, geometry of the elements, etc.) and the dependence of the error on properties of the unknown exact solution (such as its smoothness, which in many cases is already known from the theory of partial differential equations). Instead, the goals of the a posteriori error analysis are to obtain more quantitative information on the error and to develop self-adaptive methods to improve the approximation iteratively. In this paper we consider several problems related to a priori error estimates. We will deal mainly with the error analysis for flat or anisotropic elements, which arise naturally in several applications. Let us begin by recalling the basic ideas of weak formulations of differential equations and finite element approximations. A general abstract formulation for linear problems is given by B(u, v) = F (v) for all v ∈ V ,

(1.1)

where u ∈ V is the solution to be found, V is a Hilbert space, F is a continuous linear form and B is a continuous bilinear form, i.e., there exists a constant M > 0 such that |B(u, v)| ≤ Muv where · is the norm in the Hilbert space V . To approximate the solution, we want to introduce a finite dimensional space Vh ⊂ V . The usual way to do this is to introduce a partition Th of the domain where we want to solve the differential equation usually made of triangular or quadrilateral elements (or their generalizations in 3D). The parameter h is usually related to the mesh size. Then the space Vh consists of functions which restricted to each element of the partition are polynomials. The approximate solution of our problem is uh ∈ Vh that satisfies B(uh , v) = F (v) for all v ∈ Vh . Assume that the form B is coercive, namely, that there exists a constant α > 0 such that B(v, v) ≥ αv2 for all v ∈ V . (1.2) Then the classical error analysis is based on Cea’s lemma (see [14]), which states that u − uh ≤

M u − v α

for all v ∈ Vh .

(1.3)

Error estimates for anisotropic finite elements and applications

1183

Notice that (1.2) also guarantees existence and uniqueness of solution in V as well as in Vh , thanks to the well-known Lax–Milgram theorem. If this condition does not hold, but the form B satisfies the so-called inf-sup conditions, that is, there exists β > 0 such that inf sup

B(u, v) ≥ β, uv

(1.4)

inf sup

B(u, v) ≥ β, uv

(1.5)

u∈Vh v∈Vh

v∈Vh u∈Vh

then we also have u − uh ≤

M u − v β

for all v ∈ Vh .

(1.6)

If the above inf-sup conditions hold in V , we also have uniqueness and existence of solution. However, this is not sufficient to obtain (1.6), as the inf-sup conditions are not inherited by subspaces. This is the main difference between error analysis of coercive and non-coercive forms which satisfy (1.4) and (1.5). The classical example of a form B which satisfies the inf-sup conditions but is not coercive, is the form associated to the Stokes equations of fluid dynamics (see for example [13], [20]). In view of (1.3) and (1.6), in order to obtain an estimate for u − uh it is enough to bound u − v for a function v ∈ Vh . Therefore this is one of the most important problems in the theory of finite elements. Usually, the function v is taken to be a Lagrange interpolation of u. However, in some cases it is more convenient to use different approximations. In many problems it is convenient to use spaces Vh which are not contained in V . These methods are called non-conforming and in this case the right-hand sides of (1.3) and (1.6) are modified by adding the so-called “consistency terms”. One of the bestknown methods of this type is that of Crouzeix–Raviart, which is closely related to the mixed finite element methods of Raviart–Thomas (see [8], [23]). The goal of this paper is to present general ideas to obtain error estimates for different interpolations valid under very general hypotheses on the elements, in particular, allowing meshes with flat or anisotropic elements. We consider Lagrange and other kind of interpolations arising in mixed finite element methods and give some applications to the approximation of convection-diffusion equations for which anisotropic elements are needed due to the presence of boundary layers. Finally we consider the finite element approximation of the Stokes equations and recall some results for non-conforming methods.

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2. Notation and some basic inequalities The classical finite element analysis for triangular elements requires the so-called regularity assumption, i.e., hT ≤C (2.1) ρT where hT and ρT are the outer and inner diameter, respectively (see Figure 1). In other words, the constants in the error estimates depend on C (see for example [11], [14]).

hT ρT

Figure 1

The same hypothesis is also needed for the analysis of mixed and non-conforming methods (see [15] and [24]). For standard Lagrange interpolation on conforming elements, since the works of Babuska–Azis [10] and Jamet [21] it is well known that the regularity assumption can be relaxed. For example, in the case of triangles it can be replaced by the weaker maximum angle condition (i.e. angles bounded away from π ). For rectangular elements, optimal error estimates can be obtained for arbitrary rectangles (while the regularity assumption requires that the edge sizes be comparable). In the case of general quadrilaterals, the situation is more complicated and several conditions, weaker than regularity, have been introduced to prove the error estimates (see, for example, [3]). The standard method to prove error estimates is to obtain them first in a reference element and then to make a change of variables (see [14]). A different approach is to work directly in a given element and to use Poincaré type inequalities. The main idea is that the interpolation error usually has some vanishing averages (on the element, or edges, or faces, depending of the kind of interpolation considered). In this approach, the reference element is sometimes used to obtain the Poincaré type inequalities but, since one is bounding an L2 -norm, the constants appearing in the estimates are independent of the aspect ratio of the element. We will use the following notation. By H 1 () we mean the usual Sobolev space of L2 functions with distributional first derivatives in L2 and by H01 () the subspace of H 1 () of functions vanishing on the boundary. Similarly, W k,p (), for 1 ≤ p ≤ ∞, indicates the Sobolev space of Lp () functions with distributional derivatives of order k in Lp (). When p = 2 we set H k () = W k,2 ().

1185


Here ⊂ Rn , n = 2, 3, is a bounded domain. For a general triangle T , hT is its diameter, p 0 is a vertex (arbitrary unless otherwise specified), v 1 , v 2 (with v i = 1) are the directions of the edges 1 , 2 sharing p0 (see Figure 2), and νi is the exterior unit normal to the side i (with obvious generalizations to 3D). We also use the standard notation Pk for polynomials of total degree less than or equal to k, and Qk for polynomials of degree less than or equal to k in each variable. We call Tˆ the reference triangle with vertices at (0, 0), (1, 0) and (0, 1), and F : Tˆ → T the affine transformation F (x) ˆ = B xˆ + p0 with Bei = li v i , where ei are the canonical vectors. v1

v2 θ p0 Figure 2

The following two results are the classical Poincaré inequality and a generalization of it (first given in [10]) written in a convenient way for our purposes. Lemma 2.1. Let T be a triangle (resp. tetrahedron) and let f ∈ H 1 (T ) be a function with vanishing average on T . Then there exists a constant C independent of T and of f such that n ∂f |j | . (2.2) f L2 (T ) ≤ C ∂vj L2 (T ) j =1

Proof. It follows from the Poincaré inequality on Tˆ and making the change of variables F . 2 Lemma 2.2. Let T be a triangle (resp. tetrahedron) and be any of its edges (resp. faces). Let f ∈ H 1 (T ) be a function with vanishing average on . Then there exists a constant C independent of T such that f L2 (T ) ≤ C

n j =1

∂f |j |

∂vj

L2 (T )

.

(2.3)

Proof. It is enough to prove that, on the reference element Tˆ , f L2 (Tˆ ) ≤ C∇f L2 (Tˆ ) .

(2.4)

Then, for a general triangle, the result follows by making the change of variables F . The estimate (2.4) can be proved by a standard compactness argument (as was done in [10]). A different proof can be given by a using (2.2) and a trace theorem.

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Indeed, if fˆ and fTˆ denote the averages on ˆ and Tˆ , respectively, and if we assume that fˆ = 0 we have f L2 (Tˆ ) = f − fˆL2 (Tˆ ) ≤ f − fTˆ L2 (Tˆ ) + fTˆ − fˆL2 (Tˆ ) . But

1 fTˆ − fˆ = ˆ ||

ˆ

(fTˆ − f ),

and therefore an application of a standard trace theorem gives fTˆ − fˆL2 (Tˆ ) ≤ C{f − fTˆ L2 (Tˆ ) + ∇f L2 (Tˆ ) } with a constant C which depends only on the reference element. Hence (2.4) follows 2 from (2.2).

3. Error estimates for Lagrange interpolation 3.1. The two-dimensional case. To introduce the general idea we present first two simple classical cases: the Lagrange interpolation for lowest degree finite elements in triangles or rectangles. The argument is essentially that given in [10] for triangles. In the case of rectangles, an extra step is required due to the presence of a non-vanishing second derivative of the interpolating function. Given a triangle T we denote with I1 u ∈ P1 the Lagrange interpolation of u, i.e., the affine function which equals u on the vertices of T . D 2 u denotes the sum of the absolute values of second derivatives of u. Theorem 3.1. There exists a constant C such that, if θ is the maximum angle of T , ∇(u − I1 u)L2 (T ) ≤

C hT D 2 uL2 (T ) . sin θ

Proof. Observe that, for i = 1, 2, ∇(u − I1 u) · v i , has vanishing average on one side of T . Therefore, applying Lemma 2.2 and using that the second derivatives of I1 u vanish, we obtain ∂∇u · v i ∇(u − I1 u) · v i L2 (T ) ≤ C |1 |

∂v 1

L2 (T )

∂∇u · v i + |2 |

∂v 2

L2 (T )

.

Then, if we choose p 0 as the vertex corresponding to the maximum angle of T , we have C |∇(u − I1 u)| ≤ {|∇(u − I1 u) · v 1 | + |∇(u − I1 u) · v 2 |}, sin θ 2 and hence the theorem is proved.

1187


We consider now the case of rectangles. We use the same notation, I1 u, for the interpolation which now belongs to Q1 . The proof for this case is analogous to the 2I u 1 previous one, with the only difference that ∂∂x∂y does not vanish. Let R be a rectangle and let 1 , 2 be two adjacent sides. Clearly, the result of Lemma 2.2 holds for this case also. Theorem 3.2. There exists a constant C, independent of the relation between |1 | and |1 |, such that 2 2 ∂ ∂ u ∂ u (u − I u) ≤ C |1 | 2 + |2 | (3.1) 1 ∂x ∂x L2 (R) ∂x∂y L2 (R) L2 (R) and ∂ (u − I u) 1

∂y

L2 (R)

2 ∂ u ≤ C |1 |

∂x∂y

L2 (R)

2 ∂ u + |2 | 2

∂y

L2 (R)

.

(3.2)

Proof. Proceeding as in the case of triangles, we have 2 2 ∂ ∂ (u − I1 u) ∂ (u − I1 u) (u − I u) ≤ C | | + | | . 1 1 2 2 2 ∂x ∂x 2 ∂x∂y L2 (R) L (R) L (R) (3.3) ∂ 2 I1 u But, ∂x 2 = 0 and an elementary computation shows that R

i.e.,

∂ 2 I1 u ∂x∂y

is the average of

∂2u ∂x∂y

∂ 2 I1 u = ∂x∂y

R

∂ 2u , ∂x∂y

on R. Then

2 ∂ I1 u

∂x∂y

L2 (R)

2 ∂ u ≤

∂x∂y

L2 (R)

and therefore (3.1) holds. Obviously, the proof of (3.2) is analogous.

2

Remark 3.3. If the function u ∈ H 3 (R), then the last term on the right-hand side ∂2u of (3.3) is of higher order. Indeed, that term is the difference between ∂x∂y and its average. Therefore we have the estimate ∂ (u − I u) 1

∂x

L2 (R)

2 ∂ u 2

≤ C|1 |

∂x

L2 (R)

+ higher order terms.

3.2. The three-dimensional case. Many results on finite elements can be extended almost straightforward from 2D to 3D. However, this is not the case of error estimates for anisotropic elements. Indeed, counterexamples for an estimate analogous to (3.1) in the 3D case have been given in [6] and [26]. They show that the constant in the

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Ricardo G. Durán

estimate blows-up when a rectangular reference tetrahedron (or cube) is compressed in one direction. Many papers have been published considering the 3D case. For example, in the case of tetrahedra, Krízek [22] introduced a natural generalization of the maximum angle condition: if the angles between faces and the angles in the faces are bounded away from π, he obtained error estimates for smooth functions, namely, u ∈ W 2,∞ . In [16] the results of Krízek were extended to functions in W 2,p with 2 < p < ∞ (and, moreover, to functions in an intermediate Orlicz space between H 2 and W 2,p , p > 2). Therefore, although the estimate fails for functions in H 2 , it is valid for functions only slightly more regular. Let us mention that the reason why the arguments applied in 2D cannot be generalized, is that the estimate given in Lemma 2.2 is not true in 3D if is an edge instead of a face (note that the interpolation error for the Lagrange interpolation has vanishing integral on edges). On the other hand, many papers have considered error estimates for different interpolations (see for example [1], [5], [16], [17]), namely, different variants of average interpolators. This kind of interpolations have been introduced to approximate non-smooth functions (for which the Lagrange interpolation is not even defined). However, they have as well better approximation properties on anisotropic elements for functions in H 2 . Indeed, using average interpolations, the 2D results can be generalized to 3D. Observe that, in view of (1.3) and (1.6), error estimates for an average interpolation will give bounds for finite element approximations.

4. Applications to convection-diffusion equations A very important application in which anisotropic elements are needed is the approximation of convection-diffusion problems in which boundary layers arise. Consider for example the model problem −ε u + b · ∇u + cu = f in , u = 0 on ∂,

(4.1)

where = (0, 1)2 and ε > 0 is a small parameter. It is well known that the numerical approximation of this equation requires some special method in order to obtain good results when the problem is convection dominated, due to the presence of boundary or interior layers. In the case of boundary layers, one possibility is to use appropriate refined meshes near the boundary; this methodology gives rise to anisotropic elements. Using estimates (3.1) and (3.2) it is possible to obtain quasi-optimal order convergence (with respect to the number of nodes) in the ε-norm defined by v2ε = v2L2 () + ε∇v2L2 () for the standard Q1 approximation on appropriate graded meshes.

1189


This problem can be written in the general form (1.1) with V = H01 (), B(u, v) = (ε∇u∇v + b · ∇u v + c uv) dx

and F (v) =

f v dx.

Assuming that there exists a constant μ independent of ε such that c−

div b ≥ μ > 0, 2

(4.2)

the bilinear form B is coercive in the ε-norm uniformly in ε (see [25]), i.e., the constant α in (1.2) is independent of ε. However, the continuity of B is not uniform in ε and this is one of the reasons why it is not possible to apply directly the general result (1.3) to obtain error estimates valid uniformly in ε. Therefore, a special analysis is required and this was the object of [18]. It was proved in that paper that u − uh ε ≤ C

log2 (1/ε) , √ N

where N is the number of nodes and h > 0 is a parameter associated with the meshes. Observe that this order of convergence is quasi-optimal in the sense that, up to the logarithm factor, it is the same order that one obtains for a smooth solution of a problem with ε = O(1) using uniform meshes. Assuming that the coefficient b is such that the boundary layers are close to x = 0 and y = 0, the meshes Th are such that the grading in each direction is given by ⎧ ⎪ ⎪ξ0 = 0, ⎪ ⎨ξ = ihε i ⎪ ξ i+1 = ξi + hξi ⎪ ⎪ ⎩ ξM = 1,

for 1 ≤ i < h1 + 1, for h1 + 1 ≤ i ≤ M − 2,

(4.3)

where M is such that ξM−1 < 1 and ξM−1 + hξM−1 ≥ 1. We assume that the last interval (ξM−1 , 1) is not too small in comparison with the previous one (ξM−2 , ξM−1 ) (if this is not the case, we just eliminate the node ξM−1 ). Figure 3 shows the approximate solution of (4.1) for ε = 10−6 , and

b = (1 − 2ε)(−1, −1),

f (x, y) = − x −

1 − e− ε

x

1−e

− 1ε

+y−

c = 2(1 − ε) y

1 − e− ε 1−e

− 1ε

ex+y .

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1

0.8

0.6

0.4

0.2 1 0.8

0 0

0.6

0.2 0.4

0.4 0.6

x

0.2

0.8 1

y

0

Figure 3

Observe that no oscillations arise although we are using the standard Q1 finite element method. The graded meshes are an alternative to the well-known Shishkin meshes which have been widely analyzed for convection-diffusion problems (see for example [25]). From the error analysis given in [18] one can see that a graded mesh designed for a value of ε works well also for larger values of ε. This is not the case for Shishkin meshes. Table 1 shows the values of the ε-norm of the error for different values of ε, solving the problem with the mesh corresponding to ε = 10−6 , using graded meshes and Shishkin meshes. Table 1

ε 10−6 10−5 10−4 10−3 10−2 10−1

Error

0.040687 0.033103 0.028635 0.024859 0.02247 0.027278

Graded meshes, N = 10404.

ε 10−6 10−5 10−4 10−3 10−2 10−1

Error

0.0404236 0.249139 0.623650 0.718135 0.384051 0.0331733

Shishkin meshes, N = 10609.

To see the different structures, we show in Figure 4 a Shishkin mesh (on the right) and one of our graded meshes (on the left) having the same number of nodes. For the sake of clarity, we show only the part of the meshes corresponding to (0, 1/2) × 3 (0, 1/2) and ε = 10− 2 .

1191


0.5

0.5

0.45

0.45

0.4

0.4

0.35

0.35

0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Figure 4

5. Error estimates for Raviart–Thomas interpolation 5.1. The two dimensional case. The Raviart–Thomas spaces were introduced in [24] to approximate vector fields u ∈ H (div, ) where H (div, ) = {u ∈ L2 () : div u ∈ L2 }. For any integer k ≥ 0, the space RT k on a triangle T is defined by RT k (T ) = Pk2 (T ) ⊕ (x, y)Pk (T ). Calling Pk the L2 orthogonal projection on Pk (T ), it is known (see [24]) that there exists an operator RTk : H 1 (T )2 → RTk (T ) satisfying the following commutative diagram property: H 1 (T )2 RTk

div

(5.1)

Pk

RTk (T )

/ L2 (T )

div

/ Pk (T )

/ 0.

For the case of anisotropic elements, only the lowest degree case RT 0 has been considered. Error estimates for this case have been obtained in [2]. Below we will show how the arguments can be generalized to obtain error estimates for the case of RT 1 . Higher order approximations can be treated similarly although this extension is not straightforward. Let us first recall the results for RT 0 . Again, the results follow by the generalized Poincaré inequality given in Lemma 2.2 as we show in the next theorem.

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Theorem 5.1. There exists a constant C such that, if θ is the maximum angle of T , u − RT0 uL2 (T ) ≤

2 ∂u C |k | + div u 2 L (T ) . sin θ ∂v k L2 (T ) k=1

Proof. Since (u − RT0 u) · νi has zero mean value on i , it follows from Lemma 2.2 that 2 ∂(u − RT0 u) · νi (u − RT0 u) · νi L2 (T ) ≤ C |k | (5.2) 2 . ∂v k L (T ) k=1

But it is easy to check that 1 ∂(RT0 u · νi ) = (div RT0 u) v k · νi . ∂v k 2 On the other hand, using the commutative diagram property (5.1), we have div RT0 uL2 (T ) ≤ div uL2 (T ) and so it follows from (5.2) that (u − RT0 u) · νi L2 (T ) ≤ C

2 k=1

∂u |k | + div uL2 (T ) |νi · v k | . (5.3) ∂v k L2 (T )

Up to now the constant C is independent of T . If we want to bound u − RT0 uL2 (T ) , it is natural to expect that the constant will depend on the geometry of the element. In view of (5.3) it would be enough to control u − RT u in terms of its components in the directions of the normals to the edges. For a fixed triangle the estimate |u − RT0 u| ≤ C{|(u − RT0 u) · ν1 | + |(u − RT0 u) · ν2 |} holds. Moreover, for a family of triangles, the constant C will not degenerate if the angle between ν1 and ν2 does not go to 0 or π or, equivalently, if the angle between the corresponding edges does not go to 0 or π. Therefore the constant will be uniformly bounded for a family of elements with maximum angle bounded away from π . More precisely, we have u − RT0 uL2 (T ) ≤

2 C (u − RT0 u) · νi L2 (T ) sin θ

(5.4)

i=1

where θ is the maximum angle of T . Indeed, if N is the matrix which has ν1 and ν2 as its rows, then u − RT0 uL2 (T ) ≤ N −1

2 i=1

(u − RT0 u) · νi L2 (T )

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where · denotes the matrix norm associated with the euclidean norm. But since the νi are unit vectors it follows that N −1 ≤ | detC N| and | det N| = sin θ1 , where θ1 is the angle between ν1 and ν2 . If the vertex p 0 is the one corresponding to the 2 maximum angle θ1 = π − θ, then (5.4) holds and the theorem is proved. Similar arguments can be applied for the analysis of higher order elements. However the extension is not straightforward. In what follows we consider the case of RT 1 . This case requires the following generalization of the Poincaré inequality. Lemma 5.2. Let T be a triangle and one of its sides. If f ∈ H 2 (T ) satisfies fp = 0 for all p ∈ P1 () and f = 0,

T

then f L2 (T ) ≤ Ch2T D 2 f L2 (T ) with a constant C independent of the shape of the triangle. Proof. Observing that, if f ∈ P1 satisfies the three hypotheses of the lemma then f = 0, it follows by standard compactness arguments that f L2 (Tˆ ) ≤ CD 2 f L2 (Tˆ ) .

2

Then an affine change of variables concludes the proof.

To obtain the error estimate for the RT1 interpolation we will need to have a bound for the gradient of the P1 projection. This is the goal of the next lemma. Lemma 5.3. If f ∈ H 1 (T ) we have ∇P1 f L2 (T ) ≤ C∇f L2 (T ) with a constant C depending only on the maximum angle of T . Proof. We will prove that for the triangle with vertices at (0, 0), (h, 0) and (0, 1) we have ∇P1 f L2 (T ) ≤ 6∇f L2 (T ) . Then the general result follows by an affine change of variables. Let Mi , i = 1, 2, 3 be the mid-side points of T . Since the quadrature rule obtained by interpolating at these points is exact for quadratic polynomials, it is easy to see that the functions 1/2

6 φ1 = h

1/2

6 (1−2y), φ2 = h

2x 2y+ −1 h

form an orthonormal basis of P1 (T ). Then P1 f =

3 i=1

ci φi

1/2

and

6 φ3 = h

2x 1− h

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Ricardo G. Durán

with ci =

f φi . Therefore, √ 2 6 24 2x ∂P1 f = 3 − 1 dxdy. f (φ2 − φ3 ) = 2 f (x, y) y + ∂x h T h h2 T T

Now observe that, for any y ∈ (0, 1), h(1−y) 2x y+ − 1 dx = 0 h 0 h(1−y) 1 f (x, y) dx, we obtain and so, denoting f (y) = h(1−y) 0 24 ∂P1 f = 2 ∂x h

1 h(1−y) 0

0

2x (f (x, y) − f (y)) y + − 1 dxdy. h

But using the one dimensional Poincaré inequality we have h(1−y) h h(1−y) ∂f |f (x, y) − f (y))| dx ≤ (x, y) dx 2 0 ∂x 0 and, since y + 2x h − 1 ≤ 1, it follows that ∂P1 f

∂x

∂P1 f

Therefore

∂x

and consequently

1 h(1−y) ∂f ≤ 12 dxdy. (x, y)

h

0

≤ 12 ∂f

h ∂x

∂P1 f

∂x

∂x

0

L1 (T )

L2 (T )

1 ∂f 12 ≤ |T | 2 h ∂x L2 (T )

∂f ≤ 6

∂x

L2 (T )

.

Clearly, the same arguments can be applied to bound the derivative with respect to y. 2 Theorem 5.4. There exists a constant C depending only on the maximum angle of T such that u − RT1 uL2 (T ) ≤ Ch2T D 2 uL2 (T ) . Proof. From the definition of RT1 u we know that, for i = 1, 2, 3, (u − RT1 u) · νi satisfies the hypotheses of Lemma 5.2 and then (u − RT1 u) · νi L2 (T ) ≤ Ch2T D 2 (u − RT1 u)L2 (T ) . So, in order to estimate the component of u − RT1 u in the direction νi , we need to bound the second derivatives of RT1 u in terms of D 2 u.

1195


But an easy computation shows that, for any v ∈ RT 1 (T ),

∂ 2v 2 ∂(div v) ,0 , = 2 ∂x 3 ∂x and

∂ 2v 2 ∂(div v) = 0, 2 ∂y 3 ∂y

∂ 2v 1 ∂(div v) ∂(div v) . = , ∂x∂y 3 ∂y ∂x

Therefore we have (u − RT1 u) · νi L2 (T ) ≤ Ch2T D 2 uL2 (T ) + ∇(div RT1 u)L2 (T ) .

(5.5)

Now from (5.1) we know that ∇(div RT1 u) = ∇(P1 div u), hence applying Lemma 5.3 yields ∇(div RT1 u)L2 (T ) ≤ C∇(div u)L2 (T ) and using this inequality in (5.5) we obtain the estimates for the normal components of (u − RT1 u). Then, to conclude the proof of the theorem, we proceed as in the case of RT0 . 2 5.2. The three-dimensional case. As in the case of the Lagrange interpolation, the 3D case presents some important differences with the 2D one. We recall that the definition of RT k can be extended straightforwardly to the 3D case. Indeed, for T a tetrahedron we have RT k (T ) = Pk3 (T ) ⊕ (x, y, z)Pk (T ). The maximum angle condition can be generalized in different ways. The first one, introduced in [2], is the regular vertex property. We say that a tetrahedron satisfies this property with a constant c > 0 if it has a vertex p 0 such that | det M| ≥ c > 0, where M is the matrix which has v i , i = 1, 2, 3 as rows (where we are using the obvious generalization of the notation of the 2D case). Under this hypothesis, Theorem 5.1 can be generalized almost straightforwardly. Indeed, the basic result given in Lemma 2.2 is valid now for functions with vanishing average on a face of T , and using this result we can prove, arguing as in the 2D case, that (u − RT0 u) · νi L2 (T ) ≤ C

3 k=1

∂u |k | + div uL2 (T ) |νi · v k | . ∂v k L2 (T )

As a consequence we obtain the following estimate.

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Ricardo G. Durán

Theorem 5.5. Let T be a tetrahedron satisfying the regular vertex property with a constant c > 0. Then there exists a constant C depending only on c such that u − RT0 uL2 (T ) ≤ C

3 k=1

∂u |k | ∂v

k L2 (T )

+ div uL2 (T ) .

(5.6)

The other “natural” generalization of the 2D maximum angle condition is the condition introduced by Krízek [22]. We say that a family of tetrahedra satisfies the maximum angle condition with a constant ψ < π if the angles inside the faces and the angles between faces are bounded above by ψ. It is easy to see that in the 2D case the regular vertex property is equivalent to the maximum angle condition. However, the situation is different in the 3D case. In fact, the family in Figure 5, with arbitrary lengths h1 , h2 , h3 , satisfies uniformly the maximum angle condition but not the regular vertex property (take for example h1 = h3 = h2 , and h2 = h). On the other hand, the regular vertex property implies the maximum angle condition (see [2]). A natural question is whether or not error

h3

h2 h1

Figure 5

estimates for the RT0 interpolation hold under the maximum angle condition. The answer is positive. In [2] the following result was proved. Theorem 5.6. If T is a tetrahedron satisfying the maximum angle condition with a constant ψ. Then there exists a constant C depending only on ψ such that u − RT0 uL2 (T ) ≤ ChT DuL2 (T ) .

(5.7)

Again the basic tool to obtain this estimate is the generalization to 3D of Lemma 2.2. Indeed, consider the face mean average interpolator introduced in [15], namely, : H 1 (T ) → P1 (T ) given by w = w S

for any face S of T .

S

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Lemma 5.7. The following error estimates hold with a constant C independent of T : w − wL2 (T ) ≤ C

j =1

∂w

∂ξ

L2 (T )

∂(w − w)

∂ξ

where

∂ ∂ξ

L2 (T )

∂w |j |

3

∂vj

∂w

≤

≤C

∂ξ

3 j =1

L2 (T )

(5.8) (5.9)

L2 (T )

2 ∂ w |j |

∂vj ∂ξ

L2 (T )

(5.10)

is a derivative in any direction.

Proof. Since w − w has vanishing mean value on the faces of T , it follows from Lemma 2.2 that 3 ∂(w − w) w − wL2 (T ) ≤ C |j | (5.11) 2 . ∂vj L (T ) j =1

Now, it follows from the definition of that ∂w ∂w = ∂ξ T T ∂ξ ∂w or, in other words, the constant ∂w ∂ξ is the average on T of ∂ξ and so (5.9) holds and (5.10) follows from Lemma 2.1. Finally, (5.8) is a consequence of (5.11) and (5.9). 2

Now it is not difficult to check that, for any u ∈ H 1 (T )3 , RT0 u = RT0 u where is the vector version of . Consequently, u − RT0 uL2 (T ) ≤ u − uL2 (T ) + u − RT0 uL2 (T ) and therefore, in view of (5.8), to prove (5.7) it is enough to prove the error estimate for u ∈ P1 (K)3 . In this way the problem is reduced to a finite dimensional one and the error estimate (5.7) can be proved under the maximum angle condition (see [2] for details).

6. The Stokes equations The Stokes equations are given by − u + ∇p = f div u = 0 u=0

in , in , on ∂,

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Ricardo G. Durán

where u is the velocity and p the pressure of a fluid contained in . This problem can be written in the form (1.1) with V = H01 ()n × L20 () where L20 () = f ∈ L2 () : f = 0 , n ∂ui ∂vi B((u, p), (v, q)) = − p div v − q div u ∂xj ∂xj i,j =1

and F (v, q) =

f v.

Then to obtain a finite element approximation we need to use a space Wh for the velocity and a space Qh for the pressure. Note that since in this case the form B is symmetric, the two conditions (1.4) and (1.5) are exactly the same. From the classical theory for mixed finite elements of Brezzi [12] we know that to obtain (1.4) for the space Vh = Wh × Qh it is enough to prove that there exists γ > 0, independent of h, such that q div v ≥ γ. (6.1) inf sup q∈Qh v∈Wh pL2 vH 1 0

Equivalently, for any f ∈ Qh , there exists a solution u ∈ Wh of div u · q = f q for all q ∈ Qh ,

(6.2)

uH 1 ≤ Cf L2 0

(6.3)

with C depending only on the domain . A lot of work has been done to prove this inf-sup condition for different choices of spaces Wh and Qh . We refer for example to the books [13], [20]. However, most proofs require the regularity assumption (2.1) on the elements although it is not known whether it is essential or not. One of the main tools to prove (6.1) is the so-called Fortin operator introduced in [19], which in the case of the Stokes equations is an operator : H01 ()n → Wh such that q div(v − v) = 0 for all q ∈ Qh

and vH 1 ≤ CvH 1 0

0

(6.4)

with a constant C independent of h. Consider for example the non-conforming method of Crouzeix–Raviart, namely, Wh are the (P1 )n functions in each element which are also continuous at the midpoints of the edges or faces of the partition, and Qh are piecewise constant functions. Error estimates for anisotropic elements for this method have been proved in [2], [7].


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The Fortin operator for this case is the edge (or face) mean average interpolator defined in the previous section. In view of (5.9), estimate (6.4) holds with a constant independent of the geometry of the elements which can be taken to be one. However, this is a non-conforming method (because Wh ⊂ H01 ()2 ) and therefore, to obtain error estimates, some consistency terms have to be bounded. This can be done by using the RT0 interpolation analyzed in the previous section. In this way it is possible to obtain optimal error estimates for this method under the maximum angle condition (see [2]).

References [1] Acosta, G., Lagrange and average interpolation over 3D anisotropic elements. J. Comp. Appl. Math. 135 (2001), 91–109. [2] Acosta, G., Durán, R. G., The maximum angle condition for mixed and non conforming elements: Application to the Stokes equations. SIAM J. Numer. Anal. 37 (2000), 18–36. [3] Acosta, G., Durán, R. G., Error estimates for Q1 isoparametric elements satisfying a weak angle condition. SIAM J. Numer. Anal. 38 (2000), 1073–1088. [4] Apel, T., Anisotropic finite elements: Local estimates and applications. Adv. Numer. Math., Teubner, Stuttgart 1999. [5] Apel, T., Interpolation of non-smooth functions on anisotropic finite element meshes. Math. Model. Numer. Anal. 33 (1999), 1149–1185. [6] Apel, T., Dobrowolski, M., Anisotropic interpolation with applications to the finite element method. Computing 47 (1992), 277–293. [7] Apel, T., Nicaise, S., Schoeberl, J., Crouzeix-Raviart type finite elements on anisotropic meshes. Numer. Math. 89 (2001), 193–223. [8] Arnold, D. N., Brezzi, F., Mixed and nonconforming finite element methods implementation, postprocessing and error estimates. RAIRO, Modél. Math. Anal. Numér. 19 (1985), 7–32. [9] Babuška, I., Courant element: before and after. In Finite element methods, Lecture Notes Pure Appl. Math. 164, Dekker, New York 1994, 37–51. [10] Babuška, I., Aziz, A. K., On the angle condition in the finite element method. SIAM J. Numer. Anal. 13 (1976), 214–226. [11] Brenner, S. C., Scott, L. R., The Mathematical Theory of Finite Element Methods. Texts Appl. Math. 15, Springer-Verlag, New York 1994. [12] Brezzi, F., On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. RAIRO Sér. Rouge 8 (1974), 129–151. [13] Brezzi, F., Fortin, M., Mixed and Hybrid Finite Element Methods. Springer Ser. Comput. Math. 15, Springer-Verlag, New York 1991. [14] Ciarlet, P. G., The Finite Element Method for Elliptic Problems. Stud. Math. Appl. 4, North Holland, Amsterdam 1978. [15] Crouzeix, M., Raviart, P. A., Conforming and non-conforming finite element methods for solving the stationary Stokes equations. RAIRO Anal. Numér. 7 (1973), 33–76.

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[16] Durán, R. G., Error estimates for 3-d narrow finite elements. Math. Comp. 68 (1999), 187–199. [17] Durán, R. G., Lombardi, A. L., Error estimates on anisotropic Q1 elements for functions in weighted Sobolev spaces. Math. Comp. 74 (2005), 1679–1706. [18] Durán, R. G., Lombardi, A. L., Finite element approximation of convection diffusion problems using graded meshes. Appl. Numer. Math., to appear. [19] Fortin, M., An analysis of the convergence of mixed finite element methods. RAIRO Anal. Numér. 11 (1977), 341–354. [20] Girault, V., Raviart, P. A., Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer Ser. Comput. Math. 5, Springer-Verlag, Berlin 1986 [21] Jamet, P., Estimations d’erreur pour des éléments finis droits presque dégénérés. RAIRO Anal. Numér. 10 (1976), 46–71. [22] Krízek, M., On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29 (1992), 513–520. [23] Marini, L. D., An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. 22 (1985), 493–496. [24] Raviart, P. A., Thomas, J. M., A mixed finite element method for second order elliptic problems. In Mathematical aspects of the Finite Element Method (ed. by I. Galligani and E. Magenes), Lecture Notes in Math. 606, Springer-Verlag, Berlin 1977, 292–315. [25] Roos, H. G., Stynes, M., Tobiska, L., Numerical Methods for Singularly Perturbed Differential Equations. Springer Ser. Comput. Math. 24, Springer-Verlag, Berlin 1996. [26] Shenk, N. A., Uniform error estimates for certain narrow Lagrange finite elements. Math. Comp. 63 (1994), 105–119. Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina E-mail: [email protected]

Linear subdivision schemes for the refinement of geometric objects Nira Dyn

Abstract. Subdivision schemes are efficient computational methods for the design, representation and approximation of surfaces of arbitrary topology in R 3 . Subdivision schemes generate curves/surfaces from discrete data by repeated refinements. This paper reviews some of the theory of linear stationary subdivision schemes and their applications in geometric modelling. The first part is concerned with “classical” schemes refining control points. The second part reviews linear subdivision schemes refining other objects, such as vectors of Hermite-type data, compact sets in R n and nets of curves in R 3 . Examples of various schemes are presented. Mathematics Subject Classification (2000). Primary 65D07, 65D17, 65D18; Secondary 41A15, 68U07. Keywords. Subdivision schemes, geometric modelling, curves, surfaces, refinements, smoothness, arbitrary topology, nets of points, nets of curves, compact sets, approximation order.

1. Introduction Subdivision schemes in geometric applications are efficient tools for the generation of curves/surfaces from discrete data, by repeated refinements. The first subdivision schemes where devised by de Rahm [54] for the generation of functions with a first derivative everywhere and a second derivative nowhere. In geometric modelling the first schemes were proposed for easy and quick rendering of B-spline curves. A B-spline curve has the form C(t) = Pi Bm (t − i) (1) i

with {Pi } points in R d (d = 2 or 3) termed control points, and Bm a B-spline of degree m with integer knots, namely Bm |[i,i+1] is a polynomial of degree m, Bm ∈ C m−1 (R ), supp Bm = [0, m+1]. Equation (1) is a parametric representation of a B-spline curve. By using the refinement equation satisfied by a B-spline, Bm (x) =

m+1 i=0

ai[m] Bm (2x

− i),

ai[m]

=2

−m

m+1 , i

i = 0, . . . , m + 1. (2)


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Nira Dyn

C(t) in (1) has the parametric representations C(t) = Pi0 Bm (t − i) = Pi1 Bm (2t − i) = · · · i

i

=

Pik Bm (2k t − i) = · · · ,

(3)

i

where Pi+1 =

[m] ai−2j Pj ,

= 0, 1, 2, . . . ,

(4)

j

/ {0, 1, . . . , m + 1}. with the convention ai[m] = 0, i ∈ k } tend to zero as k increases, As is demonstrated in §2.3, the differences {Pik −Pi−1 and since Bm ≥ 0 and i Bm (t − i) ≡ 1 [6], the polygonal line through the control points {Pik } is close to C(t) for k large enough, and can be easily rendered. The relation (4) encompasses the refinement rule for B-spline curves. The first scheme of this type was devised by Chaikin [10] for quadratic B-spline curves, and the schemes for general B-spline curves were investigated in [14]. All other subdivision schemes can be regarded as a generalization of the spline case. original

iteration #2

iteration #1

iteration #3

Figure 1. Refinements of a polygon with Chaikin scheme.

In this paper we first review the “classical” subdivision schemes for the refinement of control points. The schemes for the generation of curves are direct generalizations of (4), in the sense that the coefficients, defined in (2), are replaced by other sets of coefficients. The “classical” schemes, and in particular those generating surfaces, are used extensively in Computer Graphics. In §2 we discuss the construction of such schemes, their approximation properties, tools for the analysis of their convergence and smoothness, and their application to the generation of surfaces from general nets of points in R 3 . Examples of important schemes are presented.

Linear subdivision schemes for the refinement of geometric objects

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Subdivision schemes for the refinement of objects other than control points are reviewed in §3. These schemes include subdivision schemes refining vectors, in particular, vectors consisting of values of a function and its consecutive derivatives, schemes refining compact sets in R n and a scheme refining nets of curves. All the schemes reviewed in this paper are linear. Recently, various non-linear schemes were devised and analyzed (see, e.g., [26] and references therein). It seems that this is one of the future directions in the study of subdivision schemes. Applications of “classical” schemes to the numerical solution of special types of PDEs is another direction. (See, e.g., [11]). New “classical” schemes are still being devised for particular applications. For example, adaptive refinements can be accomplished straightforwardly by refining according to topological rules different from the “classical” ones, therefore, corresponding linear schemes had to be devised (see, e.g. [48] and [59]).

2. Stationary linear schemes for the refinement of control points A subdivision scheme Sa for the refinement of control points is defined by a finite set of coefficients called mask a = {ai ∈ R : i ∈ σ (a) ⊂ Z s }. Here σ (a) denotes the finite support of the mask, s = 1 corresponds to curves and s = 2 to surfaces. The refinement rule is Pαk+1 = aα−2β Pβk , α ∈ Z s . (5) β∈Z s

Remark. In most of the paper we consider schemes defined on Z s , although, in geometric applications the schemes operate on finite sets of data. Due to the finite support of the mask, our considerations apply directly to closed curves/surfaces, and also to “open” ones, except in a finite zone near the boundary. In the case s = 1, a subdivision scheme is termed uniformly convergent (or convergent for geometric applications) if the sequence {P k (t)} of polygonal lines through the control points at refinement levels k = 0, 1, 2, . . . (with parametric representation as the piecewise linear interpolants to the data {(i2−k , Pik ) : i ∈ Z }, k = 0, 1, 2, . . . ), converges uniformly in bounded intervals. In the case s = 2, we require the uniform convergence of the sequence of piecewise bi-linear interpolants to the data {(α2−k , Pαk ) : α ∈ Z 2 } on bounded squares [9], [33], [24]. The convergence of a scheme Sa implies the existence of a basic-limit-function φa , being the limit obtained from the initial data, fi0 = 0 everywhere on Z s except f00 = 1. It follows from the linearity and uniformity of (5) that the limit obtained from any set of initial control points P 0 = {Pα0 ∈ R d : α ∈ Z s }, Sa∞ P 0 , can be written in terms of integer translates of φa , as Sa∞ P 0 (x) = Pα0 φa (x − α), x ∈ R s . (6) α∈Z s

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Nira Dyn

For s = 1 and d = 2 or d = 3, (6) is a parametric representation of a curve in R d , while for s = 2 and d = 3, (6) is a parametric representation of a surface in R 3 . Also, by the linearity, uniformity and stationarity of the refinement (5), φa satisfies the refinement equation (two-scale relation) aα φa (2x − α), (7) φa (x) = α∈Zs

analogous to the refinement equation (2) for B-splines. It follows from (5) or from (7) that supp(φa ) is contained in the convex hull of σ (a) [9], as is the case for the B-spline schemes. The choice of the mask in the design of good schemes is partly heuristic and partly aims at obtaining specific properties of the scheme as convergence, smoothness, locality, interpolation, shape preservation, and approximation order. For the case s = 1, the topology of Z is sufficient to describe an ordered set of control points for curve design. For the case s = 2, the topology of Z 2 , where the point (i, j ) is connected to the four points (i ± 1, j ), (i, j ± 1), is sufficient to describe a set of control points in R 3 , connected each to four neighboring points and constituting a quad-mesh. The above connectivity of Z 2 , with the additional connections of the point (i, j ) to the points (i + 1, j + 1), (i − 1, j − 1), forms the three-direction mesh which is sufficient to describe a regular triangulation (each vertex is connected to six neighboring vertices). These two types of topologies of Z 2 , are also relevant to general topologies of control points, since they are generated by most of the topological refinement rules. This is explained in §2.4. 2.1. The main construction methods of schemes. There are two main approaches to the construction of subdivision schemes. The first approach is by repeated averaging. In case s = 1, repeated averaging leads to B-spline schemes. In this approach, the refinement rule (5) consists of several simple steps. The first is the trivial refinement (8) Pαk+1,0 = Pkα , α ∈ Z s 2 with α2 the biggest integer smaller than or equal to α2 for α ∈ Z , and α21 , α22 for α = (α1 , α2 ) ∈ Z 2 . The trivial refinement is followed by a fixed number m of repeated averaging k+1,j −1

Pαk+1,j

=

Pα

k+1,j −1

+ Pα−ej 2

,

α ∈ Z s , j = 1, . . . , m,

where {e1 , . . . , em } are non-zero vectors in Z s with components in {0, 1}. The case s = 1 corresponds to the B-spline scheme of degree m, while in case s = 2 one gets the tensor-product B-spline schemes for the choice e1 = · · · = er = (1, 0), er+1 = · · · = em = (0, 1), 1 ≤ r < m, and the three-direction box-spline schemes [7] for the choice e1 = · · · = er = (1, 0), er+1 = · · · = eρ = (0, 1),

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eρ+1 = · · · = em = (1, 1), 1 ≤ r < ρ < m. One can get other box-spline schemes for more general choices of e1 , . . . , em [7]. The second construction of subdivision schemes is based on a local approximation operator A, approximating on [0, 1]s . A is defined in terms of samples of the approximated function in a set of points A ⊂ Z s , (Af )(x) = f (α)wα (x), x ∈ [0, 1]s . (9) α∈A

For geometrical applications, the set A contains the set Es of extreme points of [0, 1]s , and is symmetric relative to [0, 1]s . The operator A has to be scale and shift invariant, so that (9) can be used in any refinement level and at any location. This leads to the choice of a polynomial approximation operator A. The commonly derived refinement rule from (9) is

γ k+1 k = Pα+β wβ (10) , γ ∈ Es . P2α+γ 2 β∈A

Another possibility is

γ k+1 k P2α+γ = Pα+β wβ g + , 2

γ ∈ Es , g = {1/4}s .

(11)

β∈A

In case s = 1, with Af the interpolation polynomial based on the symmetric set of points relative to [0, 1], −N + 1, . . . , 0, 1, . . . , N , the resulting family of schemes obtained by (10) for N = 1, 2, . . . consists of the Dubuc–Deslaurier schemes [22] P2ik+1

=

Pik ,

k+1 P2i+1

=

N =−N +1

1 k w Pi+ , 2

wi (x) =

N j =−N+1 j =i

x−j i−j

(12)

The schemes in (12) are interpolatory, since the set of control points after refinement contains the control points before refinement. These schemes are convergent, and the limit curves interpolate the initial control points [22]. Interpolatory schemes in general are discussed in [34]. Recently this construction was extended to non-interpolatory schemes [30], by using (11) instead of (10) with wi (x) defined in (12). The refinement rules are N N 1 3 k+1 k+1 k k P2i = Pi+ , P2i+1 = Pi+ w w . 4 4 =−N +1

=−N+1

It is checked in [30] that, for N ≤ 10, the schemes are convergent with limit curves of higher smoothness than the limit curves of the corresponding Dubuc–Deslaurier schemes. Yet, there is no proof that this holds in general.

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Nira Dyn

In fact, (11) can be further extended to k+1 k = Pα+β wβ g + (1 − 2μ)γ , P2α+γ

γ ∈ Es , g = {μ}s

β∈A

with 0 < μ < 21 . This refinement was studied in [33], [40], [4], for s = 1 and A a linear interpolation operator at the points x = 0, x = 1. For μ = 41 , this is the Chaikin scheme for generating quadratic B-spline curves [10]. For μ = 41 it is a general corner cutting scheme. 2.2. Approximation order of subdivision schemes. A convergent subdivision scheme S, constructed by the second approach of §2.1 with refinement rule (10), has the property of reproduction of polynomials. Let the operator A map the set f |A to a unique interpolation polynomial of total degree not exceeding m, interpolating the data {(x, f (x)) : x ∈ A}. In the following, we denote by m (R s ) the space of all s-variate polynomials of degree up to m. It is easy to verify that for f ∈ m (R s ) and f 0 = {fα0 = f (αh) : α ∈ Z s }, h ∈ R + , the refinement (10) generates data on f , namely f k = S k f 0 = {fαk = f (2−k αh) : α ∈ Z s }, and therefore S ∞ f 0 = f , and the subdivision scheme reproduces polynomials in m (R s ). In case of the refinement rule (11), arguments as in [30] lead to (S ∞ f 0 )(x) = f (x + 2hg), with g as in (11). This property of the scheme S is termed reproduction with a fixed shift of polynomials in m (R s ). The reproduction of polynomials in m (R s ) (with or without a shift), the representation of S ∞ f 0 in terms of the compactly supported basic limit function φ of S, S ∞ f 0 (x) = fα0 φ(x − α), (13) α∈Z s

and classical quasi-interpolation arguments [5], lead to the error estimate

sup (S ∞ f 0 )(x) − f (x) ≤ Chm+1 .

(14)

x∈

In (14) f 0 = {fα0 = f (αh) : α ∈ Z s } for the refinement rule (10), while, for the refinement rule (11), f 0 = {fα0 = f (αh − 2gh) : α ∈ Z s }, where f is a smooth enough function, is a bounded domain in R s , and the constant C may depend on S, f , but not on h. A subdivision scheme satisfying (14) is said to have approximation order m + 1. Subdivision schemes constructed by repeated averaging reproduce constant functions and hence have approximation order 1. If the repeated averaging is done in a symmetric way relative to [0, 1]s , then the resulting scheme reproduces also linear polynomials, and the scheme has approximation order 2. For example, this property is shared by all the symmetric B-spline schemes of odd degrees. The mask of the

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scheme generating B-spline curves, based on the symmetric B-spline of degree 2+1 is 1 2 + 2 [2+1] ai = 2+1 , i = − − 1, . . . , 0, . . . , + 1. +1+i 2 The repeated averaging for such a symmetric mask takes the symmetric form 1 k k (P + Pi+1 ), i ∈ Z , 2 i 1 k+1,j −1 k+1,j −1 k+1,j −1 = Pi−1 , i ∈ Z , j = 1, . . . , , + 2Pi + Pi+1 4 Pik+1 = Pik+1, , i ∈ Z . P2ik+1,0 = Pik ,

k+1,j

Pi

k+1,0 P2i+1 =

2.3. Convergence and smoothness analysis. Given the coefficients of the mask of a scheme, one would like to be able to determine if the scheme is convergent, and what is the smoothness of the resulting basic limit function (which is the generic smoothness of the limits generated by the scheme in view of (13)). Such analysis tools are essential for the design of new schemes. We present one method for convergence analysis of the two cases s = 1, 2. The method for smoothness analysis in case s = 1 is simpler and is given in full. Its extension to s = 2 is omitted, but some special cases are discussed. There are other methods for convergence and smoothness analysis, see, e.g., [18], [19], [20], [42], [45]. An important tool in the analysis of convergence, presented here, is the symbol of a scheme Sa with the mask a = {aα : α ∈ σ (a)}, aα zα . (15) a(z) = α∈σ (a)

A first step towards the convergence analysis is the derivation of the necessary condition for uniform convergence, aα−2β = 1, α ∈ Es , (16) β∈Z s

derived easily from the refinement rule aα−2β fβk , fαk+1 =

α ∈ Zs .

β∈Z s

with f k = {fαk ∈ R : α ∈ Z s }. The necessary condition (16) implies that we have to consider symbols satisfying a(1) = 2,

a(−1) = 0

if s = 1,

(17)

or a(1, 1) = 4,

a(−1, 1) = a(1, −1) = a(−1, −1) = 0

if s = 2.

(18)

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Nira Dyn

In case s = 1, condition (17) is equivalent to a(z) = (1 + z)q(z) with

q(1) = 1.

(19)

The scheme with symbol q(z), Sq , satisfies Sq = Sa (see, e.g. [24]), where is the difference operator (20) f = (f )i = fi − fi−1 : i ∈ Z . A necessary and sufficient condition for the convergence of Sa is the contractivity of the scheme Sq , namely Sa is convergent if and only if Sq∞ f 0 = 0 for any f 0 [33]. The contractivity of Sq is equivalent to the existence of a positive integer L, such that SqL ∞ < 1. This condition can be checked for a given L by algebraic operations on the symbol q(z) (see, e.g., [24], [25]). For practical geometrical reasons, only small values of L have to be considered, since a small value of L guarantees “visual convergence” of {F k (t)} to Sa∞ P 0 , already for small k, as the distances between consecutive control points contract to zero fast. A good scheme corresponds to L = 1 as the B-spline schemes, or to L = 2 as many of the schemes constructed by the second method in §2.1 (see the following examples). For s = 2, the necessary condition (18) guarantees the existence of two decompositions of the form (1 − zi )a(z) = qi1 (z)(1 − z12 ) + qi2 (z)(1 − z22 ),

i = 1, 2,

(21)

where z = (z1 , z2 ). The above two decompositions extend to s = 2 the factorization (19) written as (1 − z)a(z) = (1 − z2 )q(z). The decompositions (21) guarantee the existence of a matrix subdivision scheme SQ , with a 2 × 2 matrix symbol Q(z) = {qij (z)}2i,j =1 , satisfying SQ (1 , 2 )T = (1 , 2 )T Sa . Here (1 , 2 )T is the vector difference operator, extending (20) to s = 2, (1 , 2 )T f = ((1 , 2 )T f )α = (fα − fα−(1,0) , fα − fα−(0,1) )T : α ∈ Z 2 . A sufficient condition for the convergence of Sa is the contractivity of SQ , which can be checked by algebraic operations on the symbol Q(z) [9], [24], [44]. Since many of the schemes have symmetries relative to Z 2 , their symbols are factorizable and have the form a(z) = (1 + z1 )(1 + z2 )q(z). As a simple extension of the case s = 1, we get that Sa is convergent if the two schemes with symbols (1 + z1 )q(z), (1 + z2 )q(z) are contractive. If a(z) is symmetric in the sense that q(z1 , z2 ) = q(z2 , z1 ), then it is sufficient to check the contractivity of (1 + z1 )q(z) (see, e.g., [25]). The smoothness analysis in the case s = 1, relies on the result that if the symbol of a scheme has a factorization 1+z ν a(z) = b(z), (22) 2


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such that the scheme Sb is convergent, then Sa is convergent and its limit functions are related to those Sb by D ν (Sa∞ f 0 ) = Sb∞ ν f 0 ,

(23)

with D the differentiation operator [33], [24]. Thus, each factor (1 + z)/2 multiplying a symbol of a convergent scheme adds one order of smoothness. This factor is termed a smoothing factor. The relation between (22) and (23) is a particular instance of the “algebra of symbols” [35]. If a(z), b(z) are two symbols of converging schemes, then Sc with the symbol c(z) = 21s a(z)b(z) is convergent, and φc = φa ∗ φb .

(24)

Example (B-spline schemes). The smoothness of the limit functions generated by m+1 the m-th degree B-spline scheme, having the symbol a [m] (z) = 2 1+z , can be 2 2

concluded easily. The factor b(z) = (1+z) corresponds to Sb generating a piecewise 2 linear interpolant to the initial data {(i, fi0 )}, which is continuous, and the factors 1+z m−1 add smoothness, so that Sa∞[m] f 0 ∈ C m−1 . Note that a [m] (z) consists of 2 smoothing factors only. In fact the B-spline schemes are optimal, in the sense that for a given support size of the mask, the limit functions generated by the corresponding B-spline scheme is of maximal smoothness. Example (the four-point scheme). Here we present the most general univariate interpolatory scheme which is based on four points [31], and describe briefly its convergence and smoothness analysis. The refinement rule is k+1 k k k f2ik+1 = fik , f2i+1 = −w(fi−1 + fi+2 ) + 21 + w (fik + fi+1 ), with w a parameter controlling the shape of the limit curves. The symbol of the scheme is 1 aw (z) = 2z (z + 1)2 1 − 2wz−2 (1 − z)2 (z2 + 1) . (25) Note that for w = 0, a0 (z) is the symbol of the two-point scheme generating the polygonal line through the initial control points, and that for w = 1/16 it coincides with the symbol of the Dubuc–Deslauriers scheme based on four points (reproducing cubic polynomials). The range of w for which Sa w is convergent is the range for which Sbw with symbol bw (z) = aw (z)/(1 + z) is√contractive. The condition Sbw ∞ < 1 holds in the range −3/8 < w < (−1 + 13)/8, while the condition Sb2w ∞ < 1 holds √ in the range −1/4 3. (37) k 4 k 2 k With the modified weights, the generated surfaces are C 1 for any valency greater than three, and are better looking in the vicinity of irregular vertices of valency between four and eight.

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Figure 6. Head. Initial control net (left), after two refinements: with Catmull–Clark scheme (middle) and with Loop scheme (right).

_

w

s3

w

sK _ 1

1 2

1 2

_

_

2w

w

_

2w

w

s2 3 4

s1

s0

Figure 7. Weights for e-vertex: butterfly scheme (left), modified butterfly scheme (right).

3. Linear extensions In this section, we review several extensions of stationary linear schemes for the refinement of points to stationary linear schemes which refine other objects. 3.1. Matrix subdivision schemes. Matrix schemes are defined by matrix masks and refine sequences of vectors. Although, in the geometric setting, the schemes of §2 refine sequences of control points in R 2 or in R 3 , the schemes operate on each component of the vectors in the same way, such that the refinement of one component is independent of the other components. This property is very important in geometric applications, since the subdivision schemes commute with affine transformations (the schemes are affine invariant). The schemes presented here are not affine invariant, and their main application is in multiwavelets constructions [12], [58] and in the

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analysis of multivariate subdivision schemes for control points as indicated in §2.3 (see, e.g., [24]). A finite set of matrices of order d × d, A = {Aα : α ∈ σ (A) ⊂ Z s }, defines a matrix subdivision scheme SA with a refinement rule Aα−2β vβ , v = {vα ∈ R d : α ∈ Z s }. (1) (SA v)α = β∈Z s

Given initial “control vectors” v 0 = {vα0 ∈ R d : α ∈ Z s }, the matrix subdivision scheme SA generates a sequence of control vectors by v k+1 = SA v k ,

k = 1, 2, . . . .

(2)

The notion of uniform convergence from §2 can be extended to this case, by considering the convergence of each of the d components of the vectors. The convergence analysis has a linear algebra component to it, in addition to the analysis component. By considering the matrices Aγ −2β , γ ∈ Es , (3) Bγ = β∈Z s

one can easily conclude a necessary condition for convergence. This condition is the analogue of condition (2.16), stating that for any initial sequence of control vectors v 0 , and any x ∈ R s , ∞ 0 v )(x) ∈ span{u ∈ R d : Bγ u = u for all γ ∈ Es }. (SA

(4)

In the extreme case of schemes with Bγ = I , γ ∈ Es , the space in (4) is R d , and no condition of linear-algebra type is imposed. Such are the schemes used in the analysis of convergence and smoothness of multivariate schemes for points. Schemes for which the space in (4) is R d , are very similar to schemes with a scalar mask [17]. In the other extreme case, the space in (4) is one dimensional with vectors of equal ∞ v 0 , has equal components. An components, implying that the limit vector function SA example of this type of schemes is provided by matrix subdivision schemes generating multiple knot B-spline curves (see, e.g., [53]). This latter extreme case is the most relevant to the construction of multiwavelets. In [13] and in [38], univariate (s = 1) matrix schemes with the space (4) of dimension m, 1 ≤ m ≤ d, are studied. An appropriate change of basis, depending on the structure of the space (4), facilitates the extension of the factorization of scalar symbols to a certain factorization of matrix symbols. This factorization is sufficient for convergence and smoothness analysis of matrix schemes, and is also necessary under an extension of the notion of L∞ -stability (see §2.3) to the matrix case. Multivariate matrix schemes with the space (4) of general dimension are considered in [56]. In the next section we discuss a special type of matrix subdivision schemes, which is relevant to curve design from locations and normals, and to the generation of

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functions from the point values of the functions and their derivatives. The use of analogous schemes for the generation of surfaces from locations and normals is not straightforward, and leads to non-linear schemes. 3.2. Hermite subdivision schemes. The first Hermite schemes to be studied were univariate and interpolatory [51]. Interpolatory Hermite subdivision schemes are matrix schemes, such that the components of the vectors are regarded as the value of a function and its consecutive derivatives up to a certain order at the points of 2−k Z s . Non-interpolatory Hermite subdivision schemes were introduced later [43]. 3.2.1. Univariate interpolatory Hermite schemes. The most common construction of interpolatory Hermite subdivision schemes is similar to the second construction method presented in §2.1. The approximation operator A is an extension of the one in (2.9). For interpolatory schemes, it is a polynomial interpolation operator of the form d−1 wα,i (x)f (i) (α), (5) (Af )(x) = α∈A i=0

= α ∈ A, i = 0, 1, . . . , d − 1. satisfying The refinement is similar to (2.10), namely D i (Af )(α)

k+1 v2α = vαk ,

f (i) (α),

k+1 v2α+1

j

=

d−1

k D j wβ,i (1/2)(vα+β )i ,

0 ≤ j ≤ d − 1. (6)

β∈A i=0

In (6), (v)i denotes the i-th component of the vector v. The refinement (6) can be written in terms of a matrix mask as, (k) vαk+1 = Aα−2β vβk , α ∈ Z s , (7) β∈Z s

where the matrices with even indices are (k)

A2α = δα,0 Id×d ,

α ∈ Z,

(8)

with δα,0 = 0 for α = 0, and δ0,0 = 1. The matrices with odd indices depend on the refinement level k, and have the form A2α+1 = d (2k )A2α+1 d (2−k ), (k)

(0)

α ∈ Z,

(9)

α ∈ A.

(10)

with d (h) = diag(1, h, h2 , . . . , hd−1 ) and

(0) A1−2α

d−1 1 = D wα,j , 2 i,j,=0 i

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The powers of 2 in (9) are due to the fact that derivatives of polynomials are not scale invariant. More precisely if q(x) = p(hx), with p a polynomial, then (D j q)(hx0 ) = hj (D j p)(x0 ). An interpolatory Hermite scheme is termed uniformly convergent if there is a limit vector function F of the form F = (D j f, 0 ≤ j ≤ d − 1)T , with f ∈ C d−1 (R ), satisfying for any closed interval [a, b], F (2−k α) − v k = 0, lim sup α k→∞ α∈2k [a,b]∩Z

with · any norm in R d . Example (a two-point Hermite interpolatory scheme). The scheme is given by the non-zero matrices of its mask: 1 1 ν2−k −ν2−k (k) (k) 2 2 A0 = I2×2 , A1 = = , A . 1−μ −1 −μ2k 1−μ μ2k 2 2 This scheme with ν = 1/8 and μ = 3/2 generates the C 1 piecewise Hermite cubic interpolant to the data {vi0 = (f (i), f (i))T : i ∈ Z }, while for ν = 0, μ = 1, it generates the piecewise linear interpolant to the given function’s values at the integers, which is only C 0 . By the analysis to be reviewed, it can be shown that for 0 < ν < 1/4, μ = 4ν + 1, the limit functions generated by the scheme are C 1 . (See, e.g. [36].) One method for the convergence analysis of such schemes is based on deriving an equivalent stationary matrix scheme, refining vectors of (d − 1)-th order divided differences, obtained from the original control vectors. The limit of such a scheme, if it exists, necessarily consists of equal components, which are the derivative of order d − 1 of the smooth function f [37]. More precisely, the divided difference vector ukn at level k is defined for each n ∈ Z by (ukn )j = [τj +1 , τj +2 , . . . , τj +d ]f, j = 0, . . . , d − 1, with τ1 = · · · = τd−1 = (n − 1)2−k , τd = τd+1 = · · · = τ2d−1 = n2−k . Here we use the definition of divided differences, allowing repeated points for functions with enough derivatives (see, e.g., [6, Chapter 1]). In our setting all integer points have k multiplicity d. The vector ukn can be derived from the vectors vn−1 and vnk . The symbol D(z) of the matrix scheme refining the control vectors uk = {ukn : (0) n ∈ Z } can be obtained recursively from the symbol D [0] (z) = α Aα zα , by algebraic manipulations, involving multiplication by certain matrix Laurent polynomials and their inverses. It is proved in [37] that the matrix symbol D(z) is a matrix Laurent polynomial if the scheme (7) reproduces polynomials of degree ≤ d − 2, and that necessarily a scheme of the form (7), which generates C d−1 functions, reproduces polynomials of degree ≤ d − 1. In (5), the degree of the interpolation polynomial is d|A| − 1,


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so the scheme (7), with the mask (8),(9),(10), reproduces polynomials of degree at least 2d − 1, as A contains at least the points 0, 1. These arguments lead to the conclusion that the Hermite subdivision scheme SA , refining the control vectors v k can be transformed into the matrix subdivision scheme SD for the control vectors uk . To determine the convergence of the scheme SD , which is equivalent to the convergence of the original Hermite subdivision scheme SA to C d−1 functions, we observe that the component (ukn )j , in case of convergence, approximates f (d−1) (2−k n) for j = 1, . . . , d. Thus as in the case of control points, a necessary condition for convergence is the contractivity of the scheme which refines the differences (ukn )j − (ukn )j −1 , j = 2, . . . , d, (ukn )1 − (ukn−1 )d , n ∈ Z . Indeed, such a scheme exists, and its symbol is a matrix Laurent polynomial when (7) reproduces polynomials of degree ≤ d − 1 [37], guaranteeing that the contractivity of this scheme can be checked by algebraic manipulations. The analysis of higher order smoothness is along the same lines. 3.3. B-spline subdivision schemes for compact sets. In the last years, the univariate B-spline schemes were extended to operate on data consisting of compact sets [27], [28]. The motivation for the study of such schemes is the problem of approximating a 3D object from a discrete set of its 2D parallel cross-sections, and the problem of approximating a 2D shape from a discrete set of its 1D parallel cross-sections. In both problems, either the 3D object or the 2D shape is regarded as a univariate setvalued function, with its parallel cross-sections as images. The B-spline subdivision schemes are adapted to this setting, so that the limit set-valued function generated by the subdivision from samples of a continuous set-valued function, approximates it. For initial data F 0 = {Fi0 ⊂ R n : i ∈ Z } consisting of convex compact sets, averages of numbers in the execution of a scheme, can be replaced by Minkowski averages n of sets. A Minkowski average of sets B1 , . . . , B ⊂ R with weights λ1 , . . . , λ ∈ R , i=1 λi = 1, is the set M

λi Bi =

λi bi : bi ∈ Bi .

i=1

i=1

Thus the m-th degree B-spline subdivision scheme (1.4) can be adapted to convex compact sets by the refinement rule [m] Fik+1 = ai−2j Fjk , i ∈ Z , (11) M

j

with a [m] = {ai[m] , i = 0, . . . , m + 1} given in (1.2). Since the coefficients of the mask are positive, the sets F k , k ≥ 1, generated by the subdivision scheme SM,a [m] with the refinement rule (11) are compact and convex [27]. By the associativity and distributivity of the Minkowski average of convex sets with positive weights, it can

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be deduced straightforwardly that the limit generated by SM,a [m] from F 0 , when F 0 consists of convex compact sets, is ∞ 0 (SM,a Fi0 Bm (t − i). (12) [m] F )(t) = i∈Z

M

In (12) the convergence is in the Hausdorff metric, defined for two sets A, B in R n , by haus(A, B) = max sup inf a − b , sup inf a − b a∈A b∈B

b∈B a∈A

with · the Euclidean norm in R n . The subdivision scheme SM,a [m] has approximation properties. It is shown in [27] that for a set-valued function G with convex compact images, which is Lipschitz continuous, namely satisfies haus(G(t), G(t + )) = O(t), and for initial data F 0h = {G(ih) : i ∈ Z } ∞ 0 (13) haus (SM,a [m] F h )(t), G(t) = O(h). The subdivision SM,a [m] fails to approximation set-valued functions with general compact images. As is shown in [29], for initial data F 0 consisting of general compact sets, ∞ 0 Fi0 Bm ( · − i) SM,a [m] F = i∈Z

0 ∞ Thus SM,a [m] F is convex even when the initial sets are non-convex, and it cannot approximate set-valued functions with general compact sets as images. There is another adaptation of the B-spline subdivision schemes to compact sets [28], which yields approximation also in case of set-valued functions with general compact sets as images. This adaptation is obtained by using the first construction in §2.1 for s = 1, and by replacing the average of two numbers by the metric average of two compact sets, introduced in [1], A ⊕t B = ta + (1 − t)b : (a, b) ∈ (A, B)

with Fi0 the convex hull of Fi0 .

with (A0 , A1 ) = (a0 , a1 ) : ai ∈ Ai , i = 0, 1,

a0 − a1 = min ai − a , j = 1 − i, for i = 0 or 1 . a∈Aj

The refinement rule of the resulting scheme SMA,m is achieved by the m + 1 steps, F2ik+1,0 = Fik , k+1,j Fi

=

k+1,j −1 Fi

⊕1 2

k+1,0 F2i+1 = Fik , k+1,j −1 Fi−1 ,

Fik+1 =

Fik+1,m ,

i ∈ Z,

i ∈ Z , j = 1, . . . , m i∈Z

(14)


1221

The refinement rule (14) is denoted formally by F k+1 = SMA,m F k . Two important properties of the metric average, which are central to its application in B-spline subdivision schemes are A ⊕t A = A,

haus(A ⊕t B, A ⊕s B) = |s − t| haus(A, B),

(15)

for (s, t) ∈ [0, 1]. Let the sequence {H k } consist of the “piecewise linear” set valued functions, k interpolating {F k = SMA,m F 0 }, k H k (t) = Fik ⊕λ(t) Fi+1 ,

2−k i ≤ t < 2−k (i + 1), i ∈ Z , k = 0, 1, 2, . . . , (16)

with λ(t) = i + 1 − 2k t. It is proved in [28], with the aid of the metric property of the metric average (the second equality in (15)) and the completeness of the metric space of compact sets with the Hausdorff metric, that the sequence {H k (t)} converges to a ∞ limit set-valued function denoted by SMA,m F 0. Moreover, for G a Lipschitz continuous set valued function with general compact sets as images, the limit generated by the scheme SMA,m starting from F 0h = {G(ih) : i ∈ Z } approximates G with “error” given by ∞ haus (SMA,m F 0h )(t), G(t) = O(h), t ∈ R . (17) 3.4. A blending-based subdivision scheme for nets of curves. The quadratic Bspline scheme (Chaikin algorithm) was extended to the refinement of nets of curves in [15]. A net of curves with parameter d > 0 consists of two families of continuous curves φi (s) : 0 ≤ i ≤ n, s ∈ [0, md] , ψj (t) : 0 ≤ j ≤ m, t ∈ [0, nd] satisfying the compatibility condition φi (j d) = ψj (id),

i = 0, . . . , n, j = 0, . . . , m.

Such a net is denoted by N (d, {φi }ni=0 , {ψj }jm=0 ). The blending-based Chaikintype scheme refines a net of curves, N0 = N (d, {φi0 }ni=0 , {ψj0 }jm=0 ) into a net of 1 2m−1 curves N1 = N d2 , {φi1 }2n−1 A repeated application of such refinei=0 , {ψj}j =0 . k k ments generates a sequence of nets Nk = N 2dk , {φik }ni=0 : k ∈ Z + , with , {ψjk }jm=0 nk = 2k (n − 1) + 1, mk = 2k (m − 1) + 1, which converges uniformly to a continuous surface [15]. The construction of the refinement rule is analogous to the second method in §2.1. The approximation operator A maps a net of curves N (d, {φi }ni=0 , {ψj }jm=0 ) into the piecewise Coons patch surface, interpolating the curves of the net, C(N )(s, t) = C(φi , φi+1 , ψj , ψj +1 ; d)(s − j d, t − id), (18) (s, t) ∈ [j d, j d + d] × [id, id + d], i = 0, . . . , n − 1, j = 0, . . . , m − 1,

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with C(φi , φi+1 , ψj , ψj +1 ; d) a Coons patch [16]. Four continuous curves φ0 , φ1 , ψ0 , ψ1 defined on [0, h] and satisfying φi (j h) = ψj (ih), i, j = 0, 1, define a Coons patch on [0, h]2 . For (s, t) ∈ [0, h]2 the Coons patch is given by C(φ0 , φ1 , ψ0 , ψ1 ; h)(s, t)

t t s s = 1− φ0 (s) + φ1 (s) + 1 − ψ0 (t) + ψ1 (t) (19) h h h h

t t t s t s 1− φ0 (0)+ φ1 (0) + 1− φ0 (h)+ φ1 (h) . − 1− h h h h h h The Coons patch is blending between two surfaces. One is interpolating linearly between corresponding points of φ0 , φ1 and the other between the corresponding points of ψ0 , ψ1 . (These two surfaces are the two first terms on the right-hand side of (19)). It is easy to verify that C(φ0 , φ1 , ψ0 , ψ1 ; h) coincides with the four curves on the boundary of [0, h]2 , namely that C(φ0 , φ1 , ψ0 , ψ1 ; h)(j h, t) = ψj (t), C(φ0 , φ1 , ψ0 , ψ1 ; h)(s, ih) = φi (s),

j = 0, 1, i = 0, 1.

Regarding the Coons patch of four curves as the analogue of a linear segment between two points, the Chaikin scheme for the refinement of control points is “extended” to nets of curves, by sampling each of the Coons patches of C(Nk ) at 1/4 and 3/4 of the corresponding parameters values. Thus the refinement rule analogous to (2.11) is 1 d 3 d k+1 k+1 φ2i (s) = C(Nk ) s, i + , φ2i+1 = C(Nk ) s, i + , 4 2k 4 2k i = 0, . . . , nk − 1, (20) 1 d 3 d k+1 k+1 ψ2j (t) = C(Nk ) j+ , t , ψ2j j+ ,t , +1 = C(Nk ) 4 2k 4 2k j = 0, . . . , mk − 1. (21) This refinement rule generates a refined net of curves after a simple reparametrization. This is written formally as Nk+1 = SBC Nk . The proof of convergence of the scheme SBC is not an extension of the analysis of §2.3, but is based on the proximity of SBC to a new subdivision scheme Sa for points, which is proved to be convergent by the analysis of §2.3. Convergence proofs by proximity to linear stationary schemes for points are employed, e.g., in [35] for the analysis of linear non-stationary schemes, and in [60] for the analysis of a certain class of non-linear schemes. Another important ingredient in the convergence proof is a property of a net of curves, which is preserved during the refinements with SBC . A net of curves


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N (d, {φi }ni=0 , {ψj }jm=0 ) is said to have the M-property if the second divided differences of all curves of the net at three points restricted to intervals of the form d, + 21 d , ∈ (1/2)Z in the domain of definition of the curves, are all bounded by a constant M. The sequence {C(Nk ) : k ∈ Z + } of continuous surfaces is shown to be a Cauchy sequence for N0 with the M-property, by comparison of one refinement of SBC with one refinement of Sa . The scheme Sa is constructed to be in proximity to SBC in the sense that 2 d E (SBC Nk ) − Sa (E (Nk )) ≤ 3 M , (22) 2 2k+1 with E (Nk ) = C(Nk ) i d2 , j d2 , 0 ≤ i ≤ 2mk , 0 ≤ j ≤ 2nk , and with M the constant in the M-property satisfied by all the nets {Nk : k ∈ Z + } which are generated by SBC . Although the limit of the Cauchy sequence {C(Nk ) : k ∈ Z + } is only C 0 , it is conjectured in [15] that SBC generates C 1 surfaces from initial curves which are C 1 . This conjecture is based on simulations. Acknowledgement. The author wishes to thank David Levin and Adi Levin for helping with the figures and the references.

References [1] Artstein, Z., Piecewise linear approximations of set-valued maps. J. Approx. Theory 56 (1989), 41–47. [2] Ball, A. A., Storry, D. J. T., Conditions for tangent plane continuity over recursively generated b-spline surfaces. ACM Transactions on Graphics 7 (1988), 83–102. [3] Ball, A. A., Storry, D. J. T., Design of an n-sided surface patch from Hermite boundary data. Comput. Aided Geom. Design 6 (1989), 111–120. [4] de Boor, C., Cutting corners always works. Comput. Aided Geom. Design 4 (1987), 125–131. [5] de Boor, C., Quasi-interpolants and approximation power of multivariate splines. In Computation of Curves and Surfaces (ed. by W. Dahmen, M. Gasca, C. Micchelli), NATO ASI Series, Kluwer Academic Press, 1990, 313–346. [6] de Boor, C., A Practical Guide to Splines. Appl. Math. Sci. 27, Springer-Verlag, New York 2001. [7] de Boor, C., Höllig, K., Riemenschneider, S., Box Splines. Appl. Math. Sci. 98, SpringerVerlag, New York 1993. [8] Catmull, E., Clark, J., Recursively generated b-spline surfaces on arbitrary topological meshes. Comput. Aided Design 10 (1978), 350–355. [9] Cavaretta, A. S., Dahmen, W., Micchelli, C. A., Stationary Subdivision. Mem. Amer. Math. Soc. 452, Amer. Math. Soc., Providence, RI, 1991.

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[10] Chaikin, G. M., An algorithm for high speed curve generation. Computer Graphics and Image Processing 3 (1974), 346–349. [11] Cirak, R., Scott, M., Schröder, P., Ortiz, M., Antonsson, E., Integrated modeling, finiteelement analysis and design for thin-shell structures using subdivision. Comput. Aided Design 34 (2002), 137–148. [12] Cohen, A., Daubechies, I., Plonka, G., Regularity of refinable function vectors. J. Fourier Anal. Appl. 3 (1997), 295–324. [13] Cohen, A., Dyn, N., Levin, D., Stability and inter-dependence of matrix subdivision schemes. In Advanced Topics in Multivariate Approximation (ed. by F. Fontanella, K. Jetter, P. J. Laurent), Ser. Approx. Decompos. 8, World Scientific Publishing Co., River Edge, NJ, 1996, 33–45. [14] Cohen, E., Lyche, T., Riesenfeld, R. F., Discrete b-splines and subdivision techniques in Computer-Aided Geometric Design and Computer graphics. Computer Graphics and Image Processing 14 (1980), 87–111. [15] Conti, C., Dyn, N., Blending-based Chaikin-type subdivision schemes for nets of curves. In Mathematical Methods for Curves and Surfaces: Tromso 2004 (ed. by M. Dahlen, K. Morken, L. Schumaker), Mod. Methods Math., Nashboro Press, Brentwood, TN, 2005, 51–68. [16] Coons, S. A., Surface for computer aided design. Tech. Rep., MIT, 1964. [17] Cotronei, M., Sauer, T., Full rank filters and polynomial reproduction. Preprint. [18] Daubechies, I., Ten Lectures on Wavelets. SIAM, Philadelphia, 1992. [19] Daubechies, I., Lagarias, J. C., Two-scale difference equations I, existence and global regularity of solutions. SIAM J. Math. Anal. 22 (1992), 1388–1410. [20] Daubechies, I., Lagarias, J. C., Two-scale difference equations II, local regularity, infinite products of matrices and fractals. SIAM J. Math. Anal. 23 (1992), 1031–1079. [21] DeRose, T., Kass, M., Truong, T., Subdivision surfaces in character animation. In SIGGRAPH 98 Conference Proceedings, Annual Conference Series, ACM SIGGRAPH, 1998, 85–94. [22] Deslauriers, G., Dubuc, S., Symmetric iterative interpolation processes. Constr. Approx. 5 (1989), 49–68; Erratum ibid. 8 (1992), 125–126. [23] Doo, D., Sabin, M., Analysis of the behaviour of recursive division surface near extraordinary points. Comput. Aided Design 10 (1978), 356–360. [24] Dyn, N., Subdivision schemes in computer aided geometric design. In Advances in Numerical Analysis II, Wavelets Subdivision Algorithms and Radial Basis Functions (ed. by W. A. Light), Oxford Sci. Publ., Oxford University Press, New York 1992, 36–104. [25] Dyn, N., Subdivision: Analysis of convergence and smoothness by the formalism of Laurent polynomials. In Tutorials on Multiresolution in Geometric Modelling (ed. by M. Floater, A. Iske, E. Quak), Math. Vis., Springer-Verlag, Berlin 2002, 51–68. [26] Dyn, N., Three families of nonlinear subdivision schemes. In Multivariate Approximation and Interpolation (ed. by K. Jetter, M. D. Buhmann, W. Haussmann, R. Schaback, J. Stöckler), Elsevier, 2005,23–38. [27] Dyn, N., Farkhi, E., Spline subdivision schemes for convex compact sets. J. Comput. Appl. Math. 119 (2000), 133–144.


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[28] Dyn, N., Farkhi, E., Spline subdivision schemes for compact sets with metric averages. In Trends in Approximation Theory (ed. by K. Kopotun, T. Lyche, M. Neamtu), Innov. Appl. Math., Vanderbilt University Press, Nashville, TN, 2001, 93–102. [29] Dyn, N., Farkhi, E., Set-valued approximations with Minkowski averages – convergence and convexification rates. Numer. Funct. Anal. Optim. 25 (2004), 363–377. [30] Dyn, N., Floater, M., Hormann, K., A C 2 four-point subdivision scheme with fourth order accuracy and its extensions. In Mathematical Methods for Curves and Surfaces: Tromso 2004 (ed. by M. Dahlen, K. Morken, L. Schumaker), Mod. Methods Math., Nashboro Press, Brentwood, TN, 2005, 145–156. [31] Dyn, N., Gregory, J. A., Levin, D., A four-point interpolatory subdivision scheme for curve design. Comput. Aided Geom. Design 4 (1987), 257–268. [32] Dyn, N., Gregory, J. A., Levin, D., A butterfly subdivision scheme for surface interpolation with tension control. ACM Transactions on Graphics 9 (1990), 160–169. [33] Dyn, N., Gregory, J. A., Levin, D., Analysis of uniform binary subdivision schemes for curve design. Constr. Approx. 7 (1991), 127–147. [34] Dyn, N., Levin, D., Interpolating subdivision schemes for the generation of curves and surfaces. In Multivariate Approximation and Interpolation (ed. by W. Haussmann, K. Jetter), Internat. Ser. Numer. Math. 94, Birkhäuser Verlag, Basel 1990, 91–106. [35] Dyn, N., Levin, D., Analysis of asymptotically equivalent binary subdivision schemes. J. Math Anal Appl 193 (1995), 594–621. [36] Dyn, N., Levin, D., Analysis of Hermite-type subdivision schemes. In Approximation Theory VIII – Wavelets and Multilevel Approximation (ed. by C. Chui, L. Schumaker), Vol. 2, Ser. Approx. Decompos. 6, World Scientific Publications, River Edge, NJ, 1995, 117–124. [37] Dyn, N., Levin, D., Analysis of Hermite-interpolatory subdivision schemes. In Spline Functions and the Theory of Wavelets (ed. by S. Dubuc), Centre de Recherches Mathématiques, CRM Proc. Lecture Notes 18, Amer. Math. Soc., Providence, RI, 1999, 105–113. [38] Dyn, N., Levin, D., Matrix subdivision-analysis by factorization. In Approximation Theory: A volume dedicated to Blagovest Sendov (ed. by B. Bojanov), Darba, Sofia 2002, 187–211. [39] Dyn, N., Levin, D., Subdivision schemes in geometric modelling. Acta Numer. 11 (2002), 73–144. [40] Gregory, J. A., Qu, R., Non-uniform corner cutting. Comp. Aided Geom. Design 13 (8) (1996), 763–772. [41] Halstead, M., Kass, M., DeRose, T., Efficient, fair interpolation using catmull-clark sutfaces. In SIGGRAPH 93 Conference Proceedings, Annual Conference Series, ACM SIGGRAPH, 1993, 35–44. [42] Han, B., Computing the smoothness exponent of a symmetric multivariate refinable function. SIAM J. Matrix Anal. Appl. 24 (2003), 693–714. [43] Han, B., Yu, T. P., Xue, Y., Noninterpolatory Hermit subdivision schemes. J. Math. Comp. 74 (2005), 1345–1367. [44] Hed, S., Analysis of subdivision schemes for surfaces. Master Thesis, Tel-Aviv University, 1990. [45] Jia, R. Q., Characterization of smoothness of multivariate refinable functions in Sobolev spaces. Trans. Amer. Math. Soc. 351 (1999), 4089–4112.

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[46] Karˆciauskas, K., Peters, J., Reif, U., Shape characterization of subdivision surfaces: case studies. Comput. Aided Geom. Design 21 (2004), 601–614. [47] Kobbelt, L., Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer Graphics Forum 15 (1996), 409–420. [48] Kobbelt, L., Sqrt(3) subdivision. In Proceedings of SIGGRAPH 2000, Annual Conference Series, ACM-SIGGRAPH, 2000, 103–112. [49] Kobbelt, L., Hesse, T., Prautzsch, H., Schweizerhof, K., Interpolatory subdivision on open quadrilateral nets with arbitrary topology. Computer Graphics Forum 15 (1996), 409–420. [50] Loop, C., Smooth spline surfaces based on triangles. Master Thesis, University of Utah, Department of Mathematics, 1987. [51] Merrien, J. L., A family of Hermite interpolants by bisection algorithms. Numer. Algorithms 2 (1992), 187–200. [52] Peters, J., Reif, U., Shape characterization of subdivision surfaces-basic principles. Comput. Aided Geom. Design 21 (2004), 585–599. [53] Plonka, G., Approximation order provided by refinable function vectors. Constr. Approx. 13 (1997), 221–244. [54] de Rahm, G., Sur une courbe plane. J. Math. Pures Appl. (9) 35 (1956), 25–42. [55] Reif, U., A unified approach to subdivision algorithms near extraordinary points. Comput. Aided Geom. Design 12 (1995), 153–174. [56] Sauer, T., Stationary vector subdivision – quotient ideals, differences and approximation power. RACSAM Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 96 (2002), 257–277. [57] Shenkman, P., Computing normals and offsets of curves and surfaces generated by subdivision schemes. Master Thesis, Tel-Aviv university, 1996. [58] Strella, V., Multiwavelets: Theory and Applications. PhD Thesis, MIT, 1996. [59] Velho, L., Zorin, D., 4-8 subdivision. Comput. Aided Geom. Design 18 (2001), 397–427. [60] Wallner, J., Dyn, N., Convergence and C 1 analysis of subdivision schemes on manifolds by proximity. Comput. Aided Geom. Design 22 (2005), 593–622. [61] Zorin, D., Schröder, P., Sweldens, W., Interpolating subdivision for meshes with arbitrary topology. In SIGGRAPH 96 Conference Proceedings, Annual Conference Series, ACMSIGGRAPH, 1996, 189–192. School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel E-mail: [email protected]

Wave propagation software, computational science, and reproducible research Randall J. LeVeque ∗

Dedicated to Germund Dahlquist (1925–2005) and Joseph Oliger (1941–2005), two of the influential mentors who have shaped my career. They each inspired generations of students with their interest in the connections between mathematics and computation. Abstract. Wave propagation algorithms are a class of high-resolution finite volume methods for solving hyperbolic partial differential equations arising in diverse applications. The development and use of the clawpack software implementing these methods serves as a case study for a more general discussion of mathematical aspects of software development and the need for more reproducibility in computational research. Sample applications discussed include medical applications of shock waves and geophysical fluid dynamics modeling volcanoes and tsunamis. Mathematics Subject Classification (2000). Primary 65Y15; Secondary 74S10. Keywords. Hyperbolic partial differential equations, software, numerical analysis, reproducible research, scientific computing, clawpack.

1. Introduction I will ultimately describe a class of numerical methods for solving hyperbolic partial differential equations, software that implements these methods, and some scientific applications. However, for the broad audience that I am honored to address in these proceedings, I would like to first make some more general comments on the topic of software development and its relation to mathematics, and on computational science and reproducible research. I begin with a quote from a 1995 paper by J. B. Buckheit and D. L. Donoho [13] about wavelet analysis and a software package they developed to aid in studying and applying their methods: An article about computational science in a scientific publication is not the scholarship itself, it is merely advertising of the scholarship. The actual scholarship is the complete software development environment and the complete set of instructions which generated the figures. ∗ This research was supported in part by several NSF and DOE grants over the past decade, including NSF grants DMS-9803442, DMS-0106511, and CMS-0245206 and DOE grants DE-FG03-96ER25292, DE-FG0300ER25292, and DE-FC02-01ER25474.


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They present this as a slogan to distill the insights of Jon Claerbout, an exploration geophysicist who has been a pioneer in this direction since the early 90s (e.g., [59]), and they give many compelling examples to illustrate its truth. I first ran across this quote on the webpage [11] of the book [12], which provides complete codes in several languages for the solutions to each of the 100-digit challenge problems proposed by Trefethen [63]. (This is set of ten computational problems, each easy to state and with a single number as the answer. The challenge was to compute at least 10 digits of each number.) In spite of some progress in the direction of reproducible research, many of the complaints of Buckheit and Donoho still ring true today, as discussed further in the recent paper by Donoho and Huo [24]. Much of my work over the past 10 years has been devoted to trying to make it easier for myself, my students, and other researchers to perform computational scholarship in the field of numerical methods for hyperbolic PDEs, and also I hope in a variety of applications areas in science and engineering where these methods are used. This work has resulted in the clawpack software [40]. This software is apparently being fairly widely used, both in teaching and research. More than 5 000 people have registered to download the code, mentioning all sorts of interesting problems they plan to tackle with it. However, I am not convinced that it is being used to the extent possible in advancing scholarship of the type described above. One goal of this paper is to encourage researchers (myself included) to work harder towards this end, in my field and more generally in computational science. I will make a distinction between software and computer programs. In my notation, software means a package of computer tools that are designed to be applied with some generality to a class of problems that may arise in many different applications. A computer program is a code that solves one particular problem. A program may be written entirely from scratch or it may employ one or more software packages as tools. This is an important distinction to make since one has different goals in developing software than in writing a specific computer program. Software is intended to be used by others and to be as general as practically possible. A program is written to solve a problem and often the author does not intend for it to be seen or used by anyone else. Pure mathematicians search for abstract structures that transcend particular examples and that often unify disparate fields. They produce theorems that apply as generally as possible and that can be used as solid and dependable building blocks for future work. In addition to developing new algorithms, some computational mathematicians also produce theorems, rigorous results guaranteeing that a particular algorithm converges or bounds on the magnitude of the error, for example. Such theorems give us the confidence to apply the algorithm to real world problems. Other computational mathematicians focus on the development of software that implements an algorithm in a dependable manner. This is perhaps an under-appreciated art in the mathematical world, compared to proving theorems, but I believe it is an analogous mathematical activity in many ways. In both cases the goal is to distill the essence of a set of particular examples into a general result, something that applies as broadly as possible while giving an interesting and nontrivial result that

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can be built upon and used as a “subroutine” in future work. In both cases the result is an encapsulation of a set of knowledge that has well defined inputs and outputs and is believed to be proved correct, and that applies in many situations. The process of developing a novel algorithm and writing software to implement it is also in some ways similar to the process involved in proving a theorem. One needs some mathematical insight to get started, but then working through many technical details is typically required to make everything fit together in a manner that produces the desired result. This part is not very glamorous but is a crucial part of the scholarship. Often frustrations arise when one little detail does not work out quite the way one hoped. Sometimes algorithms, like partially completed proofs, must be shelved for years until someone else makes a breakthrough in a related area that suddenly makes everything come together. And everything does have to fit together just right; having a nice idea that seems like it should work is not enough. Glossing over the details is not allowed, and is particularly hard to pull off in a computer program. While it may be possible to slip things by the referees in the description of an algorithm in a paper (as also sometimes happens in a shoddy proof), computers will not parse the command “and then a miracle occurs”. We are forced to be explicit in every step. Of course even once a program does work, in the sense of compiling without errors and producing results that seem reasonable, we are faced with the thorny issue of “proving” that it is in fact correct. In computer science there is a whole field devoted to developing methodologies for formally proving the correctness of computer programs. In computational science the programs are often so complex and the problem it is designed to solve so ill-defined that formal correctness proofs are generally impossible. Instead the buzzwords are Verification and Validation (V&V for short). These can be summarized by the following mnemonic: Verification: Validation:

Are we solving the problem right? Are we solving the right problem?

For a physical experiment modeled by partial differential equations, for example, we must verify that the computer program solves the PDEs accurately. A computational scientist must also validate the code by comparing it against experiments to ensure that the PDEs discretized are actually a sufficient model of reality for the situation being modeled. The Euler equations of gas dynamics are sufficient in some situations, but completely inadequate in other cases where viscosity plays a significant role. Researchers in numerical analysis and scientific computing (as defined in the next section) are generally most concerned with verification, while scientists, engineers, and applied mathematicians focusing on mathematical modeling must also be concerned with validation. Even the relatively simple task of verifying that a code solves the given equations properly can be a real challenge, however, and is often as much an art as a science. A good test problem that captures the essence of some potential difficulty while having a solution that can be checked is often hard to come by, and developing test suites for different classes of algorithms is valuable scholarship in

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itself. Numerous papers have been written on the subject of how best to test computer programs or software for scientific computing; see [17], [29], [33], [38], [53], or [56] for just a few approaches. Elegance is valued in algorithm and software design as it is in other mathematical endeavors. Often the first attempt at an algorithm is not very clean; it is a brute force approach that gets the job done. Later work is often devoted to cleaning things up, perhaps in fundamental ways that greatly reduce the computational complexity, but also often in more subtle ways that simply make it more “elegant”, a hard to define property that we recognize when we see. Perhaps I am straying too far from the topic of reproducible scientific computing, but to make progress in this direction I think it is important to recognize software development as a valid and challenging mathematical activity. It takes a slightly different type of mathematician to develop the necessary intuition and skills to excel at this than is required to prove theorems, just as doing algebra vs. analysis takes different mindsets and these are rarely done equally well by the same mathematician. But no one doubts that algebraists and analysts are both mathematicians, even if they cannot get beyond the first page of each others’ papers. Knuth [35] did an interesting study on the connections between algorithmic and mathematical thinking, a topic that he also touched on in an earlier paper [34] on “computer science and its relation to mathematics”. This paper was written in 1974, at a time when many computer science departments were just being established, often by mathematicians. It makes interesting reading today, along with similar papers of the same vintage, such as [9], [25]. Software development is a logical conclusion of algorithmic thinking, and the development of software for mathematical algorithms naturally belongs in a mathematics department. The reason I care about this topic is not for my own mathematical ego. I have been lucky to be at an institution where my work in this direction has been encouraged, or at least tolerated. It may have helped that I did not put much effort into software development until well after I was tenured.1 The main value of the tenure system is that established people do not need to worry what our colleagues think of our activities or how they choose to label them. But the future depends on bright young people. I think computational science affords a wonderful opportunity to get students involved in a host of mathematical challenges, and making significant progress on these requires computational mathematicians with solid training in a broad range of mathematical 1 However, many of my attitudes towards software development were shaped by my experiences as a graduate student in the Computer Science Department at Stanford, where students in the numerical analysis group were responsible for maintaining the library of numerical routines [10] available to physicists at the Stanford Linear Accelerator Center (SLAC) and acting as consultants, activities that were encouraged by Gene Golub and Joe Oliger. There I had the pleasure of working directly with an outstanding set of fellow students, most of whom have gone on to make software contributions of their own, including Marsha Berger, Petter Bjorstad, Dan Boley, Tony Chan, Bill Coughran, Bill Gropp, Eric Grosse, Mike Heath, Frank Luk, Stephen Nash, Michael Overton, and Lloyd Trefethen. Many of us were also shaped by Cleve Moler’s course on numerical linear algebra, where he tried out his new matlab program on us as a front end to the linpack and eispack routines that were already setting the standard for mathematical software [23], [27], [61]. I think the Computer Science students were more impressed with matlab and much more influenced by this experience than Cleve takes credit for in [48].


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tools and the ability to apply mathematical abstraction to common problems arising in multiple fields. While there are many talented computational scientists working on specific challenging problems in virtually every science and engineering department, a computational mathematician, centered in a mathematics (or applied mathematics) department, has the best chance of appreciating the common mathematical structure of the problems and producing algorithms and software that are broadly applicable. Doing so not only avoids a lot of wasted effort by scientists whose time is better spent on the peculiarities of their specialty, it also leads to the introduction of techniques into fields where they might not otherwise be invented and the discovery of new connections between existing algorithms and applications. I once heard Jim Glimm remark that “applied mathematicians are the honey bees of the mathematical world, whose job is to cross-pollinate applications.” In addition to providing a service to those in other disciplines, the process of collecting nectar can result in some sweet rewards back in our own hive. This is equally true in algorithm and software development as it is in more classical and theoretical aspects of applied mathematics. But young mathematicians will feel free to pursue such activities, and to also do the less rewarding but crucial aspects of the scholarship such as documenting their codes and making them presentable to the rest of the world, only if it is accepted as valid mathematical scholarship. If it is seen as non-mathematics, it will only be a waste of time that is best avoided by anyone seeking tenure in a mathematics department. Applied mathematics in general is becoming much more acceptable in mathematics departments than it once was, at least in the United States. However, I doubt that the careful development of software or computer programs, or the work required to turn research codes into publishable scholarship, has the same level of acceptance.

2. Numerical analysis, scientific computing, and computational science & engineering One can argue at length about the meaning of the terms in this section title. To me, “numerical analysis” has a double meaning: the analysis and solution of real-world problems using numerical methods, and the invention and analysis of the methods themselves using the techniques of mathematics. When used in the latter sense, numerical analysis belongs firmly in a mathematics (or applied mathematics) department. As just one example, analyzing the stability and convergence properties of finite difference or finite element methods is no less difficult (often more difficult) than analyzing the underlying differential equations, and relies on similar tools of analysis. Specialists in this type of numerical analysis may or may not do much computing themselves, and may be far removed from computational science. Numerical analysis in the sense of using numerical methods to solve problems, or developing software for general use, is often called “scientific computing” or “com-

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putational science & engineering” these days. One can make a further distinction between these two terms: Scientific computing is often used to refer to the development of computational tools useful in science and engineering. This is the main thrust of the SIAM Journal on Scientific Computing, for example, which contains few theorems relative to the more theoretical SIAM Journal on Numerical Analysis, but still focuses on mathematical and algorithmic developments. Computational science & engineering refers more specifically to the use of computational tools to do real science or engineering in some other field, as a complement to experimental or theoretical science and engineering. Not everyone would agree with my definitions of these terms. In particular, it can be argued that “computational science” refers to the science of doing computation and “computational engineering” to the implementation of this science in the form of software development, but for my purposes I will lump these two activities under “scientific computing”. It is important to be aware of this lack of consistency in nomenclature since, for example, many recently developed academic programs in Computational Science & Engineering stress aspects of scientific computing as well. I have been arguing that “scientific computing”, in the sense just described, is a branch of mathematics (as well as being a branch of other disciplines, such as computer science), and that other mathematicians should be more aware of the intellectual challenges and demands of this field, including the need to document and distribute code. Not only are the activities of many practitioners of scientific computing essentially mathematical, but they (and their students) benefit greatly from frequent contact with more theoretical numerical analysts and mathematicians working in related areas. Other mathematicians may also benefit from having computational experts in the department, particularly as more fields of pure mathematics develop computational sides and realize the benefits of experimental mathematics – there is even a journal (see expmath.org) now devoted to this approach.

3. Reproducible research Within the world of science, computation is now rightly seen as a third vertex of a triangle complementing experiment and theory. However, as it is now often practiced, one can make a good case that computing is the last refuge of the scientific scoundrel. Of course not all computational scientists are scoundrels, any more than all patriots are, but those inclined to be sloppy in their work currently find themselves too much at home in the computational sciences. Buckheit and Donoho [13] refer to the situation in the field of wavelets as “a scandal”. The same can be said of many other fields, and I include some of my own work in the category of scandalous. Where else in science can one get away with publishing observations that are claimed to prove a theory or illustrate the success of a technique without having to give a careful description of the methods used, in sufficient detail that others can attempt to repeat the experiment? In other branches of science it is not only expected


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that publications contain such details, it is also standard practice for other labs to attempt to repeat important experiments soon after they are published. Even though this may not lead to significant new publications, it is viewed as a valuable piece of scholarship and a necessary aspect of the scientific method. Scientific and mathematical journals are filled with pretty pictures these days of computational experiments that the reader has no hope of repeating. Even brilliant and well intentioned computational scientists often do a poor job of presenting their work in a reproducible manner. The methods are often very vaguely defined, and even if they are carefully defined they would normally have to be implemented from scratch by the reader in order to test them. Most modern algorithms are so complicated that there is little hope of doing this properly. Many computer codes have evolved over time to the point where even the person running them and publishing the results knows little about some of the choices made in the implementation. And such poor records are typically kept of exactly what version of the code was used and the parameter values chosen that even the author of a paper often finds it impossible to reproduce the published results at a later time. The idea of “reproducible research” in scientific computing is to archive and make publicly available all of the codes used to create the figures or tables in a paper in such a way that the reader can download the codes and run them to reproduce the results. The program can then be examined to see exactly what has been done. The development of very high level programming languages has made it easier to share codes and generate reproducible research. Historically, many papers and text books contained pseudo-code, a high level description of an algorithm that is intended to clearly explain how it works, but that would not run directly on a computer. These days many algorithms can be written in languages such as matlab in a way that is both easy for the reader to comprehend and also executable, with all details intact. For example, we make heavy use of this in the recent paper [15]. We present various grid mappings that define logically rectangular grids in smooth domains without corners, such as those shown later in Figure 1, and all of the matlab codes needed to describe the various mappings are short enough to fit naturally in the paper. The associated webpage contains the longer clawpack codes used to solve various hyperbolic test problems on these grids. Trefethen’s book on spectral methods [62] is a good example of a textbook along these lines, in which each figure is generated by a 1-page matlab program. These are all included in the book and nicely complement the mathematical description the methods discussed. Trefethen makes a plea for more attention to short and elegant computer programs in his recent essay [64]. However, for larger scale computer programs used in scientific publications, there are many possible objections to making them available in the form required to reproduce the research. I will discuss two of these, perhaps the primary stumbling blocks. One natural objection is that it is a lot of work to clean up a code to the point where someone else can even use it, let alone read it. It certainly is, but it is often well

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worth doing, not only in the interest of good science but also for the personal reason of being able to figure out later what you did and perhaps build on it. Those of us in academia should get in the habit of teaching good programming, documentation, and record keeping practices to our students, and then demand it of them. We owe it to them to teach this set of computational science skills, ones that I hope will be increasingly necessary in academic research environments and that are also highly valued in industrial and government labs. It will also improve the chances that we will be able to build on the work they have done once they graduate, and that future students will be able to make use of it rather than starting from scratch as is too often the case today. While ideally all published programs would be nicely structured and easily readable with ample comments, as a first step it would be valuable simply to provide and archive the working code that produced the results in a paper. Even this takes more effort than one might think. It is important to begin expecting this as a natural part of the process so that people will feel less like they have to make a choice between finishing off one project properly or going on to another where they can more rapidly produce additional publications. The current system strongly encourages the latter. As Buckheit and Donoho [13] point out, the scientific method and style of presenting experiments in publications that is currently taken for granted in the experimental sciences was unheard of before the mid-1800s. Now it is a required aspect of respectable research and experimentalists are expected to spend a fair amount of time keeping careful lab books, fully documenting each experiment, and writing their papers to include the details needed to repeat the experiments. A paradigm shift of the same nature may be needed in the computational sciences. Requiring it of our students may be a good place to start, provided we recognize how much time and effort it takes. Perhaps we should be more willing to accept an elegant and well documented computer program as a substantial part of a thesis, for example. A second objection to publishing computer code is that a working program for solving a scientific or engineering problem is a valuable piece of intellectual property and there is no way to control its use by others once it is made publicly available. Of course if the research goal is to develop general software then it is desirable to have as many people using it as possible. However, for a scientist or mathematician who is primarily interested in studying some specific class of problems and has developed a computer program as a tool for that purpose, there is little incentive to give this tool away free to other researchers. This is particularly true if the program has taken years to develop and provides a competitive edge that could potentially lead to several additional publications in the future. By making the program globally available once the first publication appears, other researchers can potentially skip years of work and start applying the program to other problems immediately. In this sense providing a program is fundamentally different than carefully describing the materials and techniques of an experiment; it is more like inviting every scientist in the world to come use your carefully constructed lab apparatus free of charge.


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This argument undoubtedly has considerable merit in some situations, but on the whole I think it is overblown. It is notoriously difficult to take someone else’s code and apply it to a slightly different problem. This is true even when people are trying to collaborate and willing to provide hands-on assistance with the code (though of course this type of collaboration does frequently occur). It is often true even when the author of the code claims it is general software that is easy to adapt to new problems. It is particularly true if the code is obscurely written with few comments and the author is not willing to help out, as would probably be true of many of the research codes people feel the strongest attachment to. Moreover, my own experience in computational science is that virtually every computational experiment leads to more questions than answers. There is such a wealth of interesting phenomena that can be explored computationally these days that any worthwhile code can probably lead to more publications than its author can possibly produce. If other researchers are able to take the code and apply it in some direction that would not otherwise be pursued, that should be seen as a positive development, both for science and for its original author, provided of course that s/he gets some credit in the process. This is particularly true for computational mathematicians, whose goals are often the development of a new algorithm rather than the solution of specific scientific problems. Even for those not interested in software development per se, anything we can do to make it easier for others to use the methods we invent should be viewed as beneficial to our own careers. Perhaps what is needed is some sort of recognized patent process for scientific codes, so that programs could be made available for inspection and independent execution to verify results, but with the understanding that they cannot be modified and used in new publications without the express permission of the author for some period of years. Permission could be granted in return for co-authorship, for example. In fact such a system already works quite well informally, and greater emphasis on reproducible research would make it function even better. It would be quite easy to determine when people are violating this code of ethics if everyone were expected to “publish” their code along with any paper. If the code is an unauthorized modification of someone else’s, this would be hard to hide.

4. Wave propagation algorithms and clawpack As a case study in software development, and its relation to mathematics, scientific computing, and reproducible research, I will briefly review some of the history behind my own work on clawpack (Conservation LAWs PACKage), software for solving hyperbolic systems of partial differential equations. This software development project began in 1994. I had just taught a graduate course on numerical methods for conservation laws and had distributed some sample computer programs to the students as a starting point for a class project. In the fall I went on sabbatical and decided to spend a few weeks cleaning up the program and

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redesigning it as a software package, in large part because I was also planning to spend much of the year revising my lecture notes [42] from a course I taught at ETH-Zürich in 1989 into a longer book, and I wanted to complement the text with programs the students could easily use. I seriously misjudged the effort involved – I spent most of that year and considerable time since developing software, which grew into something much more that I originally intended. The book [44] took several more years of work and iterations of teaching the course and did not appear until 2002. Virtually all of the figures in this book are reproducible, in the sense that the programs that generated them can each be downloaded from a website and easily run by the student. Most figure captions contain a link to the corresponding website, in the form [claw/book/chap23/advection/polar], for example, from the caption of Figure 23.3, which is easily translated into the appropriate web address. (Start at http://www.amath.washington.edu/~claw/book.html to browse through all these webpages). Each webpage contains the computer code and many also contain additional material not in the book, for example movies of the solution evolving in time. All of these examples are based on the clawpack software. This software is described briefly in the book and more completely in the User Guide [41] available on the web. Once the basic software is installed, the problem-specific code for each example in the book is quite small and easy to comprehend and modify. The reader is encouraged to experiment with the programs and observe how changes in parameters or methods affect the results. These programs, along with others on the clawpack website [40], can also form the basis for developing programs to solve similar problems. This software implements a class of methods I call “wave propagation algorithms” for solving linear or nonlinear hyperbolic problems. Hyperbolic partial differential equations are a broad class of equations that typically model wave propagation or advective transport phenomena. The classic example is the second-order wave equation ptt = c2 pxx for linear acoustics, modeling the propagation of pressure disturbances in a medium with sound speed c. The clawpack software, however, is set up to solve a different form of hyperbolic equations: systems that involve only first order derivatives in space and time. In the linear case, a first-order system of PDEs (in one space dimension and time) has the form qt + Aqx = 0, where q(x, t) is a vector of some m conserved quantities, A is an m × m matrix, and subscripts denote partial derivatives. In the nonlinear case, a system of m conservation laws takes the form qt + f (q)x = 0, where f (q) is the flux function (in the linear case, f (q) = Aq). This system is called hyperbolic if the flux Jacobian matrix f (q) is diagonalizable with real eigenvalues. The system of Euler equations for inviscid compressible gas dynamics has this form, for example, where mass, momentum, and energy are the conserved quantities. The full nonlinear equations can develop shock wave solutions in which these quantities are discontinuous, one of the primary challenges in numerical modeling. Linearizing this system gives the linear acoustics equations in the form of a first-order system of


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equations for pressure and velocity. Cross differentiating this system allows one to eliminate velocity and obtain the single second-order wave equation for p mentioned above, but the first-order formulation allows the modeling of a much broader range of phenomena. First-order hyperbolic systems arise naturally in a multitude of applications, including for example elastodynamics (linear and nonlinear), electromagnetic wave propagation (including nonlinear optics), shallow water equations (important in oceanography and atmospheric modeling), and magnetohydrodynamic and relativistic flow problems in astrophysics. The wave-propagation algorithms are based on two key ideas: Riemann solvers and limiters. The Riemann problem consists of the hyperbolic equation under study with special initial conditions at some time t¯: piecewise constant data with left state q and right state qr and a jump discontinuity in each conserved quantity at a single point in space, say x. ¯ The solution to this Riemann problem for t > t¯ is a similarity solution, a function of (x − x)/(t ¯ − t¯) alone that consists of a set of waves propagating at constant speeds away from the initial discontinuity. The definition of hyperbolicity guarantees this, and the eigenvalues of the flux Jacobian are related to the wave speeds. In the linear case the eigenvalues of the matrix A are exactly the wave speeds. In the nonlinear case the eigenvalues vary with q. The Riemann solution may then contain shock waves and rarefaction waves, but even in the nonlinear case this special problem has a similarity solution with constant wave speeds. In 1959, Sergei Godunov [30] proposed a numerical method for solving general shock wave problems in gas dynamics by using the Riemann problem as a building block. If the physical domain is decomposed into a finite number of grid cells and the solution approximated by a piecewise constant function that is constant in each grid cell, then at the start of a time step the initial data consists of a set of Riemann problems, one at each cell interface. By solving each of these Riemann problems the solution can be evolved forward in time by a small increment. The resulting solution is averaged over each grid cell to obtain a new piecewise constant approximation to the solution at the end of the time step. This procedure is repeated in the next time step. This idea of basing the numerical method on Riemann solutions turned out to be a key idea in extending the “method of characteristics” from linear hyperbolic systems to important nonlinear shock propagation problems. This also leads naturally to a software framework: the particular hyperbolic equation being solved is determined by providing a Riemann Solver. This is a subroutine that, given any two states q and qr , returns the wave speeds of the resulting waves in the similarity solution, along with the corresponding waves themselves (i.e., the jump in q across each wave). The updating formulas for the cell averages based on these waves are very simple and independent of the particular system being solved. Godunov’s method turned out to be very robust and capable of solving problems involving strong shocks where other methods failed. However, it is only first-order accurate on smooth solutions to the PDEs. This means that the error goes to zero only as the first power of the discretization steps x and t. Moreover, although

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complicated solutions involving strong shocks and their interactions could be robustly approximated without the code crashing, the resulting approximations of shock waves are typically smeared out. The process of averaging the solution over grid cells each time step introduces a large amount of “numerical dissipation” or “numerical viscosity”. During the 1970s and 1980s, a tremendous amount of effort was devoted to developing more accurate versions of Godunov’s method that better approximated smooth solutions and also captured shock waves more sharply. These methods often go by the general name of “high-resolution shock capturing methods”. A wide variety of methods of this type have been proposed and effectively used. Many of these, including the wave-propagation algorithms of clawpack, have the relatively modest goal of achieving something close to second-order accuracy on smooth solutions coupled with sharp resolution of discontinuities. Other approaches have been used that can achieve much better accuracy in certain situations, though for general nonlinear problems involving complicated shock structures, particularly in more than one dimension, it seems hard to improve very much beyond what is obtained using second-order methods. One standard second-order method for a linear hyperbolic system is the Lax– Wendroff method, first proposed in 1960, which is based on approximating the first few terms of a Taylor series expansion of the solution at time t + t about the solution at time t. This method does not work at all well for problems with discontinuities, however, as it is highly dispersive and nonphysical oscillations arise that destroy all accuracy. In the nonlinear case these oscillations around shock waves can also lead to nonlinear instabilities. The key feature in many high-resolution methods is to apply a limiter function in some manner to suppress these oscillations. In the wave-propagation algorithms this is done in the following way. The Lax–Wendroff method can be rewritten as Godunov’s method plus a correction term that again can be expressed solely in terms of the waves and wave speeds in the Riemann solutions arising at each cell interface. Where the solution is smooth, adding in these correction terms improves the accuracy. Where the solution is not smooth, for example near a shock, the Taylor series expansion is not valid and these “correction terms”, which approximate higher derivatives in the Taylor expansions, do more harm than good. We can determine how smooth the solution is by comparing the magnitude of a wave with the magnitude of the corresponding wave at the neighboring cell interfaces. If these differ greatly then the solution is not behaving smoothly and the correction terms should be “limited” in some manner. Many variants of this idea have been used. In some cases it is the Lax–Wendroff expression for the flux at the interface between cells that is limited (in so-called “flux limiter” methods). Another approach is to view the Lax–Wendroff method as a generalization of Godunov’s method in which a piecewise linear function in each grid cell is defined and the values at the edges of each cell then used to define Riemann problems. In this case the slope chosen in each cell is based on the averages in nearby cells, with some “slope limiter” applied in regions where it appears the solution is not behaving smoothly. For the special case of a scalar conservation law, a very


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nice mathematical theory was developed to guide the choice of limiter functions. The true solution to a scalar problem has its total variation non-increasing over time. By requiring that a numerical method be “total variation diminishing” (TVD) it was possible to derive methods that were essentially second order accurate, or higher, but that could be proved to not introduce spurious oscillations. The 1980s were an exciting time in this field, as robust high-resolution methods were developed for a variety of challenging applications and as computers became powerful enough that extensions of these methods and the related mathematical theory to more than one space dimension became possible and necessary. I played a modest role in some of these developments, but I think my main contribution has been in providing a formulation of these methods that lends itself well to software that is very broadly applicable, and in leading the effort to write this software. The wave-propagation formulation that I favor has the advantage that once the Riemann problem has been solved, the limiters and high-resolution correction terms are applied in a general manner that is independent of the equation being solved. Moreover a framework for doing this in two dimensions was proposed in [43] that retains this modularity, separating the process of solving a one-dimensional Riemann problem at each cell interface, along with a related “transverse Riemann problem” in the orthogonal direction, from the process of propagating these waves and correction terms with appropriate limiters. While it had long been recognized that the methods being developed were in principle broadly applicable to all hyperbolic problems, for systems of equations most methods were developed or at least presented in a form that was specific to one particular problem (often the Euler equations of gas dynamics) and a certain amount of work was required to translate them to other problems. Computer programs that I was aware of were all problem-specific, and generally not publicly available. My original motivation for developing this framework was not software development, but rather the need to teach graduate students, and the desire to write a book that explained how to apply these powerful high-resolution methods to a variety of problems. I had students coming from the Mathematics and Applied Mathematics departments, many of whom knew little about fluid dynamics, and students from several science and engineering departments who had specific interests in very diverse directions. My subsequent motivation for software development was partly educational, but also partly because I wanted to better publicize the wave-propagation framework I had developed and make it easier for others to use methods in this form. I recognized that these methods, while quite general, were also sufficiently complicated that few would bother to implement them from my descriptions in journal articles. For this research to have any impact beyond a few publications it seemed necessary to provide more than pseudo-code descriptions of the algorithms.

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5. clawpack as an environment for developing and testing methods I have provided a few basic sample programs within the clawpack package itself, and some additional test problems in a directory tree labeled applications available at the website. Complete programs also exist for each of the figures in my book. These each consist of a tarred Unix directory containing a small set of subroutines to be used together with the main software. These subroutines specify the specific problem, including the Riemann solver and the initial and boundary conditions. Many standard boundary conditions are already available in the default boundary condition routine, and a variety of Riemann solvers are also provided for many different systems of equations. As a result it is often very easy to adapt one of these examples to a new problem, particularly for some of the standard test problems that often appear in publications. Although the development of the clawpack software was originally motivated by the desire to make a set of existing methods more broadly accessible, the availability of this software has also encouraged me to pursue new algorithmic advances that I otherwise might not have. I hope that the software will also prove useful to others as a programming environment for developing and testing new algorithms, and for comparing different methods on the same problems. Since the source code is available and the basic clawpack routines are reasonably simple and well documented, it should be easy for users to modify them and try out new ideas. I encourage such use, and I certainly use it this way myself. As one example, there are many approaches to developing the “approximate Riemann solvers” that are typically used for nonlinear problems. It often is not cost effective, and may not even be possible, to solve the Riemann problem exactly at every cell interface each time step. Different versions of the Riemann solver can easily be swapped in and tested on a set of problems. See Figures 15.5 and 15.6 in [44] and the associated programs for an example of this sort of comparison. The clawpack software also comes with a set of standard limiter functions, and an input parameter specifies which one will be used. The subroutine where these are implemented can be easily modified to test out new approaches. More extensive modifications could be made to the software as well, for example by replacing the wave-propagation algorithms currently implemented by a different approach. If a new method is formulated so that it depends on a Riemann solver and boundary conditions that are specified in the form already used in clawpack, then it should be easy to test out the new method on all the test problems and applications already developed for clawpack. This would facilitate comparison of a new method with existing methods. Careful direct comparisons of different methods on the same test problems are too seldom performed in this field, as in many computational fields. One reason for this is the difficulty of implementing other peoples’ methods, so the typical paper contains only results obtained with the authors’ method. Sometimes (not always) the method has been tested on standard test problems and the results can be compared with


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others in the literature, with some work on the reader’s part, and assuming the reader is content with comparisons in the “eyeball norm” since many papers only contain contour plots of the computed solution and no quantitative results. Of course there are many exceptions to this, including some papers devoted to careful comparisons of different methods (e.g., [31], [46]), but these papers are still a minority. I hope that clawpack might facilitate this process more in the future, and that new algorithms of this same general type might be provided in a clawpack implementation that allows direct use and comparison by others.

6. clawpack extensions and infrastructure This software effort began about 10 years ago, as recounted above, and has continued in unexpected ways as the software grew beyond my initial intentions in several directions. While originally I viewed it primarily as an educational tool, it was based on my own research codes and I realized that it could perhaps be valuable for other researchers as well, perhaps even as a tool for solving real problems and not just academic test problems. To make it more useful as a general tool required several enhancements. I originally wrote subroutines for solving systems in one and two space dimensions, but started collaborating with Jan Olav Langseth that same year on the development of three-dimensional generalizations [37]. He took my course as a visiting graduate student from the University of Oslo and went on to write a thesis partly on this topic [36], and wrote the three-dimensional subroutines in clawpack. I had also been collaborating with Marsha Berger on research related to using cutcell Cartesian grids to solve the Euler equations in non-rectangular geometries (e.g., [5], [6]) and was familiar with her implementation of adaptive mesh refinement on rectangular grids, using an approach pioneered in her work with Joe Oliger and Phil Colella [4], [8], [7]. While still on sabbatical, in 1995 we started working together to modify her adaptive mesh refinement (AMR) code and make it more generally applicable to any hyperbolic system that can be solved in clawpack. In this approach the rectangular grid is refined by introducing finer grids on rectangular patches. Several levels of refinement are allowed, by an arbitrary refinement factor at each level. The program automatically refines and de-refines as the computation proceeds, based on a standard default or user-specified error criterion, and so fine grids are used only where they are needed. This approach is particularly valuable for problems involving shock waves propagating in three space dimensions. To capture the shock waves as sharp discontinuities, even with high-resolution methods, requires a fairly fine grid around the shock, much finer than is needed in regions where the solution is smooth. Without AMR many three-dimensional problems would be impractical to solve except on the largest supercomputers. This amrclaw software is now a standard part of clawpack, and was extended from two to three space dimensions with programming help from Dave McQueen and

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Donna Calhoun. The goal of the amrclaw code is primarily to facilitate research and practical problem solving, rather than to teach adaptive refinement techniques. The core library consists of about 9 000 lines of Fortran 77 code that is quite convoluted and not structured or fully documented in a way that others can easily understand. This stems in large part from its history as a merging together of two different codes, with different notation and conventions that have largely been preserved and worked around, and with many new features added over the years as afterthoughts that were not originally designed for. It may not be particularly elegant, but it has proved valuable in solving practical problems and by now has been tested to the point where it is fairly robust and reliable. All of the source code is available for others to inspect and modify, at their own peril perhaps. The clawpack framework can also be used with other AMR packages. Sorin Mitran is developing bearclaw [47], a Fortran 90 version with similar capabilities to amrclaw but designed with these capabilities in mind from the beginning. This code is still being developed and has not been tested as extensively as amrclaw, but it has been successfully used in astrophysical applications [51], [52]. Ralf Deiterding has developed a general purpose adaptive refinement code amroc [18], [19] in C++ that also allows the use of the clawpack solvers with adaptive refinement. This is a primary computational tool in the Virtual Shock Physics Test Facility at the Caltech ASC Center for Simulation of Dynamic Response of Materials [1]. Recently Donna Calhoun has written a chombo-claw interface [14] between the clawpack solvers and the C++ chombo software package developed by Colella’s group at Lawrence Berkeley Lab [16]. These newer packages have various advantages over the amrclaw software in clawpack. They are written in more advanced languages that are more suitable for the data structures and dynamic memory allocation needed in AMR codes. They also have more capabilities such as parallel implementations, the ability to handle implicit methods, and/or coupling with elliptic solvers as required in some applications. Other extensions of clawpack have also been developed, such as the clawman software [2] that solves hyperbolic systems on curved two-dimensional manifolds. This has been used to solve geophysical flow problems on the sphere and relativistic flow problems in curved space-time near a black hole [3], [54], [55]. The original software was designed for purely Cartesian grids in rectangular regions of space. Some practical problems have this form, but most are posed in more complicated physical domains, e.g., for flows around or through a physical object. There are many approaches to handling complex geometries. The cut-cell Cartesian grid approach has already been mentioned above. At the other extreme lie unstructured grids, typically composed of triangular cells in two dimensions or tetrahedra in three dimensions. These can conform to very general boundaries, but grid generation then becomes a challenging problem in itself, and implementations must deal with special data structures to keep track of what cells are adjacent to one another. For fairly simple domains, a good compromise is often possible in which the grid is logically rectangular in computational space, but is mapped to a nonrectagular


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physical domain. In two dimensions, this means that each grid cell is a quadrilateral and simple (i, j ) indexing can be used to denote the cells, with neighboring cells having indices (i ± 1, j ± 1). In three dimensions the grid cells are hexahedral but still logically rectangular. It is quite easy to apply clawpack in such situations (as described in Chapter 23 of [44]), with a standard set of additional subroutines used to specify the mapping function. One nice feature of this approach is that the AMR routines work perfectly well on mapped grids – the patches of refinement are still rectangular in computational space. The wave-propagation algorithms turn out to work quite robustly on quadrilateral grids, even if the grid is nonorthogonal and far from smooth. Rather than incorporating the grid mapping directly into the differential equations as “metric terms” that involve derivatives of the mapping function, in the wave-propagation approach one solves one-dimensional Riemann problems orthogonal to each grid interface and transverse Riemann problems based on the adjacent cell interfaces. Figure 1 shows two grids from some recent work with Calhoun [15] on the use of logically rectangular grids for solving problems in domains with smooth boundaries. The figure on the left shows a quadrilateral grid for a circle, while that on the right is a logically rectangular grid on the sphere. Each grid is simply a rectangle in computational space. Of course polar coordinates also give a logically rectangular grid in a circle, but grid lines coalesce at the center where cells have much smaller area than those near the perimeter. This presents a problem when using explicit

Figure 1. Quadrilateral grids on the circle and the sphere. In each case the computational domain is a rectangular grid. The matlab code that generates these grids is available on the webpage [39].

methods for hyperbolic problems: the disparity in cell sizes leads to an undesirable restriction on the time step. The grids in Figure 1 are far from smooth or orthogonal; in fact the images of the two orthogonal cell edges at each corner of the rectangular computational domain are nearly collinear in the physical domain. But the cell areas are nearly uniform, differing by at most a factor of 2. Standard finite difference

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methods would presumably not work well on these grids, but applying clawpack yields very nice results, as demonstrated in [15]. Writing new methods in “clawpack form” makes it possible to take advantage of the infrastructure developed for this software, for example to apply the method on a general quadrilateral grid. It may also be possible to apply adaptive mesh refinement quite easily with the new method by taking advantage of one of the AMR wrappers described above. Since implementing AMR effectively is far from trivial, and often has little to do with the particular numerical method used on each grid patch, making use of existing implementations could be worthwhile even if it takes a bit of work to formulate the method in an appropriate form. Extensive graphics routines in matlab have been written to plot the results computed by clawpack. These routines (written mostly by Calhoun and myself) deal with adaptive mesh refinement data in two or three space dimensions. This is a not easily done directly with most graphics packages, and again this infrastructure may be useful to developers of new methods. (In fact these graphics routines can take AMR data produced by any program, not just clawpack, provided it is stored in the appropriate form.) One current research project, with graduate student David Ketcheson, is to implement higher order numerical methods in clawpack, specifically the weighted essentially non-oscillatory (WENO) methods that are based on higher order interpolation in space coupled with Runge–Kutta time stepping (e.g., [31], [60]). Doing so requires reformulating these methods slightly, and in particular we are developing a version that works for linear hyperbolic systems that are not in conservation form. Systems of this form arise in many applications, such as the propagation in heterogeneous media of acoustic, elastic, or electromagnetic waves. In many linear applications the solution is smooth but highly oscillatory, and in this case higher-order methods may be beneficial.

7. Applications in computational science In addition to continuing to work on algorithm development, in the past few years I have become more directly involved in applications of computational science, driven in part by the existence of this software as a starting point. Problems that can be solved easily with the existing software have little interest to me; as a mathematician I consider them solved, though there may be plenty of interesting science to be done by judicious use of the software as a tool. This is best done, however, by scientists who are experts in a particular domain. A practical problem where clawpack fails to perform well, or where some substantial work is required to apply it, is much more interesting to me. Although the methods implemented in clawpack work well on many classical hyperbolic problems, there are a wealth of more challenging problems that are yet to be solved, and I hope that clawpack might form the basis for approaching some of these problems without the need to rewrite much of the basic infrastructure.


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In the remainder of this section I will briefly describe a few topics that are currently occupying me and my students. More details on these and other problems, along with movies, papers, and sometimes computer code, can be found by clicking on the “Research interests” link from my webpage. Shock wave therapy and lithotripsy. Focused shock waves are used in several medical procedures. Extracorporeal shock wave lithotripsy (ESWL) is a standard clinical procedure for pulverizing kidney stones noninvasively. There are several different lithotripter designs. In one model, a spark plug immersed in water generates a cavitating bubble that launches a spherical shock wave. This wave reflects from an ellipsoidal shaped reflector and refocuses at the distal focus of the ellipsoid, where the kidney stone is centered. The shock wave pulse has a jump in pressure of roughly 50 MPa (500 atmospheres) over a few nanoseconds, followed by a more slowly decaying decrease in pressure passing below atmospheric pressure before relaxing to ambient. This tensile portion of the wave is particularly important in kidney stone comminution since stones are composed of brittle material that does not withstand tensile stress well. Typically thousands of pulses are applied clinically (at a rate of 1 to 4 per second). The breakup process is not well understood and better understanding might allow clinical treatment with fewer pulses and less damage to the surrounding kidney. Together with Kirsten Fagnan and Brian MacConaghy, two graduate students in Applied Mathematics, I have recently been collaborating with researchers at the Applied Physics Laboratory and the medical school at the University of Washington to develop a computational model of nonlinear elastic wave propagation in heterogeneous media that can be used to aid in the study of this process. Preliminary computations have been performed using linear elasticity in two-dimensional axisymmetric configurations in which a cylindrical test stone is aligned with the axis of the lithotripter ellipsoid. We are currently extending these computations to an appropriate nonlinear elasticity model, and also to three dimensional calculations for non-axisymmetric configurations. We also hope to perform simulations useful in the study of extracorporeal shock wave therapy (ESWT), a relatively new application of lithotripter shock waves to treat medical conditions other than kidney stones, in which the goal is to stimulate tissue or bone without destroying it. For example, several recent clinical studies have shown that treating nonunions (broken bones that fail to heal) with ESWT can lead to rapid healing of the bone [26], [57], perhaps because it stimulates the growth of new vascular structure in regions where there is insufficient blood flow. Conditions such as tennis elbow, plantar fasciitis, and tendinitis have also been successfully treated with ESWL; see for example [32], [58]. In ESWL applications the shock wave is often focusing in a region where there is a complicated mix of bones and tissue. The interfaces between these materials cause significant reflection of wave energy. It is often crucial to insure that the shocks do not accidentally refocus in undesirable locations, such as nearby organs or nerves, which could potentially cause extensive collateral damage. Ideally one would like to be able

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to use MRI data from a patient to set material parameters and run 3d simulations of the shock wave propagation in order to adjust the angle of the beam to achieve maximal impact in the desired region with minimal focusing elsewhere. Our current work is a first step in this direction. Volcanic flows. My recent student Marica Pelanti developed a dusty gas model in which the Euler equations for atmospheric gas are coupled to another set of conservation laws for the mass, momentum, and energy of a dispersed dust phase [49], [50]. The two sets of equations are coupled together by source terms modeling viscous drag and heat transfer between the phases. The dust is assumed pressureless and requires special Riemann solvers, based on [45]. This model has been used to study the jets arising from high-velocity volcanic eruptions. The speed of sound in a dusty gas is considerably less than the sound speed in the atmosphere. As a result, volcanic jets can easily be supersonic relative to the dusty gas sound speed, leading to interesting shock wave structures within an eruption column. Pelanti explored this for 2D axi-symmetric jets for both flat topography and for topography where the jet expands through an idealized conical crater. Similar work has been performed by Augusto Neri and Tomaso Esposti Ongaro in the Earth Sciences Department at the University of Pisa and we are now collaborating with them on some comparisons of our results. We have also interacted extensively with researchers at the USGS Cascade Volcano Observatory (CVO) near Mount St. Helens (MSH), who study many aspects of volcanic flows and are charged with hazard assessment for MSH. After a long quiescent period, MSH became quite active again in October, 2004, and we worked with these researchers to try to produce a full three-dimensional model of pyroclastic flows over the topography of MSH in order to predict the possible impact of an eruption of various magnitudes. Extension of the dusty gas model to the full three-dimensional topography of MSH is underway, although it turns out there are many numerical and modeling issues still to be tackled. Considerable data is available from the 1980 eruption that can be used to validate a code. In particular, there is a set of photographs and maps that show the direction in which trees were blown down when the initial pyroclastic blast from the eruption passed over the surrounding ridges. The blown-down trees created a snapshot of the velocity vectors in the leading edge of the flow. These exhibit complex flow patterns, such as recirculation zones on the lee side of ridges where the trees were blown down in the direction pointing towards MSH instead of away. We hope to eventually compare computed velocities from our simulations with these observations. Richard Iverson and Roger Denlinger at CVO have also done extensive work on modeling debris flows, such as those that arise when water from melting glaciers on a volcano mixes with trees, boulders, and other debris, creating highly damaging and life-threatening flows [20], [21]. Their numerical work is based on equations similar to the shallow water equations but enhanced with more physics within the flow. They use Riemann solver techniques based in part on the wave-propagation framework, with the


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additional solution of an elasticity problem within each Riemann solver to compute the local stress tensor within the debris. They encounter difficulties at the edge of the flow similar to the dry-cell problems that arise in tsunami modeling (discussed below) and we have collaborated with them on solving these problems. Denlinger has also recently modeled the great Missoula floods using similar techniques [22]. Tsunami modeling. David George and I have been developing a version of amrclaw capable of modeling tsunamis, including both their global propagation over large expanses of ocean and the inundation and run-up around small scale features at the level of individual beaches or harbors. We solve the shallow water equations on rectangular grids, in which each grid cell has an elevation value for the earth surface (which is called topography if it is above sea level, or bathymetry when underwater). The components of q are the fluid depth h and momenta hu and hv. Grid cells above sea level are dry (h = 0) and cells can become wet or dry dynamically as waves move along the shore. This approach avoids the need to model the shoreline as a separate interface, but developing a robust code based on this approach requires a Riemann solver that can deal well with both wet and dry states. The bathymetry comes in as a source term in the conservation laws. Away from shore the bathymetry is varying on a scale of several kilometers (the Indian Ocean is about 4 km deep, for example) whereas a tsunami propagating over the ocean is a few meters high at most. This difference in scales leads to difficulties that I will not describe here, and requires the use of some sort of “well-balanced” scheme in which the source term is incorporated into the Riemann solver. Though of small magnitude in the ocean, a tsunami may have a wavelength of more than 100 km and so √ the shallow water equations are an appropriate model. The propagation velocity is gh, where g is the gravitational constant, and as they approach shore h decreases and the wave magnitude increases as the wavelength shortens, the same phenomenon observed in breaking waves on a beach. But in a tsunami the entire water column is set in motion by an uplifting of the ocean floor, whereas in wind-driven surface waves only the water very near the surface is moving. The enormous energy tsunamis carry gives them great destructive potential. Adaptive mesh refinement is essential for this problem. We wish to propagate waves over the ocean, where grid cells several kilometers on a side can be used, and simultaneously predict the run-up around local features, where cell sizes of a few meters are desirable. Developing a well-balanced dry-state Riemann solver that works well in the context of AMR proved to be quite challenging and many difficulties appeared at the boundaries between grids at different levels. These could only be solved by some substantial reworking of the amrclaw code. The result is a special-purpose program that incorporates these algorithmic modifications and can now be applied to many tsunami scenarios. It is currently being tested by comparing predictions with measurements made at various places around the Indian Ocean in the wake of the 26 December 2004 Sumatra earthquake. For this application we are working very closely with tsunami scientists. Our involvement in tsunami modeling arose out of a joint NSF grant with Harry Yeh in

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civil engineering at Oregon State University and Joe Hamack at Penn State, who were doing wave tank experiments and related mathematical modeling that they wished to complement with numerical simulations. Since the Sumatra event, our focus has shifted to the larger scale, and the contacts we had already established in the tsunami modeling community proved invaluable. Many Tsunami Survey Teams traveled to the Indian Ocean and surveyed different parts of the coastline, measuring the run-up and inundation observed. Yeh was on a team that mapped the region near Chennai (Madras), India, and our initial validation work is focused on comparing predictions with his observations in this area. Unfortunately fine-scale bathymetry data is hard to come by and we have resorted to digitizing navigational charts to obtain some of the necessary data. Figure 2 shows part of a simulation of the Indian Ocean tsunami, as described further in the caption. See the webpage [39] for color versions and movies, along with the computer program. The program we have developed is a research code for this particular problem, but we intend to further improve it and ultimately make it available to the community. There is far more data available than we can compare against ourselves and we hope that other researchers will be able to use it for some of this work and publish the results. If our code does not work well or does not agree well with observations in some cases then we may need to revisit it, or perhaps others will make further improvements to it. We hope that this code may ultimately be useful as a real-time tsunami prediction tool, and we are working with Vasily Titov and other scientists of the NOAA National Tsunami Hazard Mitigation Program in Seattle to compare our code with theirs and see how we can best complement their efforts (some of which are described in a recent Scientific American article [28]).

8. Conclusions I have made a case that many aspects of scientific computing and software development should be viewed as inherently mathematical, and that mathematicians can play a very important role in computational science. I have also encouraged researchers in this area to produce reproducible research, in particular by making computer programs, not just software, available to others. I have presented some of my own research activities as a case study, though I do not claim it is the best example to follow. I do hope, however, that the software we have produced will find wider use as one tool in this direction, both as a development environment for testing new methods and as a building block for solving problems in science and engineering. My hope is that others who develop methods or applications using this package will make their full code available on the web, particularly if it has been used to compute results that appear in publications.


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(a) Time 01:06:45 (525 seconds)

(b) Time 02:10:55 (4375 seconds)

(c) Time 02:43:00 (6300 seconds)

(d) Time 3:15:05 (8225 seconds)

Figure 2. Propagation of the 26 December 2004 tsunami across the Indian Ocean. Units on the axes are longitude and latitude. The top two frames show the Bay of Bengal at two early times. A coarse grid is used where nothing is yet happening and the grid cells are shown on this “Level 1 grid”, which has a mesh width of one degree (approximately 111 km). The rectangular region where no grid lines are shown is a Level 2 grid with mesh width 8 times smaller, about 14 km. Red represents water elevation above sea level, dark blue is below the undisturbed surface (see the webpage [39] for color versions of these images). Figure (c) shows a zoomed view of the southern tip of India and Sri Lanka at a later time. The original Level 1 grid is still visible along the left-most edge, but the rest of the region shown has been refined by a Level 2 grid. Part of Sri Lanka has been refined by Level 3 grids. The grid lines on Level 3 are not shown; the mesh width on this level is about 1.7 km, a factor of 8 finer than Level 2. Figure (d) shows a later time, as the wave diffracts around Sri Lanka, moving slowly through the shallow water in this coastal region. The calculation shown here was run on a single-processor 3.2 GHz PC under Linux and took about 40 minutes of wall time for the computation shown here. Movies of this simulation can be viewed on the webpage [39], which also contains pointers to finer grid calculations and more recent work on this problem.

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Even for results that are not published, it would be valuable to have more examples and test problems available on-line than what is provided on the clawpack web pages. Please let me know with a brief email if you have created such a page, or published a paper where clawpack was successfully used. I am also always interested to hear about problems that arise with the software or suggestions for improvements, though as an academic researcher with a small group of graduate students I cannot promise to provide as much technical support as I would like to. The website [39] contains the codes used to generate the two figures in this paper, two very different examples of what can be provided in conjunction with a publication. The programs for Figure 1 are each less than a page of matlab, while the program for Figure 2 is about 13,000 lines of Fortran and also requires a large set of bathymetry and earthquake source data. The webpages also contain movies that illustrate the figures much better than the static versions shown in this paper, and links to other related work. Acknowledgments. Numerous people read drafts of this paper and provided valuable comments and pointers to related work. In particular I would like to thank Marsha Berger, Donna Calhoun, Benjamin LeVeque, William LeVeque, and Nick Trefethen for their extensive comments, and David George for pulling together the necessary pieces for Figure 2.

References [1] Caltech center for simulation of dynamic response of materials. http://csdrm.caltech.edu/, 2005. [2] Bale, D., Rossmanith, J. A., and LeVeque, R. J., clawman software. http://www.amath. washington.edu/~claw/clawman.html. [3] Bale, D. S., Wave propagation algorithms on curved manifolds with applications to relativistic hydrodynamics. PhD thesis, University of Washington, 2002. http://www.amath. washington.edu/~rjl/people.html. [4] Berger, M., Adaptive mesh refinement for hyperbolic partial differential equations. PhD thesis, Computer Science Department, Stanford University, 1982. [5] Berger, M., and LeVeque, R. J., Cartesian meshes and adaptive mesh refinement for hyperbolic partial differential equations. In Hyperbolic problems. Theory, numerical methods and applications (ed. by B. Engquist and B. Gustafsson), Vol. I, Studentlitteratur, Lund 1991, 67–73. [6] Berger, M., and LeVeque, R. J., Stable boundary conditions for Cartesian grid calculations. Computing Systems in Engineering 1 (1990), 305–311. [7] Berger, M., and Oliger, J., Adaptive mesh refinement for hyperbolic partial differential equations. J. Comput. Phys. 53 (1984), 484–512. [8] Berger, M., and Colella, P., Local adaptive mesh refinement for shock hydrodynamics. J. Comput. Phys. 82 (1989), 64–84.


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[9] Birkhoff, G., Mathematics and computer science. American Scientist 63 (1975), 83–91. [10] Bolstad, J. H., Chan, T. F., Coughran, W. M., Jr., Gropp, W. D., Grosse, E. H., Heath, M. T., LeVeque, R. J., Luk, F. T., Nash,S. G., and Trefethen, L. N., Numerical Analysis Program Library User’s Guide: NAPLUG. SLAC User Note 82, http//www.slac.stanford.edu/spires/... find/hep/www?r=slac-scip-user-note-082, 1979. [11] Bornemann, F., Laurie, D., Wagon, S., and Waldvogel, J., The SIAM 100-digit challenge, a study in high-accuracy numerical computing. http://www-m3.ma.tum.de/m3old/ bornemann/challengebook/index.html. [12] Bornemann, F., Laurie, D., Wagon, S., and Waldvogel, J., The SIAM 100-Digit Challenge, A Study in High-Accuracy Numerical Computing. SIAM, Philadelphia, PA, 2004. [13] Buckheit, J. B., and Donoho, D. L., WaveLab and reproducible research. http://wwwstat.stanford.edu/~donoho/Reports/1995/wavelab.pdf, 1995. [14] Calhoun, D.A., Colella, P., and LeVeque, R. J., chombo-claw software. http://www.amath. washington.edu/~calhoun/demos/ChomboClaw. [15] Calhoun, D. A., Helzel, C., and LeVeque, R. J., Logically Rectangular Grids and Finite Volume Methods for PDEs in Circular and Spherical Domains. In preparation; http://www.amath. washington.edu/~rjl/pubs/circles, 2005. [16] Colella, P., et al., chombo software. http://seesar.lbl.gov/anag/chombo/, 2005. [17] Crowder, H., Dembo, R. S., and Mulvey, J. M., On reporting computational experiments with mathematical software. ACM Trans. Math. Software 5 (1979), 193–203. [18] Deiterding, R., AMROC software. http://amroc.sourceforge.net/, 2005. [19] Deiterding, R., , Construction and application of an amr algorithm for distributed memory computers. In Adaptive mesh refinement - theory and applications (ed. by T. Plewa), Lecture Notes in Comput. Sci. Engrg. 41, Springer-Verlag, Berlin 2005, 361–372. [20] Denlinger, R. P., and Iverson, R. M., Granular avalanches across irregular three-dimensional terrain: 1. Theory and computation. J. Geophys. Res. 109 (2004), F01014. [21] Denlinger, R. P., and Iverson, R. M., Granular avalanches across irregular three-dimensional terrain: 2. Experimental tests. J. Geophys. Res. 109 (2004), F01015. [22] Denlinger, R. P., and O’Connell, D., Two dimensional flow constraints on catastrophic outflow of glacial Lake Missoula over three dimensional terrain. Invited abstract, 3rd International Paleoflood Workshop, Hood River, OR, 2003. [23] Dongarra, J. J., Bunch, J. R., Moler, C. B., and Stewart, G. W., LINPACK Users’ Guide. SIAM, Philadelphia, PA, 1979. [24] Donoho, D. L., and Huo, X., BeamLab and reproducible research. Int. J. Wavelets Multiresolut. Inf. Process. 2 (4) (2004), 391–414. [25] Forsythe, G. E., Galler, B. A., Hartmanis, J., Perlis, A. J., and Traub, J. F., Computer science and mathematics. ACM SIGCSE Bulletin 2 (1970), 19–29. [26] Fritze, J., Extracorporeal shockwave therapy (ESWT) in orthopedic indications: a selective review. Versicherungsmedizin 50 (1998), 180–185. [27] Garbow, B. S., Boyle, J. M. Dongarra, , J. J., and Moler, C. B., Matrix Eigensystem Routines — EISPACK Guide Extensions. Lecture Notes in Comput. Sci. 51, Springer-Verlag, Berlin 1977. [28] Geist, E. L., Titov, V. V., and Synolakis, C. E., Tsunami: wave of change. Scientific American 294 (2006), 57–63.

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[29] Gentleman, R., and Lang, D. T., Statistical analyses and reproducible research. Bioconductor Project Working Papers. Working Paper 2. http://www.bepress.com/bioconductor/ paper2, 2004. [30] Godunov, S. K., A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. 47 (1959), 271–306. [31] Greenough, J. A., and Rider, W. J., A quantitative comparison of numerical methods for the compressible Euler equations: fifth-order WENO and piecewise-linear Godunov. J. Comput. Phys. 196 (2004), 259–281. [32] Hammer, A. S., Rupp, S., Ensslin, S., Kohn, D., and Seil, R., Extracorporal shock wave therapy in patients with tennis elbow and painful heel. Archives of Orthopaedic and Trauma Surgery 120 (2000), 304–307. [33] Jackson, R. H. F., Boggs, P. T., Nash, S. G., and Powell, S., Guidelines for reporting results of computational experiments. Report of the ad hoc committee. Math. Programming 49 (1991), 413–425. [34] Knuth, D. E., Computer science and its relation to mathematics. Amer. Math. Monthly, 81 (1974), pp. 323–343. [35] Knuth, D. E., Algorithmic thinking and mathematical thinking. Amer. Math. Monthly 92 (1985), 170–181. [36] Langseth, J. O., Wave Propagation Schemes, Operator Splittings, and Front Tracking for Hyperbolic Conservation Laws. PhD thesis, Department of Informatics, University of Oslo, 1996. [37] Langseth, J. O., and LeVeque, R. J., A wave-propagation method for three-dimensional hyperbolic conservation laws. J. Comput. Phys. 165 (2000), 126–166. [38] Lee, C.-Y., Bard, J., Pinedo, M., and Wilhelm, W. E., Guidelines for reporting computational results in IEE Transactions. IIE Trans. 25 (1993), 121–123. [39] LeVeque, R. J., http://www.amath.washington.edu/~rjl/pubs/icm06. [40] LeVeque, R. J., clawpack software. http://www.amath.washington.edu/~claw. [41] LeVeque, R. J., clawpack User’s Guide. http://www.amath.washington.edu/~claw/doc. html. [42] LeVeque, R. J., Numerical Methods for Conservation Laws. Lectures Math. ETH Zürich, Birkhäuser, Basel 1990. [43] LeVeque, R. J., Wave propagation algorithms for multi-dimensional hyperbolic systems. J. Comput. Phys. 131 (1997), 327–353. [44] LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems. Cambridge Texts Appl. Math., University Press, Cambridge 2002. [45] LeVeque, R. J., The dynamics of pressureless dust. J. Hyperbolic Differential Equations 1 (2004), 315–327. [46] Liska, R., and Wendroff, B., Comparison of several difference schemes on 1D and 2D test problems for the Euler equations. SIAM J. Sci. Comput. 25 (2003), 996–1017. [47] Mitran, S., bearclaw software. http://www.amath.unc.edu/Faculty/mitran/bearclaw. html. [48] Moler, C., The origins of matlab. matlab News & Notes, December, 2004. http://www. mathworks.com/company/newsletters/news_notes/clevescorner/dec04.html.


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[49] Pelanti, M., Wave Propagation Algorithms for Multicomponent Compressible Flows with Applications to Volcanic Jets. PhD thesis, University of Washington, 2005. [50] Pelanti, M., and LeVeque, R. J., High-resolution finite volume methods for dusty gas jets and plumes. SIAM J. Sci. Comput., to appear. [51] Poludnenko, A.Y., Frank, A., and Mitran, S., Clumpy flows in protoplanetary and planetary nebulae. 2003. http://xxx.lanl.gov/abs/astro-ph/0310286. [52] Poludnenko, A.Y., Frank, A., and Mitran, S., Strings in the Eta Carinae nebula: Hypersonic radiative cosmic bullets. 2003. http://xxx.lanl.gov/abs/astro-ph/0310007. [53] Roache, P. J., Verification and Validation in Computational Science and Engineering. Hermosa Publishers, Albuquerque, NM, 1998. [54] Rossmanith, J. A., A wave propagation method for hyperbolic systems on the sphere. J. Comput. Phys. 213 (2006), 629–658. [55] Rossmanith, J. A., Bale, D. S., and LeVeque, R. J., A wave propagation algorithm for hyperbolic systems on curved manifolds. J. Comput. Phys. 199 (2004), 631–662. [56] Roy, C. J., Review of code and solution verification procedures for computational simulation. J. Comput. Phys. 205 (2005), 131–156. [57] Schaden, W., Fischer, A., and Sailler, A., Extracorporeal shock wave therapy of nonunion or delayed ossesous union. Clinical Orthopaedics & Related Res. 387 (2001), 90–94. [58] Schmitt, J., Haake, M., Tosch, A., Hildebrand, R., Deike, B., and Griss, P., Low-energy extracorporeal shock-wave treatment (ESWT) for tendinitis of the supraspinatus. J. Bone & Joint Surgery 83-B (2001), 873–876. [59] Schwab, M., Karrenbach, M., and Claerbout, J., Making scientific computations reproducible. http://sepwww.stanford.edu/research/redoc/cip.html. [60] Shu, C.-W., High order ENO and WENO schemes for computational fluid dynamics. In High-Order Methods for Computational Physics (ed. by T. J. Barth and H. Deconinck), Lecture Notes in Comput. Sci. Engrg. 9, Springer-Verlag, Berlin 1999, 439–582. [61] Smith, B. T., Boyle, J. M., Dongarra, J. J., Garbow, B. S., Ikebe, Y., Klema, V. C., and Moler, C. B., Matrix Eigensystem Routines — EISPACK Guide. Lecture Notes in Comput. Sci. 6, Springer-Verlag, Berlin 1976. [62] Trefethen, L. N., Spectral Methods in Matlab. SIAM, Philadelphia, PA, 2000. [63] Trefethen, L. N., The SIAM 100-digit challenge. http://www.comlab.ox.ac.uk/nick. trefethen/hundred.html, 2004. [64] Trefethen, L. N., Ten digit algorithms. http://www.comlab.ox.ac.uk/nick.trefethen/ten_ digit_algs.htm, 2005. Department of Applied Mathematics, University of Washington, Box 352420, Seattle, WA 98195-2420, U.S.A. E-mail: [email protected]

Reduced basis method for the rapid and reliable solution of partial differential equations Yvon Maday∗

Abstract. Numerical approximation of the solution of partial differential equations plays an important role in many areas such as engineering, mechanics, physics, chemistry, biology – for computer-aided design-analysis, computer-aided decision-making or simply better understanding. The fidelity of the simulations with respect to reality is achieved through the combined efforts to derive: (i) better models, (ii) faster numerical algorithm, (iii) more accurate discretization methods and (iv) improved large scale computing resources. In many situations, including optimization and control, the same model, depending on a parameter that is changing, has to be simulated over and over, multiplying by a large factor (up to 100 or 1000) the solution procedure cost of one single simulation. The reduced basis method allows to define a surrogate solution procedure, that, thanks to the complementary design of fidelity certificates on outputs, allows to speed up the computations by two to three orders of magnitude while maintaining a sufficient accuracy. We present here the basics of this approach for linear and non linear elliptic and parabolic PDE’s. Mathematics Subject Classification (2000). 65D05, 65M60, 65N15, 65N30, 65N35. Keywords. Reduced-basis, a posteriori error estimation, output bounds, offline-online procedures, Galerkin approximation, parametrized partial differential equations.

1. Introduction Let us consider a class of problems depending on a parameter μ ∈ D set in the form: find u ≡ u(μ) ∈ X such that F (u; μ) = 0 (we do not specify much at this point what D is, it could be a subset of R, or Rp , or even a subset of functions). Such problems arise in many situations such as e.g. optimization, control or parameteridentification problems, response surface or sensibility analysis. In case F is written through partial differential equations, the problem may be stationary or time dependent but in all these cases, a solution u(μ) has to be evaluated or computed for many instances of μ ∈ D. Even well optimized, the favorite discretization method of yours ∗ This paper presents a review of results (on the definition, analysis and solution strategies) most of them first presented elsewhere and that have benefitted from the long-standing collaboration with Anthony T. Patera, Einar M. Rønquist, Gabriel Turinici, and more recently Annalisa Buffa, Emil Løvgren, Ngoc Cuong Nguyen, Georges Pau, Christophe Prud’homme. This paper is not intended to be exhaustive and is an invitation to read more complete papers that treat in depth the features presented here. Due to this, no figures nor tables synthesizing numerical results are provided.


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will lead to very heavy computations in order to approximate all these solutions and decision may not be taken appropriately due to too large computer time for reliable simulations. The approach discussed in this paper will not aim at presenting an alternative to your favorite discretization, more the contrary. The idea is that, in many cases, your discretization will help in constructing a surrogate method that will allow to mimic it or at least to do the spadework on the evaluation of the optimal or control solution. The complexity of the equations resulting from this approach will be very low, enabling very fast solution algorithms. No miracle though, the method is based on a learning strategy concept, and, for a new problem, the preliminary off-line preparation is much time consuming. It is only after this learning step that the full speed of the method can be appreciated on-line, paying off the cost of the off-line preparation step. During the first step, we evaluate accurately, based on your preferred solver, a few solutions to F (u; μ) = 0; actually, any discretization method is good enough here. In the second step, that is involved on request and on-line, the discretization method that has been used earlier is somehow forgotten and a new discretization approach is constructed based on a new ad-hoc basis set (named “reduced basis”) built out from the previous computations. In many cases the method proves very efficient and – even though the complete understanding of the reasons why it is working so well are not mastered – an a posteriori error theory allows to provide fidelity certificates on outputs computed from the reduced-basis-discretized solution. This method is valid in case the set S(D) = {u(μ), μ ∈ D} has a simple (hidden) structure, the solution u(μ) has to be regular enough in μ. We provide some explanations on the rational of the reduced basis approximation in Section 2 and present the method in the elliptic case. In Section 3 we give more rigorous explanation on the rapid convergence of the method on a particular case. This is complemented in Section 4 by an analysis of a posteriori tools that provide fidelity certificate for outputs computed from the reduced basis approximation. Section 5 tells more about the track to follow to be convinced that the method will “work” on the particular problem of yours. The efficient implementation of the reduced basis method needs some care, we present in Section 6 some of the required tools. Finally we end this paper by providing in Section 7 some of the new directions we are currently working on.

2. Basics and rational of the reduced basis approach The reduced basis method consists in approximating the solution u(μ) of a parameter dependent problem F (u; μ) = 0 by a linear combination of appropriate, preliminary computed, solutions u(μi ) for well chosen parameters μi , i = 1, . . . , N . The rational of this approach, stands in the fact that the set S(D) = {u(μ) of all solutions when μ ∈ D} behaves well. In order to apprehend in which sense the good behavior of S(D) should be understood, it is helpful to introduce the notion of n-width following Kolmogorov [8] (see also [14]).

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Definition 2.1. Let X be a normed linear space, A be a subset of X and Xn be a generic n-dimensional subspace of X. The deviation of A from Xn is E(A; Xn ) = sup inf x − yX . x∈A y∈Xn

The Kolmogorov n-width of A in X is given by dn (A, X) = inf{E(A; Xn ) : Xn an n-dimensional subspace of X} = inf sup inf x − yX .

(1)

Xn x∈A y∈Xn

The n-width of A thus measures the extent to which A may be approximated by an n-dimensional subspace of X. There are many reasons why this n-width may go rapidly to zero as n goes to infinity. In our case, where A = S(D), we can refer to regularity of the solutions u(μ) with respect to the parameter μ, or even to analyticity. Indeed, an upper bound for the asymptotic rate at which the convergence to zero is achieved is provided by this example from Kolmogorov stating that (r) (r) dn (B˜ 2 ; L2 ) = O(n−r ) where B˜ 2 is the unit ball in the Sobolev space of all 2π periodic real valued, (r −1)-times differentiable functions whose (r −1)st derivative is absolutely continuous and whose rth derivative belongs to L2 . Actually, exponential convergence is achieved when analyticity exists in the parameter dependency. The knowledge of the rate of convergence is not sufficient: of theoretical interest is the determination of the (or at least one) optimal finite dimensional space Xn that realizes the infimum in dn , provided it exists. For practical reasons, we want to restrict ourselves to finite dimensional spaces that are spanned by elements of S(D). This might increase the n-width in general Banach space, but of course it does not in Hilbert ⊥ , where X space as it follows easily from the decomposition of X into XA ⊕ XA A denotes the vectorial space spanned by A. We thus have dn (A, XA ) = dn (A, X).

(2)

We derive from this equality that the quantity inf supu∈S(D) inf y∈Xn x − yX : Xn = Span{u(μ1 ), . . . , u(μn ), μi ∈ D} (3) converges to zero (almost at the same speed as dn (S(D); X) provided very little regularity exists in the parameter dependency of the solution u(μ), and an exponential convergence is achieved in many cases since analyticity in the parameter is quite frequent. This is at the basics of the reduced basis method. Indeed we are led to choose properly a sequence of parameters μ1 , . . . , μn , . . . ∈ D, then define the vectorial space XN = Span{u(μ1 ), . . . , u(μN )} and look for an approximation of u(μ) in XN . Let us consider for example an elliptic problem: Find u(μ) ∈ X such that a(u(μ), v; μ) = f (v) for all v ∈ X.

(4)

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Here X is some Hilbert space, and a is a continuous and elliptic, bilinear form in its two first arguments, regular in the parameter dependance and f is some given continuous linear form. We assume for the sake of simplicity that the ellipticity is uniform with respect to μ ∈ D: there exists α > 0 such that a(u, u; μ) ≥ αu2X

for all μ ∈ D, u ∈ X,

and that the continuity of a is uniform with respect to μ ∈ D as well: there exists γ > 0 such that |a(u, v; μ)| ≤ γ uX vX

for all μ ∈ D, u, v ∈ X.

It is classical to state that, under the previous hypothesis, problem (4) has a unique solution for any μ ∈ D. The Galerkin method is a standard way to approximate the solution to (4) provided that a finite dimensional subspace XN on X is given. It consists in: Find uN (μ) ∈ XN such that a(uN (μ), vN ; μ) = f (vN )

for all vN ∈ XN ,

(5)

which similarly has a unique solution uN (μ). Cea’s lemma then states that

u(μ) − uN (μ)X ≤ 1 +

γ inf u(μ) − vN X . α vN ∈XN

(6)

The best choice for the basis element u(μ1 ), . . . , u(μN ) of XN would be those that realize the infimum in (3), i.e. the ones that realize the maximum of the volume VN (u(μ1 ), . . . , u(μN )) of the parallelepiped determined by the vectors u(μ1 ), . . . , u(μN ). Unfortunately, this is not a constructive method and we generally refer to a greedy algorithm such as the following one: μ1 = arg sup u(μ)X , μ∈D

μi+1 = arg sup u(μ) − Pi u(μ)X ,

(7)

μ∈D

where Pi is the orthogonal projection onto Xi = Span{u(μ1 ), . . . , u(μi )} or a variant of it that is explained at the end of Section 4. The convergence proof for the related algorithm is somehow more complex and presented in a quite general settings in [1].

3. An example of a priori analysis The previous notion of n-width is quite convenient because it is rather general, in spirit, and provides a tool to reflect the rapid convergence of the reduced basis method but it is not much constructive nor qualitatively informative. We are thus going to consider


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a particular example where the parametrized “bilinear” form a : X × X × D → R is defined as follows a(w, v; μ) ≡ a0 (w, v) + μa1 (w, v); (8) here the bilinear forms a0 : X × X → R and a1 : X × X → R are continuous, symmetric and positive semi-definite, D ≡ [0, μmax ], and we assume that a0 is coercive. It follows from our assumptions that there exists a real positive constant γ1 such that a1 (v, v) 0≤ (9) ≤ γ1 for all v ∈ X. a0 (v, v) For the hypotheses stated above, it is readily demonstrated that the problem (4) satisfies uniformly the Lax–Milgram hypothesis. Many situations may be modeled by this rather simple problem statement (4), (8). It can be the conduction in thin plates and μ represents the convective heat transfer coefficient, it can also be a variable-property heat transfer, then 1 + μ is the ratio of thermal conductivities in domains. The analysis that we did in [12] involves the eigenproblem: Find (ϕ ∈ X, λ ∈ R), satisfying a1 (ϕ, v) = λ a0 (ϕ, v) for all v ∈ X. Indeed the solution u(μ) to problem (4) can be expressed as f (ϕ) ϕ(·; λ) u(·, μ) = dλ. (10) 1 + μλ The dependency in μ is thus explicitly expressed and we can propose to approximate u(μ) by a linear combination of well chosen u(μi ). This can be done through interpolation at the μi by polynomials. It is interesting to notice at this point that we have a large choice in the variable in which the polynomial can be expressed. Indeed since we are interested through this interpolation process to evaluate the best fit, a polynomial in μ may not be the best choice but rather a polynomial in μ1 , eμ or else; in [12] we have considered a polynomial approximation in the variable τ = ln(μ+δ −1 ), where δ is some positive real number. The analysis then involves the interpolation operator at equidistant points (in the variable τ ) for which we were able to get an upper bound used, in turn, to qualify the best fit result Lemma 3.1. There exists a constant C > 0 and a positive integer Ncrit such that for N ≥ Ncrit −N inf u(μ) − wN X ≤ C exp for all μ ∈ D, wN ∈XN Ncrit where Ncrit ≡ c∗ e ln(γ μmax + 1). This analysis of [12] leads to at least three remarks: Remark 3.2. a) The analysis of the best fit done here suggests to use sample points μi that are equidistant when transformed in the τ variable. We performed some numerical

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tests to check whether this sampling gives indeed better results than more conventional ones (of course you should avoid equidistant in the original μ variable, but we tried e.g. Chebyshev points) and this was actually the case. Unfortunately, in more general situations and especially in higher parameter dimensions, we have no clue of a direct constructive best sampling method. b) For a given sampling μi , one can propose an interpolation procedure to approximate u(μ) which is more simple than referring to a Galerkin approach. Indeed, an approximation N αi (μ)u(μi ), u(μ) i=1

can be proposed by using coefficients that are the Lagrange interpolation basis in the chosen variable (above it was τ = ln(μ + δ −1 ), i.e. the mapping τ → αi (μ(τ )) is a polynomial of degree ≤ N and αi (μj ) = δij ). The problem is that the expression of αi (μ) depends on the best choice of variable which is unknown and within a set that is quite infinite providing a range of results that are quite different. Since for a given general problem we have no clue of the best interpolation system, the Galerkin approach makes sense, indeed. c) In opposition, the Galerkin approach does not require any preliminary analysis on guessing the way the solution depends upon the parameter. Its superiority over interpolation process comes from the fact stated in Cea’s lemma that the approximation that is obtained, up to some multiplicative constant, gives the optimal best fit, even if we do not know the rate at which the convergence is going. Finally, as is often the case, we should indicate that the a priori analysis helps to have confidence in developing the method but, at the end of a given computation, a computable a posteriori estimator should be designed in order to qualify the approximation. This is even more true with such new surrogate approximation in order to replace the expertise a user may have in his preferred method, e.g. his intuition on the choice of the discretization parameter to get acceptable discrete solutions. This is the purpose of the next section.

4. An example of a posteriori analysis Most of the time, the complete knowledge of the solution of the problem (4) is not required. What is required, is outputs computed from the solution s = s(u), where s is some continuous functional defined over X. In order to have a hand over this output, the reduced basis method consists first in computing uN ∈ XN solution of the Galerkin approximation (5), then propose sN = s(uN ) as an approximation of s. Assuming Lipschitz condition (ex. linear case) over s, it follows that |s − sN | ≤ cu − uN X .

(11)

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Thus any information over the error in the energy norm will allow to get verification (provided you are able to evaluate c). Actually it is well known that the convergence of sN towards s most often goes faster. This is standard but we go back over it since this will prove useful in the sequel. Let us assume we are in the linear output case where s ≡ is a linear continuous mapping over X. It is then standard to introduce the adjoint state, solution of the following problem: find ψ ∈ X such that a(v, ψ; μ) = − (v) for all v ∈ X.

(12)

The error in the output is then (remember that, for any φN ∈ XN , a(u, φN ; μ) = a(uN , φN ; μ) = (f, φN )) sN − s = (uN ) − (u) = a(u, ψ; μ) − a(uN , ψ; μ) = a(u, ψ − φN ; μ) − a(uN , ψ − φN ; μ) (for all φN ∈ XN ) = a(u − uN , ψ − φN ; μ) (for all φN ∈ XN ) ≤ cu − uN X ψ − φN X (for all φN ∈ XN ),

(13)

so that the best fit of ψ in XN can be chosen in order to improve the first error bound (11) that was proposed for |s − sN |. For instance if ψN is the solution of the Galerkin approximation to ψ in XN , we get (14) |s − sN | ≤ cu − uN X ψ − ψN X . Actually, the approximation of ψ in XN may not be very accurate since XN is well suited for approximating the elements u(μ) and – except in the case where = f named the compliant case – a separate reduced space X˜ N should be built which provides an associated approximation ψ˜ N . Then an improved approximation for (u) is given by imp = (uN ) − a(uN , ψ˜ N ) + f (ψ˜ N ) since (14) holds with ψ − ψ˜ N X for which a better convergence rate is generally observed. Even improved, this result is still a priori business and it does not allow to qualify the approximation for a given computation. In order to get a posteriori information, between (u) and (uN ) (or imp ), we have to get a hand on the residuals in the approximations of the primal and dual problems. We introduce for any v ∈ X, Rpr (v; μ) = a(uN , v; μ) − f, v,

Rdu (v; μ) = −a(v, ψ˜ N ; μ) − (v).

(15)

We then compute the reconstructed errors associated with the previous residuals. These are the solutions of the following problems: 2α ∇ eˆ pr(du) ∇v = Rpr(du) (v; μ) for all v. (16) We then have

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Theorem 4.1. Let s − = imp − α [∇(eˆ pr + eˆ du )]2 then s − ≤ s. In addition, there exists two constants 0 < c ≤ C such that c|s − sN | ≤ s − s − ≤ C|s − sN |. Proof. Let us denote by eN the difference between the exact solution and the approximated one eN = u − uN . From (16), we observe that 2α ∇ eˆpr ∇eN = −a(eN , eN ; μ) and

2α

∇ eˆdu ∇eN = −a(eN , ψ˜ N ; μ) − (eN ) = f (ψ˜ N ) − a(uN , ψ˜ N ) − (eN ).

Taking this into account allows to write pr du 2 ˜ ˜ imp − α ∇(eˆ + eˆ ) = (uN ) − a(uN , ψN ) + f (ψN ) − α ∇(eˆpr + eˆdu )2 pr du 2 = (u) − α ∇(eN + eˆ + eˆ ) − a(eN , eN ; μ) + α [∇eN ]2 , (17) and the proof follows from the uniform ellipticity of a( . , . ; μ).

2

Despite the fact that we have avoided to speak about any discretization so far, Theorem 4.1 is already informative in the sense that in order to obtain s − , the problem (16) to be solved, is parameter independent and simpler than the original one, provided that we have a good evaluation of the ellipticity constant. In Section 6 we shall explain how to transform these constructions in a method that can be implemented. Before this we should explain how the previous estimator may help in designing a good choice for the elements of the reduced basis, providing a third alternative to the greedy algorithm presented in (7). Currently indeed, we have two alternative, either a random approach (that generally works not so badly) or select out of a large number of pre-computed solution {ui }i , the best sample from a SVD approach by reducing the matrix of scalar products (ui , uj ). The former lacks of fiability, the latter is a quite expensive approach and is mostly considered in a pre analysis framework as is explained in the next section. In order to reduce the cost of the off-line stage we can propose a greedy algorithm that combines the reduced approximation and the error evaluation: • Take a first parameter (randomly). • Use a (1-dimensional) reduced basis approach over a set of parameter values (chosen randomly) and select, as a second parameter, the one for which the associated predicted error s + − s − is the largest. This gives now a 2-dimensional reduced basis method.


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• Use this (2-dimensional) reduced basis approach over the same set of parameters and select, as a third parameter, the one for which the associated error is the largest. This gives a 3-dimensional reduced basis method . . . • and proceed . . . Note that we then only compute accurately the solutions corresponding to the parameters that are selected this way. The a posteriori approach that has been presented above relies on the uniform ellipticity of the bilinear form and the knowledge of the ellipticity constant. For more general problems, where only, nonuniform inf-sup conditions are valid (e.g. the noncoercive Helmholtz acoustics problem which becomes singular as we approach resonance) smarter definitions should be considered. We refer to [18] for improved methods in this direction.

5. Some pragmatic considerations Now that some basics on the reduced basis method have been presented, it is interesting to understand if the problem you have in mind is actually eligible to this type of approximation. We are thus going to propose some pragmatic arguments that may help in the preliminary verification. First of all, let us note that we have illustrated the discretization on linear elliptic problems, of course this is just for the sake of simplicity, non linear problem [11], [19], [20] so as time dependent problems [7], [17] can be solved by these methods. Second, many things can be considered as a valid parameter: this can be the size of some simple geometric domain on which the solution is searched [16] , but it can be the shape itself [13] (the parameter in the former case is a multireal entity while in the latter it is a functional), the parameter can also be the time [7], [17] , or the position of some given singularities [2]. For all these choices, a fair regularity in the parameter is expected and wished so that the n-width goes fast to zero. An important remark has to be done here in order the size of the reduced basis be the smallest possible. Indeed, it may be smart to preprocess the precomputed solutions in order they look more similar. An example is given in [2] where quantum problem are considered; the solutions to these problems present some singularities at the position of the nuclei. If the position of the nuclei is the parameter we consider, it is useful to transform each solution in a reference configuration where the singularities/nuclei are at a unique place; the solutions are then much more comparable. Another example is given by the solution of the incompressible Stokes and Navier–Stokes problem where the shape of the computational domain is the parameter; in order to be able to compare them properly, they have to be mapped on a unique (reference) domain. This is generally done through a simple change of variable. In case of the velocity, it is a vector field that is divergence

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free and a “standard” change of variable will (generally) not preserve this feature. The Piola transform (that actually corresponds to the classical change of variable over the potential function) allows to have the velocity fields transformed over the reference domain while preserving the divergence free condition as is demonstrated in [9]. These preprocessing steps allow to diminish the n-width of S(D) and it pays to be smart! In order to sustain the intuition on the potential of the reduced basis concept, a classical way is to use a SVD approach. Let us assume that we have a bunch of solutions ui = u(μi ), snapshots of the space S(D) of solutions to our problem. Out of these, the correlation matrix (ui , uj ) which is symmetric can be reduced to its eigen-form, with positive eigenvectors that, ranked in decreasing order, go to zero. The high speed of convergence towards zero of the eigenvalues ranked in decreasing order will sustain the intuition that the reduced basis method will work. Indeed, the n-width is directly related to the size of the eigenvalues larger than the n + 1th. The idea is that if the number of eigenvectors associated with the largest eigenvalues is small, then the method is viable. In order to sustain this, you can also consider, momentarily, the space XN spanned by the eigenvectors associated with the N largest eigenvalues and analyze the norm of the difference between the snapshots in S(D) and their best fit in XN . Note that we do not claim that this is a cheap constructive method: this procedure consists in a pre-analysis of the potential of the reduced basis method to approximate the problem you consider. If the norm of the error goes to zero sufficiently fast, you know that a Galerkin approach will provide the same order of convergence and the method is worth trying. We insist on the fact that this preanalysis is not mandatory, it is only to help in understanding what you should expect, “at best” from the reduced basis approximation. In particular the greedy approach presented in Section 4 has to be preferred to the SVD approach that we discussed above for the determination of the elements that are to be incorporated in the reduced basis space, if you do not want to spend too much time during the off-line stage. Note also that the greedy approach provides solutions, that, when their number becomes large, become more and more linearly dependent (actually this is one of the aspects of the low n-width) and thus, for stability purposes it is important, through a Gram– Schmidt process, to extract, from these solutions, orthonormal elements that will be the actual elements of the reduced basis: these will be named (ζi )i . This does not change the potential approximation properties of the reduced basis but improves, to a large extent, the stability of the implementation. Finally, the preselection may be quite generous in the sense that you may be interested to select more than N basis functions, N being an evaluation of the dimension of the reduced basis for most problems. The reason for this comes from the conclusion of the a posteriori analysis that may tell you to increase the size of the reduced basis, suggesting you to work with N + 2 (say) instead of N basis functions. This again is a feature of exponentially rapid convergence that lead to a large difference between the accuracy provided by XN and XN +2 (say). It is time now to give some details on the implementation of the method.

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6. Implementation issues We start by emphasizing that any reduced basis method necessarily involves the implementation of a more “classical” approximation method. Indeed – except for very particular and uninteresting problems – the knowledge of the solutions, that we named ui , is impossible without referring to a discretization method (e.g. of finite element, spectral type, etc.). This is also the case for the ζ that are coming out from some shaping of the basis, e.g. Gram–Schmidt, as explained earlier. This is the reason why reduced basis methods should not be considered as competitor to standard approximation methods but only as surrogate approaches. This implies, though, some difficulties since the elements of the reduced basis are only known through a preliminary computation basis, which, if we want the solution ui to be well approximated, has to be very large. Knowing this, the rule of the game for the efficient implementation of any reduced basis method is to strictly prohibit any online reference to the extended basis. We allow offline precomputations of the solutions (that involves the extended basis) and some offline cross contribution of these solutions (based on their expression with respect to the extended basis) but this is forbidden online. Following [16], we explain in the next subsection how this can be done. 6.1. Black box approach. The solution procedure involves the evaluation of the elements of the stiffness matrix a(ζi , ζj ; μ), 1 ≤ i, j ≤ N that depends on the current parameter μ. This computation involves some derivatives and the evaluation of integrals, that have to be performed and this may be very lengthy. It should be stated here that the implementation of the reduced type method has to be much faster than the solution procedure that was used to compute the reduced basis, much means many order of magnitude. The O(dim XN )2 entrees of the stiffness matrix have thus to be evaluated through some smart way. Let us begin by the easy case that is named affine parametric dependance where the entries a(ζi , ζj ; μ) appear to read a(ζi , ζj ; μ) = gp (μ)ap (ζn , ζm ), (18) p

where the bilinear forms ap are parameter independent. Many simple problems where the parameter are local constitutive coefficients or local zooming isotropic or non isotropic factors, enter in this framework. The expensive computation of the ap,n,m = ap (ζn , ζm ) can be done offline, once the reduced basis is constructed; these ap,n,m are stored and, for each new problem, the evaluation of the stiffness matrix is done, 3 ) operations. These figures online, in P × N 2 operations, and solved in O(dim XN are coherent with the rapid evaluation of the reduced basis method. 6.2. A posteriori implementation. Under the same affine dependance hypothesis on a, it is easy to explain how the a posteriori analysis can be implemented, re-

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sulting in a fast on-line solution procedure, provided some off-line computations are made. First of all the computation of ψ˜ N can be implemented in the space X˜ N = Span{ξ1 , . . . , ξN )} exactly as above for the computation of uN . Taking into account (18), together with the expressions Nobtained from the inversion of problem (5) ˜ α ζ and ψ = ˜ i ξi , we can write and (12): uN = N N i=1 i i i=1 α Rpr (v, μ) =

p

and Rdu (v, μ) = −

gp (μ)αi ap (ζi , v) − (f, v),

i

p

gp (μ)α˜j ap (v, ξj ) − (v),

j

hence by solving numerically, off-line, each of the problems 2α ∇epr,p,i ∇v = ap (ζi , v), 2α ∇epr,0 ∇v = (f, v), 2α ∇edu,p,j ∇v = ap (v, ξj ), 2α ∇edu,0 ∇v = (v)

(19) (20) (21) (22)

allows to write the numerical solutions of (16) as a linear combination of the elements previously computed (e.g. eˆ pr = p i gp (μ)αi epr,p,i − epr,0 ) in O(P N) operations. 6.3. Magic points. The hypothesis of affine parametric dependency is rather restrictive, and has to be generalized. In case of quadratic or cubic dependency, this is quite straightforward but even for linear problems such as the Laplace problem, when e.g. geometry is the parameter, this is rarely the case and another approach has to be designed. In order to get a better understanding of the method, let us first indicate that, when the geometry is the parameter, the solutions have to be mapped over a ˆ Let us assume that we want to compute d(ζi , ζj ; ) where reference domain .

d(u, v; ) =

uv dA =

ˆ

ˆ uvJ d A,

ˆ onto . There is no reason where J is a Jacobian of the transformation that maps in the general case that J will be affine so that the previous approach will not work. It is nevertheless likely that there exists a sequence of well chosen transformations ∗1 , . . . , ∗M , . . . , such that J may be well approximated by an expansion J

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M

j =1 βj Jj∗ .

An approximation of d(ζi , ζj ; ) will then be given by d(ζi , ζj ; )

M

βj

j =1

ˆ

ˆ ζî ζˆj Jj∗ d A,

(23)

and again, the contributions ˆ ζî ζˆj Jj∗ d Aˆ will be pre-computed offline. We do not elaborate here on how the j∗ are selected, and refer to [9], what we want to address is the evaluation of the coefficients βj = βj () in the approximation of J above. The idea is to use an interpolation procedure as is explained in [6]. Let x 1 be the point where |J∗1 | achieves its maximum value. Assuming then that x 1 , . . . , x n have been defined, and are such that the n × n matrix with entries J∗k (x ), 1 ≤ k, ≤ n is invertible, we define x n+1 as being the point where rn+1 = |J∗n+1 − nk=1 γk J∗k | achieves it maximum value, here the scalar γk are defined so that rn+1 vanishes at any (x ) for = 1, . . . , n. The definition of the points x k is possible as long the j are chosen such that the J∗k are linearly independent (see [6]). The βj are then evaluated also through the interpolation process J (x ) =

M

βk J∗k (x )

for all 1 ≤ ≤ M.

(24)

k=1

We have not much theory confirming the very good results that we obtain (which makes us call these interpolation point “magic”). An indicator that allows to be quite confident in the interpolation process is the fact that the Lebesgue constant attached to the previously built points is, in all example we have encountered, is rather limited. Note that the same interpolation approach allows to compute the reconstructed errors with a compatible complexity as in the previous subsection. The same magic point method has to be used also for the implementation of the reduced basis method for nonlinear problems. Actually, denoting by zi = NL(ui ) the nonlinear expression involved in the problem, provided that the set ZM = Span{zi , 1 ≤ i ≤ M} has a small width, the interpolation process presented above allows both to determine a good interpolation set and a good associated interpolation nodes, we refer to [6] for more details on the implementation and to numerical results.

7. Some extensions 7.1. Eigenvalue problems. We end this paper by noticing that the reduced basis method can actually be found, at least in spirit, in many other approximations. There are indeed many numerical approaches that, in order to tackle a complex problem, use the knowledge of the solution of similar but simpler problems to facilitate the approximation. In this direction, the modal synthesis method provides a method to solve approximately eigenvalue problems on large structures based on the knowledge

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of the eigenvalues and eigenfunctions of the same problem on substructures. We refer e.g. to [4], [5] for more details on a high order implementation of these approaches. Similarly, again, one of the approaches for the resolution of Hartree–Fock problem in quantum chemistry is the L.C.A.O. method that consists in approximating the wave function of a molecule by linear combination of atomic orbitals that are nothing but solutions to the same problem on an atom, instead of a molecule. The atomic orbitals are indeed the approximations of Hydrogenoid functions (the contracted Gaussians have to be seen this way) that are the solutions of the electronic problem of one electron around a nucleus. This similarity is the guideline for the extension that is proposed in [3], [2]. At this level, it is also interesting to note that the reduced basis method, for an eigenvalue problem as the one encountered in the two previous cases, may be very appropriate since it can be proven that, letting ui denote the set of all first P eigenvectors of an eigenvalue problem depending on a parameter μ, ui ≡ (e1 (μi ), . . . , ep (μi )), then the approximation of this complete set of eigenvectors can be done with the same linear combination. More precisely it is possible to get an accurate approximation method based on u(μ)

P

αi ui ,

P

e (μ) j

i=1

αi ej (μi ) for all j

i=1

instead of ej (μ)

P

j

αi ej (μi ).

i=1

Again we refer to [2] for more details on this. 7.2. The reduced element method. In the reduced basis element method introduced in [13], we consider the geometry of the computational domain to be the generic parameter. The domain is decomposed into smaller blocks, all of them can be viewed as the deformation of a few reference shapes. Associated with each reference shape are previously computed solutions (typically computed over different deformations of the reference shapes). The precomputed solutions are mapped from the reference shapes to the different blocks of the decomposed domain, and the solution on each block is found as a linear combination of the mapped precomputed solutions. The solutions on the different blocks are glued together using Lagrange multipliers. To be more precise, we assume that the domain where the computation should be performed can be written as the non-overlapping union of subdomains k : =

K

k

,

k ∩ = ∅ for k = .

(25)

k=1

Next, we assume that each subdomain k is the deformation of the “reference” doˆ through a regular enough, and one to one, mapping. In an off-line stage, main


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this reference geometry has been “filled up” with reference reduced basis solutions uˆ 1 , uˆ 2 , . . . , uˆ N to the problem that is under evaluation. Hence, together with this geometric decomposition, a functional decomposition is proposed since every k ; this allows us to define the finite dimensional space k −1 ˆ ] , (26) YN = v ∈ L2 (), v|k = N i i=1 αi Fk [u which is a set of uncoupled, element by element, discrete functions, where Fk allows to ˆ into functions defined over k . This is generally transform functions defined over not yet adequate for the approximation of the problem of interest since some glue k at the interfaces γk, between two adjacent domains ∩ has to be added to the elements of YN , the glue depending on the type of equations we are interested to solve (it will be relaxed C 0 -continuity condition for a Laplace operator, or more generally relaxed C 1 -continuity condition for a fourth-order operator. At this stage it should be noticed that, modulo an increase of complexity in the ˆ filled with its reduced notations, there may exist not only one reference domain basis functions but a few numbers so that the user can have more flexibilities in the design of the final global shape by assembling deformed basic shapes like a plumber would do for a central heating installation. The reduced basis element method is then defined as a Galerkin approximation over the space XN being defined from YN by imposing these relaxed continuity constraints. We refer to [9], [10] for more details on the implementation for hierarchical fluid flow systems that can be decomposed into a series of pipes and bifurcations.

References [1] Buffa, A., Maday, Y., Patera, A. T., Prud’homme, C., Turinici, G. In progress, 2006. [2] Cancès, E., Le Bris, C., Maday, Y., Nguyen, N. C., Patera, A. T., Pau, G., Turinici, G. In progress, 2006. [3] Cancès, E., Le Bris, C., Maday, Y., Turinici, G., Towards reduced basis approaches in ab initio electronic structure computations. J. Sci. Comput. 17 (1–4) (2002), 461–469. [4] Charpentier, I., Maday, Y., Patera, A. T., Bounds evaluation for ouputs of eigenvalue problems approximated by the overlapping modal synthesis method. C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 909–914. [5] Charpentier, I., de Vuyst, F. , Maday, Y., The overlapping component mode synthesis: The shifted eigenmodes strategy and the case of self-adjoint operators with discontinuous coefficients. In Proceedings of the Ninth International Conference on Domain Decomposition Methods (ed. by P. E. Bjørstad, S. Magne, S. Espedal and D. E. Keyes), 1998, 583–596. [6] Grepl, M. A., Maday, Y., Nguyen, N. C., Patera, A. T., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Math. Model. Numer. Anal., submitted. [7] Grepl, M. A., Patera, A. T., A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. Math. Model. Numer. Anal. 39 (1) (2005), 157–181.

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[8] Kolmogoroff, A., Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse. Ann. of Math. 37 (1963), 107–110. [9] Løvgren, A. E., Maday, Y., Rønquist, E. M., A reduced basis element method for the steady Stokes problem. Math. Model. Numer. Anal., to appear. [10] Løvgren, A. E. , Maday, Y. , Rønquist, E. M., The reduced basis element method for fluid flows. Adv. Math. Fluid Mech., 2006, to appear. [11] Machiels, L. , Maday, M., Oliveira, I. B., Patera, A. T., Rovas, D. V., Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. C. R. Acad. Sci. Paris Sér. I Math. 331 (2000), 153–158. [12] Maday, Y., Patera, A. T., Turinici, G., A Priori Convergence Theory for Reduced-Basis Approximations of Single-Parameter Elliptic Partial Differential Equations. J. Sci. Comput. 17 (1–4) (2002), 437–446. [13] Maday, Y., and Rønquist, E. M., The reduced-basis element method: Application to a thermal fin problem. SIAM J. Sci. Comput. 26 (2004), 240–258. [14] Pinkus, A., n-Widths in Approximation Theory. Ergeb. Math. Grenzgeb. (3) 7, SpringerVerlag, Berlin 1985. [15] Prud’homme, C., Contributions aux simulations temps-réel fiables et certains aspects du calcul scientifique. Mémoire d’Habilitation à Diriger les Recherches, Université Pierre et Marie Curie, Paris, 2005. [16] Prud’homme, C., Rovas, D. V., Veroy, K., Machiels, L., Maday,Y., Patera, A. T., Turinici, G., Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bound methods. J. Fluids Engrg. 124 (2002), 70–80. [17] Rovas, D. V., Machiels, L., Maday, Y., Reduced-basis output bound methods for parabolic problems. IMA J. Numer. Anal., Advance Access published on March 6, (2006). [18] Sen, S., Veroy, K., Huynh, D. B. P., Deparis, S., Nguyen, N. C., Patera, A. T., Natural norm, a posteriori error estimators for reduced basis approximations. J. Comp. Phys., 2006, to appear. [19] Veroy, K., Prud’homme, C., Patera, A. T., Reduced-basis approximation of the viscous Burgers equation: Rigorous a posteriori error bounds. C. R. Acad. Sci. Paris Ser. I Math. 337 (2003), 619–624. [20] Veroy, K., Patera, A. T., Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: Rigorous reduced-basis a posteriori error bounds. Int. J. Numer. Meth. Fluids 47 (2005), 773–788. Université Pierre et Marie Curie-Paris6, UMR 7598 Laboratoire Jacques-Louis Lions, B.C. 187, 75005 Paris, France and Division of Applied Mathematics, Brown University, Providence, U.S.A. E-mail: [email protected]

Finite element algorithms for transport-diffusion problems: stability, adaptivity, tractability Endre Süli

Abstract. Partial differential equations with nonnegative characteristic form arise in numerous mathematical models of physical phenomena: stochastic analysis, in particular, is a fertile source of equations of this kind. We survey recent developments concerning the finite element approximation of these equations, focusing on three relevant aspects: (a) stability and stabilisation; (b) hp-adaptive algorithms driven by residual-based a posteriori error bounds, capable of automatic variation of the granularity h of the finite element partition and of the local polynomial degree p; (c) complexity-reduction for high-dimensional transport-diffusion problems by stabilised sparse finite element methods. Mathematics Subject Classification (2000). Primary 65N30; Secondary 65N12, 65N15. Keywords. Transport-dominated diffusion problems, Fokker–Planck equations, finite element methods, stability, a-posteriori error analysis, adaptivity, sparse finite elements.

1. Introduction Let be a bounded and simply-connected open set in Rd , d ≥ 2, with Lipschitz continuous boundary ∂. On , we consider the partial differential equation Lu := −∇ · (a∇u) + ∇ · (bu) + cu = f,

(1)

where f ∈ L2 () and c ∈ L∞ () are real-valued functions, b = {bi }di=1 is a vector function whose entries bi are Lipschitz continuous real-valued functions on . We shall, further, assume that a = {aij }di,j =1 is a symmetric matrix whose entries aij are bounded, Lipschitz continuous real-valued functions defined on , and that the matrix a is positive semidefinite, almost everywhere on , i.e., α(ξ ) := ξ a(x)ξ ≥ 0

for all ξ ∈ Rd and a.e. x ∈ .

(2)

Under hypothesis (2), the equation (1) is referred to as a partial differential equation with nonnegative characteristic form. Equations of this kind frequently arise as mathematical models in physics and chemistry [40] (e.g. in the kinetic theory of polymers [7], [44], [49] and coagulation-fragmentation problems [43]). They also appear in molecular biology [21], population genetics (e.g. in mathematical models of random Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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genetic drift) and in mathematical finance. Important special cases of these equations include the following: (a) when the diffusion matrix a = a is positive definite, (1) is an elliptic partial differential equation; (b) when a ≡ 0 and the transport direction b = 0, the partial differential equation (1) is a first-order hyperbolic equation; (c) when b = (0, . . . , 0, 1) ∈ Rd and

α 0 a= 0 0

where α is a (d − 1) × (d − 1) symmetric positive definite matrix, (1) is a parabolic partial differential equation, with time-like direction b. The family of partial differential equations with nonnegative characteristic form also includes a range of other linear second-order partial differential equations, such as degenerate elliptic and ultraparabolic equations. Furthermore, by a result of Hörmander [35] (cf. Theorem 11.1.10 on p. 67), second-order hypoelliptic operators with constant coefficients have nonnegative characteristic form, after possible multiplication by −1. For classical types of partial differential equations, such as those under (a), (b) and (c) above, rich families of reliable, stable and accurate numerical techniques have been developed. Yet, there have only been isolated attempts to date to explore computational aspects of the class of partial differential equations with nonnegative characteristic form as a whole (cf. [30] and [33]). In particular, only a limited amount of research has been devoted to the construction and mathematical analysis of adaptive finite element algorithms for these equations; similarly, there has been very little work on the finite element approximation of high-dimensional partial differential equations with nonnegative characteristic form (cf. [57]). The aim of this paper is to present a brief survey of some recent results in these directions. In Section 2, we state the weak formulation of a boundary-value problem for equation (1). In Sections 3 and 4, we give an overview of stabilised continuous and discontinuous finite element approximations to these equations; we shall also address the question of residual-based a posteriori error estimation for hp-version discontinuous Galerkin approximations of these equations, as well as the construction of sparse stabilised finite element methods for high-dimensional partial differential equations with nonnegative characteristic form (Section 5).

2. Boundary conditions and weak formulation For the sake of simplicity of presentation, we shall assume that is a bounded open polytope in Rd and we denote by the union of its (d − 1)-dimensional open faces; clearly, ⊂ ∂ with strict inclusion. The equation (2) will be supplemented by boundary conditions. For this purpose, let ν(x) = {νi (x)}di=1 denote the unit outward normal vector to at x ∈ . On introducing the Fichera function (cf. [48])

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Finite element algorithms for transport-diffusion problems

x ∈ → β(x) := (b · ν)(x) ∈ R, we define the following subsets of : 0 = {x ∈ : α(ν(x)) > 0}, − = {x ∈ \0 : β(x) < 0}, + = {x ∈ \0 : β(x) ≥ 0}. The set 0 is the elliptic part of , while − ∪ + represents the hyperbolic part of . The sets − and + will be referred to as the hyperbolic inflow and outflow boundary, respectively. Clearly, = 0 ∪ − ∪ + . If 0 = ∅ and has positive (d −1)-dimensional Hausdorff measure H d−1 (0 ), we shall further decompose it into disjoint subsets D and N whose union is 0 , with H d−1 (D ) > 0. We supplement the partial differential equation (1) with the boundary conditions u = gD

on D ∪ − ,

ν · (a∇u) = gN

on N ,

(3)

and adopt the (physically reasonable) hypothesis that β(x) ≥ 0 for a.e. x ∈ N , whenever N is nonempty. In addition, we assume that the following (standard) positivity hypothesis holds: there exists a positive constant cˆ0 such that c(x) +

1 2

∇ · b(x) ≥ cˆ02

a.e. x ∈ ,

(4)

and define c0 = (c + 21 ∇ · b)1/2 on . Now, consider the following boundary-value problem, corresponding to gD = 0 and gN = 0: find u such that Lu ≡ −∇ · (a∇u) + ∇ · (bu) + cu = f u=0 ν · (a∇u) = 0

in , on D ∪ − , on N .

(5) (6) (7)

Function spaces and weak formulation. The classical Sobolev space on of integer order m, m ≥ 0, will be denoted by Wqm () for q ∈ [1, ∞]; in the case q = 2 we write Hm () for W2m (); Wq0 () is simply Lq (). Wpm () is equipped with the Sobolev norm · Wqm () and seminorm | · |Wqm () . For the sake of simplicity, we shall write · instead of · L2 () , and · κ will denote · L2 (κ) for an open subset κ of . We let V = {v ∈ H1 () : γ0,∂ (v)|D = 0} where γ0,∂ (v) signifies the trace of v on ∂, and define the inner product ( · , · )H by (w, v)H := (a∇w, ∇v) + (w, v) + w, vN ∪− ∪+ . Here ( ·, · ) denotes the L2 inner product over and w, vS = S |β|wv ds, with β denoting the Fichera function b · ν, as before, and S ⊂ . We denote by H the 1/2 closure of the space V in the norm · H defined by wH := (w, w)H . Clearly, H is a Hilbert space. For w ∈ H and v ∈ V, we now consider the bilinear form B( ·, · ) : H × V → R defined by B(w, v) := (a∇w, ∇v) − (w, ∇ · (bv)) + (cw, v) + w, vN ∪+ ,

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and for v ∈ V we introduce the linear functional L : V → R by L(v) := (f, v). Note, in particular, that by (4), B(v, v) = (a∇v, ∇v) + c0 v2 + 21 v, vN ∪− ∪+ ≥ K0 v2H

for all v ∈ V,

where K0 = min cˆ02 , 21 > 0. We shall say that u ∈ H is a weak solution to the boundary-value problem (5), (6) if B(u, v) = L(v) for all v ∈ V.

(8)

We note that the boundary conditions u|− = 0 on the inflow part − of the hyperbolic boundary \ 0 = − ∪ + and the boundary condition ν · (a∇u) = 0 on the Neumann part N of the elliptic boundary 0 are imposed weakly, through (8), while the boundary condition u|D = 0 on the Dirichlet part, D , of 0 is imposed strongly, through the choice of the function space H. The existence of a unique weak solution is guaranteed by the following theorem (cf. also Theorem 1.4.1 on p. 29 of [48] and [57] for a similar result in the special case of N = ∅; for N = ∅ the proof is identical to that in [57]). Theorem 2.1. Suppose that c0 (x) ≥ cˆ0 > 0 for all x ∈ . Then, for each f ∈ L2 (), there is a unique u in a Hilbert subspace Hˆ of H such that (8) holds. Next, we shall consider the discretisation of the problem (8), first by a stabilised Galerkin method based on continuous piecewise polynomials, and then using discontinuous piecewise polynomials.

3. Continuous piecewise polynomial approximation: the streamlinediffusion method As in the previous section, we suppose that is a bounded open polytope in Rd , d ≥ 2. Let Th = {κ} be an admissible subdivision of into open element domains κ which is subordinate to the decomposition of into the subsets D , N , − and + ; here h is a piecewise constant mesh function with h(x) = hκ = diam(κ) when x is in element κ ∈ Th . We shall assume that each κ ∈ Th is the image, under a bijective affine map Fκ , of a fixed master element κ, ˆ where κˆ is either an open unit simplex d or an axiparallel open unit hypercube in R . We shall also suppose that the family of partitions {Th }h>0 is (a) regular (namely, the closures of any two elements in the subdivision are either disjoint or share a common face of dimension ≤ d − 1); and (b) shape-regular (namely, there exists a positive constant c1 , independent of h, such that c1 hdκ ≤ meas(κ) for all κ ∈ h Th ).

1275


ˆ the set of polynomials of degree at most p on κˆ For p ≥ 1, we denote by Pp (κ) when κˆ is an open unit simplex; when κˆ is an axiparallel open unit hypercube, we let Qp (κ) ˆ denote the set of all tensor-product polynomials of degree at most p in each coordinate direction. We define the finite element space Hh,p = {v ∈ H ∩ C() : v|κ ∈ Rp (κ) for all κ ∈ Th }, where Rp (κ) = {w ∈ L1 (κ) : w Fκ ∈ Rp (κ)} ˆ and Rp is either Pp or Qp . Next, we formulate the streamline-diffusion finite element approximation of (8). The method was originally introduced by Hughes and Brooks [36] in 1979 for elliptic transport-dominated diffusion equations. Its analysis was pursued by a number of authors (see [38], [39], [54], for example). The definition of the method stems from the empirical observation that standard Galerkin finite element approximations to transport-dominated diffusion problems exhibit nonphysical numerical oscillations which occur predominantly in the direction of subcharacteristic curves (i.e. the characteristic curves of the underlying hyperbolic problem); the standard Galerkin method is therefore supplemented with numerical diffusion/dissipation in the direction of the subcharacteristics through the inclusion of a streamline-diffusion stabilisation term. For a survey of recent perspectives on stabilised and multiscale finite element methods for partial differential equations, including transport-dominated diffusion problems, we refer to the survey paper of Brezzi and Marini [14]; see also [13], [15]. Here, we follow the exposition in [33] and consider the bilinear form Bδ ( · , · ) defined by Bδ (w, v) = (a∇w, ∇v) − (w, ∇ · (bv)) + (cw, v) ˆ δκ b · ∇v)κ + w, vN ∪+ + (Lw, κ∈Th

and the linear functional δ (v) = κ (f, v+δκ b·∇v)κ , where, on element κ ∈ Th , we ˆ = −∇ · (Pκ (a∇w)) + b · ∇w + cw, w ∈ H1 (κ), and Pκ signifies the define Lˆ by Lw orthogonal projection in [L2 (κ)]d onto [Rp (κ)]d . In these definitions (·, ·)κ denotes the L2 inner product over κ and the nonnegative piecewise constant function δ, called the streamline-diffusion stabilisation parameter, is defined by δ|κ = δκ for κ ∈ Th , where δκ is a nonnegative constant on element κ. The precise choice of δ will be discussed below. Now, the streamline-diffusion finite element method is defined as follows: find uSD ∈ Hh,p such that Bδ (uSD , v) = δ (v) for all v ∈ Hh,p .

(9)

Here, we shall focus on the stability and error analysis of this method. A key property is the following: from (8) and (9) we deduce that if u ∈ H ∩ H2 () then Bδ ((u − uSD ), v) = (∇ · (a∇u − Pκ (a∇u)), δκ b · ∇v)κ for all v ∈ Hh,p . (10) κ∈Th

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In particular, if a is a constant matrix, then the projection operator Pκ can be replaced by the identity operator. In this case the right-hand side in (10) is zero and this identity is then referred to as the Galerkin orthogonality property of the streamline-diffusion finite element method (9). Next we show the stability of the method (9) and state an optimal order a priori error bound. The bound will be expressed in terms of the so-called streamline-diffusion norm | · |SD defined by √ |v|2SD = ∇v2a + cˆ02 v2 + v2N ∪− ∪+ + δ b · ∇v2 , where ∇v2a = (a∇v, ∇v). The analysis requires the following results; cf. [33]. Lemma 3.1 (Inverse inequality). There exists a positive constant Cinv = Cinv (c1 ), independent of a, hκ and p such that ∇ · (Pκ (a∇v))L2 (κ) ≤ Cinv

p2 a∇vL2 (κ) for all v ∈ Hh,p , κ ∈ Th . hκ

Lemma 3.2. Suppose that M is a real d × d symmetric positive semidefinite matrix and let | · | denote the Euclidean norm on Rd ; then |Mξ |2 ≤ ρ(M)(Mξ, ξ ) for all ξ ∈ Rd , where ρ(M) = max1≤i≤d λi is the spectral radius of M and λi , i = 1, . . . , d, are the (real, nonnegative) eigenvalues of M. Now we are ready to discuss the coercivity of the bilinear form Bδ ( · , · ) over Hh,p × Hh,p . To this end we define Th = {κ ∈ Th : bL∞ (κ) = 0}. Proposition 3.3. Suppose that the streamline-diffusion parameter δκ on element κ is selected, with the convention 1/0 = ∞, so that

cˆ02 h2κ 1 0 ≤ δκ ≤ 2 min for all κ ∈ Th . , (11) (Cinv )2 ρ(a)L∞ (κ) p4 c2L∞ (κ) Then, the bilinear form Bδ ( · , · ) is coercive on Hh,p × Hh,p , i.e. Bδ (v, v) ≥ 21 |v|2SD for all v ∈ Hh,p . Proof. Integrating by parts gives √ Bδ (v, v) ≥ (a∇v, ∇v) + c02 v 2 dx + 21 v2N ∪− ∪+ + δ b · ∇v2 − δκ (∇ · (Pκ (a∇v))κ + cvκ ) b · ∇vκ ,

(12)

κ∈Th

for v ∈ Hh,p . Now, using Lemma 3.1 and Lemma 3.2 with M = a and ξ = ∇v, ∇ · (Pκ (a∇v))κ + cvκ ≤ Cinv

p2 1/2 ρ(a)L∞ (κ) ∇va,κ + cL∞ (κ) vκ , hκ

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Finite element algorithms for transport-diffusion problems 1/2

with the notation ∇va,κ = (a∇v, ∇v)κ . Thus, for any real number γ > 0, δκ (∇ · (Pκ (a∇v))κ + cvκ ) b · ∇vκ ≤ γ δκ b · ∇v2κ κ∈Th

κ∈Th

+

p4 1 1 δκ c2L∞ (κ) v2κ + δκ (Cinv )2 ρ(a)L∞ (κ) 2 ∇v2a,κ . 2γ 2γ hκ κ∈Th

κ∈Th

Choosing γ = 1/2, we deduce from (12) and the definition of c0 that √ Bδ (v, v) ≥ ∇v2a + cˆ20 v2 + 21 v2N ∪− ∪+ + 21 δ b · ∇v2 p4 − δκ c2L∞ (κ) v2κ − δκ (Cinv )2 ρ(a)L∞ (κ) 2 ∇v2a,κ . hκ κ∈Th

κ∈Th

Selecting the streamline-diffusion parameter as in (11), the result follows.

2

Corollary 3.4 (Stability). Under the hypotheses of Proposition 3.3, δκ f 2κ . |uSD |2SD ≤ 4cˆ0−2 f 2 + 4 κ∈Th

In particular, if f = 0 then uSD = 0; since Hh,p is a finite-dimensional linear space, it follows that (9) has a unique solution uSD ∈ Hh,p for any f ∈ L2 (). The next result concerns the accuracy of the method (9). Theorem 3.5. Let the streamline-diffusion parameter δκ be chosen so that

cˆ02 1 h2κ , for all κ ∈ Th , 0 < δκ ≤ min 2 (Cinv )2 ρ(a)L∞ (κ) p4 c2L (κ) ∞

with the convention 1/0 = ∞. Then, assuming that u ∈ H ∩ Hk () ∩ C(), with a k−1 (κ)]d×d , κ ∈ T , the following error bound holds: positive integer k and a ∈ [W∞ h

h2τ −1 1/2 κ 2 M (a, b, c, h , p)u |u − uSD |SD ≤ C , κ κ k H (κ) p2k−1 κ∈Th

where τ = min(p + 1, k), C is a positive constant which depends only on c1 and k, Mκ (a, b, c, hκ , p) = Aκ (p/ hκ ) + Bκ + Cκ (hκ /p), with ⎧ a2L∞ (κ) +a2 k−1 ⎨ W∞ (κ) , when ρ(a)L∞ (κ) = 0, ρ(a)L∞ (κ) Aκ = aL∞ (κ) + ⎩ 0, otherwise, bL∞ (κ) (1 + Dκ + Dκ−1 ), when bL∞ (κ) = 0, Bκ = 0, otherwise, Cκ = 1 + c0 + c0−2 c − ∇ · b2L∞ (κ) ,

Dκ = δκ bL∞ (κ) p/ hκ .

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Theorem 3.5 is an extension of classical a priori error bounds for the streamlinediffusion discretisation of a first-order hyperbolic problem and a second-order elliptic transport-diffusion problem with a isotropic and constant; see, for example, [37], [54] (h-version) and [29] (hp-version), and [33] for a proof in the case of N = ∅.

4. Discontinuous piecewise polynomial approximation: the discontinuous Galerkin method Discontinuous Galerkin finite element methods (DGFEMs, for short) date back to the early 1970s; they were simultaneously proposed by Reed & Hill [52] in 1973 for the numerical solution of the neutron transport equation and by Nitsche [45] in 1971 as a nonstandard scheme for the approximation of second-order elliptic equations. Since the early 1970s there has been extensive work on the development of these methods for a wide range of applications; for an excellent historical survey of DGFEMs up until 2000 we refer to the paper of Cockburn, Karniadakis and Shu in the volume [19]. One of the key advantages of the DGFEM in comparison with standard Galerkin finite element methods based on continuous piecewise polynomials, such as the streamline-diffusion finite method discussed in the previous section, is their high degree of locality: the computational stencil of the DGFEM remains very compact even as the degree of the approximating polynomial is increased. Hence, high-order adaptive hp- and spectral element approximations may be handled in a particularly flexible and simple manner. Indeed, hp-adaptive DGFEMs offer tremendous gains in terms of computational efficiency in comparison with standard mesh refinement algorithms which only incorporate local h-refinement with a given (fixed) polynomial degree. For discussions concerning various hp-refinement strategies see [1], [8], [32], [34], [53], [58]. A further attractive property of the discontinuous Galerkin finite element method for a transport-dominated diffusion problem is that, unlike its counterpart based on continuous piecewise polynomials, the method is stable even in the absence of streamline-diffusion stabilisation. In this section, we survey a priori and a posteriori error bounds for discontinuous Galerkin finite element approximations of second-order partial differential equations with nonnegative characteristic form. We shall then show how the a posteriori error bound can be used to drive an hp-adaptive finite element algorithm. The presentation in this section is based on the paper [25]. We consider shape-regular meshes Th = {κ} that partition the domain into open element domains κ, with possible hanging nodes. We shall suppose that the mesh is 1-irregular in the sense that there is at most one hanging node per (d − 1)-dimensional element-face, e.g. the barycenter of the face. We denote by h the piecewise constant mesh function with h(x) ≡ hκ = diam(κ) when x is in element κ. Let each κ ∈ Th be a smooth bijective image of a fixed master element κ, ˆ that is, κ = Fκ (κ) ˆ for all κ ∈ Th , where κˆ is either the open unit simplex κˆ S = {xˆ = (xˆ1 , . . . , xˆd ) ∈ Rd : 0 < x1 + · · · + xd < 1, xi > 0, i = 1, . . . , d}, or the open hypercube


1279

κˆ C = (−1, 1)d in Rd . On κˆ we define spaces of polynomials of degree p ≥ 1 as follows: Qp = span{xˆ α : 0 ≤ αi ≤ p, 1 ≤ i ≤ d}, Pp = span{xˆ α : 0 ≤ |α| ≤ p}. To each κ ∈ Th we assign an integer pκ ≥ 1; collecting the pκ and Fκ in the vectors p = {pκ : κ ∈ Th } and F = {Fκ : κ ∈ Th }, respectively, we introduce the finite element space S p (, Th , F ) = {u ∈ L2 () : u|κ Fκ ∈ Qpκ if Fκ−1 (κ) = κˆ C and u|κ Fκ ∈ Ppκ if Fκ−1 (κ) = κˆ S ; κ ∈ Th } . We assign to Th the broken Sobolev space of composite order s = {sκ : κ ∈ Th } defined by Hs (, Th ) = {u ∈ L2 () : u|κ ∈ Hsκ (κ) for all κ ∈ Th }, equipped with the broken Sobolev norm

1/2 us,Th = u2Hsκ (κ) . κ∈Th Hs (, Th )

When sκ = s for all κ ∈ Th , we write and us,Th . An interior face of Th is defined as the (non-empty) (d −1)-dimensional interior of ∂κi ∩ ∂κj , where κi and κj are two adjacent elements of Th , not necessarily matching. A boundary face of Th is defined as the (non-empty) (d − 1)-dimensional interior of ∂κ ∩ , where κ is a boundary element of Th . We denote by int the union of all interior faces of Th . Given a face e ⊂ int , shared by the two elements κi and κj , where the indices i and j satisfy i > j , we write νe to denote the (numbering-dependent) unit normal vector which points from κi to κj ; on boundary faces we put νe = ν. Further, for v ∈ H1 (, Th ) we define the jump of v across e and the mean value of v on e, respectively, by [v] = v|∂κi ∩e − v|∂κj ∩e and v = 21 (v|∂κi ∩e + v|∂κj ∩e ). On a boundary face e ⊂ ∂κ, we set [v] = v|∂κ∩e and v = v|∂κ∩e . Finally, given a function v ∈ H1 (, Th ) and an element κ ∈ Th , we denote by vκ+ (respectively, vκ− ) the interior (respectively, exterior) trace of v defined on ∂κ (respectively, ∂κ\). Since below it will always be clear from the context which element κ in the subdivision Th the quantities vκ+ and vκ− correspond to, for the sake of notational simplicity we shall suppress the letter κ in the subscript and write, respectively, v + and v − instead. Given that κ is an element in the subdivision Th , we denote by ∂κ the union of (d − 1)dimensional open faces of κ. Let x ∈ ∂κ and suppose that νκ (x) denotes the unit outward normal vector to ∂κ at x. With these conventions, we define the inflow and outflow parts of ∂κ, respectively, by ∂− κ = {x ∈ ∂κ : b(x) · νκ (x) < 0}, ∂+ κ = {x ∈ ∂κ : b(x) · νκ (x) ≥ 0}. For simplicity of presentation, we suppose that each entry of the matrix a is piecewise continuous on Th and belongs to S 0 (, Th , F ). With minor√changes only, our results can easily be extended to the case when each entry of a belongs to S q (, Th , F ), where the composite polynomial degree vector q has√nonnegative entries; for more general a, see [23]. In the following, we write a = | a |22 , where | · |2 denotes the matrix norm subordinate to the l2 -vector norm on Rd and aκ = a|κ ; by aκ˜ we denote the arithmetic mean of the values aκ over those elements κ (including κ itself) that share a (d − 1)-dimensional face with κ.

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The hp-DGFEM approximation of (1), (3) is defined as follows: find uDG in such that

S p (, Th , F )

BDG (uDG , v) = DG (v) for all v ∈ S p (, Th , F ).

(13)

Here, the bilinear form BDG ( · , · ) is defined by BDG (w, v) = Ba (w, v) + Bb (w, v) + θBe (v, w) − Be (w, v) + Bσ (w, v) , where Ba (w, v) =

κ∈Th

Bb (w, v) =

a∇w · ∇v dx, κ

(w b · ∇v − cwv) dx +

− κ

κ∈Th

∂− κ\

Bσ (w, v) =

∂+ κ

(b · νκ ) w− v + ds ,

+ Be (w, v) =

(b · νκ ) w+ v + ds

int ∪D

(a∇w) · νe [v] ds,

int ∪D

σ [w][v] ds,

and the linear functional DG ( · ) is given by DG (v) = f v dx − κ∈Th

κ

∂− κ∩(D ∪− )

(b · νκ ) gD v + ds

θ gD ((a∇v + ) · νκ ) ds + ∂κ∩D + σ gD v ds . +

+

gN v + ds

∂κ∩N

∂κ∩D

Here, σ is defined by σ |e = Cσ

ap2 h

for e ⊂ int ∪ D ,

(14)

where Cσ is a positive constant, called the discontinuity-penalisation parameter; cf. [30]. We shall adopt the convention that edges e ⊂ int ∪ D with σ |e = 0 are omitted from the integrals appearing in the definition of Bσ (w, v) and DG (v), although we shall not highlight this explicitly in our notation; the same convention is adopted in the case of integrals where the integrand contains the factor 1/σ . Thus, in particular, the definition of the DG-norm, cf. (15) below, is meaningful even if σ |e happens to be equal to zero on certain edges e ⊂ int ∪ D , given that such edges are understood to be excluded from the region of integration.


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Selecting the parameter θ = 1 gives rise to the so-called Nonsymmetric Interior Penalty (NIP) method, while setting θ = −1 yields the Symmetric Interior Penalty (SIP) scheme; in the following we write SIP/NIP to denote the symmetric/nonsymmetric versions of the interior penalty method. While a symmetric discretisation of a symmetric differential operator seems quite natural, the NIP scheme is often preferred, especially for transport-dominated problems where the underlying discretisation matrix is nonsymmetric anyway, as it is stable for any choice of the parameter Cσ > 0; see, for example, [2], [30], [51], and Theorem 4.1 below. On the other hand, the SIP scheme is only stable when Cσ > 0 is chosen sufficiently large. In terms of accuracy, both schemes converge at the optimal rate when the error is measured in terms of the DG-norm (cf. (15) below), but the lack of adjoint consistency (see, [2]) of the NIP method leads to suboptimal convergence of the error when measured in terms of the L2 norm. In this case, the SIP scheme is still optimally convergent, while the NIP method is suboptimal by a full order; however, numerical experiments indicate that in practice the L2 norm of the error arising from the NIP scheme converges to zero at the optimal rate when the polynomial degree p is odd, cf. [30]. Thereby, in practice the loss of optimality of the NIP scheme when the error is measured in terms of the L2 norm only arises for even p. However, we showed in [25] that, for p ≥ 2, the lack of adjoint consistency of the NIP scheme leads to an even more dramatic deterioration of its convergence rate when the error is measured in terms of a certain (linear) target functional J ( · ), such as J : v → v(x)ψ(x) dx, for example, where ψ is a given weight-function: for fixed p the error measured in terms of J ( · ) behaves like O(h2p ) when the SIP scheme is employed, while for the NIP scheme, in general we only have the rate of convergence O(hp ) as h tends to zero. For related work on a posteriori error estimation for DGFEMs with interior penalty, see e.g. Becker et al. [9], [10] and Rivière & Wheeler [53]. For further perspectives on the construction and postprocessing of DGFEMs, see [12], [18]. Before embarking on the analysis of the discontinuous Galerkin method (13), we define the DG-norm | · |DG by 1 ∇w2a,κ + c0 w2κ + w+ 2∂− κ∩(D ∪− ) |w|2DG = 2 κ∈Th 1 1 (15) + w+ 2∂+ κ∩ + w+ − w− 2∂− κ\ 2 2 1 σ [w]2 ds + (a ∇w) · νe 2 ds, + σ int ∪D int ∪D where ∇w2a,κ = (a∇w, ∇w)κ , · τ , τ ⊂ ∂κ, denotes the (semi)norm induced by the (semi)inner-product (v, w)τ = τ |b · νκ |vw ds, and c0 is as defined in (4). The above definition of | · |DG represents a slight modification of the norm considered in [30]; for a > 0, b ≡ 0, (15) corresponds to the norm proposed by Baumann et al. [8], [47] and Baker et al. [5], cf. [51]. With this notation, we state the following coercivity result for the bilinear form BDG ( · , · ) over S p (, Th , F ) × S p (, Th , F ).

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Theorem 4.1. With σ defined as in (14), there exists a positive constant C, which depends only on the dimension d and the shape-regularity of Th , such that BDG (v, v) ≥ C|v|2DG for all v ∈ S p (, Th , F ), provided that the constant Cσ arising in the definition of the discontinuity penalisation parameter σ is chosen so that Cσ > 0 arbitrary when θ = 1, and Cσ ≥ Cσ with Cσ sufficiently large when θ = 1. This result is an extension of the coercivity result derived by Prudhomme et al. [51] with b ≡ 0; see also [30] for the proof in the case when θ = 1. For the case of a ≡ 0, the connection of stabilisation to upwinding has been discussed in [16]. In particular, Theorem 4.1 implies that (13) has a unique solution for any f ∈ L2 (), any gD ∈ L2 (D ) and any gN ∈ L2 (N ). Theorem 4.1 also indicates that while the NIP scheme is coercive over S p (, Th , F ) × S p (, Th , F ) for any choice of the constant Cσ > 0 arising in the definition of the discontinuity-penalisation parameter σ , the SIP scheme (corresponding to θ = −1) is only coercive if Cσ is chosen sufficiently large; see [25] for details about the minimum size of Cσ . Henceforth, we shall assume that the solution u to the boundary value problem (1), (3) is sufficiently smooth: namely, u ∈ H2 (, Th ) and that u and (a∇u) · νe are continuous across each face e ⊂ ∂κ\ that intersects the subdomain of ellipticity, a = {x ∈ : ζ a(x)ζ > 0 for all ζ ∈ Rd }. If this smoothness requirement is violated, the discretisation method has to be modified accordingly, cf. [30]. We note that under these assumptions, the following Galerkin orthogonality property holds: BDG (u − uDG , v) = 0

for all v ∈ S p (, Th , F ).

(16)

1 ()]d is such that We shall assume that b ∈ [W∞

b · ∇Th v ∈ S p (, T , F )

for all v ∈ S p (, T , F ).

(17)

Let us denote by p the orthogonal projector in L2 () onto the finite element space S p (, T , F ). We remark that this choice of projector is essential in the following a priori error analysis, in order to ensure that (u − p u, b · ∇Th v) = 0 for all v in S p (, T , F ). If, on the other hand, the scheme (13) is supplemented by streamlinediffusion stabilisation, then alternative choices of p may be employed (see [29], [59], for example); in that case, hypothesis (17) is redundant. We now decompose the global error u − uDG as u − uDG = (u − p u) + (p u − uDG ) ≡ η + ξ.

(18)

Lemma 4.2. Assume that (4) and (17) hold and let β1 |κ = c/c02 L∞ (κ) ; then there exists a positive constant C that depends only on d and the shape-regularity of Th

1283


such that the functions ξ and η defined by (18) satisfy the following inequality √ |ξ |2DG ≤ C a ∇η2κ + β12 c0 η2κ + η+ 2∂+ κ∩ + η− 2∂− κ\ κ∈Th

+

int ∪D

1 (a∇η) · νe 2 ds + σ

int ∪D

σ [η]2 ds .

The proof is given in [25]. We also need the following result concerning the approximation properties of the projector p ; for simplicity, we restrict ourselves to 1-irregular, shape-regular meshes consisting of affine equivalent d-parallelepiped elements (cf. [4], [30], and also [24] for similar results in augmented Sobolev spaces). Lemma 4.3. Suppose that κ ∈ Th is a d-parallelepiped of diameter hκ and that u|κ ∈ Hkκ (κ), kκ ≥ 0, for κ ∈ Th . Then, the following approximation results hold: u − p uL2 (κ) ≤ C |u − p u|H1 (κ) ≤ C

hsκκ pκkκ

s −1/2

uHkκ (κ) ,

hsκκ −1 k

pκ κ

u − p uL2 (∂κ) ≤ C

hκκ

k −1/2

pκ κ

uHkκ (κ) ,

s −3/2

uHkκ (κ) , −3/2

|u − p u|H1 (∂κ) ≤ C

hκκ

k −5/2

pκ κ

uHkκ (κ) ,

where 1 ≤ sκ ≤ min(pκ + 1, kκ ) and C is a constant independent of u, hκ and pκ , but dependent on the dimension d and the shape-regularity of Th . For the rest of this section, we assume that the polynomial degree vector p, with pκ ≥ 1, κ ∈ Th , has bounded local variation; i.e., there exists a constant ρ ≥ 1 such that, for any pair of elements κ and κ which share a (d − 1)-dimensional face, ρ −1 ≤ pκ /pκ ≤ ρ.

(19)

On noting that η = u − p u and combining Lemmas 4.2 and 4.3, we deduce that

2(sκ −1/2) 2(sκ −1) 2sκ h h h κ κ κ α 2(k −3/2) + β2 2k + γ 2(k −1/2) u2Hkκ (κ) , |ξ |2DG ≤ C κ κ p pκ pκ κ κ κ∈Th where α|κ = aκ˜ , β2 |κ = (β1 |κ )2 c0 2L∞ (κ) , (β1 |κ = c/(c0 )2 L∞ (κ) , cf. Lemma 4.2), γ |κ = bL∞ (κ) and C is a positive constant that depends only on d, the parameter ρ in (19) and the shape-regularity of Th . The DG-norm |η|DG of η = u − p u can be estimated directly using Lemma 4.3 to show that a bound analogous to that on |ξ |DG above holds. Hence, a bound on the discretisation error u − uDG = ξ + η in the DG-norm | · |DG is obtained via the triangle inequality (see [30] for details). Very often in practice the objective of the computation is not the approximation of the analytical solution u in a given norm, but controlling the error in an output/targetfunctional J ( · ) of the solution. Relevant examples of output functionals include the lift and drag coefficients of a body immersed into a viscous fluid, the local mean value

1284

Endre Süli

of the field, or its flux through the outflow boundary of the computational domain. Here we give a brief survey of a posteriori and a priori error bounds for general linear target functionals J ( · ) of the solution; for related work, we refer to [11], [26], [28], [31], [32], [34], [42], [58], [59], for example, and to the recent monograph of Bangerth & Rannacher [6]. We shall confine ourselves to Type I (dual-weighted) a posteriori bounds; the computationally simpler, but cruder, Type II error bounds will not be discussed here (see Giles & Süli [22]). Type I a posteriori error analysis. We proceed as in [34], [58] and begin by considering the following dual or adjoint problem: find z ∈ H2 (, Th ) such that BDG (w, z) = J (w)

for all w ∈ H2 (, Th ).

(20)

Let us assume that (20) possesses a unique solution. Clearly, the validity of this assumption depends, inter alia, on the choice of the linear functional under consideration. We shall return to this issue below; see also [25], [28], [34]. For a given linear functional J ( · ) the a posteriori error bound will be expressed in terms of the finite element residual Rint defined on κ ∈ Th by Rint |κ = (f −LuDG )|κ , which measures the extent to which uDG fails to satisfy the differential equation on the union of the elements κ in the mesh Th ; thus we refer to Rint as the internal residual. Also, since uDG only satisfies the boundary conditions approximately, the differences gD − uDG and gN − (a∇uDG ) · ν are not necessarily zero on D ∪ − and N , respectively; thus we define the boundary residuals RD and RN by RD |∂κ∩(D ∪− ) = + (gD − u+ DG )|∂κ∩(D ∪− ) and RN |∂κ∩N = (gN − (a∇uDG ) · ν)|∂κ∩N , respectively. By using the divergence theorem, the Galerkin orthogonality property (16) may be rewritten as follows: 0 = BDG (u − uDG , v) = DG (v) − BDG (uDG , v) = Rint v dx − (b · νκ ) RD v + ds κ∈Th

κ

∂− κ∩

(b · νκ ) [uDG ]v + ds + θ RD ((a∇v + ) · νκ ) ds ∂ κ\ ∂κ∩D − + σ RD v + ds + RN v + ds ∂κ∩D ∂κ∩N 1 θ + + ds [uDG ](a∇v ) · νκ + [(a∇uDG ) · νκ ]v − 2 ∂κ\ 2 + − σ [uDG ]v ds +

(21)

∂κ\

for all v ∈ S p (, Th , F ). The starting point is the following result from [25]. Theorem 4.4. Let u and uDG denote the solutions of (1), (3) and (13), respectively, and suppose that the dual solution z is defined by (20). Then, the following error

1285


representation formula holds: J (u) − J (uDG ) = E (uDG , h, p, z − zh,p ) ≡

ηκ ,

(22)

κ∈Th

where ηκ =

Rint (z − zh,p ) dx − (b · νκ ) RD (z − zh,p )+ ds κ ∂− κ∩ (b · νκ ) [uDG ](z − zh,p )+ ds + ∂− κ\ θ RD ((a∇(z − zh,p )+ ) · νκ ) ds (23) + ∂κ∩D σ RD (z − zh,p )+ ds + RN (z − zh,p )+ ds + ∂κ∩D ∂κ∩N 1 θ + + [uDG ](a∇(z − zh,p ) ) · νκ + [(a∇uDG ) · νκ ](z − zh,p ) ds − 2 ∂κ\ 2 − σ [uDG ](z − zh,p )+ ds ∂κ\

for all zh,p ∈ S p (, Th , F ). Proof. On choosing w = u − uDG in (20) and recalling the linearity of J ( · ) and the Galerkin orthogonality property (21), we deduce that J (u) − J (uDG ) = J (u − uDG ) = BDG (u − uDG , z) = BDG (u − uDG , z − zh,p ), and hence (22), with ηκ defined by (23), using the definitions of the residuals.

(24)

2

Corollary 4.5. Under the assumptions of Theorem 4.4, and with ηκ defined as in (23), the following Type I a posteriori error bound holds: |ηκ | . |J (u) − J (uDG )| ≤ E|| (uDG , h, p, z − zh,p ) ≡ (25) κ∈Th

As discussed in [6], [11], [26], [58], the local weighting terms involving the difference between the dual solution z and its projection/interpolant zh,p onto S p (, Th , F ) appearing in the Type I bound (25) contain useful information concerning the global transport of the error. Therefore, we shall retain the weighting terms involving the (unknown) dual solution z in our bound and approximate them numerically, — instead of eliminating z, as one would in the derivation of a structurally simpler, but cruder, Type II a posteriori bound. However, before proceeding any further, we need to consider more carefully the dual problem defined by (20). Let us suppose, for example,

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Endre Süli

that the aim of the computation is to approximate the (weighted) mean value of the solution u; i.e., J ( · ) ≡ Mψ ( · ), where Mψ (w) = wψ dx and ψ ∈ L2 (). When θ = −1, performing integration by parts, we find that the corresponding dual solution z = zSIP is the solution of the following mesh-dependent problem: find z such that L∗ z ≡ −∇ · (a∇z) − b · ∇z + cz = ψ z=0 (b · ν)z + (a∇z) · ν = 0

in , on D ∪ + , on N .

Thus, for θ = −1 the dual problem is well-posed for this choice of target functional. We remark that, since in this case the dual problem formed by transposing the arSIP ( · , · ) involves the formal adjoint guments in the bilinear form BDG ( · , · ) = BDG SIP ( · , · ) is said to be adjoint consistent, cf. of the partial differential operator L, BDG Arnold et al. [2]; in particular, when θ = −1 and the primal and dual solutions are sufficiently smooth, the error in the functional will be seen to exhibit an optimal order of convergence. As we shall explain below by means of a priori error analysis, the situation is dramatically different when θ = −1: then, the bilinear form BDG ( · , · ) is not adjoint consistent; this, in turn, leads to degradation of the convergence rate of the error in the computed functional J ( · ) as the finite element space S p (, Th , F ) is enriched (by reducing h or by increasing the polynomial degree vector p). Once again, we refer to [25] for technical details. A priori error bounds. We continue to use the superscripts SIP and NIP to distinguish SIP between the two methods and write BDG ( · , · ) ≡ BDG ( · , · ) when θ = −1 and NIP BDG ( · , · ) ≡ BDG ( · , · ) when θ = 1. The corresponding numerical solutions uSIP DG SIP p and uNIP DG satisfy the following problems: find uDG in S (, Th , F ) such that SIP p (uSIP BDG DG , v) = DG (v) for all v ∈ S (, Th , F );

p and find uNIP DG in S (, Th , F ) such that NIP p (uNIP BDG DG , v) = DG (v) for all v ∈ S (, Th , F ),

respectively. The starting point for the a priori error analysis is the identity (24) in the proof of Theorem 4.4. Again, using the above notation, we see that SIP SIP SIP J (u) − J (uSIP DG ) = BDG (u − uDG , z − zh,p )

when the SIP scheme is employed, while for the NIP scheme, we have NIP NIP NIP ) = BDG (u − uNIP J (u) − J (uDG DG , z − zh,p )

for all zh,p in S p (, Th , F ). Here, zSIP and zNIP are the analytical solutions to the following dual problems: find zSIP ∈ H2 (, Th ) such that SIP (w, zSIP ) = J (w) BDG

for all w ∈ H2 (, Th );

1287


and find zNIP ∈ H2 (, Th ) such that NIP (w, zNIP ) = J (w) BDG

for all w ∈ H2 (, Th ).

Hence, for all zh,p ∈ S p (, Th , F ), J (u) − J (uDG ) = BDG (u − uDG , zSIP − zh,p ) − (1 + θ)Be (zSIP , u − uNIP DG ),

(26)

NIP where uDG is either uSIP DG or uDG , depending on whether θ = −1 or θ = 1, respectively. In particular, the second term on the right-hand side of (26) is absent for the SIP scheme, i.e. when θ = −1, but it is present when the NIP scheme is employed, i.e., when θ = 1. Since this second term is of lower order than the first term in (26), it will lead to suboptimal rates of convergence as the finite element space S p (, Th , F ) is enriched in the case of θ = −1.

Theorem 4.6. Let ⊂ Rd be a bounded polyhedral domain, Th = {κ} a shaperegular subdivision of into d-parallelepipeds and p a polynomial degree vector of bounded local variation. Let (4) and (17) hold, let the entries of a be piecewise constant on Th , and u|κ ∈ Hkκ (κ), kκ ≥ 2, for κ ∈ Th , zSIP |κ ∈ Hlκ (κ), lκ ≥ 2, for κ ∈ Th ; then, the solution uDG ∈ S p (, Th , F ) of (13) satisfies the error bound |J (u) − J (uDG )| ≤ C 2

2(sκ −1)

α

κ∈Th

hκ

2(kκ −3/2)

pκ

+ β3

κ h2s κ

pκ2kκ

2(sκ −1/2)

+γ

hκ

2(kκ −1/2)

pκ

u2Hkκ (κ)

2(t −1/2) 2(t −1) κ hκ κ hκ κ h2t κ α 2(l −3/2) + β4 2l + γ 2(l −1) zSIP 2Hlκ (κ) + (1 + θ) zSIP 22,Th × pκ κ pκ κ pκ κ κ∈Th for 1 ≤ sκ ≤ min(pκ + 1, kκ ), 1 ≤ tκ ≤ min(pκ + 1, lκ ), pκ ≥ 1, κ ∈ Th , where α|κ = aκ˜ , β3 |κ = (1 + (β1 |κ )2 )c0 2L∞ (κ) , (β1 |κ = c(x)/(c0 (x))2 L∞ (κ) ), β4 |κ = (c + ∇ · b)/c0 2L∞ (κ) , γ |κ = bL∞ (κ) and C is a constant depending on the dimension d, the parameter ρ from (19) and the shape-regularity of Th . If we assume uniform orders pκ = p, sκ = s, tκ = t, kκ = k, lκ = l, where s, t, k and l are positive integers, and h = maxκ∈Th hκ , then, in the diffusion-dominated case (viz. b ≈ 0), Theorem 4.6, with θ = −1 implies that for the SIP scheme |J (u) − J (uDG )| ≤ C (hs+t−2 /pk+l−2 ) p uHk () zSIP Hl () ,

(27)

where 1 ≤ s ≤ min(p + 1, k) and 1 ≤ t ≤ min(p + 1, l). This error bound is optimal with respect to h and suboptimal in p by a full order. We note, however, that ‘order-doubling’ of the rate of convergence in |J (u) − J (uDG )| observed when the SIP scheme is employed, as expressed by (27), is lost when the NIP method is used. In the hyperbolic case (a ≡ 0), the bound in Theorem 4.6 becomes |J (u) − J (uDG )| ≤ C (hs+t−1 /pk+l−1 ) p1/2 uHk () zSIP Hl () .

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Endre Süli

This error bound is optimal in h and suboptimal in p by p1/2 (cf. also [34]). Adaptive algorithm. In the light of Theorem 4.6, we now confine ourselves to the SIP scheme (θ = −1). For a user-defined tolerance TOL, we consider the problem of designing an hp-finite element space S p (, Th , F ) such that the inequality |J (u) − J (uDG )| ≤ TOL holds, subject to the constraint that the number of degrees of freedom in S p (, Th , F ) is minimized. Following [34], we use the a posteriori error bound (25) with z replaced by a discontinuous Galerkin approximation zˆ computed on the same mesh Th as for the primal solution uDG , but with a higher degree polynomial, i.e., zˆ ∈ S pˆ (, Th , F ) , pˆ = p + p inc ; in our experiments we set p inc = 1, cf. [26], [32], [58]. Thereby, in practice we enforce the stopping criterion Eˆ|| ≡ E|| (uDG , h, p, zˆ − zh,p ) ≤ TOL .

(28)

If (28) is not satisfied, then the elements are marked for refinement/derefinement according to the size of the (approximate) error indicators |ηˆ κ |; these are defined analogously to |ηκ | in (23) with z replaced by zˆ . In our experiments we use the fixed fraction mesh refinement algorithm, with refinement and derefinement fractions set to 20% and 10%, respectively. Once an element κ ∈ Th has been flagged for refinement or derefinement, a decision must be made whether the local mesh size hκ or the local degree pκ of the approximating polynomial should be altered. The choice to perform either hrefinement/derefinement or p-refinement/derefinement is based on the local smoothness of the primal and dual solutions u and z, respectively; cf. [32], [34]. Let us first consider the case when an element has been flagged for refinement. If u or z are locally smooth, then p-refinement will be more effective than h-refinement, since the error will be expected to decay quickly within the current element κ as pκ is increased. On the other hand, if both u and z have low regularity within the element κ, then h-refinement will be performed. To ensure that the desired level of accuracy is achieved efficiently, in [34] an automatic procedure was developed for deciding when to h- or p-refine, based on the smoothness-estimation strategy proposed by Ainsworth & Senior [1]. For a review of various hp-adaptive strategies as well as descriptions of new algorithms based on Sobolev index estimation via local Legendre expansions, we refer to [31], [32]. If an element has been flagged for derefinement, then the strategy implemented here is to coarsen the mesh in low-error-regions where either the primal or dual solutions u and z, respectively, are smooth and decrease the degree of the approximating polynomial in low-error-regions when both u and z are insufficiently regular, cf. [34]. Numerical experiments. We explore the performance of the hp-adaptive strategy outlined above for the symmetric version of the interior penalty method, applied to a mixed hyperbolic-elliptic problem with discontinuous boundary data (cf. [25]). We let a = ε(x)I , where ε = 21 δ(1 − tanh((r − 1/4)(r + 1/4)/γ )), r 2 = (x − 1/2)2 + (y − 1/2)2 and δ ≥ 0 and γ > 0 are constants. Let b = (2y 2 − 4x + 1, 1 + y), c = −∇ · b and f = 0. With δ > 0 and 0 < γ 1, the diffusion parameter ε

1289


is approximately equal to δ in the circular region defined by r < 1/4, where the underlying partial differential equation is uniformly elliptic. In this example, we set δ = 0.05 and γ = 0.01; a cross-section of ε along 0 ≤ x ≤ 1, y = 1/2 is shown in Figure 1. As the value of r is increased beyond 1/4, the function ε rapidly decreases through a layer of width O(γ ); for example, when r > 0.336 we have ε < 10−15 , so from the computational point of view ε is zero to within rounding error; in this 1

0.06

Point of interest

0.8

0.04 0.6

0.4

0.02 0.2

0 0

0

0.2

0.4

0.6

(a)

0.8

1

0

0.2

0.4

0.6

0.8

1

(b)

Figure 1. (a) Profile of ε along y = 0.5, 0 ≤ x ≤ 1; (b) DGFEM approximation to the primal problem on a 129 × 129 mesh with piecewise bilinear elements (p = 1); from [25].

region, the partial differential equation undergoes a change of type becoming, in effect, hyperbolic. Thus we shall refer to the part of with r > 1/4 + O(γ ) as the hyperbolic region, while the set of points in with r ≤ 1/4 will be called the elliptic region; of course, strictly speaking, the partial differential equation is elliptic in the whole of . The characteristics associated with the hyperbolic part of the operator enter the computational domain from three sides of , namely through the vertical edges placed along x = 0 and x = 1 and the horizontal edge along y = 0; the characteristics exit through the horizontal edge along y = 1. On the union of these three faces we prescribe the following boundary condition: ⎧ ⎪ for x = 0 , 0 < y ≤ 1, ⎨1 2 u(x, y) = sin (πx) for 0 ≤ x ≤ 1 , y = 0, ⎪ ⎩ −50y 4 for x = 1 , 0 < y ≤ 1. e Figure 1 shows the numerical approximation to (1) using the SIP method on a uniform 129 × 129 uniform square mesh with p = 1. Let us suppose that the objective of the computation is to calculate the value of the analytical solution u at a certain point of interest, x = (0.43, 0.9), i.e., J (u) = u(0.43, 0.9); cf. Figure 1. The true value of the functional is given by J (u) = 0.704611313375.

1290

Endre Süli

In Table 1 we show the performance of our adaptive finite element algorithm using hp-refinement. Clearly, the computed Type I a posteriori error bound (25) is very sharp in the sense that it overestimates the true error in the computed functional by a factor of about 1–8 only, and by a factor of only 3.34 on average on the meshes that arise in the course of our adaptive hp-refinement. Table 1. History of the adaptive hp-refinement. The effectivity index is defined as the ratio of the a posteriori error bound κ |ηˆ κ | and the error |J (u) − J (uDG )|; from [25].

Nodes

Elements

Degrees of freedom

|J (u) − J (uDG )|

κ

|ηˆ κ |

Effectivity index

81

64

576

1.924e-02

3.330e-02

1.73

99

76

740

1.056e-02

1.085e-02

1.03

162

130

1451

1.006e-02

2.290e-02

2.28

241

193

2483

7.400e-04

2.385e-03

3.22

302

244

3776

3.760e-05

2.754e-04

7.32

323

262

4777

1.270e-05

1.026e-04

8.08

396

325

6916

9.896e-06

2.245e-05

2.27

487

403

9941

1.224e-06

6.466e-06

5.28

577

481

13528

4.656e-07

1.163e-06

2.50

713

601

19855

2.449e-07

2.582e-07

1.05

960

820

31019

1.574e-08

3.202e-08

2.03

1313

1132

47406

6.531e-10

2.154e-09

3.30

Figure 2 shows |J (u) − J (uDG )|, using both h- and hp-refinement, against the square-root of the number of degrees of freedom on a linear-log scale. After the initial transient, the error in the computed functional using hp-refinement is seen to become (on average) a straight line, which indicates exponential convergence of J (uDG ) to J (u); this occurs since zSIP is a real analytic function in the regions of the computational domain where u is not smooth and vice versa. Figure 2 also demonstrates the superiority of the adaptive hp-refinement strategy over the standard adaptive h-refinement algorithm when TOL 10−3 . On the final mesh the error between J (u) and J (uDG ) using hp-refinement is over 4 orders of magnitude smaller than the corresponding quantity when h-refinement is used alone.

1291


|J (u) − J (uDG )|

10

10

10

hp Refinement h Refinement

2

4

6

8

10

0

50

100

150

200

sqrt(Degrees of freedom) Figure 2. Comparison between h- and hp-adaptive mesh refinement; from [25].

Figure 3 depicts the primal mesh after 11 adaptive mesh refinement steps. We display the h-mesh alone, as well as the corresponding distribution of the polynomial degree on this mesh and the percentage of elements with that degree. We see that some h-refinement of the primal mesh has taken place in the region of the computational

00 11 00 11 00 11 00 11 0000 1111 000 111 000 111 000 111 0000 1111 0000 1111 0000 000 000 000 0000 0000 00 11 00 11 00 11 00 11 1111 111 111 111 1111 1111 11 00 1 0 1 0 1 0 11 00 0 1 0 1 0000 000 000 000 0000 0000 1111 111 111 111 1111 1111 00 11 0 1 0 1 11 00 1 0 000 111 1 0 00 11 00 11 00 11 0 1 0 1 00 11 0 1 0 1 00 11 0 1 00 11 0 1 00 11 0 1 0 1 0 1 000 111 0000 1111 000 111 000 111 1 11 0 00 11 10101000 1 0 1110011 00 11 0 1 000 111 00 00 11 00 11 0000 1111 0000 1111 0 1 0 1 000 111 00 11 11001 101 11001 011 1 0010011 0 1 00 11 0 1 000 111 000 111 00 11 0 0 0 1 00 11 0 1 000 111 00 11 0 0 0 1 0 1 0 1 0000 1111 000 111 00 11 0 1 0 1 0 11 00 1 0 00 11 000 111 0000 1111 000 111 000 111 11 00 11 00 1 0 1 0 1 0 11 00 1 0 000 111 00 11 00 11 0 1 0 1 0 1 000 111 0010 1 11 01 0 1 00 11 0 1 110 00 1 11 00 0 1 0 1 00 11 0 1 000 111 0 1 0000 1111 000 111 000 111 00 11 000 111 00 11 0 1 0 0 1 00 11 0 1 0 1 000 111 0000 1111 000 111 000 111 110 11 00 1 0 1 000 111 00 11 0 00 11 0 1 00 11 0 1 0 1 00 11 0 1 1 0 00 11 0 1 11 00 11 00 010 10 0 10 111 11 00 10 0 1 1 0 00 11 00011 11 0 101 0 1 111 1111 000 0000111 1 0 11 00 11 00 1 0 1 0 1 000 00 11 1 0 11 00 1 0 11 00 000 111 000 000 111 00 11 0 1 00 11 0 0 1 11 00 11 00 1 0 11 00 1 0 11 00 1 0 1 0 1 0 1 0 1 0 11 00 00 0 1 10100110100 00011 111 00 00 000111 111 000 000 111 00 11 1 010111 011 1 0 1 0 00 11 00 11 11 00 11 00 11 00 00 11 0 1 1 00 11 0 1 100 10 0 111 00 00 11 111 00011 00 111 00011 00 11 00 11 00 1 0 111 00 0 11 011 1 0 101000 0 00 11 0 1 00 11 0 1 00 11 0 1 0000 000 1111 111 11 1 0 1 0 1 0 000 111 00 11 00 11 0 1 00 11 0 1 000 111 0 1 00 11 0 1 00 11 0 1 000 111 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 0000 000 00 11 00 11 00 11 000 00 11 00 11 000 00 11 00 1111 111 111 111 11 11 00 1 0 1 0 1 0 11 00 1 0 000 111 00 11 0 1 00 11 00 11 0 1 0 1 0 1 00 11 01 1 000 00 0111 1 010110101 11 1111 00 111 0 100 0 11 0 00 11 0 1 000 111 00 11 00 111 00 11 00 11 000 11 11 00 11 00 1 0 1 0 00 11 00 11 000 00 11 00 0 1 00 11 0 1 0 1 0 1 00 00 00 111 11 11 00 11 00 1 0 0111 1 11 10 0 111 111 0 101001100 0011 11 00 000 00 11 00 11 00 11 0 00 11 000 1 00 11 111 00 11 00 11 00 11 001 11 0 1 0 1 0 1 0 1 00 11 00 11 00 11 111 000 0011 11 00 00 11 0 1 0 1 00 11 0 1 00 11 1 0 1 0 1 0 1 0 111 000 00 11 0 1 0 1 010101011 1 00 0 1 00 11 00 111 11 000 11 0011 00 0 1 0 1 00 11 0 1 0 1 00 11 1 0 1 0 000 111 00 11 00 11 00 11 000 111 000 0000 1111 000 000 111 00 00 00 01111 00 11 000 111 00 11 00011 111 000111 11 00 11 000 111 1111 0000111 111 00011 0011 00 111 00011 00 11 1111 111 111 000 111 00 11 00 11 000 111 0000 000 00 11 00 11 000 00 00011 111 0011 00111 0001111 0000111 00011 0011 00 00011 111 00 11 0 1 00 11 00 11 0 1 00 11 00 11 0 1 00 11 00 11 00 11 11 00 11 00 00 100 0 00 11 00 11 00 11 11 0 1 0 1 00 11 1 0 11 11 00 00 11 00 11 0 1 0 1 1 0 1 0 00 11 11 0 1 0 00 11 0 1 0 1 00 11 00 11 00 11 0 0 1 00 11 00 11 0 1 000 111 0 1 00 11 00 11 00 11 11 00 1 0 1 0 1 0 1 0 00 11 11 0 1 00 11 00 11 0 1 0 1 00 1 0 1 0 00 11 00 11 00 11 0 1 000 111 0 1 00 11 00 11 0 1 1 0 11 00 11 00 00 11 00 11 00 11 0 1 0 1 0 1 0 1 0 1 0 1 00 11 00 11 0011 11 111 000 0 1 00 11 00 11 00 11 0 1 01 1 0 0 1 11 00 11 00 11 00 1 0 1 0 1 0 1 0 00 11 00 11 00 11 0 1 00 0 1 0 1 0 1 00 11 00 11 0 1 00 11 00 11 0 1 00 11 111 000 00 11 1 0 1 0 11 00 0 1 00 11 0 1 00 11 00 11 00 11 0 1 00 11 0 1 00 11 0 1 00 11 00 11 0 1 11 00 11 00 1 0 1 0 00 11 0 1 0 1 0 1 0 1 00 11 0 1 0 1 0 1 11 00 11 0 00 11 00 11 00 00 11 0 1 00 11 0 1 0011 11 00 11 0 11 1 00 0 1 00 11 0 1 00 11 0 1 0 1 0 1 00 11 0 1 0 1 01 1 0 0 0 11 1 000 111 0 1 0 1 00 11 00 00 11 1 0 11 00 11 00 1 0 11 00 1 0 00 11 0 1 0 1 00 11 0011 11 00 11 01 1 1 0 1 0 1 0 1 0 11 1 0 1 0 1 0 11 00 1 0 0 0 000 111 0 1 0 00 11 00 1 0 11 00 11 0 0 00 1 01 1 111 0 1 00 11 000 00 11 0 1 0 1 0 1 0 1 111 00 11 000 00 11 0 1 0 1 0 1 0 1 0 1 00 11 00 11 00 11 00 11 000 111 1 0 10 0 000 00 11 00 11 00 11 00 11 111 1 0 1 0 1 1 0 11 00 1 0 000 00 11 00 11 00 11 00 11 111 1 0 1 0 11 00 1 0 000 111 00 11 00 11 00 11 111 000 00 11 00 11 00 11 0 1 0 1 00 11 0 1 00 11 111 000 00 11 00 11 01 1 0 1 00 11 0 1 00 11 0 11 00 0 1 00 11 0 1 0 1 0 1 11 00 1 0 0 1 0 1 0 1 0 1 1 1 0 0 0 1 0 1 0 00 11 00 11 00 00 11 00 11 0 1 00 11 0 1 00 11 000 111 0 1 0 1 00 11 0011 11 00 11 00 11 00 11 00 11 0 1 00 11 111 000 0 1 0 1 00 0 1 00 00 11 000 111 0 1 000 111 00 11 00 11 00 11 0011 11 0 1 00 11 0 1 00 11 0 11 1 00 11 00 11 00 11 00 11 000 111 000 111 00 11 0 1 0 1 0 1 0 1 00 11 0 1 00 11 00 11 00 11 00 11 0 1 000 111 0 1 111 000 00 11 00 11 00 11 00 11 000 111 111 000 00 11 0 1 0 1 0 1 000 111 000 111 00 11 00 11 00 11 00 11 0 0 1 0 0 1 0 1 0 1 00 11 0 1 00 11 001 11 00 11 001 11 01 1 0 1 0 11 1 0 1 00 11 0 1 000 111 111 000 0 1 0 1 000 111 00 11 00 11 00 11 00 11 000 00 11 00 11 00 11 00 11 0 0 1 0 1 0 1 00 00 11 0 1 000 111 111 001 11 00 11 001 11 0 01 1 0 0 11 1 01 1 0 1 00 11 0 1 00 11 00 11 00 11 00 11 00 11 00 11 00 11 0 1 000 111 00 11 000 111 000 111 00 11 0 0 1 0 1 00 0 1 0 000 00 11 00 11 00 11 00 11 000 111 111 000 00 11 00 11 0 111 1 001 11 00 11 00 11 0 0 1 0 00 11 0 1 000 00 11 00111 11 00 11 00 11 000 111 111 000 00 11 11 00 00 11 001 11 0 1 01 1 01 1 0 111 00 11 000 11 0 1 011 1 000 111 00 11 000 00 11 00 00 11 000 111 00 11 0 1 00 11 00 11 000 111 00 11 111 000 00 11 00 11 00 11 000 111 0 1 00 11 00 11 00 11 000 111 01 1 0 1 011 1 0 1 0 1 000 111 00 11 111 000 00 11 00 11 00 11 00 11 00 00 11 00 11 000 111 00 11 0 1 0 0 1 0 1 0 00 11 00 11 000 111 0 1 01 1 0 1 00 11 0 1 0 1 00 11 00 11 000 111 00 11 0 1 0 1 0 1 0 0 1 000 111 00 11 00 11 001 11 0 01 1 01 0 1 0 1 111 11 00 1 0 1 0 1 0 1 0 1 0 00 11 00 11 000 111 00 11 00 11 000 001 11 01 01 01 01 0

11111 00000 00000 11111 7(14%) 00000 11111 00000 11111 000 111 000 111 00000 11111 111 000 000 111 6(35%) 000 111 000 111 000 111 000 111 5(36%)

111 000 000 111 000 111 000 111 00000 11111 111 000 00000 11111 4(10%) 00000 11111 00000 11111 000 111 000 111 00000 11111 000 111 000 111 111 000 3(3%) 000 111 000 111 000 111 2(1%) 1(1%)

Figure 3. h- and hp-meshes after 11 refinements, with 1313 nodes, 1132 elements and 47406 degrees of freedom: here, |J (u) − J (uDG )| = 6.531 × 10−10 ; from [25].

domain upstream of the point of interest, as well as in the circular region where the underlying partial differential equation changes type. Once the h-mesh has adequately captured the structure of the primal and dual solutions, the hp-adaptive algorithm performs p-refinement elsewhere in the domain of dependence of the point of interest.

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5. High-dimensional transport-diffusion problems We conclude by giving some pointers to recent results on stabilised sparse finite element methods for high-dimensional partial differential equations (1) with nonnegative characteristic form (cf. [57]). Such high-dimensional equations arise from a number of important applications in physics, chemistry, biology and finance. The origins of sparse tensor-product constructions and hyperbolic cross spaces can be traced back to Babenko [3] and Smolyak [56]; we refer to the papers of Temlyakov [61], DeVore, Konyagin & Temlyakov [20] for the study of high-dimensional approximation problems, to the works of Wasilkowski & Wo´zniakowski [62] and Novak & Ritter [46] for high-dimensional integration problems and associated complexity questions, to the paper of Zenger [63] for an early contribution to the numerical solution of highdimensional elliptic equations, to the articles by von Petersdorff & Schwab [50] and Hoang & Schwab [27] for the analysis of sparse-grid methods for high-dimensional elliptic multiscale problems and parabolic equations, respectively, and to the recent survey article of Bungartz & Griebel [17]. Suppose that = (0, 1)d , N = ∅ and gD = 0 in (3), and that the operator L in (1) has constant coefficients. In the simplest case, the construction of the finite element space Vˆ0L ⊂ H begins by taking the tensor product of d copies of a finite element space of univariate hierarchical continuous piecewise linear functions (p = 1) on a uniform mesh of size hL = 2−L , L ≥ 1. The resulting tensor-product space V0L has dimension dim(V0L ) = O(h−d L ). Clearly, the use of this space would lead to exponential growth of computational complexity for fixed hL , as d increases. Thus, the idea is to reduce the complexity of the computation for large d by sparsifying the space V0L ; the resulting sparse finite element space is denoted Vˆ0L and has only d−1 ) degrees of freedom. The relevant result from [57], dim(Vˆ0L ) = O(h−1 L | log hL | stated in the theorem below, is that, with a careful choice of the streamline-diffusion stabilisation parameter δL and assuming that u ∈ H 2 () ∩ H, where H 2 () = {v : D α v ∈ L2 (), |α|∞ ≤ 2} is the space of functions with L2 -bounded mixed second derivatives, one can ensure that this reduction of computational complexity is achieved at essentially no loss in accuracy in the streamline-diffusion finite element method compared to the case when the full tensor-product space V0L is used instead of the sparse space Vˆ0L . Theorem 5.1. Let f ∈ L2 (), c > 0 and u ∈ H 2 () ∩ H. Then, the following bound holds for the error u − uSD between the analytical solution u of (8) and its sparse finite element approximation uSD ∈ Vˆ0L , with L ≥ 1 and h = hL = 2−L : |u − uSD |2SD

|a| d|b| 2 4 2(d−1) ≤ C(u) |a|hL + hL | log2 hL | max , ,c h2L hL | log2 hL |d−1


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with the streamline-diffusion parameter δL defined by the formula

h2L hL | log2 hL |d−1 1 δL := Kδ min , , , d|b| c |a| with Kδ ∈ R>0 a constant, independent of hL and d, and C(u) = Const.u2H 2 () where Const. is a positive constant independent of the discretisation parameter hL . We refer to [57] for further technical details, including the proof of this result.

6. Concluding remarks We surveyed continuous stabilised and discontinuous Galerkin finite element methods for the numerical solution of second-order partial differential equations with nonnegative characteristic form. We stated a priori and residual-based a posteriori error bounds, and in the case of the discontinuous Galerkin method we showed how the a posteriori bound may be used to drive an hp-adaptive finite element algorithm. We also commented on the use of sparse stabilised finite element methods for highdimensional transport-dominated diffusion equations: stochastic analysis and kinetic theory are particularly fertile sources of Fokker–Planck equations of this kind [41]. The numerical solution of high-dimensional partial differential equations has been an active area of research in recent years [17], though the bulk of the research has been confined to self-adjoint elliptic and parabolic equations. As we have briefly indicated, extensions of these results to the, vastly richer, class of partial differential equations with nonnegative characteristic form are feasible, and we expect that activities in this direction will continue to flourish. Acknowledgments. I am grateful to Franco Brezzi, Bernardo Cockburn, Kathryn Gillow, Paul Houston, Donatella Marini, Rolf Rannacher and Christoph Schwab for numerous stimulating discussions on the ideas presented in this paper. The computational experiments in Section 4 were performed by Paul Houston.

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Ergodic control of diffusion processes Vivek S. Borkar∗

Abstract. Results concerning existence and characterization of optimal controls for ergodic control of nondegenerate diffusion processes are described. Extensions to the general ‘controlled martingale problem’ are indicated, which cover in particular degenerate diffusions and some infinite dimensional problems. In conclusion, some related problems and open issues are discussed. Mathematics Subject Classification (2000). Primary 93E20; Secondary 60H20. Keywords. Controlled diffusions, ergodic control, stationary Markov control, controlled martingale problems, dynamic programming.

1. Introduction Ergodic or ‘long run average’ control of Markov processes considers the minimization of a time-averaged cost over admissible controls. This stands apart from the usual ‘integral’ cost criteria such as finite horizon or infinite horizon discounted cost criteria because neither the dynamic programming principle nor the usual ‘tightness’ arguments for existence of optima common to these set-ups carry over easily to the ergodic problem. Thus entirely new proof techniques have to be employed. The situation gets more complicated for continuous time continuous state space processes, of which diffusion processes are a prime example, because of the additional technicalities involved. This article describes first the reasonably well-understood case of non-degenerate diffusions, and then the partly resolved case of the more general ‘controlled martingale problem’which covers degenerate diffusions and partially observed diffusions, among others. An extended account of this topic will appear in [2].

2. Ergodic control of non-degenerate diffusions 2.1. Preliminaries. The d-dimensional (d ≥ 1) controlled diffusion process X( · ) = [X1 ( · ), . . . , Xd ( · )]T is described by the stochastic differential equation t t m(X(s), u(s)) ds + σ (X(s)) dW (s), (1) X(t) = X0 + 0

0

∗ The work is supported in part by a grant from the Department of Science and Technology, Government of

India Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Vivek S. Borkar

for t ≥ 0. Here: 1. for a compact metric ‘control space’ U , m( ·, · ) = [m1 ( ·, · ), . . . , md ( ·, · )]T : Rd × U → Rd is continuous and Lipschitz in the first argument uniformly with respect to the second, 2. σ ( · ) = [[σij ( · )]]1≤i,j ≤d : Rd → Rd×d is Lipschitz, 3. X0 is an Rd -valued random variable with a prescribed law π0 , 4. W ( · ) = [W1 ( · ), . . . , Wd ( · )]T is a d-dimensional standard Brownian motion independent of X0 , 5. u( · ) : R+ → U is the ‘control process’ with measurable paths, satisfying the non-anticipativity condition: for t > s ≥ 0, W (t) − W (s) is independent of {X0 , W (y), u(y), y ≤ s}. (In other words, u( · ) does not anticipate the future increments of W ( · ).) This class of u( · ) is referred to as admissible controls. It is known that without loss of generality, one may take these to be adapted to the natural filtration of X( · ), given by Ft = the completion of s>t σ (X(y), y ≤ s). We shall say that it is a stationary Markov control if in addition u(t) = v(X(t)), t ≥ 0, for a measurable v : Rd → U . By a standard abuse of terminology, we identify this control with the map v( · ). We shall say that (1) is non-degenerate if the least eigenvalue of σ ( · )σ T ( · ) is uniformly bounded away from zero, degenerate otherwise. We use the ‘weak solution’ framework, i.e., only the law of the pair (X( · ), u( · )) is prescribed and ‘uniqueness’ is interpreted as uniqueness in law. For this section, we assume non-degeneracy. This in particular implies existence of a unique strong solution for stationary Markov controls. We shall also need the relaxation of the notion of control process u( · ) above to that of a relaxed control process. That is, we assume that U = P (U0 ), the space of probability measures on U0 with Prohorov topology, where U0 is compact metrizable (whence so is U ) and mi ( ·, · ), 1 ≤ i ≤ d, are of the form mi (x, u) = mi (x, y)u(dy), 1 ≤ i ≤ d, for some mi : Rd × U0 → R that are continuous and Lipschitz in the first argument uniformly w.r.t. the second. We may write u(t) = u(t, dy) to underscore the fact that it is a measure-valued process. Likewise for stationary Markov controls, write v( · ) = v( ·, dy). Then the original notion of U0 -valued control u0 ( · ) (say) corresponds to u(t, dy) = δu0 (t) (dy), the Dirac measure at u0 (t), for all t. We call such controls as precise controls. Precise stationary Markov controls may be defined accordingly. The objective of ergodic control is to minimize 1 T lim sup E[k(X(t), u(t))] dt (2) T →∞ T 0

Ergodic control of diffusion processes

(the average version), or to a.s. minimize 1 T k(X(t), u(t)) dt lim sup T →∞ T 0

1301

(3)

(the ‘almost sure’ version). Here k : Rd × U → R is continuous. In view of our relaxed control framework, we take it to be of the form k(x, u) = k(x, y)u(dy) for a continuous k : Rd × U0 → R. This cost criterion is popular in applications where transients are fast, hence negligible, and one is choosing essentially from among the attainable ‘steady states’. As mentioned above, we consider the non-degenerate case first. Most of the results presented in the remainder of this section have been established for bounded coefficients in the original sources, but the extension to the Lipschitz coefficients (implying linear growth) is not difficult and appears in [2]. One usually assumes (and we do) that k is bounded from below. 2.2. Existence results. Let v( · ) be a stationary Markov control such that the corresponding X( · ) is positive recurrent and therefore has a unique stationary distribution ηv ∈ P (Rd ). Define the corresponding ergodic occupation measure as μv (dx, dy) = ηv (dx)v(x, dy). Costs (2), (3) will then equal (‘a.s.’ in the latter case) kdμv . A key result is: Theorem 2.1 ([18]). The set G = {μv : v( · ) stationary Markov} is closed convex in total variation norm topology, with its extreme points corresponding to precise stationary Markov controls. We can say much more: define the empirical measures {νt } by: 1 t f dνt = f (X(s), y)u(s, dy) ds, f ∈ Cb (Rd × U0 ), t > 0. t 0 Let R = Rd ∪ {∞} = the one point compactification of Rd and view νt as a random variable in P (R × U0 ) that assigns zero mass to {∞} × U0 . Theorem 2.2 ([16]). As t → ∞, almost surely νt → {ν : ν(A) = aν (A ∩ ({∞} × U0 )) + (1 − a)ν (A ∩ (Rd × U0 )) for all A Borel in R × U0 , with a ∈ [0, 1], ν ∈ P ({∞} × U0 ), ν ∈ G}. There are two important special cases for which Theorem 2.1 allows us to reduce the control problem to the infinite dimensional linear programming problem of minimizing kdμ over G and thereby deduce the existence of an optimal precise stationary Markov control for the ‘a.s.’ version of the ergodic control problem [16]: 1. under a suitable ‘stability condition’ (such as a convenient ‘stochastic Liapunov condition’) that ensures compactness of G and a.s. tightness of {νt }, or,

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2. under a condition that penalizes escape of probability mass to infinity, such as the ‘near-monotonicity condition’: lim inf min k(x, u) > β,

||x||→∞ u

where β = the optimal cost. The latter condition is often satisfied in practice. The ‘average’ version of the ergodic cost can be handled similarly, using the average empirical measures {νt } defined via 1 t f dνt = E f (X(s), y)u(s, dy) ds, f ∈ Cb (Rd × U0 ), t > 0, t 0 in place of {νt }. 2.3. Dynamic programming. The standard approach to dynamic programming for ergodic control, inherited from earlier developments in the discrete time and state problems, is to treat it as a limiting case of the infinite horizon discounted cost problem as the discount vanishes. Hence we begin with the infinite horizon discounted cost ∞ −αt e k(X(t), u(t)) dt , E 0

where α > 0 is the discount factor. Define 1 Lf (x, u) = ∇f (x), m(x, u) + tr σ (x)σ T (x)∇ 2 f (x) 2 for f ∈ C 2 (Rd ). We may write Lu f (x) for Lf (u, x), treating u as a parameter. The Hamilton–Jacobi–Bellman (HJB) equation for the ‘value function’ ∞ α −αt V (x) = inf E e k(X(t), u(t)) dt|X(0) = x 0

(where the infimum is over all admissible controls) can be arrived at by standard dynamic programming heuristic and is min(k(x, u) − αV α (x) + LV α (x, u)) = 0 u

on the whole space. For k bounded from below, V α is its least solution in C 2 (Rd ). Define V α = V α − V α (0). Then V α satisfies min(k(x, u) − αV α (x) − αV α (0) + LV α (x, u)) = 0. u

(4)

Under suitable technical conditions (such as near-monotonicity or stability conditions mentioned above) one can show that as α ↓ 0, V α ( · ) and αV α (0) converge along

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a subsequence to some V , β in an appropriate Sobolev space and R, respectively. Letting α ↓ 0 along this subsequence in (4), these are seen to satisfy min(k(x, u) − β + LV (x, u)) = 0. u

This is the HJB equation of ergodic control. One can show uniqueness of β as being the optimal ergodic cost and of V up to an additive scalar in an appropriate function class depending on the set of assumptions one is working with. A verification theorem holds, i.e., the optimal stationary Markov control v( · ) is characterized by the condition v(x) ∈ Argmin (k(x, ·) + ∇V (x), m(x, ·) ) , a.e. See [6], [17]. Note that the minimum will be attained in particular at a precise stationary Markov control, establishing the existence of an optimal precise stationary Markov control. One also has the following stochastic representations for the ergodic value function V (modulo an additive constant):

τr V (x) = lim inf E (k(X(s), u(s)) − β) ds|X(0) = x , r↓0

0

where τr = min{t > 0 : ||X(t)|| = r} for r > 0 [17] and the infimum is over all admissible controls. Alternatively, τ

V (x) = inf inf E (k(X(s), u(s)) − β) ds|X(0) = x , τ

0

where the inner infimum is over all bounded stopping times w.r.t. the natural filtration {Ft } of X( · ), and the outer infimum is over all {Ft }-adapted controls [21].

3. Controlled martingale problems 3.1. Preliminaries. Such explicit results are not as forthcoming in the more general scenario we discuss next. We shall denote by E the Polish space that will serve as the state space of the controlled Markov process X( · ), and by U0 the compact metric ‘control’ space. U will denote the space of measurable maps [0, ∞) → U = P (U0 ) with the coarsest topology that renders continuous each of the maps T μ( · ) = μ( ·, du) ∈ U → g(t) h(u)μ(t, du) dt, 0

U0

for all T > 0, g ∈ L2 [0, T ], h ∈ Cb (U0 ). This is compact metrizable (see, e.g., [9]). The control process u( · ) can then be viewed as a U-valued random variable. def For {fk }, f ∈ B(E) = the space of bounded measurable maps E → R, say that bp

fk −−→ f (where ‘bp’ stands for ‘bounded pointwise’) if supx,k |fk (x)| < ∞ and

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Vivek S. Borkar bp

fk (x) → f (x) for all x. Q ⊂ B(E) is bp-closed if fk ∈ Q for all k and fk −−→ f together imply f ∈ Q. For Q ⊂ B(E), define bp-closure(Q) = the smallest bp-closed subset of B(E) containing Q. Let A be an operator with domain D(A) ⊂ Cb (E) and range R(A) ⊂ Cb (E×U0 ). Let ν ∈ P (E). Definition 3.1. An E × U -valued process (X( · ), π( · ) = π( ·, du)) defined on a probability space ( , F , P ) is said to be a solution to the relaxed controlled martingale problem for (A, ν) with respect to a filtration {Ft , t ≥ 0} if: • (X( · ), π( · )) is {Ft }-progressive; • L(X(0)) = ν; • for f ∈ D(A), f (X(t)) −

t Af (X(s), u)π(s, du) ds, 0

t ≥ 0,

(5)

U0

is an {Ft }-martingale. We omit explicit mention of {Ft } or ν when they are apparent from the context. The operator A is assumed to satisfy the following conditions: 1. (C1) There exists a countable subset {gk } ⊂ D(A) such that {(g, Ag) : g ∈ D(A)} ⊂ bp-closure({(gk , Agk ) : k ≥ 1}). 2. (C2) D(A) is an algebra that separates points in E and contains constant functions. Also, A1 = 0, where 1 is the constant function identically equal to 1. 3. (C3) For each u ∈ U0 , let Au f ( · ) = Af ( ·, u). Then there exists an r.c.l.l. solution to the martingale problem for (Au , δx ) for all u ∈ U0 , x ∈ E. For example, the following can be shown to fit this framework: 1. X( · ) as in (1) with or without the non-degeneracy condition. 2. An important instance of the above is the ‘separated control problem’for control of diffusions with partial observations, which we describe in some detail next. Append to (1) the ‘observation equation’ t h(X(s)) ds + W (t), Y (t) = 0

where h : Rd → Rs (s ≥ 1) is a Lipschitz observation map and W ( · ) is an s-dimensional standard Brownian motion independent of (X0 , W ( · )), representing the (integrated) observation noise. The control u( · ) is ideally

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required to be adapted to the natural filtration of Y ( · ), but a standard relaxation allows for somewhat more general ‘wide sense admissible’ controls. These require merely that under a locally (in time) absolutely continuous change of measure that retains (1) but renders Y ( · ) itself an s-dimensional standard Brownian motion independent of (X0 , W ( · )), the future increments Y (t + ·) − Y (t) should be independent of {X0 , W ( · ), u(s), Y (s), s ≤ t} for all t > 0. The correct state variable for this problem (to be precise, one choice thereof) turns out to be the P (Rd )-valued process {μt } of regular conditional laws of X(t) given {Y (s), u(s), s ≤ t} for t ≥ 0. This evolves according to the equations of nonlinear filtering: t t μs (Lu(s) f ) ds + μs (f h) − μs (f )μs (h), d Yˆ (s) μt (f ) = μ0 (f ) + 0

0

(6) for f ∈ Cb2 (Rd ), where we follow the notation ν(f ) = f dν. The products in the integrand of thestochastic integral in (6) are componentwise, and the t process Yˆ (t) = Y (t)− 0 μs (h)ds, t ≥ 0, is the so called ‘innovations process’ which is an s-dimensional standard Brownian motion that generates the same natural filtration as Y ( · ) [1]. The well-posedness of (6) can be established under additional regularity conditions on h [24]. In terms of {μt }, the ergodic cost can be rewritten as 1 t lim sup E[μs (k( ·, u(s)))] ds. t→∞ t 0 The P (Rd )-valued controlled Markov process {μt } with this cost functional can be shown to fit the above framework. This is called the ‘separated control problem’ because it separates the issues of estimation and control. 3. Certain Hilbert-space valued controlled stochastic evolution equations can also be shown to fit the above framework [7]. 3.2. The control problem. Let k : E × U0 → [0, ∞] be a continuous running cost function. The ergodic control problem is to minimize the ergodic cost 1 t lim sup E k(X(s), u)π(s, du)) ds. (7) t→∞ t 0 U0 We assume that the set of laws of (X( · ), π( · )) for which this is finite is nonempty. For a stationary (X( · ), π( · )), define the associated ergodic occupation measure ϕ ∈ P (E × U0 ) by: f (X(t), u)π(t, du) . f (x, u)ϕ(dxdu) = E

U0

Note that (7) then becomes kdϕ. Let G denote the set of all ergodic occupation measures. From [7], we then have (see [27], [33], [34] for related results):

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Theorem 3.2. G is closed convex and is characterized as G = μ ∈ P (E × U0 ) : Af dμ = 0 for all f ∈ D(A) . In particular, for each μ ∈ G, there exists a stationary pair (X( · ), π( · )) whose marginal at each time is μ. Furthermore, π( · ) may be taken to be stationary Markov. This can be made a starting point for existence results in specific cases. For example, for degenerate diffusions and the separated control problem for partially observed diffusions, somewhat stronger variants of the ‘stability’ and ‘near-monotonicity’ conditions described earlier suffice for the existence of an optimal stationary pair (X( · ), π( · )). By considering the ergodic decomposition thereof, ‘stationary’ here may be improved to ‘ergodic’ [7]. Also, in view of the above theorem, the control therein may be taken to be stationary Markov. This, however, does not imply that the process X( · ) itself is time-homogeneous Markov, or even Markov. To establish the existence of an optimal Markov solution, we assume the following: For a fixed initial law ν of X0 , the attainable laws of (X( · ), π( · )) form a tight set M(ν) ⊂ P (D([0, ∞); E) × U). Simple sufficient conditions for this can be given in specific cases mentioned above. An immediate consequence of this is that M(ν) is in fact a compact convex set. Consider the equivalence relation on M(ν) that equates two laws when the corresponding one dimensional marginals agree for a.e. t. The set of equivalence classes, called the ‘marginal classes’, then forms a convex compact set in the quotient topology. Theorem 3.3. Every representative of an extremal marginal class corresponds to a Markov process. This is proved for degenerate diffusions in [10] and for the separated control problem in [20], but the same arguments carry over to the general case. This can be combined with the above to deduce the existence of an optimal pair (X( · ), π( · )) such that π( · ) is stationary Markov and X( · ) Markov, though not necessarily timehomogeneous Markov [7]. Also, (X( · ), π( · )) need not be stationary. Our experience with the non-degenerate case, however, suggests the existence of a stationary ergodic time-homogeneous Markov solution that is optimal. Under additional technical conditions, such a result has been proved in [8] by stretching the ‘vanishing discount’ argument, but there is scope for improvement. As for dynamic programming, scattered results are available in specific cases. The degenerate problem has been approached in the viscosity solution framework [3], [4], [5]. For the separated control problem under partial observations, a martingale


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dynamic programming principle has been derived [12], [13]. Dualizing the linear programme above yields the following dual linear programme that can be interpreted as ‘dynamic programming inequalities’ [7]: Maximize z ∈ R subject to Lf (x, u) + k(x, u) ≥ z, for all x ∈ E, u ∈ U0 , f ∈ D(L).

4. Some related problems and open issues 1. ‘Ergodic control with constraints’ seeks to minimize one ergodic cost functional while imposing bounds on one or more additional ergodic cost functionals. In the linear programming formulation alluded to above, this amounts to a few additional constraints. Existence of optimal precise stationary Markov controls has been proved in the non-degenerate case under suitable stability or near-monotonicity hypotheses [11], [18]. A Lagrange multiplier formulation can be used to aggregate the costs into a single cost. 2. We did not include control in the diffusion matrix σ ( · ). The reason for this is that, for stationary Markov controls u( · ) = v(X( · )), one is in general obliged to consider at best measurable v( · ). For a merely measurable diffusion matrix, even in the non-degenerate case only the existence of a weak solution is available, the uniqueness may not hold [26] (except in one and two dimensions – see [35], pp. 192–194). It may, however, be possible to work with ‘the set of all weak solutions’ in place of ‘the’ solution, but this is not very appealing unless one has a good selection criterion that prescribes a unique choice from among the many. 3. Singularly perturbed ergodic control concerns ergodic control of diffusions wherein some components move on a much faster time scale, characterized by a perturbation parameter > 0. One can show that as ↓ 0, the slower components satisfy an ‘averaged’dynamics wherein the coefficients in their dynamics are averaged over the stationary distribution of the fast components when the latter is derived by ‘freezing’ the slower components to constant values. The ergodic control problem for this limiting case is then a valid approximation for the original problem for small > 0. See [15] for a precise statement and proofs. 4. We have not considered several related problems with a similar flavor, such as ergodic control of reflected [14] or switching diffusions [23], [30], ergodic impulse control [32], singular ergodic control [31], and stochastic differential games with ergodic payoffs [19]. The latter in particular are also of interest in risk-sensitive control problems on infinite time horizon, which effectively get converted to two person zero sum stochastic differential games with ergodic payoffs after the celebrated ‘log-transform’ of the value function [22]. 5. We have also not addressed the computational issues here. Two major strands therein are Markov chain approximations [28] and approximations of the infinite dimensional linear programmes [25].

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References [1] Allinger, D. F., Mitter, S. K., New results on the innovations problem of nonlinear filtering. Stochastics 4 (1981), 339–348. [2] Arapostathis, A., Borkar, V. S., Ghosh, M. K., Ergodic Control of Diffusion Processes. Book in preparation. [3] Arisawa, M., Ergodic problem for the Hamilton-Jacobi-Bellman equation I. Existence of the ergodic attractor. Ann. Inst. H. Poincaré Anal. Non Linéaire 14 (1997), 415–438. [4] Arisawa, M., Ergodic problem for the Hamilton-Jacobi-Bellman equation II. Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 1–24. [5] Basak, G., Borkar, V. S., Ghosh, M. K., Ergodic control of degenerate diffusions. Stochastic Anal. Appl. 15 (1997), 1–17. [6] Bensoussan, A., Frehse, J., On Bellman equations of ergodic control in R n . J. Reine Angew. Math. 429 (1992), 125–160. [7] Bhatt, A. G., Borkar, V. S., Occupation measures for controlled Markov processes: characterization and optimality. Ann. Probab. 24 (1996), 1531–1562. [8] Bhatt, A. G., Borkar, V. S., Existence of optimal Markov solutions for ergodic control of Markov processes. Sankhya 67 (2005), 1–18. [9] Borkar, V. S., Optimal Control of Diffusion Processes. Pitman Research Notes in Math. 203, Longman Scientific and Technical, Harlow, UK, 1989. [10] Borkar, V. S., On extremal solutions to stochastic control problems. Appl. Math. Optim. 24 (1991), 317–330. [11] Borkar, V. S., Controlled diffusions with constraints II. J. Math. Anal. Appl. 176 (1993), 310–321. [12] Borkar, V. S., Dynamic programmimg for ergodic control with partial observations. Stochastic Process. Appl. 103 (2003), 293–310. [13] Borkar, V. S., Budhiraja, A., A further remark on dynamic programming for partially observed Markov processes. Stochastic Process. Appl. 112 (2004), 79–93. [14] Borkar, V. S., Budhiraja, A., Ergodic control for constrained diffusions: characterization using HJB equations. SIAM J. Control Optim. 43 (2004), 1463–1492. [15] Borkar, V. S., Gaitsgory, V., Singular perturbations in ergodic control of diffusions. Submitted. [16] Borkar, V. S., Ghosh, M. K., Ergodic control of multidimensional diffusions I: the existence results. SIAM J. Control Optim. 26 (1988), 112–126. [17] Borkar, V. S., Ghosh, M. K., Ergodic control of multidimensional diffusions II: adaptive control. Appl. Math. Optim. 21 (1990), 191–220. [18] Borkar, V. S., Ghosh, M. K., Controlled diffusions with constraints. J. Math. Anal. Appl. 152 (1990), 88–108. [19] Borkar, V. S., Ghosh, M. K., Stochastic differential games: an occupation measure based approach. J. Optim. Theory Appl. 73 (1992), 359–385; erratum ibid. 88, 251–252. [20] Borkar, V. S., Kumar, S., On extremal solutions of controlled non-linear filtering equations. SIAM J. Control Optim. 33 (1995), 718–724.


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[21] Borkar, V. S., Mitter, S. K., A note on stochastic dissipativeness. In Directions in Mathematical Systems Theory and Optimization (ed. by A. Rantzer and C. I. Byrnes), Lecture Notes in Control and Inform. Sci. 286, Springer-Verlag, Berlin, Heidelberg 2003, 41–49. [22] Fleming, W. H., McEneaney, W. M., Risk-sensitive control on an infinite horizon. SIAM J. Control Optim. 33 (1995), 1881–1915. [23] Ghosh, M. K., Arapostathis, A., and Marcus, S. I., Optimal control of switching diffusions with applications to flexible manufacturing systems. SIAM J. Control Optim. 31 (1993), 1183–1204. [24] Haussmann, U. G., L’Equation de Zakai et le Problème Séparé du Contrôle Optimal Stochastique. In Seminaire de Probabilites XIX (ed. by J. Azéma and M. Yor), Lecture Notes in Math. 1123, Springer-Verlag, Berlin, Heidelberg 1985, 37–62. [25] Helmes, K., Stockbridge, R. H., Numerical comparison of controls and verification of optimality for stochastic control problems. J. Optim. Theory Appl. 106 (2000), 107–127. [26] Krylov, N. V., Controlled Diffusion Processes. Applications of Mathematics 14, SpringerVerlag, New York, Heidelberg, Berlin 1980. [27] Kurtz, T. G., Stockbridge, R., Existence of Markov controls and characterization of Markov controls. SIAM J. Control Optim. 36 (1998), 609–653; erratum ibid. 37 (1999), 1310–1311. [28] Kushner, H. J., Dupuis, P., Numerical Methods for Stochastic Control Problems in Continuous Time. 2nd edition, Appl. Math. (N.Y.) 24, Springer-Verlag, New York 2001. [29] Menaldi, J.-L., Robin, M., Ergodic control of reflected diffusions with jumps. Appl. Math. Optim. 35 (1997), 117–137. [30] Menaldi, J.-L., Perthame, B., Robin, M., Ergodic problem for optimal stochastic switching. J. Math. Anal. Appl. 147 (1990), 512–530. [31] Menaldi, J.-L., Robin, M., Taksar, M. I., Singular ergodic control for multidimensional Gaussian processes. Math. Control Signals Systems 5 (1992), 93–114. [32] Robin, M., On some impulse control problems with long run average cost. SIAM J. Control Optim. 19 (1981), 333–358. [33] Stockbridge, R. H., Time-average control of martingale problems: existence of a stationary solution. Ann. Probab. 18 (1990), 190–205. [34] Stockbridge, R. H., Time-average control of martingale problems: a linear programming formulation. Ann. Probab. 18 (1990), 206–217. [35] Stroock, D. W., Varadhan, S. R. S., Multidimensional Diffusion Processes. Grundlehren Math. Wiss. 233, Springer-Verlag, Berlin, New York 1979. School of Technology and Computer Science, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India. E-mail: [email protected]

Convex optimization of graph Laplacian eigenvalues Stephen Boyd ∗

Abstract. We consider the problem of choosing the edge weights of an undirected graph so as to maximize or minimize some function of the eigenvalues of the associated Laplacian matrix, subject to some constraints on the weights, such as nonnegativity, or a given total value. In many interesting cases this problem is convex, i.e., it involves minimizing a convex function (or maximizing a concave function) over a convex set. This allows us to give simple necessary and sufficient optimality conditions, derive interesting dual problems, find analytical solutions in some cases, and efficiently compute numerical solutions in all cases. In this overview we briefly describe some more specific cases of this general problem, which have been addressed in a series of recent papers. • Fastest mixing Markov chain. Find edge transition probabilities that give the fastest mixing (symmetric, discrete-time) Markov chain on the graph. • Fastest mixing Markov process. Find the edge transition rates that give the fastest mixing (symmetric, continuous-time) Markov process on the graph. • Absolute algebraic connectivity. Find edge weights that maximize the algebraic connectivity of the graph (i.e., the smallest positive eigenvalue of its Laplacian matrix). The optimal value is called the absolute algebraic connectivity by Fiedler. • Minimum total effective resistance. Find edge weights that minimize the total effective resistance of the graph. This is same as minimizing the average commute time from any node to any other, in the associated Markov chain. • Fastest linear averaging. Find weights in a distributed averaging network that yield fastest convergence. • Least steady-state mean-square deviation. Find weights in a distributed averaging network, driven by random noise, that minimizes the steady-state mean-square deviation of the node values.

Mathematics Subject Classification (2000). Primary 05C35; Secondary 90C25. Keywords. Graph theory, Laplacian matrix, convex optimization, semidefinite programming, Markov chain, distributed averaging, effective resistance.

∗ Supported in part by the MARCO Focus Center for Circuit and System Solutions (C2S2, www.c2s2.org) under contract 2003-CT-888, by AFOSR grant AF F49620-01-1-0365, by NSF grant ECS-0423905, and by DARPA/MIT grant 5710001848. This paper reports work with co-authors Persi Diaconis, Arpita Ghosh, SeungJean Kim, Sanjay Lall, Pablo Parrilo, Amin Saberi. Jun Sun, and Lin Xiao.


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1. Introduction Let G = (V , E) be an undirected graph with n = |V | nodes and m = |E| edges, with weights w1 , . . . , wm ∈ R on the edges. Suppose edge l connects vertices (or nodes) i and j . We define al ∈ Rn as (al )i = 1, (al )j = −1, with other entries 0. The weighted Laplacian (matrix) is the n × n matrix defined as L=

m

wl al alT = A diag(w)AT ,

l=1

where diag(w) ∈ Rm×m is the diagonal matrix formed from w = (w1 , . . . , wm ) ∈ Rm , and A ∈ Rn×m is the incidence matrix of the graph, A = [a1 · · · am ]. We assume that the weights are such that L is positive semidefinite, which we write as L 0. This is always the case when the weights are nonnegative. Since L1 = 0, where 1 is the vector with all components one, L has smallest eigenvalue 0, corresponding to the eigenvector 1. We denote the eigenvalues of the Laplacian matrix L as 0 = λ1 ≤ λ2 ≤ · · · ≤ λn . Let φ be a symmetric closed convex function defined on a convex subset of Rn−1 . Then ψ(w) = φ(λ2 , . . . , λn ) (1) is a convex function of w [2, §5.2]. Thus, a symmetric convex function of the positive Laplacian eigenvalues yields a convex function of the edge weights. As a simple example, consider φ(u1 , . . . , un−1 ) = n−1 i=1 ui , i.e., the sum. In this case we have ψ(w) =

n i=2

λi =

n

λi = Tr L = 21T w,

i=1

twice the sum of the edge weights, which is linear and therefore also convex. As another example, the function φ(u1 , . . . , un−1 ) = maxn−1 i=1 ui (which is convex and symmetric) yields the function ψ(w) = λn , the largest eigenvalue (or spectral radius) of the Laplacian matrix (and a convex function of the edge weights). We consider optimization problems with the general form minimize ψ(w) subject to w ∈ W ,

(2)

where W is a closed convex set, and the optimization variable here is w ∈ Rm . The problem (2) is to choose edge weights on a graph, subject to some constraints, in order to minimize a convex function of the positive eigenvalues of the associated Laplacian matrix. We can also handle the case of maximizing a concave function φ of the positive Laplacian eigenvalues, by minimizing −ψ over w ∈ W .

Convex optimization of graph Laplacian eigenvalues

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The problem (2) is a convex optimization problem. Roughly speaking, this means that the analysis of the problem is fairly straightforward, and that the problem is easily solved numerically; see, e.g., [6]. In the cases we will consider, the problem (2) can be formulated even more specifically as a semidefinite program (SDP), which has the form minimize cT x (3) n subject to i=1 xi Ai B. Here x ∈ Rn is the variable, and the problem data are c ∈ Rn and the symmetric matrices A1 , . . . , An , B ∈ Rk×k . The inequality symbol between symmetric matrices refers to inequality with respect to the cone of positive semidefinite matrices. The constraint ni=1 xi Ai B is called a linear matrix inequality (LMI). The SDP (3) can be thought of as a generalization of a linear program (LP), minimize cT x n subject to i=1 xi ai ≤ b, where here, a1 , . . . , an , b are vectors, and the inequality symbol between vectors means componentwise. Many results for LPs have analogs for SDPs; moreover, in the last 15 years or so, effective algorithms for numerically solving SDPs have been developed, and are now widely used in many application areas.

2. Fastest mixing Markov chain In this section we briefly describe the problem of finding the fastest mixing symmetric Markov chain on a given graph. Many more details (and additional references) can be found in [4, 5]. We consider a symmetric Markov chain on the graph G, with transition matrix P ∈ Rn×n , where Pij = Pj i is the probability of a transition from vertex i to vertex j . Since P is symmetric, the uniform distribution πunif = (1/n)1T is an equilibrium distribution. The rate of convergence of the distribution π(t) to uniform is governed by μ(P ), the second largest eigenvalue magnitude (SLEM) of P , with smaller μ(P ) meaning faster asymptotic convergence. To find the fastest mixing symmetric Markov chain on the graph, we must choose the transition matrix P to minimize μ(P ), subject to the following conditions: P = PT,

P 1 = 1,

Pij ≥ 0, i, j = 1, . . . , n,

Pij = 0 for (i, j ) ∈ E.

The first three conditions state that P is a symmetric stochastic matrix; the last states that transitions can only occur over the graph edges. Identifying the graph edge weights with edge transition probabilities, we find that P can be expressed as P = I − L. The conditions above are equivalent to the conditions w ≥ 0, diag(L) ≤ 1

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Stephen Boyd

imposed on the edge weight vector w. (Here diag(L) is the vector consisting of the diagonal entries of L, and both inequalities above are vector inequalities, i.e., componentwise.) The eigenvalues of P are 1 − λ1 , . . . , 1 − λn . Since 1 − λ1 = 1, and |1 − λi | ≤ 1 (since P is stochastic), its SLEM is given by μ(P ) = max{|1 − λ2 |, . . . , |1 − λn |} = max{1 − λ2 , λn − 1}.

(4)

This has the general form (1), with φ(u1 , . . . , un−1 ) = maxn−1 i=1 |1−ui |. In particular, the SLEM μ(P ) is a convex function of the edge transition probabilities. Thus, the fastest mixing symmetric Markov chain problem can be expressed as our general problem (2), with W = {w | w ≥ 0, diag(L) ≤ 1}, a polyhedron. The semidefinite programming formulation of the problem is minimize γ subject to −γ I I − L − (1/n)11T γ I,

w ≥ 0,

diag(L) ≤ 1,

with variables w ∈ Rm and γ ∈ R. Since the fastest mixing symmetric Markov chain problem is convex, indeed, equivalent to an SDP, it can be solved effectively. Generic methods can be used for problems with only a few thousand edges; far larger problems, with millions of edges, can be solved using subgradient optimization techniques, exploiting Lanczos methods to efficiently compute a few extreme eigenvalues and eigenvectors of I − L − (1/n)11T ; see [5]. The optimal transition probabilities can be quite interesting; for example, a graph can have many edges with optimal transition probability zero. This means (roughly) that those edges are not needed to achieve fastest mixing on the given graph. We also note that the optimal transition probabilities can yield a mixing rate that is unboundedly better than some simple standard schemes for assigning transition probabilities for fast mixing, such as the maximum-degree method, or the Metropolis–Hastings method [5]. Standard methods can be used to construct various dual problems for the fastest mixing symmetric Markov chain problem. One such dual is maximize 1T z (5) subject to Y 1 = 0, Y = Y T , Y ∗ ≤ 1 (zi + zj )/2 ≤ Yij , (i, j ) ∈ E, with variables z ∈ Rn and Y ∈ Rn×n . Here Y ∗ = ni=1 |λi (Y )|, the sum of the singular values of Y , which is the dual norm of the spectral norm. This dual problem is convex, since the objective, which is maximized, is linear, hence concave, and the constraints are all convex. We have the following: • Weak duality. If Y , z are feasible for the dual problem (5), then we have 1T z ≤ μ , where μ is the optimal value of the fastest mixing symmetric Markov chain problem.

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• Strong duality. There exist Y , z that are optimal for the dual problem, and satisfy 1T z = μ . This means that the optimal values of the primal and dual problems are the same, and that the dual problem yields a sharp lower bound on the optimal SLEM. Both of these conclusions follow from general results for convex optimization problems (see, e.g., [10, 1, 6]). We can conclude strong duality using (a refined form of) Slater’s condition (see, e.g., [1, §3.3] and [6, §5.2]), since the constraints are all linear equalities and inequalities.

3. Fastest mixing Markov process Here we briefly describe the problem of finding the fastest mixing continuous-time symmetric Markov process on a given graph [11]. Consider a continuous-time Markov process on the graph G, with transition rate (or intensity) wl across edge l. The probability density π(t) ∈ R1×n at time t ≥ 0 is given by π(t) = π(0)e−tL . It follows that the asymptotic rate of convergence to the uniform distribution is governed by λ2 , the smallest positive eigenvalue of the Laplacian matrix. The deviation from uniform distribution decays, in the worst case, as e−λ2 t . We can express λ2 as λ2 = min{λ2 , . . . , λn }, which has the standard form (1), with φ(u1 , . . . , un−1 ) = minn−1 i=1 ui . Since the minimum function is concave, we see that λ2 is a concave function of the edge weights w. It is evidently homogeneous in w, so to get a sensible problem we must normalize the weights in some way, for example, as 1T w = 1. To find the transition rates that give fastest convergence (among weights that sum to one), we pose the problem maximize λ2 subject to w ≥ 0,

1T w = 1,

with variable w ∈ Rm . This is a convex optimization problem, which can be formulated as the SDP maximize γ subject to γ I L + β11T ,

w ≥ 0,

1T w = 1,

with variables γ , β ∈ R, w ∈ Rm . The same problem, allocating a fixed total edge weight across the graph edges so as to maximize the smallest positive Laplacian eigenvalue, arises in other areas. For example, λ2 arises in graph theory, and is called the algebraic connectivity of the graph. Fiedler refers to the maximum value of λ2 that can be obtained by allocating a fixed total weight to the edges of a graph, as its absolute algebraic connectivity [7].

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Stephen Boyd

The dual of the fastest mixing Markov process problem can be given a very interesting interpretation. It is equivalent to the following problem. We are given some distances d1 , . . . , dm on the graph edges. The goal is find a configuration of points x1 , . . . , xn ∈ Rn that satisfy xi − xj 2 ≤ dl , whenever edge l connects vertices i and j , and in addition maximizes the total variance, given by i=j xi − xj 2 . This problem was recently formulated in the machine learning literature as a method for identifying low dimensional structure in data; see, e.g., [12].

4. Minimum total effective resistance Here we describe the problem of choosing the edge weights to minimize the total effective resistance of a graph, subject to some given total weight [8]. We consider the graph as an electrical circuit or network, with the edge weight representing the conductance (inverse of resistance) of the associated electrical branch. We define Rij as the resistance in the network seen between nodes i and j . The total effective resistance is defined as R = i<j Rij . The total effective resistance comes up in several applications beyond circuit theory. For example, it is proportional to the average commute time, over all pairs of vertices, in the random walk on the graph defined by the weights wl [8]. (The probability of a transition from vertex i to vertex j is wl , the associated edge weight, divided by the total weight of all edges adjacent to vertex i.) It can be shown that n 1 1/λi , R= n i=2

i.e., it is proportional to the sum of the inverses of the positive Laplacian eigenvalues. This follows our general form (1), with φ(u1 , . . . , un−1 ) = n−1 i=1 1/ui , with domain n−1 R++ . (R++ is the set of positive reals.) In particular, the total effective resistance is a convex function of the weight vector w. Minimizing total effective resistance, subject to w ≥ 0 and 1T w = 1, is thus a convex optimization problem. The problem can be formulated as the SDP minimize n Tr Y subject to 1T w = 1, w ≥ 0, L + (1/n)11T I 0, I Y with variables w ∈ Rm and the (slack) matrix Y = Y T ∈ Rn×n (see [8]).

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5. Fast averaging Here we describe the problem of choosing edge weights that give fastest averaging, using a classical linear iteration [13]. The nodes start with value x(0) ∈ Rn , and at each iteration we update the node values as x(t + 1) = (I − L)x(t). The goal is to choose the edge weights so that xi (t) converges, as rapidly as possible, to the average value, i.e., x(t) → (1/n)11T x(0). This iteration can be given a very simple interpretation. At each step, we replace each node value with a weighted average of its previous value and its neighbors’ previous values. The weights used to form the average are taken from the graph edge weights, with the self-weight chosen so that the sum of the adjacent edge weights, plus the self-weight, equals one. The weights used to carry out this local averaging sum to one at each node, but can be negative. When the weights are symmetric (which we assume here), the convergence rate of this averaging process is determined by the SLEM of I − L, i.e., (4), exactly as in the Markov chain problem. The difference here is that the weights can be negative; in the Markov chain, of course, the weights (transition probabilities) must be nonnegative. The optimal weights can be found by solving the unconstrained problem minimize maxni=2 |1 − λi |, which evidently is a convex optimization problem. It can be posed as the SDP minimize γ subject to −γ I L − (1/n)11T γ I, with variables γ ∈ R, w ∈ Rm . Without loss of generality, we can assume that L 0. The problem is the same as the fastest mixing symmetric Markov chain problem, but without the nonnegativity requirement on w. It often happens that some of the optimal weights are negative [13].

6. Minimum RMS consensus error Here we describe a variation on the fastest linear averaging problem described above, in which an additive random noise perturbs the node values [15]. The iteration is x(t + 1) = (I − L)x(t) + v(t), where v(t) ∈ Rn are uncorrelated zero mean unit variance random variables, i.e., E v(t) = 0,

E v(t)v(t)T = I,

E v(t)v(s)T = 0, t = s.

This iteration arises in noisy averaging, distributed data fusion, and load balancing applications; see the references in [15].

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Stephen Boyd

We can measure the effectiveness of the averaging iteration at countering the effects of the additive noises by the steady-state mean-square deviation, defined as

1 δss = lim E (xi (t) − xj (t))2 . t→∞ n i<j

The steady-state mean-square deviation can be expressed as δss =

n i=2

1 , λi (2 − λi )

provided 0 < λi < 2 for i = 2, . . . , n, and is infinite otherwise. (The condition 0 < λi < 2 for i = 2, . . . , n is the same as max{1−λ2 , λn −1} < 1, which is the condition that the linear iteration for averaging, without the additive noise, converges.) Once again, this has the standard form (1), with φ(u1 , . . . , un−1 ) = n−1 i=1 1/(ui (2 − ui )), with domain (0, 2)n−1 . In particular, we see that δss is a convex function of the edge weights. To find the weights that yield the smallest steady-state mean-square deviation, we simply minimize the convex function δss over w ∈ Rm .

7. Methods All the problems described above can be effectively solved numerically, by a variety of standard methods for convex optimization, including interior-point methods for modest sized problems (with a few thousand weights) and subgradient-based methods for larger problems. We can exploit structure in the problems (such as sparsity of the underlying graph) to increase the efficiency of these methods. We can also exploit symmetry in solving the problems. Two edges are symmetric if there exists an automorphism of the graph that maps one edge to the other. Whenever two edges are symmetric, we can assume without loss of generality that the corresponding edge weights are equal. (This follows from a basic result in convex optimization: there is always a solution that is invariant under the group of permutations that leave the objective function and constraint set fixed.) If the symmetry group of the graph is large, this can considerably reduce the size of the optimization problem that needs to be solved. As an extreme example, consider an edge-transitive graph, i.e., one in which any two edges are symmetric. For such a graph, we can assume that all edge weights are equal, i.e., there is only one common edge weight to be determined. This reduces the problem to one with at most one scalar variable (the common edge weight); if there is an equality constraint, such as 1T w = 1, we conclude that an optimal solution is given by uniform edge weights, w = (1/m)1 [3]. This idea is used in [9] to reduce some specific weight optimization problems to ones with a handful of variables, which can be solved analytically.


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References [1] Bertsekas, D. P., Nonlinear Programming. Second edition, Athena Scientific, Belmont, MA, 1999. [2] Borwein, J., and Lewis, A., Convex Analysis and Nonlinear Optimization, Theory and Examples. CMS Books Math./Ouvrages Math. SMC 3, Springer-Verlag, New York 2000. [3] Boyd, S., Diaconis, P., Parrilo, P., and Xiao, L., Symmetry analysis of reversible Markov chains. Internet Math. 2 (1) (2005), 31–71. [4] Boyd, S., Diaconis, P., Sun, J., and Xiao, L., Fastest mixing Markov chain on a path. Amer. Math. Monthly 113 (1) (2006), 70–74. [5] Boyd, S., Diaconis, P., and Xiao, L., Fastest mixing Markov chain on a graph. SIAM Review 46 (4) (2004), 667–689. [6] Boyd, S., and Vandenberghe, L., Convex Optimization. Cambridge University Press, Cambridge 2004. [7] Fiedler, M., Absolute algebraic connectivity of trees. Linear and Multilinear Algebra 26 (1990), 85–106. [8] Ghosh, A., and Boyd, S., Minimizing effective resistance of a graph. SIAM Review, to appear; www.stanford.edu/~boyd/eff_res. [9] Ghosh, A., and Boyd, S., Upper bounds on algebraic connectivity via convex optimization. Linear Algebra Appl., to appear; www.stanford.edu/~boyd/eigbounds_laplacian. [10] Rockafellar, R. T., Convex Analysis. Princeton University Press, Princeton, NJ, 1970. [11] Sun, J., Boyd, S., Xiao, L., and Diaconis, P., The fastest mixing Markov process on a graph and a connection to a maximum variance unfolding problem. SIAM Review, to appear 2006; www/~boyd/fmmp. [12] Weinberger, K., and Saul, L., Unsupervised learning of image manifolds by semidefinite programming. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR-04), Washington D.C., 2004. [13] Xiao, L., and Boyd, S., Fast linear iterations for distributed averaging. Systems Control Lett. 53 (2004), 65–78. [14] Xiao, L., and Boyd, S., Optimal scaling of a gradient method for distributed resource allocation. J. Optim. Theory Appl. 129 (3) (2006), to appear. [15] Xiao, L., Boyd, S., and Kim, S.-J., Distributed average consensus with least-mean-square deviation. Journal of Parallel and Distributed Computing, summitted May 2005; www/ ~boyd/lms_consensus. Electrical Engineering Department, Stanford University, Packard 264, Stanford, CA 94305, U.S.A. E-mail: [email protected]

Controllability of evolution equations of fluid dynamics Oleg Yu. Emanouilov (Imanuvilov)

Abstract. In this paper we will discuss recent developments in controllability of evolution equations of fluid mechanics. The control is assumed to be distributed either on a part of the boundary or locally distributed in some subdomain. We will present some ideas of proof of main theorems. Special attention will be paid to the technique based on Carleman estimates. Mathematics Subject Classification (2000). 93C20, 35B37. Keywords. Carleman estimates, exact controllability, Navier–Stokes system.

1. Introduction This paper is concerned with the problem of exact controllability of partial differential equations with control concentrated either on the part of the boundary or locally distributed inside of the boundary in some subdomain. The typical statement of general controllability problem, which we are going to discuss in this paper, can be formulated as follows: let a function y(t, x), which describes the state of a system, satisfy a semilinear partial differential equation ∂t y + A(x, D)y + F (x, y, ∇y) = χω u B(x, D)y = 0

on (0, T ) × ∂,

in (0, T ) × , y(0, · ) = y0 ,

(1.1) (1.2)

where A(x, D) is a linear operator, F is a nonlinear term, B(x, D) is a boundary operator, χω is the characteristic function of the domain ω ⊂ where the control function u(t, x) is supported. The initial conditions, the function y0 (x) and another function y1 (x) called target function, are given. Let us choose some time moment T . Then the exact controllability problem may be formulated as follows: Find the control u and the state function y such that y(T , · ) = y1 . (1.3) The solvability properties of the controllability problem (1.1)–(1.3) are completely different from the properties of the initial boundary value problem (1.1)–(1.2) with fixed u. For initial value problems with a reasonable choice of boundary conditions and for smooth initial conditions we usually expect the uniqueness of the solution. Moreover, if a priori estimates are obtained, this solution typically may be extended Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Oleg Yu. Emanouilov (Imanuvilov)

globally in time. On the other hand, for most boundary/locally distributed controllability problems of equations of mathematical physics, solutions are not unique and a priori estimates typically are absent. In case of control of linear equations these difficulties do not produce a huge problem, since the controllability problem typically can be reduced to an observability problem which can be formulated as follows. Suppose two Banach spaces X and Y are given. For the solution of the adjoint linear equation −∂t z + A∗ (x, D)z = 0 B ∗ (x, D)z = 0

in (0, T ) × , on (0, T ) × ∂

(1.4) (1.5)

one needs to obtain the a priori estimate zX ≤ Cχω zY .

(1.6)

The initial conditions at t = T for problem (1.4), (1.5) are not assumed to be known. This creates the main difficulty in proving estimate (1.6). There are several methods to deal with such observability problems: 1. The method based on the theorem on propagation of singularities (see Bardos– Lebeau–Rauch [2]); 2. Multipliers method (see [31], [28], [27], [29], [39]); 3. Carleman estimates (see [21], [22], [25], [26], [37], [38]). The first two methods are effective for the wave and Schrödinger equations. As for the equations of parabolic type and the generalized Stokes system, the Carleman type estimates with the singular weight functions appears to be more effective method compared to methods 1 and 2.

2. Controllability of parabolic equations and the Burgers equation In a bounded domain ∈ RN with ∂ ∈ C 2 we consider the semilinear parabolic equation G(y) = ∂t y − y + f (t, x, y) = χω u + g (2.7) with given initial condition and zero Dirichlet boundary conditions y|(0,T )×∂ = 0,

y(0, · ) = y0 .

(2.8)

Here ω is an arbitrary but fixed subdomain, χω is the characteristic function of the domain ω and u(t, x) is the control locally distributed in ω. Suppose that the target function y1 (x) is given and some moment of time T is fixed. We are looking for control u such that (2.9) y(T , · ) = y1 . Since the solution y(t, x) of the heat equation with zero right-hand side is analytic as a function of x for any positive t, we cannot solve in general problem (2.7)–(2.9) for an arbitrary smooth target function y1 .

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Controllability of evolution equations of fluid dynamics

Let us assume that ¯ × R1 ), f ∈ C 1 ([0, T ] ×

f (t, x, 0) = 0

for all (t, x) ∈ (0, T ) × , (2.10)

and that the function f (t, x, y) satisfies the Lipschitz condition |f (t, x, ζ1 ) − f (t, x, ζ2 )| ≤ K|ζ1 − ζ2 |

for all (t, x) ∈ (0, T ) × , ζ1 , ζ2 × R1 , (2.11) where the constant K is independent of t, x, ζ . We have Theorem 2.1 ([22]). Let y1 ≡ 0 and conditions (2.10), (2.11) hold true. Suppose that 1

there exists δ > 0 such that e (T −t)1+δ g ∈ L2 ((0, T ) × ). Then for any y0 ∈ W 12 (), there exists a solution (y, u) ∈ W21,2 ((0, T ) × ) × L2 ((0, T ) × ω) to problem (2.7)–(2.9). β

Here W 1,2 ((0, T ) × ) = {y(t, x) | ∂t y, ∂x y ∈ L2 ((0, T ) × ) for all |β| ≤ 2}. Thanks to assumption (2.11) by standard methods of functional analysis the proof of Theorem 2.1 may be reduced to the question of solvability of the controllability problem for the linear parabolic equation ∂t v − v + c(t, x)v = χω u˜ + g˜ v|(0,T )×∂ = 0, v(0, · ) = v0 ,

in (0, T ) × , v(T , · ) = 0,

(2.12)

where c ∈ L∞ ((0, T ) × ). The solvability of problem (2.12) is equivalent to obtaining the observability estimate for the adjoint parabolic equation: −∂t z − z + c(t, x)z = q

in [0, T ] × ,

z|(0,T )×∂ = 0.

(2.13) (2.14)

The observability estimate for (2.13)–(2.14) can be proved using the technique of Carleman estimates. First we need to introduce some weight functions. Let ψ ∈ C 2 () be such that ψ(x) > 0 for all x ∈ ,

ψ|∂ = 0,

|∇ψ(x)| > 0 for all x ∈ \ ω0 , (2.15)

where ω0 ⊂⊂ ω is some open set. Using the function ψ we construct three more functions: ϕ(t, x) = eλψ(x) /(t (T − t)), α(t, x) = (eλψ − e2λ||ψ||C() )/(t (T − t)), ∞ and η(t, x) = (eλψ − e2λ||ψ||C() )/((t)(T 3T − t)) where ∈ C [0, T ], (t) > 0 for any t ∈ [0, T ] and (t) = t for t ∈ 4 , T . The following holds: Lemma 2.1 ([22]). There exists a number λˆ > 0 such that for an arbitrary λ ≥ λˆ there exists s0 (λ) such that for each s ≥ s0 (λ) solutions to problem (2.13)–(2.14)

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satisfy the following inequality: 2 1 ∂z 2 2 3 3 2 2sα ∂t + |z| + sϕ|∇z| + s ϕ z e dx dt (0,T )× sϕ (2.16) 2 2sα 3 3 2 2sα |q| e dxdt + s ϕ z e dxdt , ≤C [0,T ]×ω

(0,T )×

where the constant C is independent of s. This estimate, combined with the standard energy estimate for equation (2.13), implies that for any v0 ∈ W 12 () and e−s0 η g˜ ∈ L2 ((0, T ) × ) there exists a solution to problem (2.12): a pair (y, u) ∈ W 1,2 ((0, T ) × ) × L2 ((0, T ) × ω) such that 3 e−s0 η u˜ ∈ L2 ((0, T ) × ), e−s0 η v/(T − t) 2 ∈ L2 ((0, T ) × ). A different approach, still based on Carleman estimates, was proposed by G. Lebeau and L. Robbiano in [30] for linear parabolic equations with time independent coefficients. In [8], [11], [35] solutions for the controllability problem of the linear heat equation were constructed directly by solving a moment problem. In [36] the solution to the controllability problem for the heat equation was obtained from a solution of the corresponding problem for the wave equation. In [34] another method was proposed, essentially based on the solvability of the Cauchy problem for the one dimensional heat equation. Later this method was applied to the semilinear parabolic equation in [33]. The approximate controllability for equation (2.7) was proved in [9]. Next we consider the problem of exact controllability of equation (2.7) with boundary control. Let 0 be an arbitrary subdomain of ∂. Suppose that the control u is distributed over 0 : G(y) = g, y|(0,T )×0 = u, y|(0,T )×∂\0 = 0, y(0, · ) = y0 , y(T , · ) = y1 . (2.17) We have Theorem 2.2 ([22]). Let y1 ≡ 0 and conditions (2.10), (2.11) hold true. Suppose 1

that there exists δ > 0 such that e (T −t)1+δ g ∈ L2 ((0, T ) × ). Then for any y0 ∈ 1 1,2 W 12 () there exists a solution (y, u) ∈ W2 ((0, T ) × ) × L2 (0, T ; H 2 (0 )) to problem (2.17). Theorem 2.2 will easily follow from Theorem 2.1 if we enlarge the domain up ˜ in such a way that to ˜ ω ⊂ ,

˜ \ , ω=

∂ω ∪ ∂ ⊂ 0 .

˜ with the control locally distributed in ω. Then we consider problem (2.7)–(2.9) in Since the existence of the solution y is guaranteed by Theorem 2.1 we consider the restriction of y on and put u = y|0 . Next we consider the situation when the target function is not zero. In order to solve the controllability problem we need some conditions on the functions y1 and g.

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Condition 2.1. There exists a constant τ > 0 and a function u˜ ∈ L2 ((0, T ) × ω) such that the boundary value problem G(y) ˜ = χω u˜ + g in [T − τ, T ] × ,

y| ˜ [T −τ,T ]×∂ = 0,

y(T ˜ , · ) = y1

has a solution y˜ ∈ W 1,2 ((0, T ) × ). We have

Theorem 2.3 ([22]). Let y0 ∈ W 12 () and g ∈ L2 ((0, T ) × ). Suppose that (2.10), (2.11) hold true. Let the functions y1 and g satisfy Condition 2.1. Then there exists a solution (y, u) ∈ W 1,2 ((0, T ) × ) × L2 ((0, T ) × ω) of problem (2.7)–(2.9). Theorem 2.3 provides necessary and sufficient conditions for solvability of problem (2.7)–(2.9). A similar result holds true for the situation when the control is locally distributed over the boundary. Condition 2.2. There exists a constant τ > 0 and a function u˜ ∈ L2 ((0, T ) × ω) such that the boundary value problem G(y) ˜ = g in [T − τ, T ] × , y| ˜ [T −τ,T ]×0 = u, ˜ y| ˜ [T −τ,T ]×∂\0 = 0,

y(T ˜ , · ) = y1

has a solution y˜ ∈ W 1,2 ((0, T ) × ). The following holds:

Theorem 2.4 ([22]). Let y0 ∈ W 12 () and g ∈ L2 ((0, T ) × ). Suppose that (2.10), (2.11) hold true. Let the functions y1 and g satisfy Condition 2.2. Then there exists 1 a solution (y, u) ∈ W 1,2 ((0, T ) × ) × L2 (0, T ; H 2 (∂)) of problem (2.17). In case when the nonlinear term of the parabolic equation is superlinear the situa¯ and a time moment Tˆ which tion is different. For example, there exists y0 ∈ C ∞ () depends on only, such that any solution for the initial value problem ∂t y − y + y 2 = 0 in ,

y(0, · ) = y0 ,

y|(0,T )×∂ = u

will blow up at some time τ (u) < Tˆ . Hence we even cannot prevent a blowup by the boundary control. The similar question for the nonlinearity f (t, x, y) = −y 3 is open. If the nonlinear term has the form f (t, x, y) = y 3 for any y0 ∈ W21 () ∩ L6 () and sufficiently regular u (which satisfies the compatibility condition) a solution to the initial value problem ∂t y − y + y 3 = 0 in ,

y(0, · ) = y0 ,

y|(0,T )×∂ = u

exists and satisfies the a priori estimate d 1 ρ 7 (x)y 2 (t, x)dx + ρ 7 (x)y 4 (t, x)dx ≤ C, dt 8

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¯ is an arbitrary function such that ρ(x) > 0 for each x ∈ , where ρ ∈ C 2 () ρ|∂ = 0, |∇ρ|∂ = 0 and the constant C depends on ρ only. This estimate immediately implies that for some open set of target functions y1 (x) in L2 () there is no solution to problem (2.17). Let us consider the Burgers equation ∂t y − ∂x2 y + ∂x y 2 = χω u(t, x),

(t, x) ∈ [0, T ] × [0, L],

(2.18)

with zero Dirichlet boundary conditions and the initial condition y(t, 0) = y(t, L) = 0,

y(0, · ) = y0 .

(2.19)

Here ω ⊂ [0, L] is an arbitrary but fixed open set. We are looking for a control u such that y(T , · ) = y1 (2.20) The following holds:

Theorem 2.5 ([16]). Let y1 ∈ W 12 (0, L) be a steady-state solution to the Burgers equation and y0 ∈ W 12 (0, L). Then there exists a time moment T (y1 ) such that the controllability problem (2.18)–(2.20) has a solution (y, u) ∈ W 1,2 ((0, T ) × [0, L]) × L2 ((0, T ) × [0, L]). Suppose that ω satisfies the following condition: there exists b > 0 such that ω ⊂ (b, L).

(2.21)

We have Lemma 2.2 ([16]). Let y(t, x) be a solution to problem (2.18), (2.19). Denote y+ (t, x) = max(y(t, x), 0). Then for arbitrary N > 5 the following estimate holds true: b d 4 (b − x)N y+ (t, x)dx < γ (N)bN−5 . (2.22) dt 0 Here γ (N) > 0 is a constant depending on N only. The immediate consequence of (2.22) is the existence of an open set of target functions which is unreachable by means of the locally distributed control satisfying (2.21) or by means of the boundary control concentrated at x = L. If condition (2.21) fails, we of course do not have the a priori estimate (2.22). In terms of the boundary control this situation corresponds to the case when the control is located at both endpoints of the segment [0, L]. By Hopf’s transformation this problem might be reduced to the controllability problem of the one-dimensional heat equation with control located at both endpoints of the segment [0, L] but with one additional constraint: control functions are nonnegative. Then from results of [1] it follows that for some initial condition y0 the set of all reachable functions is not dense in L2 (0, L). Later we will see that the controllability properties of the Burgers equation and the Navier–Stokes system are completely different.

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3. Local controllability of the Navier–Stokes system In [32] J.-L. Lions conjectured that the Navier–Stokes system with boundary or locally distributed control is globally approximately controllable. This paper inspired intensive research in the area. In this section we discuss the local controllability results for the Navier–Stokes system and the Boussinesq system. Let us consider the Navier–Stokes system defined on the bounded domain ⊂ RN (N = 2, 3) with boundary ∂ ∈ C 2 ∂t y(t, x) − y(t, x) + (y, ∇) y + ∇p = f + χω u in , y|(0,T )×∂ = 0,

div y = 0,

y(0, · ) = y0 ,

(3.23) (3.24)

where y(t, x) = (y1 (t, x), . . . , yN (t, x)) is the velocity of fluid, p is the pressure. The density of external forces f (t, x) = (f1 (t, x), . . . , fN (t, x)) and the initial velocity y0 are given, u(t, x) is a control distributed in some arbitrary but fixed subdomain ω of the domain . Let (y(t, ˆ x), p(t, ˆ x)) be a solution of the Navier–Stokes equations with the righthand side f exactly the same as in (3.23): ∂t yˆ − yˆ + (y, ˆ ∇)yˆ + ∇ pˆ = f in (0, T ) × ,

div yˆ = 0,

y| ˆ (0,T )×∂ = 0 (3.25)

close enough to the initial condition y0 at the moment t = 0 ˆ · )V ≤ ε, y0 − y(0,

(the parameter ε is sufficiently small)

(3.26)

where V = {y(x) = (y1 , . . . , yN ) ∈ (W21 ())N : div y = 0 in , y|∂ = 0}. We are looking for a control u such that, for a given T > 0, the following equality holds y(T , · ) = y(T ˆ , · ). (3.27) In order to formulate our results, we introduce the following functional spaces: H = {y(x) = (y1 , . . . , yN ) ∈ (L2 ())N : div y = 0, in , (y, n)|∂ = 0}, V 1,2 ((0, T ) × ) = {y(t, x) ∈ (W21,2 ((0, T ) × ))N : div y = 0 in , y|∂ = 0}, where n = n(x) = (n1 (x), . . . , nN (x)) is the outward unit normal to ∂. Suppose that the function yˆ has the following regularity properties: yˆ ∈ L∞ ((0, T ) × ), ∂t yˆ ∈ L2 (0, T ; Lσ ()),

σ > 6/5 for N = 3, σ > 1 for N = 2.

(3.28)

The following result in particular gives us a positive answer to the question of the possibility of stabilization of the flow near an unstable steady state solution by means of locally distributed control.

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Theorem 3.1 ([12]). Let y0 ∈ V , f ∈ L2 (0, T ; H ) and suppose that the pair (y, ˆ p) ˆ solves (3.25) and satisfies condition (3.28). Then for sufficiently small ε > 0 there exists a solution (y, p, u) ∈ V 1,2 ((0, T ) × ) × L2 (0, T ; W21 ()) × (L2 ((0, T ) × ω)))N to problem (3.23), (3.24), (3.26), (3.27). This result first has been proved in [15] for the control distributed over the whole boundary ∂. In [21] the case of control distributed over an arbitrary small subdomain ω, but with some assumptions on the geometry of was considered. Finally, in [23], these assumptions on were removed under the regularity condition on the function yˆ which is stronger then (3.28). Since the existence theorem 3.1 is local, in order to prove this existence result one first proves the solvability of the controllability problem for the Navier–Stokes equation linearized at trajectory y: ˆ ⎧ ⎪ ∂ y˜ − y˜ + (y, ˆ ∇)y˜ + (y, ∇)yˆ + ∇ p˜ = f + χω u, ˜ div y˜ = 0 in (0, T ) × , ⎪ ⎨ t y˜ = 0 on (0, T ) × ∂, ⎪ ⎪ ⎩ ˜ ,·) = 0 in . y(0, ˜ · ) = y0 , y(T (3.29) After the solvability of (3.29) is established in appropriate functional spaces the conclusion of the Theorem 3.1 follows from the standard implicit function theorem. The typical way to solve (3.29) is to reduce it to the observability problem for the operator adjoint to the operator of the linearization of the Navier–Stokes system at trajectory y. ˆ More precisely, let the function z ∈ L2 (0, T ; H ) satisfy the equations −∂t z − z − Dzyˆ = ∇π + g in (0, T ) × , z|(0,T )×∂ = 0,

div z = 0, where the function Dz = ∇z + ∇zt . Denote α(t, x) =

e

λψ(x)+8ψL∞ ()

10λψL∞ ()

−e (t (T −t))4

ˆ x) = maxx∈ α(t, x), ϕ(t,

e

8λψL∞ () (t (T −t))4

, ϕ(t, x) =

(3.30) (3.31)

, α ∗ (t) = minx∈ α(t, x), α(t) ˆ = e

8λψL∞ () +ψ(x) (t (T −t))4

. The function ψ is

introduced in (2.15). For the system (3.30)–(3.31), we have the following observability estimate: ˆ C depending on and Theorem 3.2 ([12]). There exist three positive constants sˆ , λ, 2 ω such that for every z0 ∈ H , g ∈ L ((0, T ) × ) the corresponding solution to (3.30), (3.31) verifies: 2 n ∂ 2 z 2 1 ∂z 2 2 3 4 3 2 2sα + ∂x ∂x + sλ ϕ|∇z| + s λ ϕ |z| e dxdt sϕ ∂t (0,T )×

i,j =1

15

i

≤ C(1 + T 2 ) s 2 λ20 +

j

15

∗

ˆ |g|2 ϕˆ 2 e4s α−2sα dxdt (0,T )×

(0,T )×ω

∗

ˆ s 16 λ40 ϕˆ 16 |z|2 e8s α−6sα dxdt

(3.32) for all s ≥ s0 ,

1329


λˆ T y ˆ L∞ ((0,T )×) for all λ ≥ λˆ (1 + y ˆ 2L∞ ((0,T )×) + ∂t y ˆ 2L2 (0,T ;Lσ ()) + e ) and 2

s ≥ sˆ (T 4 + T 8 ).

The strategy of the proof of (3.32) is as follows. First we apply the Carleman estimate (2.16) to equations (3.31). Next we need to eliminate the norm of the function ∇π on the right-hand side. In order to do that we observe that the pressure π for each t ∈ [0, T ] satisfies the Laplace equation −π = div(Dzy) ˆ + div g

in .

(3.33)

Since the velocity field z satisfies the zero Dirichlet boundary conditions, there are no explicit boundary conditions for the pressure π . Therefore to equation (3.33) we apply the Carleman estimates for elliptic equations obtained in [24] with weights which minimize the contribution of the boundary terms. Finally we eliminate the norms of the functions π |∂ and χω π using some a priori estimates for the initial value problems for the Stokes system and the heat equation. In many controllability problems in addition to be locally distributed in a subdomain, the control u is required to satisfy some additional constraints. Below we discuss the situation when in problems (3.23), (3.24), (3.27) the control satisfies the following constraint: one of the components of the vector function u(t, x) is identically equal zero on (0, T ) × . Suppose that ω satisfies the following condition: ˜ ∩ ∂. there exists x 0 ∈ ∂, δ˜ > 0 such that ω ∩ ∂ ⊃ B(x 0 ; δ)

(3.34)

˜ is the ball centered at x 0 of radius δ.) ˜ (B(x 0 ; δ) 4 Let E = H for N = 2 and E = H ∩ L () for N = 3. Assume that the initial condition y0 is close to y(0, ˆ · ) in the norm of the space E: ˆ · )E ≤ ε. y0 − y(0,

(3.35)

We have Theorem 3.3. Assume that ω satisfies (3.34). Let y0 ∈ E, f ≡ 0 and suppose that the pair (y, ˆ p) ˆ solves (3.25) and satisfies condition (3.28). Then for sufficiently small ε > 0 there exists a solution (y, p, u) to problem (3.23), (3.24), (3.35), (3.27) with control u ∈ (L2 ((0, T ) × ω))N having one component identically zero. In the case of locally distributed control with zero component uk for the corresponding observability problem, associated with (3.30),(3.31) we do not have any information on the k-th component of the function z in (0, T ) × ω. This means that the function zk should not appear in the right-hand side of the inequality (3.32). This difficulty can be overcome if we recall that z is divergence free function and therefore

its k-th component satisfies the equation ∂xk zk = jN=1,j =k ∂xj zj . From this ordinary differential equation, thanks to zero Dirichlet boundary conditions and assumption (3.34), in some subdomain of ω we can estimate zk by the remaining components of the function z and then apply (3.32).

1330


Next we consider the similar controllability problem of the Boussinesq system. ⎧ ∂t y − y + (y, ∇)y + ∇p = χω u + θ eN , div y = 0 in (0, T ) × , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨∂t θ − θ + (y, ∇θ) = χω h in (0, T ) × , ⎪ ⎪ y = 0, θ = 0 ⎪ ⎪ ⎪ ⎪ ⎩ y(0, · ) = y0 , θ(0, · ) = θ0

on (0, T ) × ∂, in .

(3.36) In the domain (0, T ) × ω we control both the density of external forces u and the density of external heat sources h. ˆ be a sufficiently regular solution to the Boussinesq system: Let (y, ˆ p, ˆ θ) ∂t yˆ − yˆ + (y, ˆ ∇)yˆ + ∇ pˆ = θˆ eN , div yˆ = 0 ˆ =0 ∂t θˆ − θˆ + (y, ˆ ∇ θ)

in (0, T ) × ,

(3.37)

in (0, T ) × ,

(3.38)

yˆ = 0, θˆ = 0

on (0, T ) × ∂,

(3.39)

ˆ · ) = θˆ0 y(0, ˆ · ) = yˆ0 , θ(0,

in .

(3.40)

Assume that yˆ satisfies (3.28) and the temperature θˆ has the following regularity θˆ ∈ L∞ ((0, T ) × ), ∂t θˆ ∈ L2 (0, T ; Lσ ()),

σ > 1 if N = 2, σ > 6/5 if N = 3.

(3.41)

In addition to condition (3.34) we assume that there exists k < N, such that nk (x 0 ) = 0.

(3.42)

Our goal is to prove that for some ε > 0 , whenever (y0 , θ0 ) ∈ E × L2 () and (y0 , θ0 ) − (yˆ0 , θˆ0 )E×L2 () ≤ ε,

(3.43)

we can find L2 controls u and h with uk ≡ uN ≡ 0 such that ˆ , · ) in . y(T , · ) = y(T ˆ , · ) and θ(T , · ) = θ(T

(3.44)

We note that for dimension N = 2 we are trying to control both the velocity field and the temperature by choosing the density of external heat sources in the subdomain ω. The following holds: Theorem 3.4. Assume that ω satisfies (3.34) and (3.42). Let y0 ∈ E, θ0 ∈ L2 () and ˆ p) suppose that the pair (y, ˆ θ, ˆ solves (3.37)–(3.40) and satisfies conditions (3.28), (3.41). Then for sufficiently small ε > 0 there exists a solution (y, θ, p, u, h) to problem (3.36), (3.43), (3.44) such that (u, h) ∈ (L2 ((0, T ) × ))N+1 and uk ≡ uN ≡ 0. In particular, if N = 2, we have local exact controllability with controls u ≡ 0 and h ∈ L2 ((0, T ) × ω).

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4. Global controllability of the Navier–Stokes and the Boussinesq system In this section we will discuss the global controllability of the Boussinesq and the Navier–Stokes systems. We start with the controllability problem for the Boussinesq system with periodic boundary conditions: ∂t y − y + (y, ∇)y + ∇p = f + θeN + χω u in K = jN=1 [0, 2π ], ∂t θ − θ + (y, ∇θ) = g + χω h in K,

div y = 0, (4.45) (4.46)

y(t, . . . xi + 2π, . . . ) = y(t, x), θ(t, . . . xi + 2π, . . . ) = θ(t, x) for all i ∈ {1, . . . , N }, y(0, · ) = y0 ,

θ(0, · ) = θ0 ,

y(T , · ) = y(T ˆ , · ),

ˆ , · ). θ(T , · ) = θ(T

(4.47) (4.48)

ˆ yˆ is some solution to the Boussinesq system with the same right-hand side: Here θ, ˆ N ˆ ∇)yˆ + ∇ pˆ = f + θe ∂t yˆ − yˆ + (y,

in (0, T ) × K,

ˆ =g ˆ ∇ θ) ∂t θˆ − θˆ + (y, y(t, ˆ . . . xi + 2π, . . . ) = y(t, ˆ x),

div yˆ = 0,

(4.49)

in (0, T ) × K,

(4.50)

ˆ . . . xi + 2π, . . . ) = θ(t, ˆ x) θ(t, for all i ∈ {1, . . . , N }.

(4.51)

A very essential role in controllability problems for the Navier–Stokes system and the Boussinesq system is played by the type of boundary conditions. For the case of periodic boundary conditions the situation is understood much better than for the case of Dirichlet boundary conditions. One reason for this striking difference is that for the periodic case we can construct explicitly a set of nonzero solutions of the Boussinesq system ˜ N + χω u˜ in K, div y˜ = 0, ∂t y˜ − y˜ + (y, ˜ ∇)y˜ = ∇ p˜ + θe y(t, ˜ . . . xi + 2π, . . . ) = y(t, ˜ x),

y(0, ˜ · ) = y(T ˜ , · ) = 0, (4.52) ˜θ(t, . . . xi + 2π, . . . ) = θ(t, ˜ x) (4.53) for all i ∈ {1, . . . , N },

˜ = 0 in K, ∂t θ˜ − θ˜ + (y, ˜ ∇ θ)

˜ · ) = θ(T ˜ ,·) = 0 θ(0,

˜ x) ≡ 0, y(t, ˜ x) = m(t, x), θ(t, m(t, . . . , xi + 2π, . . . ) = m(t, x)

for all i ∈ {1, . . . , N },

(4.54)

in the form (4.55)

where m(t, x) = ∇γ (t, x) and γ (t, · ) = 0 in K \ ω for all t ∈ [0, T ] and γ (0, · ) = γ (T , · ) = 0. (Obviously for the Dirichlet boundary conditions the function γ ≡ 0 ˜ N˜ p) is the only possible choice!) Note that for any N˜ the functions (N˜ y, ˜ N˜ θ, ˜ also

1332


solve (4.52)–(4.54) with some u˜ N˜ . If we are looking for a solution of the problem (4.45), (4.46), (4.47), (4.48) in the form (y, θ) = (Y + N˜ m, ˜ θ) then in new equations for (Y, θ) the large parameter N˜ will appear. Therefore the next logical step in finding (Y, θ) is to solve a controllability problem associated to the transport equation. In order to do that we need to make a special choice of the vector field m. The following holds: Lemma 4.1 ([17]). There exists a vector field m(t, x) = (m1 (t, x), . . . , mN (t, x)) ∈ C ∞ ([0, T ] × K) such that div m = 0 in [0, T ] × K,

m(t, x) = ∇γ (t, x) and γ = 0 in [0, T ] × (K \ ω),

for arbitrary k ∈ N m(0, x) ≡ m(T , x) ≡ 0,

∂ k m(t, x) ∂ k m(t, x) | = |t=T = 0, t=0 ∂t k ∂t k

and the relation {(t, x(t, x0 ), t ∈ (0, T )} ∩ [0, T ] × ω = ∅ is valid for every x0 ∈ K, where x(t, x0 ) is solution to the Cauchy problem d x(t, x0 ) = m(t, x(t, x0 )), dt

x(t, x0 )|t=0 = x0 .

Moreover, x(T , x0 ) = x0 for each x ∈ K. Furthermore there exist a finite cover {Oj j = 1, . . . , J } of K by open sets Oj and a number δˆ > 0 such that for each j all ˆ the curves x(t, x0 ), x0 ∈ Oj lie in ω for some time interval δ. In case we choose the vector field m as in Lemma 4.1 the following controllability problem may be solved for all regular initial data y0 , θ0 : ¯ div r = 0, ∂t r + (m, ∇)r + (r, ∇)m − ∇q1 = χω u, ¯ ∂t z + (m, ∇z) = χω h, r(t, . . . , xi + 2π, . . . ) = r(t, x), z(t, . . . , xi + 2π, . . . ) = z(t, x), i ∈ {1, . . . , N }, r(0, · ) = y0 ,

z(0, · ) = z0 ,

r(T , · ) = y(εT ˆ , · ),

ˆ z(T , · ) = θ(εT , · ).

Finally one can construct an approximation for the solution to problem (4.45)–(4.48) in the form t t 1 t (4.56) y(t, x) = m , x + r , x + yε , θ(t, x) = z , x + θε , ε ε ε ε 1 t 1 t 1 t (4.57) u(t, x) = u¯ , x − χω m , x , h = h¯ , x . ε ε ε ε ε ε Here the terms yε , θε are small provided that ε > 0 is small. Of course, we do not have the exact equality y(εT , · ) = y(εT ˆ , · ) but the difference y(εT , · ) − y(εT ˆ ,·)


1333

can be made arbitrarily small.( This proves Lions global controllability conjecture for the Boussinesq and the Navier–Stokes system with control distributed on the whole boundary.) Then the local controllability result similar to Theorem 3.4 could be ˆ applied in order to switch to the trajectory (y, ˆ θ). The idea to construct a solution to the controllability problem in the form (4.56), (4.57) was proposed by J. M. Coron in [3], [4] for the two dimensional Navier–Stokes system and Euler equation. In particular, Coron proved that if the control acts on an arbitrary open subset of the boundary which meets any connected component of this boundary, then the 2-D Euler equations are exactly controllable. Later the proof was extended to 3-D Euler equation by Glass [18], [19]. In [6], Coron constructed explicitly the feedback laws which globally asymptotically stabilize the fluid flow described by the Euler equation. The global exact controllability of the Navier–Stokes system on a manifold without boundary was studied in [7]. In order to formulate controllability results for the Boussinesq system rigorously we introduce the functional spaces V 0 (K) = {y(x) ∈ (L2 (K))N : div y = 0, y(t, . . . xi + 2π, . . . ) = y(t, x) for all i ∈ {1, . . . , N }}, V 1 (K) = {y ∈ V 0 (K) ∩ (W21 (K))N }, V 1,2 = {y(t, x) ∈ (W21,2 ((0, T )×K))N : div y = 0, y(t, . . . xi +2π, . . . ) = y(t, x)}. We have the following result: Theorem 4.1 ([17]). Let y0 ∈ V 1 (K), θ0 ∈ W21 (K), f ∈ L2 (0, T ; V 0 (K)), g ∈ ˆ p) L2 ((0, T ) × K) and suppose that for some β ∈ (0, 1) the function (y, ˆ θ, ˆ ∈ 1 1 0 2,β N 1 2,β 2 C (0, T ; V (K) ∩ (C (K)) ) × C (0, T ; C (K)) × L (0, T ; W2 (K)) is a given solution of the Boussinesq system (4.49)–(4.51). Then there exists a solution (y, θ, p, u, h) ∈ V 1,2 ×W 1,2 ((0, T )×K)×L2 (0, T ; W21 (K))×(L2 ((0, T )×ω))N+1 to problem (4.45), (4.46), (4.47), (4.48). Now we consider the problem of global controllability for the 2-D Navier–Stokes system with zero Dirichlet boundary conditions and control distributed over a part of the boundary. We need to introduce a “large parameter” in this problem but the analog of (4.55) for a general domain is hard to find. Therefore below we consider the Navier– Stokes system in the special domain = {(x1 , x2 ) : x1 ∈ (0, 1), x2 ∈ (0, 1)}. Let us consider the following controllability problem: ⎧ ⎪ ∂ y − y + (y, ∇)y = ∇p + f, div y = 0 (t, x) ∈ (0, T ) × , ⎪ ⎪ t ⎨ (t, x2 ) ∈ (0, T ) × (0, 1), (4.58) y(t, 0, x2 ) = 0 ⎪ ⎪ ⎪ ⎩ x = (x1 , x2 ) ∈ . y(0, · ) = y0 , y(T , · ) = 0 The initial condition y0 satisfies div y0 = 0, x ∈ ,

and

y0 (0, x2 ) = 0, x2 ∈ (0, 1).

(4.59)

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Observe that in system (4.58) we did not fix traces of y on ({1} × (0, 1)) ∪ ((0, 1) × {0, 1}). They can be chosen arbitrarily and considered as a boundary control. Next we construct an analog of the vector field m. Let the function U (t, x) have the form U (t, x) = (0, z(t, x1 )) where z = z(t, x1 ) solves the following problem associated to a linear heat equation: ⎧ ⎪ ∂t z − ∂x21 x1 z = c(t) (t, x1 ) ∈ (0, T ) × (0, 2), ⎪ ⎪ ⎨ (4.60) z(t, 0) = 0, z(t, 1) = w(t) t ∈ (0, T ), ⎪ ⎪ ⎪ ⎩ x1 ∈ (0, 2). z(0, x1 ) = 0 Here c(t) is a constant for each t such that c(0) = 0,

w(t) ∈ C ∞ [0, T ],

w(0) = 0, w (0) = c(0), w (0) = c (0).

Using this function, we construct U (t, x) = (0, z(t, x1 )) and q = x2 c(t) for t ∈ (0, T ), x ∈ K˜ = [0, 1] × [0, 2], which for an arbitrary N˜ ∈ R1 solves ⎧ ˜ ) − NU ˜ + (NU, ˜ ∇)(NU ˜ ) = ∇(Nq), ˜ ∂t (NU ⎪ ⎪ ⎪ ⎪ ˜ ˜ ⎪ (t, x) ∈ (0, T ) × K, ⎨div(NU ) = 0 (4.61) ˜ (t, 0, x2 ) = 0 ⎪ NU (t, x ) ∈ (0, T ) × R1 , 2 ⎪ ⎪ ⎪ ⎪ ⎩˜ ˜ NU (0, x) = 0 x ∈ K. We have Theorem 4.2 ([20]). Let f ∈ L2 ((0, T ) × ) and let y0 ∈ W21 () satisfy (4.59). Then there exists a sequence of functions fε such that fε → f in Lp0 (0, T ; V ),

p0 ∈ (1, 8/7),

and there exists at least one solution to the controllability problem ⎧ ⎪ ∂ y − yε + (yε , ∇)yε + ∇pε = fε , div yε = 0 (t, x) ∈ (0, T ) × , ⎪ ⎪ t ε ⎨ (t, x2 ) ∈ (0, T ) × (0, 1), yε (t, 0, x2 ) = 0 ⎪ ⎪ ⎪ ⎩ x ∈ . yε (0, x) = y0 , yε (T , x) = 0 (4.62) The sequence of the functions yε can be constructed in the following way: First let us choose a sufficiently small number δ = δ(ε) > 0 such that f Lp0 (T −3δ,T ;V ) ≤ ε/10. • On the interval between t = 0 and t = T − 3δ, we do not exert any control. So in this interval our function yε is given by the solution to the Navier–Stokes system with homogeneous Dirichlet boundary conditions.

1335


• Next, on the interval [T −3δ, T −2δ], we consider a function y˜0,ε ∈ V ∩C0∞ () close to y(T − 3δ, x) in V . In particular, y˜0,ε − y(T − 3δ, · )V ≤ δ 3 . On the interval [T − 3δ, T − 2δ] we set (t − T + 3δ) (t − T + 2δ) y˜0,ε (x) − y(T − 3δ, x), δ δ (t, x) ∈ [T − 3δ, T − 2δ] × .

yε (t, x) =

˜ we look for the • As the next step, on the segment [T − 2δ, T − 2δ + 2/N], solution uε in the form yε (t, x) = N˜ 2 U˜ (t, x) + y(t, x) − V˜ (t, x),

pε (t, x) = r˜ (t, x),

where U˜ (t, x) = U (t − T + 2δ, x), y(t, x) = y(t ˜ − T + 2δ, x), V˜ (t, x) = θ(t − T + 2δ)V (t − T + 2δ, x), r˜ (t, x) = θ(t)r(t − T + 2δ, x). The function y˜ solves the following controllability problem for the transport equation: ⎧ ˜ ⎪ ∂t y˜ + N˜ 2 (U, ∇)y˜ + N˜ 2 (y, ˜ ∇)U = 0 (t, x) ∈ (0, T ) × K, ⎪ ⎪ ⎨ (4.63) (t, x2 ) ∈ (0, T ) × R1 , y(t, ˜ 0, x2 ) = 0 ⎪ ⎪ ⎪ ⎩ ˜ x) = 0 ˜ ˜ N, x ∈ K. y(0, ˜ x) = y˜0,ε , y(1/ The function V˜ is a correction term, which ensure that the vector field yε is divergence free: ⎧ ˜ ∂ V − V = ∇r, div V = div y˜ (t, x) ∈ (0, T ) × K, ⎪ ⎪ ⎪ t ⎪ ⎪ ⎪ ⎨V (t, 0, x2 ) = V (t, 1, x2 ) = 0 (t, x2 ) ∈ (0, T ) × R1 , (4.64) ⎪ ⎪ (t, x1 , x2 ) ∈ (0, T ) × (0, 2) × R1 , V (t, x1 , x2 ) = V (t, x1 , x2 + 2) ⎪ ⎪ ⎪ ⎪ ⎩ ˜ V (0, x) = 0 x ∈ K. There exists a positive constant C > 0 independent of N˜ such that V C([0,2/N˜ ];L2 (K)) 2 (K)) ˜ + Vx2 C([0,2/N];L ˜ ˜ ≤

C . 1 N˜ 8

(4.65)

This estimate is the consequence of the global version of sharp regularity result for the ˜ is an arbitrary function pressure obtained in [10]. Finally θ = θ(t) ∈ C 2 ([0, 2/N]) such that ˜ θ (t) = 1, t ∈ [0, 1/N],

and θ(t) = 0 in a neighborhood of 2/N˜ .

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˜ × . We set fε = ∂t yε − yε + Let y ε = 0 for (t, x) ∈ (T − 2δ, T − 2δ + 2/N) ˜ (yε , ∇)yε for all (t, x) ∈ [T − 2δ, T − 2δ + 2/N] × . A short computation and (4.65) imply ˜ 7/8−1/p0 . f˜ε Lp0 (T −2δ,T −2δ+2/N;V ˜ ) ≤ CN Thanks to our choice of p0 , this constant tends to zero as N˜ → +∞. ˜ T ], we take fε ≡ 0 and we try to find a • Finally, on the interval [T − 2δ + 1/N, boundary control which drives the associated solution of (4.62) which starts at time ˜ x) to zero at time t = T . t = T − 2δ + 2/N˜ from the initial condition N˜ 2 U (2/N, 2 ˜ ˜ ˜ x) since θ(2/N) ˜ = 0. Observe that we have yε (T − 2δ + 2/N, x) = N U (2/N, 2 By Theorem 2.1 for any z0 ∈ L (0, 1), there exists a boundary control ρ = ρ(t) ∈ L2 (0, 2/N˜ − 2δ) such that the solution of ⎧ ⎪ (t, x1 ) ∈ (0, T ) × (0, 1), ∂t z − ∂x21 x1 z = 0 ⎪ ⎪ ⎨ z(t, 0) = 0, z(t, 1) = ρ(t) t ∈ (0, T ), ⎪ ⎪ ⎪ ⎩ z(0, x1 ) = z0 x1 ∈ (0, 1). satisfies

˜ x1 ) = 0, z(2δ − 2/N,

x1 ∈ (0, 1).

Then it suffices to take ˜ x1 )), yε (t, x) = (0, z(t − T + 2δ − 2/N,

˜ T ) × , (t, x) ∈ (T − 2δ + 2/N,

with z the solution of the previous null controllability problem with initial condition ˜ x1 ) z0 (x1 ) = z(2/N,

x1 ∈ (0, 1).

The construction of the function yε is finished.

References [1] Belishev, M. I., On approximating properties of Solutions of the heat equation. In Control theory of partial differential equations, Lect. Notes Pure Appl. Math. 242, Chapman & Hall/CRC, Boca Raton, FL, 2005, 43–50. [2] Bardos, C., Lebeau, G., Rauch, J., Sharp sufficient conditions for the observation, control, and stabilization of wave from boundaries. SIAM J. Control Optim. 30 (1992), 1024–1065. [3] Coron, J.-M., On the controllability of the 2-D incompressible Navier-Stokes equations with the Navier-Slip boundary conditions. ESAIM Contrôle Optim. Calc. Var. 1 (1996), 35–75. [4] Coron, J.-M., On the controllability of 2-D incompressible perfect fluids. J. Math. Pures Appl. 75 (1996), 155–188.


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[5] Coron, J.-M., Contrôlabilité exacte frontière de l’équation d’Euler des fluids parfaits incompresibles bidimensionnesls. C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 271–276. [6] Coron, J.-M., On null assymtotic stabilization of the 2-D Euler equation of incompressible fluids on simply connected domains. SIAM J. Control Optim. 37 (1999), 1874–1896. [7] Coron, J.-M., Fursikov, A. V., Global exact controllability of the 2-D Navier-Stokes equations on manifold without boundary. Russian J. Math. Phys. 4 (1996), 1–20. [8] Egorov, Y. V., Some problems in the theory of optimal control. Ž. Vyˇcisl. Mat. i Mat. Fiz. 3 (1963), 887–904. [9] Fabre, C., Puel, J.-P., Zuazua, E., Approximate controllability of the semilinear heat equation. Proc. Roy. Soc. Edinburgh Sect. A 125 (1995), 31–61. [10] Fabre, C., Lebeau, G., Prolongement unique des solutions de l’équation de Stokes. Comm. Partial Differential Equations 21 (1996), 573–596. [11] Fattorini, H. O., Boundary control of temperature distributions in a parallepipedon. SIAM J. Control Optim. 13 (1975), 1–13. [12] Fernandez-Cara, E., Guerrero, S., Imanuvilov, O., Puel, J.-P., Local exact controllability of Stokes and Navier-Stokes system. J. Math. Pures Appl. 83 (2005), N12, 1501–1542. [13] Fernandez-Cara, E., Guerrero, S., Imanuvilov, O., Puel, J.-P., Some controllability results for the N-dimentional Navier-stokes and Boussinesque system with N-1 scalar controls. SIAM J. Control Optim., to appear. [14] Fursikov, A. V., Imanuvilov, O. Yu., Local exact controllability of two dimensional NavierStokes system with control on the part of the boundary. Sb. Math. 187 (1996), 1355–1390. [15] Fursikov, A. V., Imanuvilov, O. Yu., Local exact boundary controllability of the Boussinesq equation. SIAM J. Control Optim. 36 (1988), 391–421. [16] Fursikov, A. V., Imanuvilov, O. Yu., Controllability of evolution equations. Lecture Notes Ser. 34, Seoul National University, Seoul 1996. [17] Fursikov, A. V., Imanuvilov, O. Yu., Exact controllability of the Navier-Stokes equations and the Boussinesq system. Russian Math. Surveys 54 (1999), 565–618. [18] Glass, O., Contrôlabilité de l’équation d’Euler tridimensionelle pour les fluides parfaits incompressibles. Séminaire sur les Équations aux Dérivees Partielles, 1997–1998, Exp. No XV, 11 pp, École Polytechnique, Palaiseau 1998. [19] Glass, O., Contrôlabilité exacte frontière de l’équation d’Euler des fluides parfaits incompressibles en dimension 3. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), 987–992. [20] Guerrero, S., Imanuvilov, O.Yu., Puel, J. P., Remarks on global approximate controllability for the 2-D Navier-Stokes system with Dirichlet boundary conditions. Submitted. [21] Imanuvilov, O. Yu., On exact controllability for the Navier-Stokes equations. ESAIM Contrôle Optim. Calc. Var. 3 (1998), 97–131. [22] Imanuvilov, O. Yu., Boundary controllability of parabolic equations. Sb. Math. 186 (1995), 879–900. [23] Imanuvilov, O. Yu., Remarks on exact controllability for Navier-Stokes equations. ESAIM Contrôle Optim. Calc. Var. 6 (2001), 39–72. [24] Imanuvilov, O. Yu., Puel, J. P., Global Carleman estimates for weak solutions of elliptic nonhomogeneous Dirichlet problems. Internat. Math. Res. Notices 2003 (16) (2003), 883–913.

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[25] Imanuvilov, O. Yu., Yamamoto, M., Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations. Publ. Res. Inst. Math. Sci. 39 (2) (2003), 227–274. [26] Kazemi, M. V., Klibanov, M. A., Stability estimates for ill-posed Cauchy problems involving hyperbolic equations and inequalities. Appl. Anal. 50 (1993), 93–102. [27] Komornik, V., Exact controllability and stabilization. RAM Res. Appl. Math., Masson, Paris 1994. [28] Lagnese, J. E., Lions, J.-L., Modelling, Analysis and Control of Thin Plates. Rech. Math. Appl. 6, Masson, Paris 1988. [29] Lasiecka, I., Triggiani, R., Yao, P. F., Inverse observability estimates for second order hyperbolic equations with variable coefficients. J. Math. Anal. Appl. 235 (1999), 13–57. [30] Lebeau, G., Robbiano, L., Contrôle exact de l’équation de la chaleur. Comm. Partial Differential Equations 20 (1995), 336–356. [31] Lions, J.-L., Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Rech. Math. Appl. 8, Masson, Paris 1988. [32] Lions, J.-L., Are there connections between turbulence and controllability? 9e Conférence internationale de l’INRIA, Antibes, 12–15 juin 1990. [33] Lin Guo, Y.-J., Littman, W., Null boundary controllability for semilinear heat equations. Appl. Math. Optim. 32 (1995), 281–316. [34] Littman, W., Boundary control theory for hyperbolic and parabolic equations with constant coefficients. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) (1978), 567–580. [35] Russell, D. L., Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions. SIAM Rev. 20 (1978), 639–739. [36] Russell, D. L.,A unified boundary controllability theory for hyperbolic and parabolic partial differential equations. Studies in Appl. Math. 52 (1973), 189–212. [37] Tataru, D., Boundary controllability for conservative PDE’s. Appl. Math. Optim. 31 (1995), 257–296. [38] Tataru, D., A priori estimates of Carleman’s type in domains with boundary. J. Math. Pure Appl. 73 (1994), 355–387. [39] Triggiani, R., Yao, P. F., Carleman estimates with no lower order terms for general Riemannian waves equations. Global uniqueness and observability in one shot. Appl. Math. Optim. 46 (2002), 331–375. Department of Mathematics, Colorado State University, 101 Werber Building, Fort Collins, CO 80523-1874, U.S.A. E-mail: [email protected]

Port-Hamiltonian systems: an introductory survey Arjan van der Schaft

Abstract. The theory of port-Hamiltonian systems provides a framework for the geometric description of network models of physical systems. It turns out that port-based network models of physical systems immediately lend themselves to a Hamiltonian description. While the usual geometric approach to Hamiltonian systems is based on the canonical symplectic structure of the phase space or on a Poisson structure that is obtained by (symmetry) reduction of the phase space, in the case of a port-Hamiltonian system the geometric structure derives from the interconnection of its sub-systems. This motivates to consider Dirac structures instead of Poisson structures, since this notion enables one to define Hamiltonian systems with algebraic constraints. As a result, any power-conserving interconnection of port-Hamiltonian systems again defines a port-Hamiltonian system. The port-Hamiltonian description offers a systematic framework for analysis, control and simulation of complex physical systems, for lumped-parameter as well as for distributed-parameter models. Mathematics Subject Classification (2000). Primary 93A30,70H05,70H45,70Q05, 70G45, 93B29, 37J60; Secondary 93C10, 93C15, 93C20, 37K05. Keywords. Interconnection, Dirac structures, constrained systems, Hamiltonian DAEs, stabilization, boundary control, conservation laws.

1. Introduction Historically, the Hamiltonian approach has its roots in analytical mechanics and starts from the principle of least action, and proceeds, via the Euler-Lagrange equations and the Legendre transform, towards the Hamiltonian equations of motion. On the other hand, the network approach stems from electrical engineering, and constitutes a cornerstone of mathematical systems theory. While most of the analysis of physical systems has been performed within the Lagrangian and Hamiltonian framework, the network point of view is prevailing in modelling and simulation of (complex) physical engineering systems. The framework of port-Hamiltonian systems combines both points of view, by associating with the interconnection structure of the network model a geometric structure given by a (pseudo-) Poisson structure, or more generally a Dirac structure. The Hamiltonian dynamics is then defined with respect to this Dirac structure and the Hamiltonian given by the total stored energy. Furthermore, port-Hamiltonian systems are open dynamical systems, which interact with their environment through ports. ReProceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Arjan van der Schaft

sistive effects are included by terminating some of these ports on energy-dissipating elements. Dirac structures encompass the geometric structures which are classically being used in the geometrization of mechanics (that is, Poisson structures and pre-symplectic structures), and allow to describe the geometric structure of dynamical systems with algebraic constraints. Furthermore, Dirac structures allow to extend the Hamiltonian description of distributed-parameter systems to include variable boundary conditions, leading to distributed-parameter port-Hamiltonian systems with boundary ports. Acknowledgements. This survey is based on joint work with several co-authors. In particular I thank Bernhard Maschke and Romeo Ortega for fruitful collaborations.

2. Finite-dimensional port-Hamiltonian systems In this section we recapitulate the basics of finite-dimensional port-Hamiltonian systems. For more details we refer e.g. to [19], [17], [20], [33], [34], [30], [36], [12], [5].

2.1. From classical Hamiltonian equations to port-Hamiltonian systems. The standard Hamiltonian equations for a mechanical system are given as ∂H (q, p), ∂p ∂H (q, p) + F p˙ = − ∂q q˙ =

(1)

where the Hamiltonian H (q, p) is the total energy of the system, q = (q1 , . . . , qk )T are generalized configuration coordinates for the mechanical system with k degrees of freedom, p = (p1 , . . . , pk )T is the vector of generalized momenta, and the input F is the vector of external generalized forces. The state space of (1) with local coordinates (q, p) is called the phase space. One immediately derives the following energy balance: d ∂T H ∂T H ∂T H H = (q, p)q˙ + (q, p)p˙ = (q, p)F = q˙ T F, dt ∂q ∂p ∂p

(2)

expressing that the increase in energy of the system is equal to the supplied work (conservation of energy). This motivates to define the output of the system as e = q˙ (the vector of generalized velocities).

Port-Hamiltonian systems: an introductory survey

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System (1) is more generally given in the following form ∂H (q, p), (q, p) = (q1 , . . . , qk , p1 , . . . , pk ), ∂p ∂H (q, p) + B(q)f, f ∈ Rm , p˙ = − ∂q ∂H ˙ e ∈ Rm , (q, p) (= B T (q)q), e = B T (q) ∂p q˙ =

(3)

with B(q)f denoting the generalized forces resulting from the input f ∈ Rm . In case m < k we speak of an underactuated system. Similarly to (2) we obtain the energy balance dH (q(t), p(t)) = eT (t)f (t). (4) dt A further generalization is to consider systems which are described in local coordinates as ∂H x˙ = J (x) (x) + g(x)f, x ∈ X, f ∈ Rm , ∂x (5) ∂H T m (x), e∈R , e = g (x) ∂x where J (x) is an n × n matrix with entries depending smoothly on x, which is assumed to be skew-symmetric, that is J (x) = −J T (x), and x = (x1 , . . . , xn ) are local coordinates for an n-dimensional state space manifold X (not necessarily evendimensional as above). Because of skew-symmetry of J we easily recover the energyT balance dH dt (x(t)) = e (t)f (t). We call (5) a port-Hamiltonian system with structure matrix J (x), input matrix g(x), and Hamiltonian H ([17], [19], [18]). Remark 2.1. In many examples the structure matrix J will additionally satisfy an integrability condition (the Jacobi-identity) allowing us to find by Darboux’s theorem “canonical coordinates”. In this case J is the structure matrix of a Poisson structure on X. Example 2.2. An important class of systems that naturally can be written as portHamiltonian systems, is constituted by mechanical systems with kinematic constraints [22]. Consider a mechanical system locally described by k configuration variables q = (q1 , . . . , qk ). Suppose that there are constraints on the generalized velocities q, ˙ described as AT (q)q˙ = 0, (6) with A(q) an r × k matrix of rank r everywhere. The constraints (6) are called holonomic if it is possible to find new configuration coordinates q¯ = (q¯1 , . . . , q¯k ) such that the constraints are equivalently expressed as q˙¯k−r+1 = q˙¯n−r+2 = · · · = q˙¯k = 0, in which case the kinematic constraints integrate to the geometric constraints q¯k−r+1 = ck−r+1 , . . . , q¯k = ck

(7)

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for certain constants ck−r+1 , . . . , ck determined by the initial conditions. Then the system reduces to an unconstrained system in the remaining configuration coordinates (q¯1 , . . . , q¯k−r ). If it is not possible to integrate the kinematic constraints as above, then the constraints are called nonholonomic. The equations of motion for the mechanical system with constraints (6) are given by the constrained Hamiltonian equations ∂H (q, p), ∂p ∂H (q, p) + A(q)λ + B(q)f, p˙ = − ∂q ∂H (q, p), e = B T (q) ∂p ∂H (q, p). 0 = AT (q) ∂p q˙ =

(8)

The constrained state space is therefore given as the following subset of the phase space: ∂H (q, p) = 0 . (9) Xc = (q, p) | AT (q) ∂p One way of proceeding is to eliminate the constraint forces, and to reduce the equations of motion to the constrained state space, leading (see [32] for details) to a portHamiltonian system (5). The structure matrix of this reduced port-Hamiltonian system satisfies the Jacobi identity if and only if the constraints (6) are holonomic [32]. An alternative way of approaching the system (8) is to formalize it directly as an implicit port-Hamiltonian system (with respect to a Dirac structure), as will be the topic of Section 2.3. 2.2. From port-based network modelling to port-Hamiltonian systems. In this subsection we take a different point of view by emphasizing how port-Hamiltonian systems directly arise from port-based network models of physical systems.

L1

L2 C Q

ϕ1 V

Figure 1. Controlled LC-circuit.

ϕ2


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In network models of complex physical systems the overall system is regarded as the interconnection of energy-storing elements via basic interconnection (balance) laws such as Newton’s third law or Kirchhoff’s laws, as well as power-conserving elements like transformers, kinematic pairs and ideal constraints, together with energydissipating elements [3], [14], [13]. The basic point of departure for the theory of port-Hamiltonian systems is to formalize the basic interconnection laws together with the power-conserving elements by a geometric structure, and to define the Hamiltonian as the total energy stored in the system. This is already illustrated by the following simple example. Example 2.3 (LCTG circuits). Consider a controlled LC-circuit (see Figure 1) consisting of two inductors with magnetic energies H1 (ϕ1 ), H2 (ϕ2 ) (ϕ1 and ϕ2 being the magnetic flux linkages), and a capacitor with electric energy H3 (Q) (Q being the charge). If the elements are linear then H1 (ϕ1 ) = 2L1 1 ϕ12 , H2 (ϕ2 ) = 2L1 2 ϕ22 and 1 Q2 . Furthermore let V = u denote a voltage source. Using Kirchhoff’s H3 (Q) = 2C laws one obtains the dynamical equations ⎡ ⎤ ⎤ ∂H ⎡ ⎤ ⎡ ⎡ ⎤ ˙ 0 1 −1 ⎢ ∂Q ⎥ 0 Q ⎣ϕ˙1 ⎦ = ⎣−1 0 0 ⎦ ⎢ ∂H ⎥ + ⎣1⎦ u, ⎣ ∂ϕ1 ⎦ 1 0 0 0 ∂H ϕ˙2 (10)

∂ϕ2 J

∂H y= ∂ϕ1

(= current through voltage source)

with H (Q, ϕ1 , ϕ2 ) := H1 (ϕ1 ) + H2 (ϕ2 ) + H3 (Q) the total energy. Clearly (by Tellegen’s theorem) the matrix J is skew-symmetric. In this way every LC-circuit with independent elements can be modelled as a portHamiltonian system. Similarly any LCTG-circuit with independent elements can be modelled as a port-Hamiltonian system, with J now being determined by Kirchhoff’s laws and the constitutive relations of the transformers T and gyrators G. 2.3. Dirac structures and implicit port-Hamiltonian systems. From a general modeling point of view physical systems are, at least in first instance, often described as DAE’s, that is, a mixed set of differential and algebraic equations. This stems from the fact that in network modeling the system under consideration is regarded as obtained from interconnecting simpler sub-systems. These interconnections usually give rise to algebraic constraints between the state space variables of the sub-systems; thus leading to implicit systems. Therefore it is important to extend the framework of port-Hamiltonian systems to the context of implicit systems; that is, systems with algebraic constraints. 2.3.1. Dirac structures. In order to give the definition of an implicit port-Hamiltonian system we introduce the notion of a Dirac structure, formalizing the concept of

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a power-conserving interconnection, and generalizing the notion of a structure matrix J (x) as encountered before. Let F be an -dimensional linear space, and denote its dual (the space of linear functions on F ) by F ∗ . The product space F × F ∗ is considered to be the space of power variables, with power defined by P = f ∗ |f ,

(f, f ∗ ) ∈ F × F ∗ ,

(11)

where f ∗ |f denotes the duality product. Often we call F the space of flows f , and F ∗ the space of efforts e, with the power of an element (f, e) ∈ F × F ∗ denoted as e|f . Example 2.4. Let F be the space of generalized velocities, and F ∗ be the space of generalized forces, then e|f is mechanical power. Similarly, let F be the space of currents, and F ∗ be the space of voltages, then e|f is electrical power. There exists on F × F ∗ the canonically defined symmetric bilinear form (f1 , e1 ), (f2 , e2 )F ×F ∗ := e1 |f2 + e2 |f1

(12)

for fi ∈ F , ei ∈ F ∗ , i = 1, 2. Definition 2.5 ([6], [8], [7]). A constant Dirac structure on F is a linear subspace D ⊂ F × F ∗ such that (13) D = D⊥ where ⊥ denotes the orthogonal complement with respect to the bilinear form , F ×F ∗ . It immediately follows that the dimension of any Dirac structure D on an -dimensional linear space is equal to . Furthermore, let (f, e) ∈ D = D ⊥ . Then by (12) 0 = (f, e), (f, e)F ×F ∗ = 2e | f .

(14)

Thus for all (f, e) ∈ D we obtain e | f = 0. Hence a Dirac structure D on F defines a power-conserving relation between the power variables (f, e) ∈ F × F ∗ , which moreover has maximal dimension. Remark 2.6. For many systems, especially those with 3-D mechanical components, the Dirac structure is actually modulated by the energy or geometric variables. Furthermore, the state space X is a manifold and the flows fS = −x˙ corresponding to energy-storage are elements of the tangent space Tx X at the state x ∈ X, while the efforts eS are elements of the co-tangent space Tx∗ X. Modulated Dirac structures often arise as a result of kinematic constraints. In many cases, these constraints will be configuration dependent, causing the Dirac structure to be modulated by the configuration variables, cf. Section 2.2.

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In general, a port-Hamiltonian system can be represented as in Figure 2. The port variables entering the Dirac structure D have been split in different parts. First, there are two internal ports. One, denoted by S, is corresponding to energy-storage and the other one, denoted by R, is corresponding to internal energy-dissipation (resistive elements). Second, two external ports are distinguished. The external port denoted by C is the port that is accessible for controller action. Also the presence of sources may be included in this port. Finally, the external port denoted by is the interaction port, defining the interaction of the system with (the rest of) its environment. R

S

D

C

Figure 2. Port-Hamiltonian system.

2.3.2. Energy storage port. The port variables associated with the internal storage port will be denoted by (fS , eS ). They are interconnected to the energy storage of the system which is defined by a finite-dimensional state space manifold X with coordinates x, together with a Hamiltonian function H : X → R denoting the energy. The flow variables of the energy storage are given by the rate x˙ of the energy variables x. Furthermore, the effort variables of the energy storage are given by the co-energy variables ∂H ∂x (x), resulting in the energy balance d ∂H ∂T H H = (x) x˙ = (x)x. ˙ dt ∂x ∂x

(15)

(Here we adopt the convention that ∂H ∂x (x) denotes the column vector of partial derivatives of H .) The interconnection of the energy storing elements to the storage port of the Dirac structure is accomplished by setting fS = −x, ˙ ∂H eS = (x). ∂x

(16)

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Hence the energy balance (15) can be also written as d ∂T H H = (x)x˙ = −eST fS . dt ∂x

(17)

2.3.3. Resistive port. The second internal port corresponds to internal energy dissipation (due to friction, resistance, etc.), and its port variables are denoted by (fR , eR ). These port variables are terminated on a static resistive relation R. In general, a static resistive relation will be of the form R(fR , eR ) = 0,

(18)

with the property that for all (fR , eR ) satisfying (18) eR | fR ≤ 0.

(19)

In many cases we may restrict ourselves to linear resistive relations. This means that the resistive port variables (fR , eR ) satisfy linear relations of the form Rf fR + Re eR = 0.

(20)

The inequality (19) corresponds to the square matrices Rf and Re satisfying the properties of symmetry and semi-positive definiteness Rf ReT = Re RfT ≥ 0,

(21)

together with the dimensionality condition rank[Rf |Re ] = dim fR . Without the presence of additional external ports, the Dirac structure of the portT f = 0 which leads to Hamiltonian system satisfies the power-balance eST fS + eR R d T fR ≤ 0. H = −eST fS = eR dt

(22)

An important special case of resistive relations between fR and eR occurs when the resistive relations can be expressed as an input-output mapping fR = −F (eR ), where the resistive characteristic F : Rmr → Rmr satisfies T eR F (eR ) ≥ 0,

eR ∈ Rmr .

(23)

˜ R , for some positive semiFor linear resistive elements this specializes to fR = −Re T ˜ ˜ definite symmetric matric R = R ≥ 0. 2.3.4. External ports. Now, let us consider in more detail the external ports to the system. We distinguish between two types of external ports. One is the control port C, with port variables (fC , eC ), which are the port variables which are accessible for controller action. Other type of external port is the interaction port , which denotes the interaction of the port-Hamiltonian system with its environment. The

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port variables corresponding to the interaction port are denoted by (fI , eI ). By taking both the external ports into account the power-balance extends to T T eST fS + eR fR + eC fC + eIT fI = 0

(24)

d T T fR + eC fC + eIT fI . H = eR dt

(25)

whereby (22) extends to

2.3.5. Port-Hamiltonian dynamics. The port-Hamiltonian system with state space X, Hamiltonian H corresponding to the energy storage port S, resistive port R, control port C, interconnection port , and total Dirac structure D will be succinctly denoted by = (X, H, R, C, , D). The dynamics of the port-Hamiltonian system is specified by considering the constraints on the various port variables imposed by the Dirac structure, that is (fS , eS , fR , eR , fC , eC , fI , eI ) ∈ D, ˙ eS = ∂H and to substitute in these relations the equalities fS = −x, ∂x (x). This leads to the implicitly defined dynamics ∂H − x(t), ˙ (26) (x(t)), fR (t), eR (t), fC , (t), eC (t), fI (t), eI (t) ∈ D ∂x with fR (t), eR (t) satisfying for all t the resistive relation (18): R(fR , eR ) = 0.

(27)

In many cases of interest the dynamics (26) will constrain the allowed states x, depending on the values of the external port variables (fC , eC ) and (fI , eI ). Thus in an equational representation port-Hamiltonian systems generally will consist of a mixed set of differential and algebraic equations (DAEs). Example 2.7 (General LC- circuits). Consider an LC-circuit with general network topology. Kirchhoff’s current and voltage laws take the general form AL T IL + AC T IC + AP T IP = 0, VL = AL λ,

VC = AC λ,

VP = AP λ

for some matrices AL , AC , AS . Here IL , IC , IP denote the currents, respectively through the inductors, capacitors and external ports. Likewise, VL , VC , VP denote the voltages over the inductors, capacitors and external ports. Kirchhoff’s current and voltage laws define a Dirac structure between the flows and efforts: ˙ −φ, ˙ IP ), f = (IC , VL , IP ) = (−Q, ∂H ∂H e = (VC , IL , VP ) = , , VP ∂Q ∂φ

1348


with Hamiltonian H (φ, Q) the total energy. This leads the to port-Hamiltonian system in implicit form −φ˙ = AL λ, ∂H = AC λ, ∂Q VP = AP λ, ∂H ˙ + AP T IP . 0 = AL T − AC T Q ∂φ Example 2.8 (Electro-mechanical system). Consider the dynamics of an iron ball in the magnetic field of a controlled inductor: The port-Hamiltonian description of this V R I

g

q

φ

m

Figure 3. Magnetically levitated ball.

system (with q the height of the ball, p the vertical momentum, and ϕ the magnetic flux of the inductor) is given as ⎡ ⎤ ⎡ ⎤ ⎡ ⎡ ⎤ ⎤ ∂H 0 1 0 ⎢ ∂q ⎥ q˙ 0 ⎣p˙ ⎦ = ⎣−1 0 0 ⎦ ⎢ ∂H ⎥ + ⎣0⎦ V , ⎣ ∂p ⎦ ϕ˙ 1 0 0 − R1 ∂H (28) ∂ϕ

∂H I= . ∂ϕ This is a typical example of a system where the coupling between two different physical domains (mechanical and magnetic) takes place via the Hamiltonian H (q, p, ϕ) = mgq +

p2 ϕ2 + 2m 2k1 (1 −

q k2 )


1349

where the last term depends both on a magnetic variable (in this case ϕ) and a mechanical variable (in this case the height q). 2.4. Input-state-output port-Hamiltonian systems. An important special case of port-Hamiltonian systems is the class of input-state-output port-Hamiltonian systems, where there are no algebraic constraints on the state space variables, and the flow and effort variables of the resistive, control and interaction port are split into conjugated input–output pairs. Input–state–output port-Hamiltonian systems without interaction port are of the form x˙ = [J (x) − R(x)] y = g T (x)

∂H (x) + g(x)u, ∂x

∂H (x) ∂x

(29)

where u, y are the input–output pairs corresponding to the control port C. Here the matrix J (x) is skew-symmetric, while the matrix R(x) = R T (x) ≥ 0 specifies the ˜ R (x) for some linear resistive resistive structure, and is given as R(x) = gRT (x)Rg T ˜ R , R˜ = R˜ ≥ 0, with gR representing the input matrix correrelation fR = −Re sponding to the resistive port. The underlying Dirac structure of the system is then given by the graph of the skew-symmetric linear map ⎛ ⎞ −J (x) −gR (x) −g(x) ⎝ g T (x) 0 0 ⎠. (30) R T 0 0 g (x)

3. Control by interconnection of port-Hamiltonian systems The basic property of port-Hamiltonian systems is that the power-conserving interconnection of any number of port-Hamiltonian systems is again a port-Hamiltonian system. To be explicit, consider two port-Hamiltonian systems A and B with Dirac structures DA and DB and Hamiltonians HA and HB , defined on state spaces XA , respectively XB . For convenience, split the ports of the Dirac structures DA and DB into the internal energy storage ports and all remaining external ports whose port-variables are denoted respectively by fA , eA and fB , eB . Now, consider any interconnection Dirac structure DI involving the port-variables fA , eA , fB , eB possibly together with additional port-variables fI , eI . Then the interconnection of the systems A and B via DI is again a port-Hamiltonian system with respect to the composed Dirac structure DA DI DB , involving as port-variables the internal storage port-variables of DA and DB together with the additional port-variables FI , eI . For details we refer to [5], [34], [30].

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Furthermore, the state space of the interconnected port-Hamiltonian system is the product of the two state spaces XA × XB , while its Hamiltonian is simply the sum HA + HB of the two Hamiltonians. This basic statement naturally extends to the interconnection of any number of port-Hamiltonian systems via an interconnection Dirac structure. Control by port-interconnection is based on designing a controller system which is interconnected to the control port with port-variables (fC , eC ). In principle this implies that we only consider collocated control, where the controller will only use the information about the plant port-Hamiltonian system that is contained in the conjugated pairs (fC , eC ) of port variables of the control port, without using additional information about the plant (e.g. corresponding to observation on other parts of the plant system). In the second place, we will restrict attention to controller systems which are themselves also port-Hamiltonian systems. There are two main reasons for this. One is that by doing so the closed-loop system is again a port-Hamiltonian system, allowing to easily ensure some desired properties. Furthermore, it will turn out that the port-Hamiltonian framework suggests useful ways to construct port-Hamiltonian controller systems. Second reason is that port-Hamiltonian controller systems allow in principle for a physical system realization (thus linking to passive control and systems design) and physical interpretation of the controller action. Since we do not know the environment (or only have very limited information about it), but on the other hand, the system will interact with this unknown environment, the task of the controller is often two-fold: 1) to achieve a desired control goal (e.g. setpoint regulation or tracking) if the interaction with the environment is marginal or can be compensated, 2) to make sure that the controlled system has a desired interaction behavior with its environment. It is fair to say that up to now the development of the theory of control of port-Hamiltonian systems has mostly concentrated on the second aspect (which at the same time, is often underdeveloped in other control theories). Most successful approaches to deal with the second aspect of the control goal are those based on the concept of “passivity”, such as dissipativity theory [38], impedance control [13] and Intrinsically Passive Control (IPC) [36]. In fact, the port-Hamiltonian control theory can be regarded as an enhancement to the theory of passivity, making a much closer link with complex physical systems modeling at one hand and with the theory of dynamical systems (in particular, Hamiltonian dynamics) at the other hand. As said above, we will throughout consider controller systems which are again port-Hamiltonian systems. We will use the same symbols as above for the internal and external ports and port-variables of the controller port-Hamiltonian system, with an added overbar¯or a superscript c in order to distinguish it from the plant system. (The interaction port of the controller system may be thought of as an extra possibility for additional controller action (outer-loop control).) In order to further distinguish the plant system and the controller we denote the state space of the plant system by Xp with coordinates xp , the Dirac structure by Dp and its Hamiltonian by Hp , while we will denote the state space manifold of the controller system by Xc with coordinates xc , its Dirac structure by Dc and its Hamiltonian by Hc : Xc → R. The interconnection


1351

of the plant port-Hamiltonian system with the controller port-Hamiltonian system is obtained by equalizing the port variables at the control port by fC = −f¯C , eC = e¯C

(31)

where f¯C , e¯C denote the control port variables of the controller system. Here, the minus sign is inserted to have a uniform notion of direction of power flow. Clearly, T f + e¯ T f¯ = 0. this ’synchronizing’ interconnection is power-conserving, that is eC C C C Remark 3.1. A sometimes useful alternative is the gyrating power-conserving interconnection fC = −e¯C , eC = f¯C .

(32)

In fact, the standard feedback interconnection can be regarded to be of this type. For both interconnection constraints it directly follows from the theory of composition of Dirac structures that the interconnected (closed-loop) system is again a port-Hamiltonian system with Dirac structure determined by the Dirac structures of the plant PH system and the controller PH system. The resulting interconnected PH system has state space Xp × Xc , Hamiltonian Hp + Hc , resistive ports (fR , eR , f¯R , e¯R ) and interaction ports (fI , eI , f¯I , e¯I ), satisfying the power-balance d T T ¯ (Hp + Hc ) = eR fR + eIT fI + e¯IT f¯I ≤ eIT fI + e¯IT f¯I fR + e¯R dt

(33)

T f ≤ 0 and e¯ T f¯ ≤ 0. Hence we immediately recover the state since both eR R R R space formulation of the passivity theorem, see e.g. [31], if Hp and Hc are both nonnegative, implying that the plant and the controller system are passive (with respect to their controller and interaction ports and storage functions Hp and Hc ), then also the closed -loop system is passive (with respect to the interaction ports and storage function Hp + Hc .) Furthermore, due to the Hamiltonian structure, we can go beyond the passivity theorem, and we can derive conditions which ensure that we can passify and/or stabilize plant port-Hamiltonian systems for which the Hamiltonian Hp does not have a minimum at the desired equilibrium.

3.1. Stabilization by Casimir generation. What does the power-balance (33)mean for the stability properties of the closed-loop system, and how can we design the controller port-Hamiltonian system in such a way that the closed-loop system has desired stability properties? Let us first consider the stability of an arbitrary portHamiltonian system = (X, H, R, C, , D) without control or interaction ports,

1352


that is, an autonomous port-Hamiltonian system = (X, H, R, D). Clearly, the power-balance (33) reduces to d T fR ≤ 0. H = eR dt

(34)

Hence we immediately infer by standard Lyapunov theory that if x ∗ is a minimum of the Hamiltonian H then it will be a stable equilibrium of the autonomous portHamiltonian system = (X, H, R, D), which is actually asymptotically stable if T f is negative definite outside x ∗ , or alternatively if some sort the dissipation term eR R of detectability condition is satisfied, guaranteeing asymptotic stability by the use of LaSalle’s Invariance principle (see for details e.g. [31]). However, what can we say if x ∗ is not a minimum of H , and thus we cannot directly use H as a Lyapunov function? A well-known method in Hamiltonian systems, sometimes called the EnergyCasimir method, is to use in the Lyapunov analysis next to the Hamiltonian other conserved quantities (dynamical invariants) which may be present in the system. Indeed, if we may find other conserved quantities then candidate Lyapunov functions can be sought within the class of combinations of the Hamiltonian H and those conserved quantities. In particular, if we can find a conserved quantity C : X → R such that V := H + C has a minimum at the desired equilibrium x ∗ then we can still infer stability or asymptotic stability by replacing (34) by d T fR ≤ 0 V = eR dt

(35)

and thus using V as a Lyapunov function. For the application of the Energy-Casimir method one may distinguish between two main cases. First situation occurs if the desired equilibrium x ∗ is not a stationary point of H , and one looks for a conserved quantity C such that H + C has a minimum at x ∗ . This for example happens in the case that the desired set-point x ∗ is not an equilibrium of the uncontrolled system, but only a controlled equilibrium of the system. Second situation occurs when x ∗ is a stationary point of H , but not a minimum. Functions that are conserved quantities of the system for every Hamiltonian are called Casimir functions or simply Casimirs. Casimirs are completely characterized by the Dirac structure of the port-Hamiltonian system. Indeed, a function C : X → R is a Casimir function of the autonomous port-Hamiltonian system (without energy T dissipation) = (X, H, D) if and only if the gradient vector e = ∂∂xC satisfies eT fS = 0

for all fS for which there exists eS such that (fS , eS ) ∈ D.

(36)

Indeed, (36) is equivalent to d ∂T C ∂T C C= (x(t))x(t) ˙ = (x(t))fS = eT fS = 0 dt ∂x ∂x

(37)


1353

for every port-Hamiltonian system (X, H, D) with the same Dirac structure D. By the generalized skew-symmetry of the Dirac structure (36) is equivalent to the reT quirement that e = ∂∂xC satisfies (0, e) ∈ D. Similarly, we define a Casimir function for a port-Hamiltonian system with dissipation = (X, H, R, D) to be any function C : X → R satisfying (0, e, 0, 0) ∈ D.

(38)

d ∂T C ∂T C C= (x(t))x(t) ˙ = (x(t))fp = eT fp = 0 dt ∂x ∂x

(39)

Indeed, this will imply that

for every port-Hamiltonian system (X, H, R, D) with the same Dirac structure D. (In fact by definiteness of the resistive structures the satisfaction of (39) for a particular resistive structure R implies the satisfaction for all resistive structures R.) Now let us come back to the design of a controller port-Hamiltonian system such that the closed-loop system has desired stability properties. Suppose we want to stabilize the plant port-Hamiltonian system (Xp , Hp , R, C, Dp ) around a desired equilibrium xp∗ . We know that for every controller port-Hamiltonian system the closed-loop system satisfies d T T ¯ fR ≤ 0. fR + e¯R (40) (Hp + Hc ) = eR dt What if x ∗ is not a minimum for Hp ? A possible strategy is to generate Casimir functions C(xp , xc ) for the closed-loop system by choosing the controller port-Hamiltonian system in an appropriate way. Thereby we generate candidate Lyapunov functions for the closed-loop system of the form V (xp , xc ) := Hp (xp ) + Hc (xc ) + C(xp , xc ) where the controller Hamiltonian function Hc : Xc → R still has to be designed. The goal is thus to construct a function V as above in such a way that V has a minimum at (xp∗ , xc∗ ) where xc∗ still remains to be chosen. This strategy thus is based on finding all the achievable closed-loop Casimirs. Furthermore, since the closed-loop Casimirs are based on the closed-loop Dirac structures, this reduces to finding all the achievable ¯ closed-loop Dirac structures D D. Another way to interpret the generation of Casimirs for the closed-loop system is to look at the level sets of the Casimirs as invariant submanifolds of the combined plant and controller state space Xp × Xc . Restricted to every such invariant submanifold (part of) the controller state can be expressed as a function of the plant state, whence the closed-loop Hamiltonian restricted to such an invariant manifold can be seen as a

1354


shaped version of the plant Hamiltonian. To be explicit (see e.g. [31], [24], [25] for details) suppose that we have found Casimirs of the form xci − Fi (xp ),

i = 1, . . . , np

where np is the dimension of the controller state space, then on every invariant manifold xci − Fi (xp ) = αi , i = 1, . . . , np , where α = (α1 , . . . , αnp ) is a vector of constants depending on the initial plant and controller state, the closed-loop Hamiltonian can be written as Hs (xp ) := Hp (xp ) + Hc (F (xp ) + α), where, as before, the controller Hamiltonian Hc still can be assigned. This can be regarded as shaping the original plant Hamiltonian Hp to a new Hamiltonian Hs . 3.2. Port Control. In broad terms, the Port Control problem is to design, given the plant port-Hamiltonian system, a controller port-Hamiltonian system such that the behavior at the interaction port of the plant port-Hamiltonian system is a desired one, or close to a desired one. This means that by adding the controller system we seek to shape the external behavior at the interaction port of the plant system. If the desired external behavior at this interaction port is given in input–output form as a desired (dynamic) impedance, then this amounts to the Impedance Control problem as introduced and studied by Hogan and co-workers [13]; see also [36] for subsequent developments. The Port Control problem, as stated in this generality, immediately leads to two fundamental questions: 1). Given the plant PH system, and the controller PH system to be arbitrarily designed, what are the achievable behaviors of the closed-loop system at the interaction port of the plant? 2). If the desired behavior at the interaction port of the plant is not achievable, then what is the closest achievable behavior? Of course, the second question leaves much room for interpretation, since there is no obvious interpretation of what we mean by ’closest behavior’. Also the first question in its full generality is not easy to answer, and we shall only address an important subproblem. An obvious observation is that the desired behavior, in order to be achievable, needs to be the port behavior of a PH system. This leads already to the problem of characterizing those external behaviors which are port behaviors of port-Hamiltonian systems. Secondly, the Port Control problem can be split into a number of subproblems. Indeed, we know that the closed-loop system arising from interconnection of the plant PH system with the controller PH system is specified by a Hamiltonian which is just the sum of the plant Hamiltonian and the controller Hamiltonian, and a resistive structure which is the “product” of the resistive structure of the plant and of the controller system, together with a Dirac structure which is the composition of the plant Dirac structure and the controller Dirac structure. Therefore an important subproblem is again to characterize the achievable closed-loop Dirac structures. On the other hand, a fundamental problem in addressing the Port Control problem in general theoretical terms is the lack of a systematic way to specify ’desired behavior’.


1355

The problem of Port Control is to determine the controller system in such a way that the port behavior in the port variables fI , eI is a desired one. In this particular (simple and linear) example the desired behavior can be quantified e.g. in terms of a desired stiffness and damping of the closed-loop system, which is easily expressed in terms of the closed-loop transfer function from fI to eI . Of course, on top of the requirements on the closed-loop transfer function we would also require internal stability of the closed-loop system. For an appealing example of port control of port-Hamiltonian systems within a context of hydraulic systems we refer to [15]. 3.3. Energy Control. Consider two port-Hamiltonian systems i (without internal dissipation) in input–state–output form ∂Hi + gi (xi )ui , ∂xi ∂Hi , i = 1, 2, yi = giT (xi ) ∂xi

x˙i = Ji (xi )

(41)

d both satisfying the power-balance dt Hi = yiT ui . Suppose now that we want to transfer the energy from the port-Hamiltonian system 1 to the port-Hamiltonian system 2 , while keeping the total energy H1 + H2 constant. This can be done by using the following output feedback u1 0 −y1 y2T y1 = . (42) u2 y2 y2 y1T 0

Since the matrix in (42) is skew-symmetric it immediately follows that the closed-loop system composed of systems 1 and 2 linked by the power-conserving feedback is d energy-preserving, that is dt (H1 + H2 ) = 0. However, if we consider the individual energies then we notice that d H1 = −y1T y1 y2T y2 = − y1 2 y2 2 ≤ 0 dt

(43)

implying that H1 is decreasing as long as y1 and y2 are different from 0. Conversely, as expected since the total energy is constant, d H2 = y2T y2 y1T y1 = y2 2 y1 2 ≥ 0 dt

(44)

implying that H2 is increasing at the same rate. In particular, if H1 has a minimum at the zero equilibrium, and 1 is zero-state observable, then all the energy H1 of 1 will be transferred to 2 , provided that y2 is not identically zero (which again can be guaranteed by assuming that H2 has a minimum at the zero equilibrium, and that 2 is zero-state observable). If there is internal energy dissipation, then this energy transfer mechanism still works. However, the fact that H2 grows or not will depend on the balance between the energy delivered by 1 to 2 and the internal loss of energy in 2 due to dissipation.

1356


We conclude that this particular scheme of power-conserving energy transfer is accomplished by a skew-symmetric output feedback, which is modulated by the values of the output vectors of both systems. Of course this raises, among others, the question of the efficiency of the proposed energy-transfer scheme, and the need for a systematic quest of similar power-conserving energy-transfer schemes. We refer to [9] for a similar but different energy-transfer scheme directly motivated by the structure of the example (control of a snakeboard). 3.4. Achievable closed-loop Dirac structures. In all the control problems discussed above the basic question comes up what are the achievable closed-loop Dirac structures based on a given plant Dirac structure and a controller Dirac structure, which still is to be determined. Theorem 3.2 ([5]). Given any plant Dirac structure Dp , a certain interconnected D = Dp Dc can be achieved by a proper choice of the controller Dirac structure Dc if and only if the following two equivalent conditions are satisfied: Dp0 ⊂ D 0 , D π ⊂ Dpπ where Dp0 := {f1 , e1 ) | (f1 , e1 , 0, 0) ∈ Dp }, Dpπ := {(f1 , e1 ) | there exists (f2P , e2P ) with (f1 , e1 , f2P , e2P ) ∈ Dp }, D 0 := {(f1 , e1 ) | (f1 , e1 , 0, 0) ∈ D}, D π := {(f1 , e1 ) | there exists (f3 , e3 ) with (f1 , e1 , f3 , e3 ) ∈ D}. An important application of the above theorem concerns the characterization of Casimir functions which can be achieved by interconnecting a given plant portHamiltonian system with a controller port-Hamiltonian system.

4. Distributed-parameter port-Hamiltonian systems The treatment of infinite-dimensional Hamiltonian systems in the literature is mostly confined to systems with boundary conditions such that the energy exchange through the boundary is zero. On the other hand, in many applications the interaction with the environment (e.g. actuation or measurement) will actually take place through the boundary of the system. In [35] a framework has been developed to represent classes of physical distributed-parameter systems with boundary energy flow as infinite-dimensional port-Hamiltonian systems. It turns out that in order to allow the inclusion of boundary variables in distributed-parameter systems the concept of (an


1357

infinite-dimensional) Dirac structure provides again the right type of generalization with respect to the existing framework [23] using Poisson structures. As we will discuss in the next three examples, the port-Hamiltonian formulation of distributed-parameter systems is closely related to the general framework for describing basic distributed-parameter systems as systems of conservation laws, see e.g. [11], [37]. Example 4.1 (Inviscid Burger’s equation). The viscous Burger’s equation is a scalar parabolic equation defined on a one-dimensional spatial domain (interval) Z = [a, b] ⊂ R, with the state variable α(t, z) ∈ R, z ∈ Z, t ∈ I , where I is an interval of R, satisfying the partial differential equation ∂α ∂α ∂ 2α +α − ν 2 = 0. ∂t ∂z ∂z

(45)

The inviscid (ν = 0) Burger’s equations may be alternatively expressed as ∂ ∂α + β=0 ∂t ∂z

(46)

where the state variable α(t, z) is called the conserved quantity and the function 2 β := α2 the flux variable. Eq. (46) is called a conservation law, since by integration one obtains the balance equation b d α dz = β(a) − β(b). (47) dt a Furthermore, according to the framework of Irreversible Thermodynamics [27], one may express the flux β as a function of the generating force which is the variational derivative of some functional H (α) of the state variable. The variational derivative δH δα of a functional H (α) is uniquely defined by the requirement b δH H (α + εη) = H (α) + ε η dz + O(ε2 ) (48) a δα for any ε ∈ R and any smooth function η(z, t) such that α + εη satisfies the same boundary conditions as α [23]. For the inviscid Burger’s equation one has β = δH δα , where b 3 α H (α) = dz. (49) a 6 Hence the inviscid Burger’s equation may be also expressed as ∂α ∂ δH =− . ∂t ∂z δα

(50)

This defines an infinite-dimensional Hamiltonian system in the sense of [23] with ∂ respect to the skew-symmetric operator ∂z that is defined on the functions with support contained in the interior of the interval Z.

1358


From this formulation one derives that the Hamiltonian H (α) is another conserved quantity. Indeed, by integration by parts b δH d ∂ δH 1 2 (51) H = ·− dz = β (a) − β 2 (b) . dt ∂z δα 2 a δα We note that the right-hand side is a function of the flux variables evaluated at the boundary of the spatial domain Z. The second example consists of a system of two conservations laws, corresponding to the case of two physical domains in interaction. Example 4.2 (The p-system, cf. [11], [37]). The p-system is a model for e.g. a one-dimensional gas dynamics. Again, the spatial variable z belongs to an interval Z ⊂ R, while the dependent variables are the specific volume v(t, z) ∈ R+ , the velocity u(t, z) and the pressure functional p(v) (which for instance in the case of an ideal gas with constant entropy is given by p(v) = Av −γ where γ ≥ 1). The p-system is then defined by the following system of partial differential equations ∂v ∂u − = 0, ∂t ∂z ∂u ∂ p(v) + =0 ∂t ∂z

(52)

representing respectively conservation of mass and of momentum. By defining the state vector as α(t, z) = (v, u)T , and the vector-valued flux β(t, z) = (−u, p(v))T the p-system is rewritten as ∂α ∂ + β = 0. (53) ∂t ∂z Again, according to the framework of Irreversible Thermodynamics, the flux vector may be written as function of the variational derivatives of some functional. Inb deed, consider the energy functional H (α) = a H(v, u)dz where the energy density H(v, u) is given as the sum of the internal energy and the kinetic energy densities H(v, u) = U(v) +

u2 2

(54)

with −U(v) a primitive function of the pressure. (Note that for simplicity the mass density has been set equal to 1, and hence no difference is made between the velocity and the momentum.) The flux vector β may be expressed in terms of the variational derivatives of H as δH 0 −1 δv β= . (55) δH −1 0 δu

The anti-diagonal matrix represents the canonical coupling between two physical domains: the kinetic and the potential (internal) domain. Thus the variational derivative

1359


of the total energy with respect to the state variable of one domain generates the flux variable for the other domain. Combining eqns. (53) and (55), the p-system may thus be written as the Hamiltonian system ∂α δH ∂ 1 0 − ∂z δα1 ∂t = . (56) δH ∂ ∂α2 − 0 δα ∂z ∂t 2 Using again integration by parts, one may derive the following energy balance equation: d H = β1 (a)β2 (a) − β1 (b)β2 (b). (57) dt Notice again that the right-hand side of this power-balance equation is a quadratic function of the fluxes at the boundary of the spatial domain. The last example is the vibrating string. It is again a system of two conservation laws representing the canonical interdomain coupling between the kinetic energy and the elastic potential energy. However in this example the classical choice of the state variables leads to express the total energy as a function of some of the spatial derivatives of the state variables. Example 4.3 (Vibrating string). Consider an elastic string subject to traction forces at its ends, with spatial variable z ∈ Z = [a, b] ⊂ R. Denote by u(t, z) the displacement of the string and the velocity by v(t, z) = ∂u ∂t . Using the vector of state variables x(t, z) = (u, v)T , the dynamics of the vibrating string is described by the system of partial differential equations ∂x v (58) = 1 ∂ ∂u T ∂t μ ∂z ∂z where the first equation is simply the definition of the velocity and the second one is Newton’s second law. Here T denotes the elasticity modulus, and μ the mass density. The total energy is H (x) = U (u) + K(v), where the elastic potential energy U is a function of the strain ∂u ∂z (t, z)

b

U (u) = a

1 T 2

∂u ∂z

2 (59)

dz

and the kinetic energy K depends on the velocity v(t, z) = b 1 μv(t, z)2 dz. K(v) = 2 a Thus the total system (58) may be expressed as δH 1 0 ∂x δu μ = δH − μ1 0 ∂t δv

∂u ∂t

as (60)

(61)

1360


∂u δU ∂ δH δK where δH δu = δu = − ∂z T ∂z is the elastic force and δv = δv = μv is the momentum. In this formulation, the system is not anymore expressed as a system of conservation laws since the time-derivative of the state variables is a function of the variational derivatives of the energy directly, and not the spatial derivative of a function of the variational derivatives as before. Instead of being a simplification, this reveals a drawback for the case of non-zero energy flow through the boundary of the spatial domain. Indeed, in this case the variational derivative has to be completed by a boundary term since the Hamiltonian functional depends on the spatial derivatives of the state. For example, in the computation of the variational derivative of the elastic potential energy U one obtains by integration by parts that U (u + εη) − U (u) equals b ∂u ∂u b ∂ T η dz + ε η T + O(ε2 ) (62) −ε ∂z ∂z a a ∂z and the second term in this expression constitutes an extra boundary term. Alternatively we now formulate the vibrating string as a system of two conservation laws. Take as alternative vector of state variables α(t, z) = (ε, p)T , where ε denotes the strain α1 = ε = ∂u ∂z and p denotes the momentum α2 = p = μv. Recall that in these variables the total energy is written as b 1 1 2 2 T α1 + α2 dz (63) H0 = μ a 2 and directly depends on the state variables and not on their spatial derivatives. Fur0 thermore, one defines the flux variables to be the stress β1 = δH δα1 = T α1 and the

velocity β2 =

δH0 δα1

=

α2 μ.

In matrix notation, the flux vector β is thus expressed as a

0 function of the variational derivatives δH δα by 0 −1 δH0 . β= −1 0 δα

(64)

Hence the vibrating string may be alternatively expressed by the system of two conservation laws ∂ δH ∂α 0 ∂z 0 = ∂ (65) 0 ∂t δα ∂z satisfying the power balance equation (57). 4.1. Systems of two conservation laws in interaction. Let us now consider the general class of distributed-parameter systems consisting of two conservation laws with the canonical coupling as in the above examples of the p-system and the vibrating string. Let the spatial domain Z ⊂ Rn be an n-dimensional smooth manifold with smooth (n − 1)-dimensional boundary ∂Z. Denote by k (Z) the vector space of (differential) k-forms on Z (respectively by k (∂Z) the vector space of k-forms on ∂Z).

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! Denote furthermore by = k≥0 k (Z) the algebra of differential forms over Z and recall that it is endowed with an exterior product ∧ and an exterior derivation d. Definition 4.4. A system of conservation laws is defined by a set of conserved quantities " αi ∈ ki (Z), i ∈ {1, . . . , N } where N ∈ N, ki ∈ N, defining the state space X = i=1,...,N ki (Z), and satisfying a set of conservation laws ∂αi + dβi = gi ∂t

(66)

where βi ∈ ki −1 (Z) denote the set of fluxes and gi ∈ ki (Z) denote the set of distributed interaction forms. In general, the fluxes βi are defined by so-called closure equations (67) βi = J (αi , z) , i = 1, . . . , N leading to a closed form for the dynamics of the conserved quantities αi . The integral form of the conservation laws yields the following balance equations d αi + (68) βi = gi dt S ∂S S for any surface S ⊂ Z of dimension equal to the degree of αi . Remark 4.5. A common case is that Z = R3 and that the conserved quantities are 3-forms, that is, the balance equation is evaluated on volumes of the 3-dimensional space. In this case () takes in vector calculus notation the familiar form ∂αi (z, t) + divz βi = gi , ∂t

i = 1, . . . , n.

(69)

However, systems of conservation laws may correspond to differential forms of any degree. Maxwell’s equations are an example where the conserved quantities are differential forms of degree 2. In the sequel, as in our examples sofar, we consider a particular class of systems of conservation laws where the closure equations are such that fluxes are linear functions of the variational derivatives of the Hamiltonian functional. First recall the general definition of the variational derivative of a functional H (α) with respect to the differential form α ∈ p (Z) (generalizing the definition given before). Definition 4.6. Consider a densityfunction H : p (Z) × Z → n (Z) where p ∈ {1, . . . , n}, and denote by H := Z H ∈ R the associated functional. Then the n−p (Z) which satisfies for all α ∈ p (Z) uniquely defined differential form δH δα ∈ and ε ∈ R δH H (α + εα) = ∧ α + O ε2 H (α) + ε Z Z δα is called the variational derivative of H with respect to α ∈ p (Z).

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Definition 4.7. Systems of two conservation laws with canonical interdomain coupling are systems involving a pair of conserved quantities αp ∈ p (Z) and αq ∈ q (Z), differential forms on the n-dimensional spatial domain Z of degree p and q respectively, satisfying p+q = n+1 (’complementarity of degrees’). The closure equations generated by a Hamiltonian density function H : p (Z)×q (Z)×Z → n (Z) resulting in the Hamiltonian H := Z H ∈ R are given by δH 0 (−1)r βp δαp =ε (70) δH βq 1 0 δα q

where r = pq +1, ε ∈ {−1, +1}, depending on the sign convention of the considered physical domain. Define the vector of flow variables to be the time-variation of the state, and the effort variables to be the variational derivatives ∂αp δH fp ep δα ∂t = ∂α = δHp . , (71) q fq eq δα ∂t

q

Their product equals again the time-variation of the Hamiltonian dH = (ep ∧ fp + eq ∧ fq ). dt Z

(72)

Using the conservation laws (4.5) for gi = 0, the closure relations (70) and the properties of the exterior derivative d and Stokes’ theorem, one obtains dH = εβq ∧ (−dβp ) + (−1)r βp ∧ ε(−dβq ) dt Z (73) = −ε βq ∧ dβp + (−1)q βq ∧ dβp = −ε βq ∧ βp . Z

∂Z

Finally, as before we define the power-conjugated pair of flow and effort variables on the boundary as the restriction of the flux variables to the boundary ∂Z of the domain Z: f∂ βq |∂Z = . (74) e∂ βp |∂Z On the total space of power-conjugated variables, that is, the differential forms (fp , ep ) and (fq , eq ) on the domain Z and the differential forms (f∂ , e∂ ) defined on the boundary ∂Z, one defines an interconnection structure by Eqn. (74) together with fq eq 0 (−1)r d =ε . (75) d 0 fp ep This interconnection can be formalized as a special type of Dirac structure, called Stokes–Dirac structure, leading to the definition of distributed-parameter port-Hamiltonian systems [35].


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5. Concluding remarks We have surveyed some of the recently developed theory of port-Hamiltonian systems; for further applications towards modeling, analysis, simulation and control we refer to the literature cited below. From the geometric point of view many questions regarding port-Hamiltonian systems are waiting to be investigated. A theory of symmetry and reduction of portHamiltonian systems has been explored in [29], [1], while some questions concerning integrability of Dirac structures have been studied in [7]. A main question for distributed-parameter port-Hamiltonian systems concerns the relation with variational calculus.

References [1] Blankenstein, G., van der Schaft, A. J., Symmetry and reduction in implicit generalized Hamiltonian systems. Rep. Math. Phys. 47 (2001), 57–100. [2] Bloch, A. M. and Crouch, P. E., Representations of Dirac structures on vector spaces and nonlinear LC circuits. In Differential geometry and control (ed. by G. Ferreyra, R. Gardner, H. Hermes, H. Sussmann), Proc. Sympos. Pure Math. 64, Amer. Math. Soc., Providence, RI, 1999, 103–117. [3] Breedveld, P. C., Physical systems theory in terms of bond graphs. PhD thesis, University of Twente, Faculty of Electrical Engineering, 1984. [4] R. W. Brockett, Control theory and analytical mechanics. In Geometric Control Theory (ed. by C. Martin, R. Hermann), Lie Groups: History, Frontiers and Applications VII, Math. Sci. Press, Brookline 1977, 1–46. [5] Cervera, J., van der Schaft, A. J., Banos, A., Interconnection of port-Hamiltonian systems and composition of Dirac structures. Automatica, submitted. [6] Courant, T. J., Dirac manifolds. Trans. Amer. Math. Soc. 319 (1990), 631–661. [7] Dalsmo, M., and van der Schaft, A. J., On representations and integrability of mathematical structures in energy-conserving physical systems. SIAM J. Control Optim. 37 (1999), 54–91. [8] Dorfman, I., Dirac Structures and Integrability of Nonlinear Evolution Equations. John Wiley, Chichester 1993. [9] Duindam, V., Blankenstein, G., Stramigioli, S., Port-Based Modeling and Analysis of Snakeboard Locomotion. In Proceedings 16th International Symposium on Mathematical Theory of Networks and Systems (MTNS2004), Leuven 2004. [10] Escobar, G., van der Schaft, A. J., and Ortega, R., A Hamiltonian viewpoint in the modelling of switching power converters. Automatica 35 (Special Issue on Hybrid Systems) (1999), 445–452. [11] Godlewsky, E., and Raviart, P., Numerical Approximation of Hyperbolic Systems of Conservation Laws. Appl. Math. Sci. 118, Springer-Verlag, New York 1996. [12] Golo, G., van der Schaft, A. J., Breedveld, P. C., Maschke, B. M., Hamiltonian formulation of bond graphs. In Nonlinear and Hybrid Systems in Automotive Control (ed. by R. Johansson, A. Rantzer), Springer-Verlag, London 2003, 351–372.

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[13] Hogan, N., Impedance Control: An approach to manipulation. J. Dyn. Systems, Measurements Control 107 (1985), 1–24. [14] Karnopp, D. C., Margolis, D. L., and Rosenberg, R. C., System Dynamics, A Unified Approach. John Wiley and Sons, 1990. [15] Kugi, A., Kemmetmüller, W., Impedance control of Hydraulic piston actuators. In Proceedings 6th IFAC Symposium on Nonlinear Control Systems (NOLCOS 2004), 1–3 September, Stuttgart. [16] Marsden, J. E., and Ratiu, T. S., Introduction to Mechanics and Symmetry. Texts Appl. Math. 17, Springer-Verlag, New York 1994. [17] Maschke, B. M., and van der Schaft, A. J., Port-controlled Hamiltonian systems: Modelling origins and system-theoretic properties. In Proc. 2nd IFAC NOLCOS, Bordeaux 1992, 282–288. [18] Maschke, B. M., and van der Schaft, A. J., System-theoretic properties of port-controlled Hamiltonian systems. In Systems and Networks: Mathematical Theory and Applications, Vol. II, Akademie-Verlag, Berlin 1994, 349–352. [19] Maschke, B. M., van der Schaft, A. J., and Breedveld, P. C., An intrinsic Hamiltonian formulation of network dynamics: non-standard Poisson structures and gyrators. J. Franklin Inst. 329 (1992), 923–966. [20] Maschke, B. M., van der Schaft, A. J., and Breedveld, P. C., An intrinsic Hamiltonian formulation of the dynamics of LC-circuits. IEEE Trans. Circuits Systems I Fund. Theory Appl. 42 (1995), 73–82. [21] Maschke, B. M., Ortega, R., and van der Schaft, A. J., Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation. IEEE Trans. Automat. Control 45 (2000), 1498–1502. [22] Neimark, J. I., and Fufaev, N. A., Dynamics of Nonholonomic Systems. Transl. Math. Monogr. 33, Amer. Math. Soc., Providence, RI, 1972. [23] Olver, P. J., Applications of Lie Groups to Differential Equations. Second edition, Grad. Texts in Math. 107, Springer-Verlag, New York 1993. [24] Ortega, R., van der Schaft, A. J., Maschke, B. M., and Escobar, G., Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems. Automatica 38 (2002), 585–596. [25] Ortega, R., van der Schaft, A. J., Mareels, I., and Maschke, B. M., Putting energy back in control. IEEE Control Syst. Mag. 21 (2001), 18–33. [26] Paynter, H. M., Analysis and design of engineering systems. M.I.T. Press, Cambridge, MA, 1960. [27] Prigogine, I., Introduction to Thermodynamics of Irreversible Processes. John Wiley and Sons, 1962. [28] van der Schaft, A. J., System theoretic properties of physical systems. CWI Tract 3, Centre for Mathematics and Informatics, Amsterdam 1984. [29] van der Schaft, A. J., Implicit Hamiltonian systems with symmetry. Rep. Math. Phys. 41 (1998), 203–221. [30] van der Schaft, A. J., Interconnection and geometry. In The Mathematics of Systems and Control, From Intelligent Control to Behavioral Systems (ed. by J. W. Polderman, H. L. Trentelman), Groningen 1999, 203–218.


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[31] van der Schaft, A. J., L2 -Gain and Passivity Techniques in Nonlinear Control. 2nd revised and enlarged edition, Comm. Control Engrg. Ser., Springer-Verlag, London 2000 (first edition; Lecture Notes in Control and Inform. Sci. 218, Springer-Verlag, London 1996). [32] van der Schaft, A. J., and Maschke, B. M., On the Hamiltonian formulation of nonholonomic mechanical systems. Rep. Math. Phys. 34 (1994), 225–233. [33] van der Schaft, A. J., and Maschke, B. M., The Hamiltonian formulation of energy conserving physical systems with external ports. AEÜ—Arch. Elektron. Übertragungstech. 49 (1995), 362–371. [34] van der Schaft, A. J., and Maschke, B. M., Interconnected Mechanica5a Systems, Part I: Geometry of Interconnection and implicit Hamiltonian Systems. In Modelling and Control of Mechanical Systems (ed. by A. Astolfi, D. J. N. Limebeer, C. Melchiorri, A. Tornambe, R. B. Vinter), Imperial College Press, London 1997, 1–15. [35] van der Schaft, A. J., Maschke, B. M., Hamiltonian representation of distributed parameter systems with boundary energy flow. J. Geom. Phys. 42 (2002), 166–194. [36] Stramigioli, S., Modeling and IPC control of Interactive Mechanical Systems: a coordinate free approach. Lecture Notes in Control and Inform. Sci. 266, Springer-Verlag, London 2001. [37] Serre, D., Systems of Conservation Laws. Cambridge University Press, Cambridge 1999. [38] Willems, J. C., Dissipative dynamical systems - Part I: General Theory. Arch. Rational Mech. Anal. 45 (1972), 321–351. Department of Mathematics and Computing Science, University of Groningen, P.O. Box 800, 9700 AV Groningen, The Netherlands E-mail: [email protected]

Passive linear discrete time-invariant systems Olof J. Staffans∗

Abstract. We begin by discussing linear discrete time-invariant i/s/o (input/state/output) systems that satisfy certain ‘energy’ inequalities. These inequalities involve a quadratic storage function in the state space induced by a positive self-adjoint operator H that may be unbounded and have an unbounded inverse, and also a quadratic supply rate in the combined i/o (input/output) space. The three most commonly studied classes of supply rates are called scattering, impedance, and transmission. Although these three classes resemble each other, we show that there are still significant differences. We then present a new class of s/s (state/signal) systems which have a Hilbert state space and a Kre˘ın signal space. The state space is used to store relevant information about the past evolution of the system, and the signal space is used to describe interactions with the surrounding world. A s/s system resembles an i/s/o system apart from the fact that inputs and outputs are not separated from each other. By decomposing the signal space into a direct sum of an input space and an output space one gets a standard i/s/o system, provided the decomposition is admissible, and different i/o decompositions lead to different i/o supply rates (for example of scattering, impedance, or transmission type). In the case of non-admissible decompositions we obtain right and left affine representations, both of the s/s system itself, and of the corresponding transfer function. In particular, in the case of a passive system we obtain right and left coprime representations of the generalized transfer functions corresponding to nonadmissible decompositions of the signal space, and we end up with transfer functions which are, e.g., generalized Potapov or Nevanlinna class functions. Mathematics Subject Classification (2000). 93A05, 47A48, 47A67, 47B50. Keywords. Passive, storage function, supply rate, scattering, impedance, transmission, input/state/output, state/signal, Schur function, Carathéodory function, Nevanlinna function, Potapov function, behavior.

1. H -passive discrete time i/s/o systems The evolution of a linear discrete time-invariant i/s/o (input/state/output) system i/s/o with a Hilbert input space U, a Hilbert state space X, and a Hilbert output space Y is described by the system of equations x(n + 1) = Ax(n) + Bu(n), y(n) = Cx(n) + Du(n), x(0) = x0 ,

n ∈ Z+ = {0, 1, 2, . . .},

(1.1)

∗ This article is based on recent joint work with Prof. Damir Arov [AS05], [AS06a], [AS06b], [AS06c]. Thank you, Dima, for everything that I have learned from you!


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Olof J. Staffans

where the initial state x0 ∈ X may be chosen arbitrarily and A : X → X, B : U → X, C : X → Y, and D : U → Y are bounded linear operators. Equivalently, A B x(n) x(n + 1) , n ∈ Z+ , x(0) = x0 , = (1.2) C D u(n) y(n) A B X ; X where C ∈ B .1 We call u = {u(n)}∞ Y n=0 the input sequence, D U ∞ ∞ x = {x(n)}n=0 the state trajectory, and y = {y(n)}n=0 the output sequence, and we refer to the triple (u, x, y) as a trajectory of i/s/o . The operators appearing in (1.1) and (1.2) are usually called as follows: A is the main operator, B is the control operator, C is the observation operator, and D is the feedthrough operator. The transfer function or characteristic function D of this system is given by2 D(z) = zC(1X − zA)−1 B + D,

z ∈ (A),

where (A) is the set of points z ∈ C for which 1X − zA has a bounded inverse, plus the point at infinity if A has a bounded inverse. Note that D is analytic (A), and

A B on that D = D(0). We shall denote the above system by i/s/o = C D ; U, X, Y . Since all the systems in this paper will be linear and time-invariant and have a discrete time variable we shall in the sequel omit the words “linear discrete time-invariant” and refer to a system of the above type by simply calling it an i/s/o system. The i/s/o system i/s/o is controllable if the sets of all states x(n), n ≥ 1, which appear in some trajectory (u, x, y) of i/s/o with x0 = 0 (i.e., an externally generated trajectory) is dense in X. The system i/s/o is observable if there do not exist any nontrivial trajectories (u, x, y) where both u and y are identically zero. Finally, i/s/o is minimal if i/s/o is both controllable and observable. In this work we shall primarily be concerned with i/s/o systems which are passive or even conservative. To define these notions we first introduce the notions of a storage function EH which represents the (internal) energy of the state, and a supply rate j which describes the interchange of energy between the system and its surroundings. In the classical case the storage (or Lyapunov) function EH is bounded, and it is given by EH (x) = x, H xX , where H is a bounded positive self-adjoint operator on X (positivity of H means that x, H xX > 0 for all x = 0). However, we shall also consider unbounded storage functions induced by some √ (possibly unbounded) positive self-adjoint operator H on X. In this case we let H be the positive selfadjoint square root of H , and define the storage function EH by √ √ (1.3) EH (x) = H x2X , x ∈ D( H ). Clearly, this is equivalent to the earlier definition of EH if H is bounded. The supply rate j will always be a bounded (indefinite) self-adjoint quadratic form on Y ⊕ U,

1 Here X is the cartesian product of X and U, and B(U; Y) is the set of bounded linear operators from U U

to Y. 2 1 is the identity operator in X. X

Passive linear discrete time-invariant systems

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i.e., it can be written in the form

where J =

J11 J12 J21 J22

y y , j (u, y) = ,J u u Y⊕U

(1.4)

is a bounded self-adjoint operator in Y ⊕ U. For simplicity we

throughout require J to have a bounded inverse. Often J is taken to be a signature operator (both self-adjoint and unitary), so that J = J ∗ = J −1 . In the sequel we shall always use one and the same supply rate j for a given system i/s/o and include this

Asupply rate in the notation of the system, thus denoting the system by B ; U, X, Y; j whenever the supply rate is important. i/s/o = C D

A B forward H -pasDefinition 1.1. The i/s/o system i/s/o = C D ; U, X, Y; j is√ sive, where H is a positive self-adjoint operator in X, if x(n) ∈ D( H ) and √ √ H x(n + 1)2X − H x(n)2X ≤ j (u(n), y(n)), n ∈ Z+ , (1.5) √ for every trajectory (u, x, y) of i/s/o with x0 ∈ D( H ). If the above inequality holds as an equality then i/s/o is forward H -conservative. It is not difficult to see that i/s/o is forward H -passive if and only if3 H > 0 is a solution of the (forward) generalized i/s/o KYP (Kalman–Yakubovich–Popov) inequality4 √ √ √ H (Ax + Bu)2X − H x2X ≤ j (u, Cx + Du), x ∈ D( H ), u ∈ U, (1.6) and that i/s/o is forward H -conservative if and only if H > 0 is a solution of the corresponding equality. This inequality is named after Kalman [Kal63], Yakubovich [Yak62], and Popov [Pop61] (who at that time restricted themselves to the finitedimensional case). There is a rich literature on the finite-dimensional version of the KYP inequality and the corresponding equality; see, e.g., [PAJ91], [IW93] and [LR95], and the references mentioned there. In the seventies the classical results on the KYP inequalities were extended to infinite-dimensional systems by V. A. Yakubovich and his students and collaborators (see [Yak74], [Yak75], and [LY76] and the references listed there). There is now also a rich literature on this infinite-dimensional case; see, e.g., the discussion in [Pan99] and the references cited there. However, until recently it was assumed throughout that either H itself is bounded or H −1 is bounded. The first study of this inequality which permits both H and H −1 to be unbounded was done by Arov, Kaashoek and Pik in [AKP05]. Above we have defined forward H -passivity and forward H -conservativity. The corresponding backward notions are defined by means of the adjoint i/s/o system 3 The notation H > 0 means that H is a (possibly unbounded) self-adjoint operator satisfying x, H x > 0 X

for all nonzero x ∈ D(H ). √ 4 In particular, in order for the first term in this inequality to be well-defined we require A to map D( H ) √ into itself and B to map U into D( H ).

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A∗ ∗ i/s/o = B ∗ of equations

Olof J. Staffans C∗ D∗

; Y, X, U; j∗ whose trajectories (y∗ , x∗ , u∗ ) satisfy the system x∗ (n + 1) = A∗ x∗ (n) + C ∗ y∗ (n), u∗ (n) = B ∗ x∗ (n) + D ∗ y∗ (n), x∗ (0) = x∗0 .

n ∈ Z+ ,

(1.7)

Note that this system has the same state space X, but the input and output have been interchanged, so that Y is the input space and U is the output space. The appropriate ∗ storage function and supply rates for the adjoint system i/s/o differ from those of the −1 primal system i/s/o : H is replaced by H , and the primal supply rate j is replaced by the dual supply rate u∗ u , J∗ ∗ , (1.8) j∗ (y∗ , u∗ ) = y∗ y∗ U⊕Y 0 −1U −1 0 −1Y J . (1.9) J∗ = 1Y 0 1U 0

A B Definition 1.2. Let i/s/o = C D ; U, X, Y; j be an i/s/o system, and let H be a positive self-adjoint operator in X. where

∗ (i) i/s/o is backward H -passive if the adjoint system i/s/o is forward H −1 passive. ∗ (ii) i/s/o is backward H -conservative if the adjoint system i/s/o is forward H −1 conservative.

(iii) i/s/o is H -passive if it is both forward and backward H -passive. (iv) i/s/o is H -conservative if it is both forward and backward H -conservative. (v) By passive or conservative (with or without the attributes “forward” or “backward”) we mean 1X -passive or 1X -conservative, respectively. ∗ The generalized KYP inequality for the adjoint i/s/o system i/s/o with storage 5 function EH −1 is given by

H −1/2 (A∗ x∗ + C ∗ y∗ )2X − H −1/2 x∗ 2X ≤ j∗ (y∗ , B ∗ x∗ + D ∗ y∗ ),

√ x∗ ∈ H , y∗ ∈ Y.

(1.10)

Thus, i/s/o is backward H -passive if and only if H is a solution of (1.10), and i/s/o is backward H -conservative if and only if H is a solution of the corresponding equality. √

5 In particular, in order for the first term in this inequality to be well-defined we require A∗ to map R( H ) √ into itself and C ∗ to map Y into R( H ).

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2. Scattering, impedance and transmission supply rates The three most common supply rates are the following: (i) The scattering supply rate jsca (u, y) = −y, yY + u, uU with signature −1Y 0 . The signature operator of the dual supply rate is operator Jsca = 0 1U −1U 0 Jsca∗ = 0 1Y . (ii) The impedance 0 supply rate jimp (u, y) = 2y, uU with signature operator Jimp = ∗ 0 , where is a unitary U → Y. The signature operator operator ∗ of the dual supply rate is Jimp∗ = 0 0 . (iii) The transmission supply rate jtra(u, y) = −y, JY yY + u, JU uU with sig−J

0

Y , where JY and JU are signature operators nature operator Jtra = 0 JU in Y and The signature operator of the dual supply rate is U, respectively. −JU 0 Jtra∗ = 0 JY .

In the sequel when we talk about scattering H -passive or impedance H -conservative, etc., we mean that the supply rate is of the corresponding type. It turns out that although Definition 1.1 and 1.2 can be applied to all three types of supply rates, these three cases still differ significantly from each other. 2.1. Scattering supply rate. In the case of scattering supply rate forward H -passivity is equivalent to backward H -passivity, hence This iseasy to see

AtoBpassivity. in the case where H = 1X : the system ; U, X, Y; jsca is forward = i/s/o C D A B passive if and only if the operator C is a contraction, which is true if and only D A∗ C ∗ is a contraction, which is true if and only if the adjoint system if its adjoint B

A∗ C∗∗ D ∗ ∗ i/s/o = B ∗ D ∗ ; U, X, Y; jsca∗ is forward passive. The case where H is bounded and has a bounded inverse is almost as easy, and the general case is proved in [AKP05, Proposition 4.6]. The existence of an operator H > 0 such that i/s/o is scattering H -passive is related to the properties of the transfer function i/s/o . To formulate this result we first recall some definitions. The Schur class S(U, Y; D) is the unit ball in H ∞ (U, Y, D), i.e., each function in S(U, Y; D) is an analytic function on the open unit disk D = {z ∈ C | |z| < 1} whose values are contractions in B(U, Y). The restricted Schur class S(U, Y; ), where ⊂ D, contains all functions θ which are restrictions to of some function in S(U, Y; D). In other words, θ ∈ S(U, Y; ) if the (Nevanlinna– Pick) extension (or interpolation) problem with the (possibly infinite) set of data points (z, θ(z)), z ∈ , has a solution in S(U, Y; D). It is known that this problem has a solution if and only if the kernel θ Ksca (z, ζ ) =

1Y − θ(z)θ(ζ )∗ 1 − zζ

,

z, ζ ∈ ,

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Olof J. Staffans

is nonnegative definite on × , or equivalently, if and only if the kernel θ∗ (z, ζ ) = Ksca

1U − θ(ζ )∗ θ(z) 1 − ζz

,

z, ζ ∈ ,

is nonnegative definite on × (see [RR82]). We shall here be interested in the case where is an open subset of D, which implies that the solution of this Nevanlinna– Pick extension problem is unique (if it exists).

A B Theorem 2.1. Let i/s/o = C D ; U, X, Y; jsca be an i/s/o system with scattering supply rate and transfer function D, and let 0 (A) be the connected component of (A) ∩ D which contains the origin. (i) If i/s/o is forward H -passive for some H > 0, then i/s/o is H -passive and D|0 (A) ∈ S(U, Y; 0 (A)). (ii) Conversely, if i/s/o is minimal and D|0 (A) ∈ S(U, Y; 0 (A)), then i/s/o is H -passive for some H > 0. In statement (ii) it is actually possible to choose the operator H to satisfy an additional minimality requirement. We shall return to this question in Theorem 3.5. 2.2. Impedance supply rate. Also in the case of impedance supply rate forward H -passivity is equivalent to backward H -passivity, hence to passivity. This is well known in the case where H = 1X (see, e.g., [Aro79a]). One way to prove this is to reduce the impedance case to the scattering case by means of the following simple transformation.

A B Suppose that i/s/o = C D ; U, X, Y;jimp is a forward impedance H -passive system with signature operator Jimp = 0∗ 0 . Let (u, x, y) be a trajectory of i/s/o . We define a new input u× by u× = √1 (u + ∗ y) and a new output y × by y × =

√1 (u 2

2

− y), after which we solve (1.2) for x and y × in terms of x0

and u× . It turns out that for this to be possible we need + D to have a bounded inverse. However, this is always the case, since (1.6) (with x = 0) implies that × × shows ∗ D + D ∗ ≥ 0. A direct ×computation that (y , x, u ) is a trajectory of × × A B another system i/s/o = C × D × ; U, X, Y , called the external Cayley transform of i/s/o , whose coefficients are given by √ A× = A − B( + D)−1 C, B × = 2 B( + D)−1 , (2.1) √ C × = − 2 ( + D)−1 C, D × = ( − D)( + D)−1 . The transfer functions of the two systems are connected by D× (z) = ( − D(z))( + D(z))−1 ,

z ∈ (A) ∩ (A× ).

(2.2)

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The external Cayley transform is its own inverse in the sense that + D × = 2( + D)−1 always has a bounded inverse, and if we apply the external Cayley transform × to the system i/s/o , then we recover the original system i/s/o . The main reason for defining the external Cayley transform in the way that we did above is that it ‘preserves the energy exchange’ in the sense that jimp (u, y) = × jsca (y × , u× ). This immediately implies that i/s/o is forward scattering H -passive 6 whenever i/s/o is forward impedance H -passive. According to the discussion in × is equivalent to backward scatterSection 2.1, forward scattering H -passivity of i/s/o × ing H -passivity of i/s/o , and this in turn is equivalent to the backward (impedance) H -passivity of i/s/o . Thus, we get the desired conclusion, namely that forward impedance H -passivity implies backward impedance H -passivity, hence impedance H -passivity. The same argument can be used to convert all the results mentioned in Section 2.1 into an impedance setting. For simplicity we below take Y = U and = 1U (this amounts to replacing the output sequence y with values in Y by the new output sequence ∗ y with values in U)). The Carathéodory class C(U; D) (also called the Carathéodory–Nevanlinna class, or Nevanlinna class, or Weyl class, or Titchmarsh– Weyl class, etc.) consists of all analytic B(U)-valued functions ψ on D with nonnegative ‘real part’, i.e., ψ(z) + ψ(z)∗ ≥ 0 for all z ∈ D. The restricted Carathéodory class C(U; ), where ⊂ D, contains all functions θ which are restrictions to of some function in C(U; D). In other words, θ ∈ C(U; ) if the extension problem with the set of data points (z, θ(z)), z ∈ , has a solution in C(U; ). This is equivalent to the requirement that the kernel ψ

Kimp (z, ζ ) =

ψ(z) + ψ(ζ )∗ 1 − zζ

,

z, ζ ∈ ,

is nonnegative definite on × (this can be proved by reducing the impedance case to the scattering case as explained above).

A B Theorem 2.2. Let i/s/o = C D ; U, X, U; jimp be an i/s/o system with impedance supply rate, signature operator Jimp = 10U 10U , and transfer function D. Let 0 (A) be the connected component of (A) ∩ D which contains the origin.

(i) If i/s/o is forward H -passive for some H > 0, then i/s/o is H -passive and D|0 (A) ∈ C(U, Y; 0 (A)). (ii) Conversely, if i/s/o is minimal and D|0 (A) ∈ C(U, Y; 0 (A)), then i/s/o is H -passive for some H > 0. This theorem follows from Theorem 2.1 as explained above. 6 It is also true that × i/s/o is forward impedance H -passive if i/s/o is forward scattering H -passive, provided × ( + D) has a bounded inverse so that i/s/o exists.

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Above we have reduced the impedance passive case to the scattering passive case. Historically the development went in the opposite direction: the impedance version is older than the scattering version. It is related to Neumark’s dilation theorem for positive operator-valued measures (see [Bro71, Appendix 1]). In many classical and also in some recent works (especially those where the functions are defined on a halfplane instead of the unit disk) the impedance version is used as ‘reference system’ from which scattering and other results are derived (see, e.g., [Bro78]). Thus, one easily arrives at the (in my opinion incorrect) conclusion that it does not really matter which one of the two classes is used as the basic corner stone on which the theory is built. However, there is a significant difference between the two classes: the external Cayley transformation that converts one of the classes into the other is well-defined for every impedance H -passive system, but not for every scattering H -passive system. In other words, the external Cayley transform maps the class of impedance H -passive systems into but not onto the class of scattering H -passive systems (even if we restrict the input and output dimensions of the scattering system to be the same). What happens if we try to apply the external Cayley transform to a scattering H -passive system for which this transform is not defined (i.e., +D is not invertible)? In this case the formal transfer function of the resulting system may take its values in the space of closed unbounded operators in U, and it may even be multi-valued. To be able to study this class of ‘generalized Carathéodory functions’ we need some other more general type of linear systems than the i/s/o systems we have considered so far. This was one of the motivations for the introduction of the notion of a state/signal system in [AS05], to be discussed in Section 3. 2.3. Transmission supply rate. In the case of transmission supply rate forward H -passivity is no longer equivalent to backward H -passivity. For simplicity, let us take H to be the identity. Arguing in the same way as in the scattering case we find that A B i/s/o = C D ; U, X, Y; jtra is forward (transmission) passive if and only if the A B 7 operator C D is a contraction between two Kre˘ın spaces, namely from the space X 1X 0 X 0 JU to the space Y with the signature operaU with the signature operator 1 0 tor 0X JY . In the same way we find that i/s/o is backward (transmission) passive A B ∗ 1X 0 X if C is a contraction from the space with the signature operator Y 0 JY D X 1X 0 to the space U with the signature operator 0 JU . However, in a Kre˘ın space setting the contractivity of an operator does not imply that the adjoint of this operator is contractive, and hence forward transmission passivity does not imply backward transmission passivity without any further restrictions on the system. One necessary condition for the system i/s/o to be both forward and backward (transmission) H -passive is that the dimensions of the negative eigenspaces of JU and JY are the 7An operator A ∈ B(U; Y), where U and Y are Kre˘ın spaces, is a contraction if [Au, Au] ≤ [u, u] for Y U all u ∈ U.

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same. If these dimensions are the same and finite, then it is true that forward H passivity is equivalent to backward H -passivity, hence to passivity. To prove these statements one can use the following transformation that maps the transmission supply rate into a scattering supply rate.

A B Suppose that i/s/o = C D ; U, X, Y; jtra is a forward transmission H -

passive system with signature operator Jtra =

JY 0 0 JY

. We begin by splitting the

output space Y into the orthogonal direct sum Y = −Y− [] Y+ , where Y− is the negative and Y+ is the positive eigenspace of JY . In the same we split the input Away B accordingly into space U into U = −U− [] U+ , and we split the operator C D ⎡ ⎤ A B1 B1 A B ⎢ ⎥ = ⎣ C1 D11 D12 ⎦ . C D C2 D21 D22 Let (u, x, y) be a trajectory of i/s/o , and split y and u into y = yy−+ and u= uu−+ , so that y− is a sequence in Y− ,etc. We define a new input u by u = uy−+ and a new output y by y = uy+− , so that u is a sequence in U = Y− ⊕ U+ and y is a sequence in Y = U− ⊕ Y+ . We then solve (1.2) for x and y in terms of x0 and u . It turns out that for this to be possible we need D11 to have a bounded inverse. The forward H -passivity of i/s/o implies that D11 is injective and has a closed range, but it need not be surjective. However, let us suppose that D11 has a bounded inverse. Then a direct shows that (u , x, y ) is a trajectory of computation A B another system i/s/o = C , called the Potapov–Ginzburg D ; U , X, Y (or chain scattering) transform of i/s/o , whose coefficients are given by

⎡

A

A B ⎢ =⎣ 0 C D C2 ⎡ 1X ⎢ =⎣ 0 0

B1

B2

⎤⎡

1X

0

0

⎤−1

⎥⎢ ⎥ 1Y− 0 ⎦ ⎣ C1 D11 D12 ⎦ D21 D22 0 0 1U+ ⎤−1 ⎡ ⎤ −B1 0 0 B2 A ⎥ ⎢ ⎥ −D11 0 ⎦ ⎣ C1 −1U− D12 ⎦ . −D21 1Y+ C2 0 D22

(2.3)

The transfer functions of the two systems are connected by (D11 (z))−1 D11 (z) D −(D11 (z))−1 D12 (z) 12 (z) = , D D21 (z)(D11 (z))−1 D22 (z) − D21 (z)(D11 (z))−1 D12 (z) 21 (z) D22 (z)

z ∈ (A) ∩ A . (2.4) = D −1 The Potapov–Ginzburg transform is its own inverse in the sense that D11 11 always has a bounded inverse, and if we apply the Potapov–Ginzburg transform to , then we recover the original system the system i/s/o i/s/o .

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The Potapov–Ginzburg transform has been designed to ‘preserve the energy exchange’ in the sense that jtra (u, y) = jsca (u , y ). This immediately implies that i/s/o is forward scattering H -passive whenever i/s/o is forward transmission H passive, provided that D11 is invertible so that the transform is defined. As in the impedance case we conclude that the forward transmission H -passive system i/s/o is also backward H -passive, i.e., H -passive, if D11 has a bounded inverse (where D11 is the part of the feedthrough operator D that maps the negative part of the input space U into the negative part of the output space Y). The converse is also true: if i/s/o is (transmission) H -passive, then D11 has a bounded inverse. Thus, a forward transmission H -passive system i/s/o is H -passive if and only if D11 has a bounded inverse, or equivalently, if and only if the Potapov–Ginzburg transform of i/s/o is defined. The analogue of Theorems 2.1 and 2.2 is more complicated to formulate than in the scattering and impedance cases. In particular, it is not immediately clear how to define the appropriate class of transfer functions. Above we first defined the Schur class S(U, Y; D) and the Carathéodory class C(U; D) in the full unit disk, and then restricted these classes of functions to some subset ⊂ D. Here it is easier to proceed in the opposite direction, and to directly define the restricted Potapov class P (U, Y; ) for some ⊂ D. We now interpret U and Y as Kre˘ın spaces, i.e., we replace the original Hilbert space inner products in Y and U by the Kre˘ın space inner products [y, y ]Y = y, JY y Y , [u, u ]U = u, JU u U . In the sequel we compute all adjoints with respect to these Kre˘ın space inner products, and we also interpret positivity with respect to these inner products (so that, e.g., an operator D is nonnegative definite in U if [u, Du]U ≥ 0 for all u ∈ U). A function ϕ : → B(U; Y) belongs to P (U, Y; ) if both the kernels ϕ

Ktra (z, ζ ) =

1Y − ϕ(z)ϕ(ζ )∗

, 1 − zζ 1U − ϕ ∗ (ζ )ϕ(z) ϕ∗ , Ktra (z, ζ ) = 1 − ζz

z, ζ ∈ , (2.5) z, ζ ∈ ,

are nonnegative definite on × .

A B Theorem 2.3. Let i/s/o = C D ; U, X, Y; jtra be an i/s/o system with transmisJ

0

sion supply rate, signature operator Jtra = 0Y J , and transfer function D. Let U 0 (A) be the connected component of (A) ∩ D which contains the origin. (i) If i/s/o is H -passive for some H > 0, then D|0 (A) ∈ P (U, Y; 0 (A)). (ii) Conversely, if i/s/o is minimal and D|0 (A) ∈ P (U, Y; 0 (A)), then i/s/o is H -passive for some H > 0.


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This theorem follows from Theorem 2.1 via the Potapov–Ginzburg transformation. Note that (2.5) with z = ζ = 0 implies that both D and D ∗ are Kre˘ın space contractions, so that D11 is invertible and the Potapov–Ginzburg transform is defined. From what we have said so far it seems to follow that the transmission case is not that different from the scattering and impedances cases. However, this impression is not correct. One significant difference is that the Potapov–Ginzburg transformation is not always defined for a forward transmission H -passive i/s/o system. Another even more serious problem is that a function in the Potapov class may have singularities inside the unit disk D, which means that in the definition of the (full) Potapov class P (U, Y; D) we must take into account that the function in this class need not be defined everywhere on D. If the negative dimensions of U and Y are the same and finite, then this is not a serious problem, because in this case it is possible to define the Potapov class P (U, Y; D) to be the set of all meromorphic functions on D whose values in B(U; Y) are contractive with respect to the Kre˘ın space inner products in U and Y at all points where the functions are defined. However, in the general case the set of singularities of a function in P (U, Y; D) may be uncountable, and the domain of definition of a function in P (U, Y; D) need not even be connected. For this reason we prefer to define P (U, Y; D) in a different way. We say that a function ϕ belongs to the (full) Potapov class P (U, Y; D) if it belongs to P (U, Y; ) where the domain is maximal in the sense that the function ϕ does not have an extension to any larger domain ⊂ D with the property that the two kernels in (2.5) are still nonnegative on × . The existence of such a maximal domain is proved in [AS06b]. This maximal domain need not be connected, but it is still true that if we start from an open set ⊂ D, then the values of ϕ on define the extension of ϕ to its maximal domain uniquely. Moreover, as shown in [AS06b], if ϕ ∈ P (U, Y; D), then ϕ does not have an analytic extension to any boundary point of its domain contained in the open unit disk D. Taking a closer look at Theorem 2.3 we observe that it puts one artificial restriction on the transfer function D, namely that the domain of definition must contain the origin. Not every function in the Potapov class is defined at the origin, so the class of transfer functions covered by Theorem 2.3 is not the full Potapov class. In addition it is possible to extend the Potapov class so that the values of the functions in this class may be unbounded, even multivalued, operators (as in the impedance case) by taking the formal Potapov transforms of functions in S(U, Y, D). Thus, we again see the need of a more general class of systems than the i/s/o class that we have discussed up to now.

3. State/signal systems It is possible to develop a linear systems theory where the differences between the three different types of supply rates, namely scattering, impedance, and transmission, more or less disappear. Both the basic transforms that we have presented above,

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namely the external Cayley transform which is used to pass from an impedance H passive system to a scattering H -passive system and back, and the Potapov–Ginzburg transform that is used to pass from a transmission H -passive system to a scattering H -passive system and back, can be regarded as simple ‘changes of coordinates in Y the signal space W = U ’. The main idea is not to distinguish between the input sequence u and the output sequence y, but to simply regard these as components of the general ‘signal sequence’ w = yu . We start by combining the input space U and the output space Y into one signal Y space W = U . This signal space has a natural Kre˘ın space8 inner product obtained from the supply rate j in (1.4), namely y y y y , = ,J . u u W u Y⊕U u If we combine the input sequence u and the output sequence y into one signal sequence w = yu , then the basic i/s/o relation (1.1) can be rewritten in the form ⎡

⎤ x(n + 1) ⎣ x(n) ⎦ ∈ V , w(n) where V is the subspace of K := V =

z x w

n ∈ Z+ = {0, 1, 2, . . .}, x(0) = x0 , X X W

(3.1)

given by

X z = Ax + Bu, ∈ X W y = Cx + Du,

w=

y u

, x ∈ X, u ∈ U .

(3.2)

It is not difficult to show that the subspace V obtained in this way has the following four properties: (i) V is closed in K. (ii) For every x ∈ X there is some (iii) If

z 0 0

z w

∈

X W

such that

z x w

∈ V.

∈ V , then z = 0.

(iv) The set

x w

∈

X z x ∈ V for some z ∈ X is closed in X . W W w

Definition 3.1. A triple = (V ; X, W ), where the (internal) state space X is a Hilbert space and the (external) signal space W is a Kre˘ın space and V is a subspace 8 Both [BS05] and [AS06a] contain short sections on the geometry of a Kre˘ın space. For more detailed treatments we refer the reader to one of the books [ADRdS97], [AI89] and [Bog74].

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X of the product space K := X is called a s/s (state/signal) node if it has properties W (i)–(iv) listed above. We interpret K as a Kre˘ın space with the inner product z z z z x , x x , x ∈ K, (3.3) = −z, z X + x, x X + [w, w ]W , w

w

w

K

w

and we call K the node space and V the generating subspace. By a trajectory of we mean a pair of sequences (x, w) satisfying (3.1). We call x the state component and w the signal component of this trajectory. By the s/s system we mean the s/s node together with all its trajectories. The conditions (i)–(iv) above have natural interpretations in terms of the trajectories of : for each x0 ∈ X condition (ii) gives forward existence of at least one trajectory (x, w) of with x(0) = x0 . Condition (iii) implies that a trajectory (x, w) is determined uniquely by x0 and w, and conditions (i) and (iv) imply that the signal + + part x depends continuously in XZ on x0 ∈ X and w ∈ W Z . A s/s system is controllable if the set of all states x(n), n ≥ 1, which appear in some trajectory (x, w) of with x(0) = 0 (i.e., an externally generated trajectory) is dense in X. The system is observable if there do not exist any nontrivial trajectories (x, w) where the signal component w is identically zero. Finally, is minimal if is both controllable and observable. Above we explained how to interpret an i/s/o system i/s/o as a s/s system. Conversely, from every s/s system it is possible to create not only one, but infinitely many i/s/o systems. The representation (3.2) is characterized by the fact that it is a graph where U is one of the two components in a direct sum representation of V over X U decomposition of W = Y U (not necessarily orthogonal) of W . Indeed, splitting w into w = yu and reordering the components we find that (3.2) is equivalent to V =

z y x u

∈

X Y X U

z X x A B x . ∈ , = y U u C D u

(3.4)

As shown in [AS05], the generating subspace of every s/s system has at least one (hence infinitely many) graph representation of this type. A direct sum decomposition W = Y U of W is called an admissible i/o (input/output) decomposition of W for , or simply an admissible decomposition, if it leads to a graph representation of the generating subspace of described

Aabove. From each such graph representation B ; U, X, Y of , which we call an i/s/o of V we get an i/s/o system i/s/o = C D representation of . The above definitions are taken from [AS05], [AS06a], and [AS06b]. It turns out that a very large part of the proof of the H -passivity theory covered in Section 2 can be carried out directly in the s/s setting, rather than applying the same arguments separately with the scattering, impedance, and transmission supply rates. This leads to both a simplification and to a unification of the whole theory. Below we present the

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most basic parts of the H -passive s/s theory, and refer the reader to [AS05]–[AS06c] for details. Let = (V ; X, W ) be a s/s node. The adjoint ∗ = (V∗ ; X, W∗ ) of (introduced in [AS06a, Section 4]) is another s/s node, with the same state space X as , and with the signal space W∗ = −W .9 The generating subspace V∗ of ∗ is given by x∗ z∗ V∗ = wz∗ wx∗ ∈ V [⊥] , ∗

∗

where V [⊥] is the orthogonal companion to V with respect to the Kre˘ın space inner product of K.10 The adjoint system ∗ is determined by the property that −x(n+1), x∗ (0)X +x(0), x∗ (n+1)X +

n [w(k), w∗ (n−k)]W = 0,

n ∈ Z+ ,

k=0

for all trajectories (x, w) of . The following definition is the s/s version of Definitions 1.1 and 1.2. Definition 3.2. Let H be a positive self-adjoint operator in the Hilbert space X. A s/s system is √ (i) forward H -passive if x(n) ∈ D( H ) and √ √ H x(n + 1)2X − H x(n)2X ≤ [w(n), w(n)]W , n ∈ Z+ , √ for every trajectory (x, w) of with x(0) ∈ D( H ), (ii) forward H -conservative if the above inequality holds as an equality, (iii) backward H -passive or H -conservative if ∗ is forward H −1 -passive or H −1 -conservative, respectively, (iv) H -passive or H -conservative if it is both forward and backward H -passive or H -conservative, respectively, (v) passive or conservative if it is 1X -passive or 1X -conservative. To formulate a s/s version of Theorems 2.1, 2.2 and 2.3 we need a s/s analogue of the transfer function of an i/s/o system. Such an analogue is most easily obtained in the time domain (as opposed to the frequency domain), and it amounts to the introduction of a behavior11 on the signal space W . By this we mean a closed right-shift invariant + subspace of the Fréchet space W Z . Thus, in particular, the set W of all sequences w 9Algebraically −W is the same space as W , but the inner product in −W is obtained from the one in W by multiplication by the constant factor −1. 10 Thus, V [⊥] = {k ∈ K | [k, k ] = 0 for all k ∈ V }. Note that V differs from V [⊥] only by the order of ∗ ∗ K ∗ the first two components. 11 Our behaviors are what Polderman and Willems call linear time-invariant mainfest behaviors in [PW98, Definitions 1.3.4, 1.4.1, and 1.4.2]. We refer the reader to this book for further details on behaviors induced by systems with a finite-dimensional state space and for an account of the extensive literature on this subject.

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that are the signal parts of externally generated trajectories of a given s/s system is a behavior. We call this the behavior induced by , and refer to as a s/s realization of W, or, in the case where is minimal, as a minimal s/s realization of W. A behavior is realizable if it has a s/s realization. Two s/s systems 1 and 2 with the same signal space are externally equivalent if they induce the same behavior. This property is related to the notion of pseudo-similarity. Two s/s systems = (V ; X, W ) and 1 = (V1 ; X1 , W ) are called pseudo-similar if there exists an injective densely defined closed linear operator R : X → X1 with dense range such that the following conditions hold: If (x(·), w(·)) is a trajectory of on Z+ with x(0) ∈ D(R), then x(n) ∈ D(R) for all n ∈ Z+ and (Rx(·), w(·)) is a trajectory of 1 on Z+ , and conversely, if (x1 (·), w(·)) is a trajectory of 1 on Z+ with x1 (0) ∈ R(R), then x1 (n) ∈ R(R) for all n ∈ Z+ and (R −1 x1 (·), w(·)) is a trajectory of on Z+ . In particular, if 1 and 2 are pseudo-similar, then they are externally equivalent. Conversely, if 1 and 2 are minimal and externally equivalent, then they are necessarily pseudo-similar. Moreover, a realizable behavior W on the signal space W has a minimal s/s realization, which is determined uniquely by W up to pseudo-similarity. (See [AS05, Section 7] for details.) The adjoint of the behavior W on W is a behavior W∗ on W∗ defined as the set of sequences w∗ satisfying n

[w(k), w∗ (n − k)]W = 0,

n ∈ Z+ ,

k=0

for all w ∈ W. If W is induced by , then W∗ is (realizable and) induced by ∗ , and the adjoint of W∗ is the original behavior W. The following definition is a s/s analogue of our earlier definitions of the Schur, Carathéodory, and Potapov classes of transfer functions. Definition 3.3. A behavior W on W is (i) forward passive if n

[w(k), w(k)]W ≥ 0,

w ∈ W, n ∈ Z+ ,

k=0

(ii) backward passive if W∗ is forward passive, (iii) passive if it is realizable12 and both forward and backward passive. It is not difficult to see that a s/s system = (V ; X, W ) is forward H -passive if and only if H > 0 is a solution of the generalized s/s KYP (Kalman–Yakubovich– 12 We do not know if the realizability assumption is redundant or not.

1382 Popov) inequality13 √ √ H z2X − H x2X ≤ [w, w]W ,

Olof J. Staffans

z x w

√ ∈ V , x ∈ D( H ),

(3.5)

and that it is forward H -conservative if and only if the above inequality holds as an equality. The following proposition is a s/s version of parts (i) of Theorems 2.1, 2.2, and 2.3. Proposition 3.4. Let W be the behavior induced by a s/s system . (i) If is forward H -passive for some H > 0, then W is forward passive. (ii) If is backward H -passive for some H > 0, then W is backward passive. (iii) If is forward H1 -passive for some H1 > 0 and backward H2 -passive for some H2 > 0, then is both H1 -passive and H2 -passive, and W is passive. The following theorem generalizes parts (ii) of Theorems 2.1, 2.2, and 2.3. Theorem 3.5. Let W be a passive behavior on W . Then (i) W has a minimal passive s/s realization. (ii) Every H -passive realization of W is pseudo-similar to a passive realiza√ tion H with pseudo-similarity operator H . The system H is determined uniquely by and H . (iii) Every minimal realization of W is H -passive for some H > 0, and it is possible to choose H in such a way that the system H in (ii) is minimal. Assertion (ii) can be interpreted in the following way: we can always convert an H -passive s/s system into a passive one by simply √ replacing the original norm the state space by the new norm x = H xX , which is finite for all ·X in√ H √ x ∈ D( H ), and then completing D( H ) with respect to this new norm. We shall end this section with a result that says that a suitable subclass of all operators H > 0 for which a s/s system is H -passive can be partially ordered. Here we use the following partial ordering of nonnegative self-adjoint operators on X: if H1 and H2 are two nonnegative self-adjoint operators on the Hilbert space X, 1/2 1/2 1/2 1/2 then we write H1 H2 whenever D(H2 ) ⊂ D(H1 ) and H1 x ≤ H2 x 1/2 for all x ∈ D(H2 ). For bounded nonnegative operators H1 and H2 with D(H2 ) = D(H1 ) = X this ordering coincides with the standard ordering of bounded selfadjoint operators. For each s/s system we denote the set of operators H > 0 for which is H -passive by M , and we let Mmin be the set of H ∈ M for which the system H in assertion (ii) of Theorem 3.5 is minimal. √ 13 for the first term in this inequality to be well-defined we require z ∈ D( H ) whenever z In particular, in order √ x ∈ V and x ∈ D( H ). w


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Theorem 3.6. Let be a minimal s/s system with a passive behavior. Then Mmin contains a minimal element H and a maximal element H• , i.e., H H H• for every H ∈ Mmin . The two extremal storage functions EH and EH• correspond to Willems’[Wil72a], [Wil72b] available storage and required supply, respectively (there presented in an i/s/o setting). In the terminology of Arov [Aro79b], [Aro95], [Aro99] (likewise in an i/s/o setting), H is the optimal and H• is the ∗-optimal realization of W.

4. Scattering, impedance and transmission representations of s/s systems The results presented in Section 2 can be recovered from those in Section 3, together with a number of additional results. This is done by studying different i/s/o representations of a s/s system. Depending on the admissible i/o decomposition of the signal space W into an input space U and an output space Y we get different supply rates (inherited from the Kre˘ın space inner product in W ). Let = (V ; X, W ) be a s/s system, and decompose W into the direct sum of an input space U and an output space Y. Furthermore, suppose that this decomposition is admissible, so that it gives rise to an i/s/o representation i/s/o of . In the case of a fundamental decomposition W = −Y [] U, where Y and U are Hilbert spaces (i.e., −Y is an anti-Hilbert space) and −Y and U are orthogonal in W , the inner product in W is given by y y , = −y, y Y + u, u U , u u W which leads to a scattering supply rate for the i/s/o representation i/s/o . In this case we call i/s/o an admissible scattering representation of . In the case of a (nonorthogonal) Lagrangian decomposition, where both Y and U are Lagrangian14 subspaces of W we get an impedance supply rate and an admissible impedance representation of . Finally, if W = −Y [] U is an arbitrary orthogonal decomposition of W (not necessarily fundamental), then we get a transmission supply rate and an admissible transmission representation of . Thus, in the s/s setting the external Cayley transform and the Potapov–Ginzburg transform that we presented in Section 2 are simply two different ways at looking at the same s/s system, via different i/o decompositions of the signal space W into an input space U and an output space Y. The following proposition is related to the discussions at the beginning of Sections 2.1 and 2.2. Proposition 4.1. Let = (V ; X, W ) be a forward H -passive s/s system for some H > 0. Then the following claims hold. 14A subspace of a Kre˘ın space is Lagrangian if it coincides with its own orthogonal companion.

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(i) is H -passive if and only if has an admissible scattering representation, in which case every fundamental decomposition of W is admissible. (ii) If has an admissible impedance representation, then is H -passive. The converse of (ii) is not true: there do exist passive s/s systems which do not have any admissible impedance representation, even if we require the positive and negative dimensions of W to be the same. Every H -passive s/s system does have some admissible transmission representations (for example, every scattering representation can be interpreted as a transmission representation), but in general there also exist orthogonal decompositions of the signal space that are not admissible. One way to prove many of the results listed above is to pass to some particular i/s/o representation i/s/o of the s/s system , to prove the corresponding result for i/s/o , and to reinterpret the result for the s/s system . In many cases the most convenient choice is to use a scattering representation, corresponding to some admissible fundamental decomposition of the signal space. We recall from Proposition 4.1 that if is H -passive for some H > 0, then every fundamental decomposition is admissible. However, this is not the only possible choice. If W = YU is an arbitrary admissible i/o decomposition for , then is forward or backward H -passive if and only if the corresponding i/s/o system i/s/o is forward or backward H -passive with respect to the supply rate on Y U inherited product [ ·, ·]W . Thus, in the family

A B from the inner of i/s/o systems i/s/o = C D ; U, X, Y that we get from by varying the i/o A B vary, and so do the supply rates decomposition W = Y U the coefficients C D j (u, y), but the set of solutions of the generalized KYP inequalities (1.6) and (1.10) stay the same. Up to now we have only considered admissible i/o decompositions of the signal space W of a s/s system . As we commented earlier, not every Lagrangian or orthogonal decomposition need be admissible for , even if is H -passive for some H > 0. However, it is still possible to study also these non-admissible decompositions by replacing the i/s/o representations by left or right affine representations of . These are defined for arbitrary decompositions W = Y U (not only for the admissible ones). By a right affine i/s/o representation of we mean an i/s/o system15 r i/s/o

=

A B CY DY CU DU

D

; L, X,

DY

Y U

with the following two properties: 1) = D has a bounded left-inverse, and

U

y r for 2) x, u is a trajectory of if and only if , x, yu is a trajectory of i/s/o some sequence with values in L. By a left affine i/s/o representation of we mean 15 Here the new input space L is an auxiliary Hilbert space called the driving variable space.

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an i/s/o system16 l i/s/o

=

B A BY U D C DY U

Y ; U , X, K

with the following two properties: 1) D = DY DU has a bounded right-inverse,

l and 2) x, yu is a trajectory of if and only if yu , x, 0 is a trajectory of i/s/o (i.e., the output is identically zero in K). The transfer functions of these systems are called the right, respectively left, affine transfer functions of corresponding to the i/o decomposition W = Y U. Note, in particular, the right and left affine that D transfer functions are now decomposed into D = DY and D = DY DU , U

respectively. Let

r (i/s/o ) = {z ∈ A | DU (z) has a bounded inverse}, l ) = {z ∈ A | DY (z) has a bounded inverse}, (i/s/o

and let

r ), r (; U, Y) be the union of the above sets (i/s/o l ). l (; U, Y) be the union of the above sets (i/s/o

We can now define the notions of right and left generalized transfer functions of with input space U and output space Y on the sets r (; U, Y) and l (; U, Y), respectively, by the formulas Dr (z) = DY (z)DU (z)−1 , Dl (z) =

−DY (z)−1 DU (z),

(4.1) (4.2)

respectively. r Theorem 4.2. The right-hand side of (4.1) does not depend on the choice of i/s/o r as long as (i/s/o ) z, and the right-hand side of (4.2) does not depend on the l l ) z. choice of i/s/o as long as (i/s/o

Theorem 4.3. The right and left generalized transfer functions defined by (4.1) and (4.2), respectively, coincide on (; U, Y) = r (; U, Y) ∩ l (; U, Y) (whenever this set is nonempty). If the i/o decomposition W = Y U is admissible, and if A is the main operator of the corresponding i/s/o representation of , then r (; U, Y) = l (; U, Y) = A , and the left and right generalized transfer functions coincide with the ordinary transfer function corresponding to the decomposition W = Y U 16 Here the new output space K is an auxiliary Hilbert space called the error variable space.

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In the case where the s/s system is H -passive for some H > 0 we can say more. In this case it is possible to choose the different affine representations of in such a way that the right and left transfer functions are defined in the whole unit disk D and belong to H ∞ , and they will even be right and left coprime in H ∞ , respectively. In this way we obtain right and left coprime transmission representations of , and in the case that the positive and negative dimensions of the signal space W are the same we also obtain right and left coprime impedance representations. The corresponding right and left coprime affine transfer functions will be generalized Potapov and Carathéodory class functions, respectively.

5. Further extensions The results of Sections 3 and 4 are taken primarily from [AS05], [AS06a]–[AS06c]. At present they do not yet make up a complete theory that would be ready to replace the classical i/s/o theory. However, the following additional discrete part ingredients of the s/s theory are presently under active development: • The study of the interconnection of two s/s systems (this is the s/s analogue of feedback). • Lossless behaviors and bi-lossless extensions of passive behaviors (including the s/s analogue of Darlington synthesis). • Additional representations of generalized Carathéodory and Potapov class functions. • External and internal symmetry of s/s systems (including reciprocal systems). • Further studies of the stability properties of passive s/s systems. • Conditions for ordinary similarity (as opposed to pseudo-similarity) of minimal passive realizations. An even larger project is still in its infancy, namely the extension of the s/s theory to continuous time systems. Some preliminary results in this direction have been obtained in [BS05] and [MS06a], [MS06b].

References [ADRdS97] Alpay, Daniel, Dijksma, Aad, Rovnyak, James, and de Snoo, Henrik, Schur functions, operator colligations, and reproducing kernel Hilbert spaces. Oper. Theory Adv. Appl. 96, Birkhäuser, Basel 1997. [Aro79a]

Arov, Damir Z., Optimal and stable passive systems. Dokl. Akad. Nauk SSSR 247 (1979), 265–268; English translation in Soviet Math. Dokl. 20 (1979), 676–680.


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[Aro79b]

—, Stable dissipative linear stationary dynamical scattering systems. J. Operator Theory 1 (1979), 95–126; English translation in Interpolation theory, systems theory and related topics, Oper. Theory Adv. Appl. 134, Birkhäuser, Basel 2002, 99–136.

[Aro95]

—, A survey on passive networks and scattering systems which are lossless or have minimal losses. Archiv für Elektronik und Übertragungstechnik 49 (1995), 252–265.

[Aro99]

—, Passive linear systems and scattering theory. In Dynamical Systems, Control Coding, Computer Vision (Padova, 1998), Progr. Systems Control Theory 25, Birkhäuser, Basel 1999, 27–44.

[AKP05]

Arov, Damir Z., Kaashoek, Marinus A., and Pik, Derk R., The Kalman– Yakubovich–Popov inequality and infinite dimensional discrete time dissipative systems. J. Operator Theory (2005), 46 pages, to appear.

[AS05]

Arov, Damir Z., and Staffans, Olof J., State/signal linear time-invariant systems theory. Part I: Discrete time systems. In The State Space Method, Generalizations and Applications, Oper. Theory Adv. Appl. 161, Birkhäuser, Basel 2005, 115–177.

[AS06a]

—, State/signal linear time-invariant systems theory. Part II: Passive discrete time systems. Submitted, manuscript available at http://www.abo.fi/~staffans/, 2006.

[AS06b]

—, State/signal linear time-invariant systems theory. Part III: Transmission and impedance representations of discrete time systems. In preparation, 2006.

[AS06c]

—, State/signal linear time-invariant systems theory. Part IV:Affine representations of discrete time systems. In preparation, 2006.

[AI89]

Azizov, Tomas Ya., and Iokhvidov, Iosif S., Linear operators in spaces with an indefinite metric. Pure Appl. Math. (N. Y.), John Wiley, Chichester 1989.

[BS05]

Ball, Joseph A., and Staffans, Olof J., Conservative state-space realizations of dissipative system behaviors. Integral Equations Operator Theory 54 (2) (2006), 151–213.

[Bog74]

Bognár, János, Indefinite inner product spaces. Ergeb. Math. Grenzgeb. 78, Springer-Verlag, Berlin, Heidelberg, New York 1974.

[Bro71]

Brodski˘ı, M. S., Triangular and Jordan representations of linear operators. Transl. Math. Monogr. 32, Amer. Math. Soc., Providence, RI, 1971.

[Bro78]

—, Unitary operator colligations and their characteristic functions. Russian Math. Surveys 33 (4) (1978), 159–191.

[IW93]

Ionescu, Vlad, and Weiss, Martin, Continuous and discrete-time Riccati theory: a Popov-function approach. Linear Algebra Appl. 193 (1993), 173–209.

[Kal63]

Kalman, Rudolf E., Lyapunov functions for the problem of Luré in automatic control. Proc. Nat. Acad. Sci. U.S.A. 49 (1963), 201–205.

[LR95]

Lancaster, Peter, and Rodman, Leiba, Algebraic Riccati equations. Oxford Science Publications, The Clarendon Press, Oxford University Press, New York 1995.

[LY76]

Lihtarnikov, Andrei L., and Yakubovich, Vladimir A., A frequency theorem for equations of evolution type, Sibirsk. Mat. Ž. 17 (5) (1976), 1069–1085, 1198; English translation in Siberian Math. J. 17 (1977), 790–803.

[MS06a]

Malinen, Jarmo, and Staffans, Olof J., Conservative boundary control systems. Submitted, manuscript available at http://www.abo.fi/~staffans/, 2006.

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[MS06b]

—, Internal well-posedness of impedance passive boundary control systems. In preparation, 2006.

[Pan99]

Pandolfi, Luciano, The Kalman-Yakubovich-Popov theorem for stabilizable hyperbolic boundary control systems. Integral Equations Operator Theory 34 (4) (1999), 478–493.

[PAJ91]

Petersen, Ian R., Anderson, Brian D. O., and Jonckheere, Edmond A., A first principles solution to the non-singular H ∞ control problem. Internat. J. Robust Nonlinear Control 1 (1991), 171–185.

[PW98]

Polderman, Jan Willem, and Willems, Jan C., Introduction to mathematical systems theory: A behavioral approach. Texts Appl. Math. 26, Springer-Verlag, New York 1998.

[Pop61]

Popov, Vasile-Mihai, Absolute stability of nonlinear systems of automatic control. Avtomat. i Telemeh. 22 (1961), 961–979; English translation in Automat. Remote Control 22 (1961), 857–875.

[RR82]

Rosenblum, Marvin, and Rovnyak, James, An operator-theoretic approach to theorems of the Pick–Nevanlinna and Loewner types, II. Integral Equations Operator Theory 5 (1982), 870–887.

[Wil72a]

Willems, Jan C., Dissipative dynamical systems Part I: General theory. Arch. Rational Mech. Anal. 45 (1972), 321–351.

[Wil72b]

—, Dissipative dynamical systems Part II: Linear systems with quadratic supply rates. Arch. Rational Mech. Anal. 45 (1972), 352–393.

[Yak62]

Yakubovich, Vladimir A., The solution of some matrix inequalities encountered in automatic control theory. Dokl. Akad. Nauk SSSR 143 (1962), 1304–1307.

[Yak74]

—, The frequency theorem for the case in which the state space and the control space are Hilbert spaces, and its application in certain problems in the synthesis of optimal control. I. Sibirsk. Mat. Ž. 15 (1974), 639–668, 703; English translation in Siberian Math. J. 15 (1974), 457–476 (1975).

[Yak75]

—, The frequency theorem for the case in which the state space and the control space are Hilbert spaces, and its application in certain problems in the synthesis of optimal control. II. Sibirsk. Mat. Ž. 16 (5) (1975), 1081–1102, 1132; English translation in Siberian Math. J. 16 (1974), 828–845 (1976).

Åbo Akademi University, Department of Mathematics, 20500 Åbo, Finland E-mail: [email protected] URL: http://www.abo.fi/~staffans/

Control and numerical approximation of the wave and heat equations Enrique Zuazua∗ Ama, Magaly, Oihane eta Ainarari.

Abstract. In recent years important progress have been done in the context of numerical approximation of controllability problems for PDEs. It is by now well known that, often, numerical approximation schemes that are stable for solving initial-boundary value problems, develop instabilities when applied to controllability problems. This is due to the presence of spurious high frequency numerical solutions that the control mechanisms are not able to control uniformly as the mesh-size tends to zero. However, the theory is far from being complete. In this article we present some new results in this framework for the wave and the heat equations, which also raise a number of open questions and future directions of research. We first prove that a two-grid method, introduced by R. Glowinski, that is by now well-known to guarantee convergence for the 1 − d wave equation, also converges in the semilinear setting for globally Lipschitz nonlinearities. This result provides a further evidence of the robustness of the two-grid method. We then show that boundary controls for finite-difference space semi-discretizations of the heat equation converge when applied all along the boundary of the domain, a fact that does not hold for wave-like equations. This confirms that the strong irreversibility of the heat equation enhances the control properties of its numerical approximation schemes. This result fails when the control is restricted to some subsets of the boundary because of the lack of unique continuation of some high frequency eigenvectors of the underlying discrete eigenvalue problem. Mathematics Subject Classification (2000). Primary 93B05; Secondary 35A35, 35K05, 35L05. Keywords. Partial differential equations, finite-difference approximation schemes, controllability, wave equation, heat equation, two-grid method.

1. Introduction In recent years important progresses have been done in the context of numerical approximation of controllability problems for PDEs. It is by now well known that, often, numerical approximation schemes that are stable for solving an initial-boundary value problem, develop instabilities when applied to controllability problems. This is due to the presence of spurious high frequency numerical solutions that the control mechanisms are not able to control uniformly as the mesh-size tends to zero. ∗ Partially supported by grant MTM2005-00714, the DOMINO Project CIT-370200-2005-10 in the PROFIT program of the MEC (Spain), the SIMUMAT Project S-0505/ESP/0158 of the CAM (Spain) and the European network “Smart Systems”.


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To cure these instabilities a number of methods have been introduced in the literature. We refer to [30] for a recent survey article on the topic. In this context and in an effort to build a general theory, there are two prototypical equations that need to be understood first of all: the wave equation and the heat equation. In the framework of the linear wave equation, R. Glowinski [6] introduced a twogrid control mechanism that allows filtering the high frequency numerical spurious solutions and guarantee the convergence of controls. There are clear numerical evidences of the convergence of the method whose proof has been successfully carried out in [17] in the 1 − d case by using discrete multipliers. More recently the same result has been proved, with a better estimate on the minimal control time, in [16] by using Ingham type inequalities. Other methods have also been developed for avoiding these instabilities to occur: Tychonoff regularization, Fourier filtering, mixed finite elements,…(see [30]). But most of the existing theory is devoted to linear problems. The first part of this article is devoted to show how the convergence result of the twogrid algorithm can be extended to semilinear systems too, with globally Lipschitz nonlinearities. This result adds one more evidence of the robustness and efficiency of the two-grid algorithm for the control of wave problems. The high frequency spurious numerical solutions for the wave equation are due to the existence of wave-packets that travel with a vanishing group velocity (see [21], [30]). This can be understood by analyzing the symbol of the operator and the dynamics of the Hamiltonian system generating the bicharacteristic rays. However, one expects that the heat equation, because of its intrinsic time-irreversibility and strong damping should escape to those pathologies and that most common numerical approximation schemes should be controllable, uniformly with respect to the mesh-size. This holds indeed in the 1 − d setting (see [14]). But, surprisingly enough, this property may fail to hold in 2 − d even for the simplest finite-difference semi-discretization scheme for the heat equation in the square. This is due to the fact that there are some high-frequency numerical solutions that do not fulfill the classical property of unique continuation of the continuous heat equation. Thus, at the control level, the numerical approximation schemes may generate some solutions which are insensitive to the action of controls. Strictly speaking this happens when the control acts on some (small enough) subsets of the boundary where the equation holds. However, this fact clearly indicates a major difference in the control theoretical behavior of the continuous and the semi-discrete heat equation since the first one is controllable from any open and non-empty subset of the boundary while the second one is not. Characterizing completely the subsets of the boundary for which these pathologies arise is probably a difficult problem. In this article we prove that convergence occurs when the controls act everywhere on the boundary of the domain. This confirms that heat equations are better behaved than wave ones. Indeed, for the wave equation, even if controls act everywhere on the boundary of the domain, the uniform controllability property for numerical approximation schemes may fail because of the existence of spurious numerical solutions that are trapped in the interior of the mesh without reaching the

Control and numerical approximation of of the wave and heat equations

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boundary in an uniform time. Our positive result for the heat equation shows that this kind of spurious solutions are ruled out due to the strong dissipativity of the heat equation and its numerical approximation schemes but, so far, only when the control is distributed everywhere on the boundary. The lack of unique continuation for semi-discrete heat equations is due to the fact that the property fails to hold for the spectrum of the discrete Laplacian. Indeed, for the Dirichlet spectrum of the continuous Laplacian, unique continuation holds in the sense that, when the normal derivative of an eigenfunction vanishes in a subset of the boundary, the eigenfunction vanishes everywhere. This property fails to hold for the eigenvectors of the discrete Laplacian. The main method to prove unique continuation properties in the continuous framework are the so-called Carleman inequalities. But the discrete analogue is still to be developed. An alternate natural way of addressing this issue, in the spirit of the classical theory of numerical analysis, would consist in viewing the solutions of the discrete problem as a perturbation of those of the continuous one and applying the continuous Carleman inequalities. This approach has been successfully applied in [31] to elliptic equations with irregular coefficients in the principal part. Developing this program in the context of discrete elliptic equations is an interesting open problem. The two topics we address in this article also raise a number of interesting open problems and future directions of research that we mention briefly. Some of them, in our opinion, are deep and will require important research efforts. The interest of these problems goes much beyond Control Theory since they mainly concern the way classical numerical analysis and the existing theory of partial differential equations have to be melt to address subtle qualitative aspects of numerical solutions. We hope that this article will serve to stimulate research in this area.

2. Controllability of the two-grid approximation scheme for the 1 − d semilinear wave equation One of the main drawbacks of the existing theory to analyze the controllability of numerical approximation schemes for PDE is that it often relies on Fourier analysis. This makes it of little use for nonlinear problems. However there is by now an extensive literature on the controllability of semilinear PDE and, in particular, of wave and heat equations. Therefore, it is natural to develop numerical methods allowing to address these nonlinear models and to build convergent numerical schemes for their control. In this section we consider the 1 − d semilinear wave equation with boundary control: ⎧ ⎪ x ∈ (0, 1), 0 < t < T , ⎨ytt − yxx + f (y) = 0, (2.1) y(0, t) = 0, y(1, t) = v(t), 0 < t < T, ⎪ ⎩ 0 1 y(x, 0) = y (x), yt (x, 0) = y (x), x ∈ (0, 1).

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Here the control v = v(t) enters into the system through the extreme x = 1 of the boundary. This semilinear wave equation is known to be controllable under sharp growth conditions on the nonlinearity. Namely, if |f (s)| ≤ C log2 (1 + |s|) for all s ∈ R

(2.2)

for some C > 0, system (2.1) is exactly controllable in any time T > 2. This means that for all (y 0 , y 1 ) ∈ L2 (0, 1) × H −1 (0, 1) and (z0 , z1 ) ∈ L2 (0, 1) × H −1 (0, 1) there exists a control v ∈ L2 (0, T ) such that the solution y of (2.1) satisfies y(x, T ) = z0 (x), yt (x, T ) = z1 (x)

in (0, 1).

(2.3)

This result was proved in [26] for C > 0 sufficiently small in (2.2) and, without restrictions on the size of the constant C, in [1]. This growth condition is sharp since blow-up phenomena may occur for nonlinearities growing faster at infinity and, due to the finite speed of propagation, boundary controls are unable to avoid blow-up to occur. In that case controllability fails. The most common method to derive the exact controllability property of semilinear equations is based on the following ingredients: • a fixed point argument; • sharp estimates on the dependence of controls for the underlying linear equation perturbed by a potential. We refer to [25] where this method was introduced in the context of the wave equation (see also [22] for further developments, and [23] for an updated survey on this problem) and to [4] where the same technique was applied to semilinear heat equations. Knowing that the semilinear wave equation (2.1) is controllable under the growth condition (2.2) it is natural to analyze whether the controls can be obtained as limits of controls of numerical approximation schemes. As we have explained in the introduction this issue is delicate even for linear problems, and it is necessarily more complex for nonlinear ones. Among the possible remedies to the lack of convergence of the standard conservative schemes the two-grid method introduced in [6] seems to be the one that is better adapted to semilinear problems. In this section we confirm this assertion by proving its convergence in this nonlinear setting for globally Lipschitz nonlinearities. The two-grid scheme is, roughly, as follows. Given an integer N ∈ N we introduce the partition {xj = j h}j =0,..., N+1 of the interval (0, 1) with h = 1/(N + 1) so that x0 = 0 and xN+1 = 1. We then consider the conservative finite-difference semi-discretization of the semilinear wave equation (2.1) as follows: ⎧ 2yj −yj +1 −yj −1 ⎪ + f (yj ) = 0, j = 1, . . . , N, 0 < t < T , ⎨yj + h2 (2.4) y0 (t) = 0, yN +1 (t) = v(t), 0 < t < T, ⎪ ⎩ 0 1 yj (0) = yj , yj (0) = yj , j = 0, . . . , N + 1.


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The scheme is conservative in the sense that, in the absence of control (i.e. for v ≡ 0) the energy of solutions is conserved. The same property holds for the continuous version (2.1). In that case the energy is given by 1 E(t) = 2

1 0

yt2 (x, t) + yx2 (x, t)

where F is a primitive of f , i.e. F (z) = corresponding energy is

z 0

dx +

1

F (y(x, t)) dx, 0

f (s)ds. In the semi-discrete case the

N N

yj +1 − yj 2 h

2 Eh (t) = F (yj ). |yj | + + h 2 h j =0

j =0

The goal of this section is to analyze the controllability of (2.4) and whether, as h → 0, the controls of (2.4) converge to those of (2.1). The controls being, in general, non unique, one has to be precise when discussing their convergence. Here, in the linear context, we shall always refer to the controls of minimal L2 (0, T )-norm which are given by the so called Hilbert Uniqueness Method (HUM) ([13]). As we mentioned above, in the nonlinear case, the controls we shall deal with are obtained by fixed point methods on the basis of the HUM controls for the linearized problems. But, even in the linear case, to guarantee convergence as h → 0, the final control requirement has to be relaxed, or the numerical scheme modified. In [29] it was proved that, if the exact controllability condition is relaxed to the approximate controllability one (in which the state is required to reach an εneighborhood of the target), then convergence occurs in the linear framework. But it is convenient to deal with other relaxation criteria that do not introduce extra parameters since the controls may depend on them in a very sensitive way. The two-grid method is a very natural way of introducing such relaxation. It is based on the idea of relaxing the final condition to avoid the divergence of controls due to the need of controlling high frequency spurious oscillations. To be more precise, the semi-discrete analogue of the exact controllability final condition (2.3) is yj (T ) = zj0 , yj (T ) = zj1 ,

j = 0, . . . , N + 1.

(2.5)

But, as it is by now well-known (see [30]), under the final requirement (2.5), controls diverge as h → 0 even for the linear wave equation. In the two-grid algorithm, the final condition (2.5) is relaxed to h Y (T ) = h (Z 0 ), h Y (T ) = h (Z 1 ), (2.6) where Y (t) and Y (t) stand for the vector-valued unknowns Y (t) = y0 (t), . . . , yN +1 (t) , Y (t) = y0 (t), . . . , yN+1 (t) .

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We shall also use the notation Yh for Y when passing to the limit to better underline the dependence on the parameter h. h is the projection operator so that

1 1 1 g2j +1 + g2j + g2j +2 h (G) = 2 2 2

j =0,...,

N+1 2 −1

,

(2.7)

with G = (g0 , g1 , . . . , gN , gN +1 ). Note that the projection h (G) is a vector of dimension (N + 1)/2. Thus, roughly speaking, the relaxed final requirement (2.6) only guarantees that half of the state of the numerical scheme is controlled. Despite this fact, the formal limit of (2.6) as h → 0 is still the exact controllability condition (2.3) on the continuous wave equation. The main result of this section is as follows: Theorem 2.1. Assume that the nonlinearity f : R → R is such that f is globally Lipschitz.

(2.8)

Let T0 > 0 be such that the two-grid algorithm for the control of the linear wave equation converges for all T > T0 . Then, the algorithm converges for the semilinear system (2.1) too for all T > T0 . More precisely, for all (y 0 , y 1 ) ∈ H s (0, 1) × H s−1 (0, 1) with s > 0, there exists a family of controls {vh }h>0 for the semi-discrete system (2.4) such that the solutions of (2.4) satisfy the relaxed controllability condition (2.6) and vh (t) → v(t) (Yh , Yh )

→ (y, yt )

in L2 (0, T ), h → 0 in L (0, T ; L (0, 1) × H 2

2

(2.9) −1

(0, 1))

(2.10)

where y is the solution of the semilinear wave equation (2.1) and v is a control such that the state y satisfies the final requirement (2.3). Remark 2.2. Several remarks are in order. • The controllability of the semilinear wave equation (2.1) under the globally Lipschitz assumption (2.8) on the nonlinearity was proved in [25] in 1 − d and in the multi-dimensional case. The proof of Theorem 2.1 is based on an adaptation of the arguments in [25] to the two-grid approximation scheme. Whether the two-grid algorithm applies under the weaker and sharp growth condition (2.2) is an open problem. The difficulty for doing that is that the two existing proofs allowing to deal with the semilinear wave equation under the weaker growth condition (2.2) are based, on a way or another, on the sidewise solvability of the wave equation, a property that the semi-discrete scheme fails to have. • Theorem 2.1 holds for a sufficiently large time T0 . The requirement on T0 is that, in the linear case (f ≡ 0), the two-grid algorithm converges for all T > T0 . This was proved to hold for T > 4 in [17]. The proof in [17] is based on the obtention of the corresponding observability inequality for the solutions of the adjoint semi-discrete wave equation by multiplier techniques. Later on this result was improved in [16]


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using a variant √ of the classical Ingham inequality obtaining the sharp minimal control time T0 = 2 2. Note that the minimal time for controllability of the continuous wave equation (2.1) is T = 2.1 However this minimal time may not be achieved by the two-grid algorithm as described here since, despite it filters the spurious high frequency numerical solutions, it is compatible with the existence of wave packets travelling with velocity smaller than 1, and this excludes the controllability in the minimal time T = 2. The two-grid algorithm can be further improved to get smaller minimal times by considering other projection operators h , obtained by means of the two-grid approach we shall describe below but with ratio 1/2 , for some ≥ 2, instead of the ratio 1/2. This idea has been used successfully in [7] when proving dispersive estimates for conservative semi-discrete approximation schemes of the Schrödinger equation. When diminishing the ratio between grids, the filtering that the two-grid algorithm introduces concentrates the solutions of the numerical problem on lower and lower frequencies for which the velocity of propagation becomes closer and closer to that of the continuous wave equation. In that way the minimal controllability time may be made arbitrarily close to that of the wave equation T = 2 by means of the two-grid approach. • In the statement of Theorem 2.1 we have chosen initial data for (2.1) in the space (y 0 , y 1 ) ∈ H s (0, 1) × H s−1 (0, 1), but we have not explained how the initial data for the semi-discrete system (2.4) have to be taken. The simplest way for doing that is taking as initial data for (2.4) the truncated Fourier series of the continuous initial data (y 0 , y 1 ), involving only the first N Fourier modes. One can also define the discrete initial data by taking averages of the continuous ones on the intervals [xj − h/2, xj + h/2] around the mesh-points. • The meaning of the convergence property (2.10) needs also to be made precise. This may be done by extending the semi-discrete state (Yh (t), Yh (t)) into a continuous one (yh (x, t), yh (x, t)) and then proving convergence (2.10) for the extended one. This extension may be defined at least in two different ways. Either by extending the Fourier representation of Yh or rather by using a standard piecewise linear and continuous extension. We refer to [28] and [19] where these two extensions have been used in similar limit processes. • In the statement of Theorem 2.1 the initial data are assumed to be in H s (0, 1) × −1+s H (0, 1) for some s > 0, which is a slightly stronger regularity assumption than the one needed for the semilinear wave equation (2.1) to be controllable (L2 (0, 1) × H −1 (0, 1)). This is probably a purely technical assumption but it is needed for the method we develop here to apply. The same difficulty arises in the context of the continuous semilinear wave equation [25]. This extra regularity condition for the continuous wave equation was avoided in [1] and [26] but using the very special property of the 1 − d wave equation of being well-posed in the sidewise sense. 1 By minimimal control time we mean that the controllability property holds for all time T which is greater than 2. Thus, this does not necessarily mean that controllability occurs for time T = 2.

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In the context of the problem of numerical approximation we are working this difficulty seems hard to avoid even at the level of passing to the limit as h → 0 on the state equations. Indeed, this requires passing to the limit, in particular, on the nonlinear terms and this seems hard to achieve in the L2 (0, 1) × H −1 (0, 1)-setting because the corresponding states Yh would then be merely bounded in C([0, T ]; L2 (0, 1)) ∩ C 1 ([0, T ]; H −1 (0, 1)), which seems to be insufficient to guarantee compactness and the convergence of the nonlinear term. Proof of Theorem 2.1. To simplify the presentation we assume that the final target is the null trivial state z0 ≡ z1 ≡ 0, although the same proof applies in the general case. We proceed in several steps. Step 1. Two-grid controllability of the semi-discrete system (2.4). First of all, following the standard fixed point argument ([25]), we prove that, for h > 0 fixed, the semilinear system (2.4) is controllable. In fact this argument allows proving that (2.4) is exactly controllable for all T > 0. But, as we mentioned above (see [30]), the controls fail to be bounded as h → 0. It is precisely to guarantee that the controls are bounded that we need to relax the final condition to the weaker two-grid one (2.6) and the time T is needed to be large enough as in the statement of Theorem 2.1. To simplify the presentation we assume that f ∈ C 1 (R; R) and f (0) = 0, although the proof can be easily adapted to globally Lipschitz nonlinearities. We then introduce the continuous function f (z)/z, z = 0, g(z) = (2.11) f (0), z = 0. Given any semi-discrete function Z = Z(t) ∈ C([0, T ]; RN+2 ) we consider the linearized wave equation ⎧ 2yj −yj +1 −yj −1 ⎪ + g(zj )yj = 0, j = 1, . . . , N ; 0 < t < T , ⎨yj + h2 0 (2.12) y0 (t) = yj , yN +1 (t) = v(t), 0 < t < T, ⎪ ⎩ 1 yj (0) = yj (0) = yj , j = 0, . . . , N + 1. We proceed by a classical fixed point argument (see [25]). This requires essentially proving that: a) For all Z = Z(t) as above (2.12) is two-grid controllable in the sense of (2.6); b) The mapping N (Z) = Y has a fixed point. To be more precise, we shall identify uniquely a control of minimal L2 (0, T )-norm v (which, obviously, depends on Z. Thus, in some cases we shall also denote it as vZ ). In this way the controlled trajectory Y = YZ will also be uniquely determined and the nonlinear map N well defined. The problem is then reduced to proving that the

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map N has a fixed point. Indeed, if Z = Y , and, consequently, g(zj )yj = f (yj ) for all j = 1, . . . , N , then Y is also solution of the semilinear semi-discrete equation (2.1) and, of course, satisfies the two-grid relaxed final requirement (2.6). The existence of the fixed point of N is consequence of Schauder’s fixed point Theorem. The key point to apply it is to show a bound on the two-grid control for the linearized equation (2.12) which is independent of Z, i.e. the existence of C > 0 such that vZ L2 (0, T ) ≤ C for all Z ∈ C([0, T ]; RN+2 ). (2.13) Here and in the sequel we denote by vZ the control of the linearized system (2.12) to underline the fact that the control depends on the potential g(Z) and thus on Z. To do that we argue as in [17], reducing the problem to the obtention of a suitable observability inequality for the adjoint system: ⎧ 2ϕj −ϕj +1 −ϕj −1 ⎪ + g(zj )ϕj = 0, j = 1, . . . , N, 0 < t < T , ⎨ϕj + h2 (2.14) ϕ0 = ϕN +1 = 0, 0 < t < T, ⎪ ⎩ ϕj (T ) = ϕj0 , ϕj (T ) = ϕj1 , j = 1, . . . , N. For doing that, however, system (2.14) has to be considered only in the class of slowly oscillating data obtained as extensions to the fine grid (the original one, of size h) of data defined on a coarse grid of size 2h. In other words, we consider the class of data

Vh = = (ϕ0 , . . . , ϕN +1 ) : ϕ2j +1

ϕ2j + ϕ2j +2 N −1 = . , j = 0, . . . , 2 2

(2.15)

Note that any vector in Vh is completely determined by its values on the grid of mesh-size 2h. Implicitly we are assuming that 1/2h is an integer number so that (N − 1)/2 = 1/2h − 1 is an integer too. In [17] and [16] it was proved that for T > T0 , where T0 is as in Remark 2.2, the following observability inequality holds:

T

E0 ≤ C 0

ϕ 2 N dt, h

(2.16)

with C > 0 independent of h > 0 and for all solution = (ϕ0 , . . . , ϕN+1 ) of (2.14) with data (0 , 1 ) ∈ Vh × Vh when g ≡ 0. Here E0 stands for the total energy of solutions at time t = T , which is constant in time when g ≡ 0: N ϕ h

j +1 − ϕj 2 2 E(t) = |ϕj | + . h 2

(2.17)

j =0

At this point it is important to emphasize that the key ingredient of the proof of convergence for the two-grid algorithm for the linear wave equation is precisely that the observability constant C in (2.16) is uniform, independent of h.

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Let us now address the perturbed problem (2.14). We first observe that, because of the globally Lipschitz assumption on f , the function g is uniformly bounded, i.e. g L∞ (R) ≤ L,

(2.18)

L being the Lipschitz constant of f . Therefore for all Z ∈ C([0, T ]; RN+2 ) it follows that (2.19) g(Z) L∞ (0, T ; RN+2 ) ≤ L. System (2.14) can then be viewed as a family of perturbed semi-discrete wave equations, the perturbations with respect to the conservative wave equation being a family of zero order bounded potentials. Then a standard perturbation argument allows showing that (2.16) holds for system (2.14) too, with, possibly, a larger observability constant, depending on L, but independent of Z. In fact, arguing by contradiction, since h > 0 is fixed and we are therefore dealing with finite-dimensional dynamical systems, the problem is reduced to show that the following unique continuation property holds for all Z: If ϕN (t) = 0, 0 < t < T , then ≡ 0.

(2.20)

This property is easy to prove by induction. Indeed, using the boundary condition ϕN +1 ≡ 0 and the fact that ϕN ≡ 0, and writing the equation (2.14) for j = N we deduce that ϕN −1 2ϕN − ϕN −1 = ϕN + + g(zN )ϕN = 0. (2.21) 2 h h2 This implies that ϕN −1 = 0. Repeating this argument we deduce that ≡ 0. Once (2.16) holds for the solutions of (2.14) with initial data in Vh ×Vh , uniformly on Z, system (2.14) turns out to be controllable in the sense of (2.6) with an uniform bound on the control, independent of Z, i.e. v L2 (0, T ) ≤ C(h, (Y 0 , Y 1 ) , L, T ) for all Z.

(2.22)

We emphasize that the bound (2.22), in principle, depends on the time of control T , the mesh-size h, the Lipschitz constant L of the nonlinearity f and the norm of the initial data to be controlled, but is independent of Z. As a consequence of (2.22) a similar estimate can be obtained for the state Y solution of (2.12), i.e. Y C 1 ([0, T ]; RN+2 ) ≤ C

for all Z.

(2.23)

This allows applying the Schauder’s fixed point theorem to the map N , which turns out also to be compact from L2 (0, T ; RN +2 ) into itself, thanks to (2.23). In this way we conclude that, for all h > 0, system (2.4) is controllable in the sense of (2.6). Step 2. Uniform controllability with respect to h. In the previous step we have proved the controllability of (2.4) but with estimates on controls and states depending on h.


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In order to pass to the limit as h → 0 we need to get a bound on controls and states which is independent of h. For doing that we need to assume that T > T0 (so that the linear unperturbed numerical schemes are uniformly, with respect to h > 0, two-grid controllable) and that the initial data (y 0 , y 1 ) belong to H s (0, 1) × H s−1 (0, 1). The last requirement is important to get the compactness of the nonlinear term. Indeed, in that setting the control for the continuous wave equation (2.1) belongs to H s (0, T ) rather than L2 (0, T ) and the controlled trajectory y then belongs to C([0, T ]; H s (0, 1)) ∩ C 1 ([0, T ]; H s−1 (0, 1)). This guarantees the required compactness properties to deal with the nonlinear term f (y) in (2.1). Indeed, when passing to the limit, the pointwise convergence of the state in (0, 1) × (0, T ) is needed and this is achieved by means of the extra H s regularity imposed on the initial data (see [25]). This is necessary both when treating the continuous equation (2.1) by fixed point arguments and also when dealing with numerical approximation issues and limit processes as h → 0. To analyze the controllability of the systems under consideration in H s (0, 1) × s−1 H (0, 1), we first need to analyze the H −s -version of the observability inequality (2.16), namely: ϕ 2 N . (2.24) E0, −s ≤ Cs h H −s (0, T ) Inequality (2.24) may be proved for the adjoint system (2.14) in the absence of the potential induced by the nonlinearity, i.e. for ⎧ 2ψj −ψj +1 −ψj −1 ⎪ = 0, j = 1, . . . , N, 0 < t < T , ⎨ψj + h2 (2.25) ψ0 = ψN +1 = 0, 0 < t < T, ⎪ ⎩ 0 1 ψj (T ) = ψj , ψj (T ) = ψj , j = 1, . . . , N. More precisely, for 0 < s < 1/2 and T > T0 as in Theorem 2.1, there exists a constant Cs such that (2.24) holds for all solution ψ of (2.25) with initial data in Vh × Vh and all h > 0. We emphasize that the constant Cs is independent of h. In (2.24) E0, −s stands for the H −s version of the energy of system (2.25), which is constant in time. It can be defined easily by means of the Fourier expansion of solutions and it is then the discrete analogue of the continuous energy E0, −s =

2 1 ψ 0 2 1−s + ψ 1 H −s (0, 1) (0, 1) H 2

(2.26)

which is constant in time for the solutions of the unperturbed adjoint wave equation ⎧ ⎪ 0 < x < 1, 0 < t < T , ⎨ψ − ψxx = 0, (2.27) ψ(0, t) = ψ(1, t) = 0, 0 < t < T, ⎪ ⎩ 0 1 ψ(x, T ) = ψ (x), ψt (x, T ) = ψ (x), 0 < x < 1. The inequality (2.24) may be obtained, as (2.16), by the two methods mentioned above:

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• It can be proved as a consequence of (2.16) directly using interpolation arguments (see, for instance, [25]); • It can also be obtained by the variant of the Ingham inequality in [16]. Once (2.24) is proved for the unperturbed system (2.25), uniformly on h > 0, we are in conditions to prove it for the perturbed system (2.14) uniformly on h > 0 and Z too. To do it we use a classical perturbation and compactness argument (see [25]). We decompose the solution of (2.14) as = + where solves the unperturbed system (2.25) with the same data (0 , 1 ) as itself and where the remainder = (σ0 , . . . , σN +1 ) solves ⎧ 2σj −σj +1 −σj −1 ⎪ = −g(zj )ϕj , j = 1, . . . , N, 0 < t < T , ⎨σj + h2 (2.28) 0 < t < T, σ0 = σN +1 = 0, ⎪ ⎩ σj (T ) = σj (T ) = 0, j = 1, . . . , N. In view of (2.24), which is valid for , we deduce that σ 2 ϕ 2 N N . + −s E0, −s ≤ 2Cs −s h H (0, T ) h H (0, T ) Using discrete multipliers (see [8]) it follows that σ N ≤ C {g(zj )ϕj } L2 (0, T ; 2 ) 2 h h L (0, T )

(2.29)

(2.30)

with a constant C which depends on T but is independent of h. In (2.30) we use the notation {pj } L2 (0, T ; 2 ) = h h

N T

0

1/2 pj2 (t) dt

(2.31)

j =1

which is simply the L2 -norm, scaled to the mesh-size h > 0. Combining (2.30)–(2.31) and using that the nonlinearity g is uniformly bounded we deduce that ϕ 2 N E0, −s ≤ C (2.32) + 2L2 (0, T ; 2 ) , h H −s (0, T ) h for every solution of (2.14) with data (0 , 1 ) in Vh × Vh , every h > 0 and Z. To conclude we can apply a compactness-uniqueness argument whose details may be found in [28] where it was fully developed in the context of the 2 − d semi-discrete wave equation. It consists simply in showing, by contradiction, that there exists an uniform constant C > 0 such that ϕ N L2 (0, T ; 2 ) ≤ C −s (2.33) h h H (0, T )


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for every solution of (2.14), every h > 0 and Z. To do it we assume that there exists a sequence h → 0, potentials of the form g(Zh ) and initial data in Vh × Vh for which (2.33) fails and, consequently, ϕ N → 0, h → 0 (2.34) −s h H (0, T ) L2 (0, T ; 2 ) = 1. h

(2.35)

Combining (2.32), (2.34) and (2.35), the corresponding sequence of data (0h , 1h ) turns out to be bounded in H 1−s (0, 1) × H −s (0, 1) (at this point we are implicitly working with the piecewise linear extension of the data). By the well-posedness of (2.14) in these spaces the corresponding solutions h turn out to be bounded in L∞ (0, T ; H s−1 (0, 1)) ∩ W 1, ∞ (0, T ; H −s (0, 1)). Therefore, they are relatively compact in L2 (0, T ; L2 (0, 1)). Passing to the limit as h → 0 we obtain a solution ϕ of an adjoint wave equation of the form ⎧ ⎪ 0 < x < 1, 0 < t < T , ⎨ϕ − ϕxx + a(x, t)ϕ = 0, (2.36) ϕ(0, t) = ϕ(1, t) = 0, 0 < t < T, ⎪ ⎩ 0 1 ϕ(x, T ) = ϕ (x), ϕt (x, 0) = ϕ (x), 0 < x < 1, such that ∂x ϕ(1, t) = 0,

0 2. In this argument the bounded limit potential a = a(x, t) in (2.36) arises as weak-∗ limit of the discrete ones g (Zh ), (of its piecewise linear extension to 0 < x < 1, 0 < t < T , to be more precise). Therefore a also fulfills the bound a ∞ ≤ L, L being the Lipschitz constant of f . For this argument to apply one needs to pass to the limit in the potential perturbation g (Zh ) h in (2.14). This can be done because of the strong convergence of h (of its extension to 0 < x < 1) in L2 ((0, 1) × (0, T )). Once (2.24) is known to hold for all h > 0 and all data in Vh × Vh this allows proving the uniform controllability of (2.4) in the spaces H s (0, 1) × H −1+s (0, 1), in the two-grid sense (2.6). This can be done applying the fixed point argument in Step 1. More precisely, it follows that there exists a family of controls vh ∈ H s (0, T ), with an uniform bound vh H s (0, T ) ≤ C Yh0 , Yh1 H s (0,1)×H s−1 (0,1) (2.39) such that the solutions Yh of (2.4) satisfy (2.6).

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Step 3. Two-grid observability ⇒ Two-grid controllability. For the sake of completeness, let us show how the two-grid control of (2.12) can be obtained as a consequence of the observability inequality (2.24) for the solutions of the adjoint wave equation (2.14) with initial data in the class Vh × Vh in (2.15) of slowly oscillating data. We introduce the functional

0

1

Jh ,

N

0 1 ϕN 2 = −s +h yj ϕj (0) − yj1 ϕj (0) , 2 h H (0, T )

(2.40)

j =1

which is continuous and convex. Moreover, in view of (2.24), the functional Jh : Vh × ∗ 1, ∗ Vh → R is uniformly coercive. Let us denote by (0, h , h ) the minimizer of Jh over Vh × Vh . Then, ∗ 1, ∗ 0 DJh 0, , , 1 = 0 (2.41) h , h for all (0 , 1 ) ∈ Vh × Vh . This implies that ϕ∗

ϕN +h yj0 ϕj (0) − yj1 ϕj (0) = 0 −s h H (0, T ) N

N

,

h

(2.42)

j =1

∗ 1, ∗ where ∗h stands for the solution of (2.14) with the minimizer (0, h , h ) as data 0 1 and the solution with data ( , ). We now choose the control ϕ∗ vh = Is N , (2.43) h where Is : H −s (0, T ) → H s (0, T ) is the canonical duality isomorphism. Equation (2.42) then reads

vh 0

ϕN yj0 ϕj (0) − yj1 ϕj (0) = 0. dt + h h N

T

(2.44)

j =0

On the other hand, using (2.43) as control in (2.12), multiplying by the solution of the adjoint system (2.14), adding on j = 1, . . . , N and integrating by parts with respect to t ∈ (0, T ), we deduce that 0

T

ϕN v dt + h yj0 ϕj (0) − yj1 ϕj (0) − h yj (T )ϕj1 − yj (T )ϕj0 = 0. (2.45) h N

N

j =1

j =1

Combining (2.44)–(2.45) we deduce that the solution Yh of (2.12) satisfies h

N

yj (T )ϕj1 − yj (T )ϕj0 = 0

j =1

for all (ϕ 0 , ϕ 1 ) ∈ Vh × Vh .

(2.46)


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This means that both Yh (T ) and Yh (T ) are perpendicular to Vh . This is equivalent to the two-grid control requirement (2.6) with z0 ≡ z1 ≡ 0. In view of this construction and using the observability inequality (2.24), which is uniform with respect to h > 0 and Z, we can obtain uniform bounds on the controls. Indeed, by (2.43) we have ϕ∗ vh H s (0, T ) = N −s . (2.47) h H (0, T ) ∗ 1, ∗ On the other hand, the minimizer (0, h , h ) is such that ∗ 1, ∗ ≤0 J 0, h , h

(2.48)

and this implies N

∗ 0 ∗ 1 ϕN 2 H −s (0, T ) ≤ h yj ϕj (0) − yj1 ϕj∗ (0) 2 h j =1 ∗ ≤ (Y 0 , Y 1 ) H s (0,1)×H s−1 (0,1) E0, −s

(2.49)

0, ∗ 1, ∗ ∗ where E0, −s denotes the E0, −s energy of the minimizer (h , h ). Combining (2.48), (2.49) and the observability inequality (2.24) we deduce that vh H s (0, T ) ≤ 2 Cs (Y 0 , Y 1 ) H s (0,1)×H s−1 (0,1) , (2.50)

where Cs is the same constant as in (2.24). In particular, the bound (2.50) on the control is independent of h > 0 and Z. Step 4. Passing to the limit. Using the uniform bound (2.39) it is easy to pass to the limit and get the null-controllability of the semilinear wave equation (2.1). Indeed, as a consequence of (2.39) and by the well-posedness of (2.4) we deduce that the controlled state Yh is uniformly bounded in L∞ (0, T ; H s (0, 1)) ∩ W 1, ∞ (0, T ; H s−1 (0, 1)). By extracting subsequences we have vh v weakly in H s (0, T ) Yh y weakly in L2 (0, T ; H s (0, 1)) ∩ H 1 (0, T ; H s−1 (0, 1)).

(2.51) (2.52)

Consequently, in particular, vh → v strongly in L2 (0, T )

(2.53)

Yh → y strongly in L2 ((0, 1) × (0, T )).

(2.54)

These convergences suffice to pass to the limit in (2.4) and to get (2.1). The strong convergence (2.54) is particularly relevant when doing that since it allows passing to the limit in the nonlinearity.

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Here the convergence of the states Yh may be understood in the sense that its extensions to functions defined for all 0 < x < 1 converge. One can also check that the limit state y = y(x, t) satisfies the final exact controllability requirement (2.3) as a consequence of the two-grid relaxed version (2.6) that the semi-discrete state Yh satisfies. This can be done either by transposition or by compactness. This concludes the sketch of the proof of Theorem 2.1. Remark 2.3. Several remarks are in order: • The proof we have given can be adapted to other equations and schemes. In particular it applies to the two-grid finite element approximation of (2.1). • In [2] a mixed finite-element discretization scheme has been introduced for which the uniform controllability property holds without requiring any filtering or two-grid adaptation. The arguments we have developed here can also be adapted to prove convergence of that method in the semilinear case under the globally Lipschitz assumption on the nonlinearity f . • In [12] it was proved that the standard finite-difference semi-discretization for the exact controllability of the following beam equation converges without filtering or two-grid adaptation: ytt + yxxxx = 0. The method of proof of Theorem 2.1 allows showing that the same is true in the semilinear context too. • The proof of convergence of the two-grid control algorithm is still to be developed for numerical approximations of the wave equation in the multi-dimensional case. But, in view of the proof of Theorem 2.1, which can be easily adapted to the multi-dimensional framework, we can say that, if convergence is proved in the linear case, the same will hold in the semilinear one too, for globally Lipschitz nonlinearities. • For the semilinear wave equation (2.1) the local null-controllability can be proved in wider classes of nonlinearities satisfying f (0) = 0. Here by local null-controllability we refer to the property that sufficiently small initial data can be driven to the null state, i.e. to the existence of δ > 0 such that the control driving the solution to the final equilibrium {0, 0} exists for all initial data {y 0 , y 1 } such that y 0 L2 (0, 1) + y 1 H −1 (0, 1) ≤ δ. It can be proved as a consequence of the controllability of the linear wave equation applying the inverse function theorem around the null state. In order to guarantee the well-posedness of the semilinear wave equation (2.1) in


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L2 (0, 1) × H −1 (0, 1) one also needs to impose a growth condition on the nonlinearity of the form |f (s)| ≤ C|s|

for all s ∈ R.

But this allows proving local controllability for quadratic nonlinearities, for instance (see [24]). The method of proof of Theorem 2.1 can be used to prove the convergence of the two-grid method in that context of local controllability too. • In [24] it was also observed that for nonlinearities with the good sign property, for instance, for f (s) = |s|p−1 s for all s ∈ R with 1 < p ≤ 2, every initial datum may be driven to zero for a sufficiently large time. For doing that one first uses the exponential decay of solutions with boundary feedback ([9]) to later apply the local controllability property when the solution becomes small enough. To adapt that result to the framework of the two-grid scheme one could need a uniform (with respect to h) stabilization result for the numerical schemes with boundary feedback. However it is well known that, due to the lack of uniform boundary observability as h → 0, the uniform stabilization property fails. In [19] and [20] (see also [18]) the uniform (with respect to h) exponential decay property was proved but by adding a viscous damping term distributed all along the mesh. In view of the efficiency of the two-grid approach at the level of controllability, one would expect the uniform (with respect to h > 0) exponential decay property to hold for initial data in the space Vh × Vh without the extra viscous damping term. But this property does not seem to happen. Indeed, when trying to obtain the uniform exponential decay from the uniform observability inequality a technical difficulty appears since the space Vh × Vh is not invariant under the flow of the semi-discrete wave equation. Thus, the observability inequality we have obtained in the time interval [0, T ] with T > T0 can not be extended for all t ≥ 0, a fact that would be needed for proving the exponential decay. But in fact, the situation is much worse and the uniform exponential decay fails to hold since, despite of the fact that the two-grid initial data have a distribution of energy so that most of it is concentrated on the low frequencies, as time evolves, this partition of energy is lost because, precisely, high frequency components are weakly dissipated. The apparently purely technical difficulty for proving the uniform decay turns out to be in fact the reason for the lack of uniform decay. • Recently it has also been proved that 1 − d semilinear wave equations are controllable in the sense that two different equilibria can be connected by a controlled trajectory provided they belong to the same connected component of

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the set of stationary solutions (see [3]). This holds without any sign restriction on the nonlinearity and therefore without excluding blow-up phenomena to occur. Proving the convergence of the two-grid algorithm in what concerns that result is an open problem too. • The main drawback of the arguments we have used in the proof of Theorem 2.1 is that they do not provide any explicit estimate on the cost of controlling the system with respect to the Lipschitz constant L of the nonlinearity. This is due to the use of compactness-uniqueness arguments in the obtention of the uniform (with respect to h) observability estimates (2.24) in the class of initial data Vh × Vh . Therefore we may not recover by this method the property of controllability of the semilinear wave equation under the sharp superlinear growth condition (2.2). • The existence of the convergent controls of the semi-discrete semilinear system has been proved by means of a fixed point method. This adds extra technical difficulties for its efficient computation. The most common tool to deal with such problem, for h fixed and after a suitable time discretization, is the Newton method with variable step. It is applicable in the present situation since the nonlinear map N under consideration is differentiable when the nonlinearity f in the equation is C 1 . In each iteration of the Newton method, one is lead to solve a linearized control problem. But this one is solvable by means of a standard conjugate gradient algorithm ([6]) because of the uniform observability properties that are guaranteed to hold, as we have seen, due to the two-grid relaxation we have introduced. A complete numerical study of these issues is yet to be developed. • The limit control v we have obtained can be proved to be a fixed point of the nonlinear map N that corresponds to the controllability of the semilinear continuous wave equation, based on the HUM controls of minimal L2 (0, T )norm for the linearized wave equations. Thus, the controls we are dealing with both, for the continuous and the semi-discrete equation, belong to the same category.

3. Boundary control of the finite-difference space semi-discretizations of the heat equation 3.1. Problem formulation. This section is devoted to analyze the null controllability of the space semi-discretizations, by means of finite differences, of the heat equation in multi-dimensional domains. To simplify the presentation we focus on the 2 − d case although the same results, with similar proofs, apply in any dimension d ≥ 2. The heat equation in bounded domains is known to be null-controllable from any open, non-empty subset of the domain or its boundary [5]. Thus, it is natural to


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analyze whether the control is the limit of the controls of the semi-discrete systems as the mesh-size tends to zero. But this turns out not to be the case even for the heat equation in the square when the control is applied on a strict subset of one of the segments constituting its boundary (see [30]). In this section we prove a positive counterpart of that result. More precisely, we prove that convergence holds when the control acts on a whole side of the boundary. The proof uses the Fourier series development of solutions, which allows reducing the problem to a one-parameter family of controllable 1 − d heat equations. As a consequence of that result we can prove convergence for general domains when the control is applied on the whole boundary. For that it is sufficient to extend the initial data in the original domain to data in a square containing it and then obtaining the controls on the boundary of the original domain as restrictions to the boundary of the states defined in the extended square. The same results hold in any space dimension. To be more precise, let be the square = (0, π) × (0, π) of R2 . Let 0 be one side of its boundary, say 0 = {(x1 , 0) : 0 < x1 < π}. Consider the heat equation with control on 0 : ⎧ yt − y = 0 in × (0, T ), ⎪ ⎪ ⎪ ⎨y = 0 on [∂ \ 0 ] × (0, T ), (3.1) ⎪ y=v on 0 × (0, T ), ⎪ ⎪ ⎩ y(x, 0) = y 0 (x) in . Here y = y(x, t), with x = (x1 , x2 ), is the state and v = v(x1 , t) is the control. System (3.1) is well-known to be null-controllable in any time T > 0 (see Fursikov and Imanuvilov [5] and Lebeau and Robbiano [11]). More precisely, the following holds: For any T > 0 and any y 0 ∈ L2 () there exists v ∈ L2 (0 × (0, T )) such that the solution y = y(x, t) of (3.1) satisfies y(x, T ) ≡ 0.

(3.2)

Moreover, there exists a constant C > 0 depending on T but independent of the initial datum y 0 such that v L2 (0 ×(0,T )) ≤ C y 0 L2 ()

for all y 0 ∈ L2 ().

(3.3)

In fact the same result holds in a general smooth bounded domain and with controls in any open non-empty subset 0 of its boundary. In the present setting, this result is equivalent to an observability inequality for the adjoint heat equation: ⎧ ⎪ in × (0, T ), ⎨ϕt + ϕ = 0 (3.4) ϕ=0 on ∂ × (0, T ), ⎪ ⎩ 0 ϕ(x, T ) = ϕ (x) in .

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More precisely, it is equivalent to the existence of a positive constant C > 0 such that ϕ(0) 2L2 () ≤ C

T 0

2 ∂ϕ dσ dt 0 ∂n

for all ϕ 0 ∈ L2 ().

(3.5)

Here and in the sequel by n we denote the unit exterior normal vector field and by ∂ · /∂n the normal derivative. In this case, over 0 , ∂ · /∂n = −∂ · /∂x2 . Let us now consider the finite-difference space semi-discretizations of (3.1) and (3.4). Given N ∈ N we set h = π/(N + 1) and we consider the mesh xi,j = (ih, j h),

i, j = 0, . . . , N + 1.

(3.6)

We now introduce the finite-difference semi-discretizations: ⎧ + 1 (4y − y ⎪ yj,k j,k j +1,k − yj −1,k − yj,k+1 − yj,k−1 ) = 0, (j, k) ∈ h , 0 < t < T , ⎪ h2 ⎪ ⎪ ⎨y = 0, (j, k) ∈ [∂ \ ] , 0 < t < T , j,k 0 h (3.7) ⎪ 0 < t < T, ⎪yj,0 = vj , j = 0, . . . , N + 1, ⎪ ⎪ ⎩y (0) = y 0 , (j, k) ∈ , j,k

j,k

h

and ⎧ 1 ⎪ ⎨ϕj,k − h2 (4ϕj,k − ϕj +1,k − ϕj −1,k − ϕj,k+1 − ϕj,k−1 ) = 0, (j, k) ∈ h , 0 < t < T , ϕj,k = 0, (j, k) ∈ [∂]h , 0 < t < T , ⎪ ⎩ (3.8) 0 ϕj,k (T ) = ϕj,k , (j, k) ∈ h . To simplify the notations, we have denoted by h (resp. ∂h ) the set of interior (resp. boundary) nodes, and by [∂ \ 0 ]h the set of indices (j, k) so that the corresponding nodes belong to ∂ \ 0 . Here and in the sequel yj,k = yj,k (t) (resp. ϕj,k = ϕj,k (t)) stands for an approximation of the solution y of (3.1) (resp. ϕ of (3.4)) at the meshpoints xi,j . On the other hand, vj denotes the control that acts on the semi-discrete system (3.7) through the subset [0 ]h of the boundary. Note that the control does not depend of the index k since the subset of the boundary [0 ]h where the control is being applied corresponds to k = 0. In order to simplify the notation we introduce the vector unknowns and control Yh = (yj,k )0≤j,k≤N +1 ,

h = (φj,k )0≤j,k≤N+1 ,

that we shall often denote simply by Y , and V . Accordingly, systems (3.7) and (3.8) read as follows: Yh + Ah Yh = Bh Vh , Yh (0) = Yh0 ,

Vh = (vj )1≤j ≤N ,

(3.9)

(3.10)


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h − Ah h = 0, h (T ) = 0h .

(3.11)

We denote by Ah the usual positive-definite symmetric matrix associated with the fivepoint finite-different scheme we have employed in the discretization of the Laplacian so that 1 (Ah W )j,k = 2 (4wj,k − wj +1,k − wj −1,k − wj,k+1 − wj,k−1 ), (3.12) h for the inner nodes. In (3.11) the homogenous boundary conditions have been integrated by assuming simply that their values in the expression (3.12) have been replaced by the zero one. On the other hand the linear operator Bh in (3.10) is such that the action of the control vj enters on those nodes which are neighbors to those of [0 ]h , i.e. for k = 1, so that [Bh V ]j,k = 0 whenever 2 ≤ k ≤ N but [Bh V ]j,1 = −vj / h2 . The null-controllability problem for system (3.10) reads as follows: Given Y 0 ∈ N +2×N+2 R to find V ∈ L2 (0, T ; RN ) such that the solution Y of (3.10) satisfies Y (T ) = 0.

(3.13)

On the other hand, the problem of observability for system (3.11) consists in proving the existence of C > 0 such that T

N φj,1 2 2 (0) h ≤ Ch (3.14) dt h 0 j =1

for every solution of (3.11). In (3.14) · h stands for the scaled Euclidean norm +1 1/2 N

h = h2 |φj,k |2

(3.15)

j,k=0

and the right hand side term of inequality (3.14) represents the discrete version of the L2 -norm of the normal derivative in (3.5). A similar problem can be formulated in general bounded smooth domains . In that case, obviously, the domain needs to be approximated by domains h whose boundaries are constituted by mesh-points. We first address the case of the square domain by Fourier series to later derive some consequences for general domains. All this section is devoted to the problem of null control. Obviously the situation is different if the final requirement is relaxed to an approximate controllability condition. In that context, as a consequence of the null controllability of the limit heat equation and the convergence of the numerical algorithm it can be proved that the state Yh at time t = T can be driven to a final state of norm εh such that εh → 0 as h → 0. But, as mentioned above, this property fails in general in the framework of null controllability. At this point the work in [10] is also worth mentioning. There it was proved that, in the context of analytic semigroups, one can also get uniform bounds on the number of iterations needed for computing controls using conjugate gradient algorithms.

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3.2. The square domain. The goal of this subsection is to prove that, as h → 0, the controls Vh of (3.10) are uniformly bounded and converge in L2 (0 × (0, T )) to the control of (3.1). All along this section we deal with controls of minimal L2 -norm, the so-called HUM controls. In order to make this convergence result more precise it is convenient to take the following facts into account: • To state and analyze the convergence of the discrete states Yh it is convenient to extend them to continuous functions yh (x, t) with respect to the space variable x = (x1 , x2 ). This can be done, as in the previous section, in two different ways either by considering a piecewise linear and continuous extension or extending the discrete Fourier expansion of solutions by keeping exactly the same analytic expression. The control Vh has to be extended as well to a function depending on the continuous variable 0 < x1 < π . This can be done in the same two ways. • To state the convergence of controls as h → 0 the initial data Yh0 in (3.10) have to be chosen in connection with the initial data y 0 of the PDE (3.1). This may be done in several ways. When y 0 is continuous, Yh0 can be taken to be the restriction of y 0 to the mesh-points. Otherwise, one can take average values over cells, or simply truncate the Fourier expansion of the continuous initial datum y 0 by taking the first N × N terms. This being made precise, the following result holds: Theorem 3.1. Let T > 0 be any positive control time. Let y 0 ∈ L2 () and Yh0 be as above. Then, the null controls Vh for the semi-discrete problem (3.10) are uniformly bounded, with respect to h and converge in L2 (0 × (0, T )) towards the null control of the heat equation (3.1). The semi-discrete controlled states Yh also converge to the controlled state y of the heat equation in L2 (0, T ; H −1 ()) satisfying the null final condition (3.2). Remark 3.2. The result is sharp in what concerns the support 0 of the control. Indeed, as pointed out in [30] this result fails when [0 ]h is replaced by the set of indices [0∗ ]h in which the first node corresponding to the index j = 1 is removed. In that case the observability inequality (3.14) fails because of the existence of a nontrivial solution (3.11) such that vanishes on [0∗ ]h . This is so in fact because of the existence of a non-trivial eigenvector of the discrete Laplacian Ah with eigenvalue λh = 4/ h2 , taking alternating values ±1 along the diagonal and vanishing out of it. The main elements of the proof of this result are the following. The key point is precisely proving that the observability inequality (3.14) is uniform with respect to the mesh-size h > 0. Once this is done standard variational methods allow proving that the controls are uniformly bounded and then passing to the limit as h → 0. We refer to [27] where the same issue was addressed for the heat equation in thin cylindrical


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domains by similar tools and to [14] where the limit process was described in detail in the context of the finite-difference semi-discrete approximation of the 1 − d heat equation. The method of proof of the uniform estimate (3.14) depends heavily on the Fourier decomposition of solutions. To develop it we need some basic facts about the Fourier decomposition of the discrete Laplacian. The eigenvalue problem associated with the semi-discrete system (3.11) is as follows: 1 4wj,k − wj +1,k − wj −1,k − wj,k+1 − wj,k−1 = λwj,k , (j, k) ∈ h , h2 (j, k) ∈ [∂]h . wj,k = 0, (3.16) Its spectrum may be computed explicitly: 4 h mh λ,m (h) = 2 sin2 + sin2 (3.17) h 2 2 (3.18) W ,m (h) = w,m (x) x=(j h,kh),j,k=0,··· ,N+1 for , m = 1, . . . , N , where w,m (x) are the eigenfunctions of the continuous Laplacian: 2 w,m (x) = sin(x1 ) sin(mx2 ). π In particular, in view of (3.18) the eigenvectors of the discrete system (3.16) are simply the restrictions of the eigenfunctions of the continuous Laplacian to the mesh points. Of course, this is a very particular fact that is not true for general domains . It is also easy to check that λ,m (h) → λ,m = 2 + m2

as h → 0

(3.19)

for all , m ≥ 1, where λ,m stand for the eigenvalues of the continuous Laplacian. This confirms that the 5-point finite-difference scheme provides a convergent numerical scheme. The eigenvectors {W ,m },m=1,...,N constitute an orthonormal basis of RN×N with respect to the scalar product N 1/2

f, fh = h2 fj,k fj,k ,

(3.20)

j,k=1

associated with the norm (3.15). The solution of the semi-discrete adjoint system (3.11) can also be easily developed in this basis: N

,m h (t) = a ,m e−λ (h)(T −t) W ,m (3.21) ,m=1

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where {a ,m } are the Fourier coefficients of the datum at time t = T : 0h

=

N

a ,m = 0h , W ,m h .

a ,m W ,m ,

(3.22)

,m=1

Solutions may also be rewritten in the form h (t) =

N

ψ m (t) ⊗ σ m ,

(3.23)

=1

√

where σm =

2 √ sin(mkh) π

, k=0,...,N+1

so that W ,m = σ ⊗σ m , and each vector-valued function ψ m (t) = (ψjm (t))j =0,...,N+1 is a solution of the 1 − d semi-discrete problem: ⎧ 2 m ⎪ ⎨ψj − 2ψj − ψj +1 − ψj −1 / h + μ ψj = 0, j = 1, . . . , N, 0 < t < T , ψ0 = ψN +1 = 0, 0 < t < T, ⎪ ⎩ 0 j = 1, . . . , N, ψj (T ) = ψj , (3.24) mh 4 2 m where μ = h2 sin 2 . The observability inequality (3.14) is equivalent to proving the 1 − d analogue for (3.24), uniformly with respect to the index m ≥ 1, i.e. T 2 ψ1 dt, ψ(0) 2h ≤ C (3.25) h 0 for all ψ 0 , ψ being the solution of (3.24), with a constant C > 0 which is independent of m. The proof of this 1 − d uniform estimate can be developed easily following the arguments in [14]. In fact that inequality is an immediate consequence of the explicit form of the spectrum together with a technical result on series of real exponentials that we recall for the sake of completeness. Consider the class L(ξ, M) constituted by increasing sequences of positive real numbers {νj }j ≥1 such that νj +1 − νj ≥ ξ > 0

1 ≤δ νk

for all j ≥ 1,

(3.26)

for all δ > 0.

(3.27)

k≥M(δ)

Here ξ is any positive number and M : (0, ∞) → N is a function such that M(δ) → ∞ as δ → 0. Obviously, different values of ξ and M determine different classes of sequences L(ξ, M). The following holds (see [14]):

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Proposition 3.3. Given a class of sequences L(ξ, M) and T > 0, there exists a positive constant C > 0 such that 0

T

∞

2 C ak e−νk t dt ≥ !

k≥1 1/νk

k=1

|ak |2 e−2νk T k≥1

νk

,

(3.28)

for all {νk }k≥1 ∈ L(ξ, M) and all bounded sequence {ak }k≥1 . Note that the sequences of eigenvalues of problems (3.24) belong to the same class L(ξ, M) for all h > 0 and m ≥ 1. Thus, the constant C in (3.28) is uniform and, consequently, (3.24) holds, with an observability constant independent of h > 0 and m ≥ 1 as well. Remark 3.4. The same result holds for the case in which the control acts as a right hand side external force applied on a band, i.e. on a set of the form ω = {(x1 , x2 ) : 0 < x1 < γ , 0 < x2 < π} with 0 < γ < π. The corresponding continuous model reads ⎧ ⎪ ⎨yt − y = f 1ω in × (0, T ), (3.29) y=0 on ∂ × (0, T ), ⎪ ⎩ 0 y(x, 0) = y (x) in , where f = f (x1 , x2 , t) is the control and 1ω is the characteristic function of the set ω where the control is applied. The corresponding observability inequality is ϕ(0) 2L2 () ≤ C

T 0

ϕ 2 dxdt

for all ϕ 0 ∈ L2 ().

(3.30)

ω

The problems can be formulated similarly for the semi-discrete scheme we have considered. The observability inequality (3.30) and the corresponding semi-discrete versions hold uniformly with respect to the mesh-size parameter h > 0. Consequently the heat equation (3.29) and the corresponding semi-discretizations are uniformly (with respect to h > 0) null controllable. Convergence of controls and states holds as well. In this case the most natural functional setting is the following one. The initial data y 0 belongs to L2 (), the control f lies in L2 (ω × (0, T )) and the solutions then belong to C([0, T ]; L2 ()) ∩ L2 (0, T ; H01 ()). Convergences hold in these classes as well. 3.3. General domains. The methods of proof of the previous section, based on Fourier series expansions, do not apply to general domains. In fact, even in the context of the continuous heat equation, the existing proofs of null controllability require obtaining the observability estimates by Carleman inequalities (see [5] and [11]). So far the discrete or semi-discrete version of these Carleman inequalities

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and its possible applications to observability estimates for numerical approximation schemes for the heat equation is a completely open subject of research. However, in view of the results of the previous section, and using a classical argument, based on extending the control domain and then getting the controls as restrictions to the original boundary of the controlled states, one can derive similar results for general domains but provided the controls are supported everywhere on the boundary of the domain. The problem of determining sharp conditions on the subsets of the boundary so that the semi-discrete systems are uniformly controllable is completely open. As we have mentioned above, even in the simplest geometry of the square domain of the previous subsection, the result fails to hold without some restrictions on the support of the control that are not needed for the continuous heat equation. The following holds: Theorem 3.5. For all bounded smooth domain , all time T > 0 and all initial data y 0 ∈ L2 (), there exists a uniformly bounded sequence of discrete controls Vh ∈ L2 (∂h × (0, T )) ensuring the null controllability of the finite-difference semidiscrete approximation in h . These controls can be chosen so that the solutions Yh converge weakly in L2 (0, T ; H 1 ()) to the solution y of the heat equation satisfying the null final condition (3.2). Proof. Let us briefly explain how this classical extension-restriction method can be implemented in this framework. Without loss of generality we can assume that is contained in the square domain ˜ = (0, π) × (0, π). We discretize the square as in the previous sections, and define the approximating domains h as those that, having their boundary constituted by mesh-points, better approximate the domain . For the sake of simplicity we assume ˜ that h contains . We also consider a band-like control subdomain ω in the square so that the results of the previous sections apply and h ∩ ω = ∅ for all h > 0. Given initial data y 0 ∈ L2 () for the continuous heat equation we define approximating discrete data Yh0 in h in a standard way, for instance, by simply taking on each mesh-point the average of y 0 on the neighboring square of sides of size h. This data can be easily extended by zero to discrete data Y˜h0 defined in the whole mesh of the square. In view of the results of the previous section (Remark 3.4 ) this generates controls Fh with support in ω, which are uniformly bounded in L2 (ω × (0, T )) and converging, ˜ This yields as h → 0, to the control of the heat equation (3.29) in the square . 2 2 ˜ ∩ L (0, T ; H 1 ()). ˜ also uniformly bounded states Y˜h in the space C([0, T ]; L ()) 0 Obviously, here, as in previous sections, these bounds hold in fact for the piecewise linear continuous extensions of the discrete solutions. More precisely, the corresponding solutions Y˜h converge to the solution y of the ˜ heat equation in the space L2 (0, T ; H01 ()). We can then restrict these solutions to the domains h and obtain the solutions Yh of the semi-discrete system in h , which, by construction, satisfy the final null condition (3.13) and converge to the solution of the heat equation. These solutions satisfy non-homogeneous boundary conditions.


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We read their trace as the boundary controls Vh in ∂h (resp. ∂) for the semi-discrete (resp. continuous) heat equations. These controls are bounded in L2 (∂h × (0, T )) ˜ because they are traces of solutions of bounded energy in L2 (0, T ; H01 ()). Their weak convergence can also be proved. However, at this point one has to be careful since the controls are defined on boundaries ∂h that depend on h. A possible way of stating that convergence rigorously is considering smooth test functions θ(x) defined everywhere in the square and ensuring that ∂h Vh θdσ tends to ∂ vθdσ , as h → 0 for all smooth test functions θ. This convergence property of controls holds as well. 2 Remark 3.6. The method of proof we have presented based on the extension of the domains and using the previously proved results on the square has two main drawbacks: • The first one is that the control is required to be supported everywhere on the boundary of the domain. We emphasize however that, despite the fact that no geometric restrictions are needed for the continuous heat equation, in the sense that null controllability holds from an arbitrarily small open subset of the boundary, that is not the case for the semi-discrete one. Thus, the class of subsets of the boundary for which passing to limit on the null-controllability property is possible is still to be clarified, and the result above showing that the whole boundary always suffices is the first positive one in this direction. • The second one is that it is based on the results obtained in the square by Fourier series techniques. As we have mentioned above, the main tool to deal with continuous heat equations are the Carleman inequalities. As far as we know there is no discrete counterpart of those inequalities and this would be essential to deal with more general heat equations with variable coefficients, or semilinear perturbations. The methods described in Section 2 showing the twogrid controllability of the semilinear wave equation by compactness-uniqueness arguments do not apply for heat-like equations because of their very strong timeirreversibility. Thus, the Carleman approach seems to be the most promising one. However, the fact that observability fails for the semi-discrete system for some observation subdomains indicates that the problem is complex in the sense that the discrete version of the continuous Carleman inequality does not hold. This is a widely open subject of research. Remark 3.7. Similar results hold for a semi-discrete regular finite-element approximation of the heat equation, as long as solutions can be developed in Fourier series, allowing to reduce the problem in the square to a one-parameter family of 1 − d problems, and then apply the extension-restriction method to address general domains. Acknowledgements. I am grateful to E. Trélat and X. Zhang for fruitful comments that contributed to improve the first version of this article. My thanks also go to

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C. Simó for his valuable comments on the use of Newton’s method for efficiently computing the controls obtained in Section 2 by means of fixed point techniques.

References [1] Cannarsa, P., Komornik, V., Loreti, P., Controllability of semilinear wave equations with infinitely iterated logarithms. Control Cybernet. 28 (3) (1999), 449–461. [2] Castro, C., Micu, S., Boundary controllability of a linear semi-discrete 1-D wave equation derived from a mixed finite elements method. Numer. Math. 102 (3) (2006), 413–462. [3] Coron, J.-M., Trélat, E., Global steady-state stabilization and controllability of 1-D semilinear wave equations. Commun. Contemp. Math., to appear. [4] Fernández-Cara, E., Zuazua, E., Null and approximate controllability for weakly blowingup semilinear heat equations. Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (5) (2000), 583–616. [5] Fursikov, A., Imanuvilov, O., Controllability of evolution equations. Lectures Notes Series 34, Research Institute of Mathematics, Global Analysis Research Center, Seoul National University, Seoul 1996. [6] Glowinski, R., Ensuring well-posedness by analogy: Stokes problem and boundary control of the wave equation. J. Comput. Phys. 103 (1992), 189–221. [7] Ignat, L., Zuazua, E., A two-grid approximation scheme for nonlinear Schrödinger equations: Dispersive properties and convergence. C. R. Acad. Sci. Paris Sér. I Math. 341 (6) (2005), 381–386. [8] Infante, J. A., Zuazua, E., Boundary observability for the space-discretizations of the onedimensional wave equation. M2AN Math. Model. Numer. Anal. 33 (2) (1999), 407–438. [9] Komornik, V., Zuazua, E., A direct method for the boundary stabilization of the wave equation. J. Math. Pures Appl. 69 (1) (1990), 33–55. [10] Labbé, S., Trélat, E., Uniform controllability of semidiscrete approximations of parabolic control systems. Systems Control Lett. 55 (7) (2006), 597–609. [11] Lebeau, G., Robbiano, L., Contrôle exact de l’équation de la chaleur. Comm. Partial Differential Equations 20 (1995), 335–356. [12] León, L., Zuazua, E., Boundary controllability of the finite-difference space semidiscretizations of the beam equation. ESAIM Control Optim. Calc. Var. 8 (2002), 827–862. [13] Lions, J. -L., Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, Contrôlabilité exacte, Rech. Math. Appl. 8, Masson, Paris 1988. [14] López, A., Zuazua, E., Some new results related with the null-controllability of the 1 − d heat equation. In Séminaire sur les Equations aux Dérivées Partielles, École Polytechnique, Palaiseau 1997–1998. [15] López, A., Zuazua, E., Uniform null controllability for heat equations with rapidly oscillating coefficients. Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (5) (2002), 543–580. [16] Loreti, P., Mehrenberger, M., A Ingham type proof for a bigrid observability theorem. Preprint #2005-012, Institut de Recherche Mathématique Avancée, Université Strasbourg, Strasbourg 2005; ESAIM Control Optim. Calc. Var., to appear.


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[17] Negreanu, M., Zuazua, E., Convergence of a multigrid method for the controllability of a 1-d wave equation. C. R. Acad. Sci. Paris Sér. I Math. 338 (4) (2004), 413–418. [18] Ramdani, K., Takahashi, T., Tucsnak, M., Uniformly exponentially stable approximations for a class of second order evolution equations. Preprint 27/2003, Institut Elie Cartan, Nancy 2003; ESAIM Control Optim. Calc. Var., to appear. [19] Tcheugoue, L. R., Zuazua, E., Uniform exponential long time decay for the space semidiscretizations of a damped wave equation with artificial numerical viscosity. Numer. Math. 95 (3) (2003), 563–598. [20] Tcheugoue, L. R., Zuazua, E., Uniform boundary stabilization of the finite difference space discretization of the 1 − d wave equation. Adv. Comput. Math., to appear. [21] Trefethen, L. N., Group velocity in finite difference schemes. SIAM Rev. 24 (2) (1982), 113–136. [22] Zhang, X., Explicit observability estimate for the wave equation with potential and its application. Royal Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), 1101–1115. [23] Zhang, X., Zuazua, E., Exact controllability of the semi-linear wave equation. In Sixty Open Problems in the Mathematics of Systems and Control (ed. by V. D. Blondel and A. Megretski), Princeton University Press, Princeton, N.J., 2004, 173–178. [24] Zuazua, E., Exact controllability for the semilinear wave equation. J. Math. Pures Appl. 69 (1) (1990), 1–32. [25] Zuazua, E., Exact boundary controllability for the semilinear wave equation. In Nonlinear Partial Differential Equations and their Applications (ed. by H., Brezis and J.-L. Lions), Collège de France Seminar, Vol. X, Pitman Res. Notes Math. Ser. 220, Longman Scientific & Technical, Harlow 1991, 357–391. [26] Zuazua, E., Exact controllability for the semilinear wave equation in one space dimension. Ann. Inst. H. Poincaré Anal. Non Linéaire 10 (1993), 109–129. [27] Zuazua, E., Null controllability of the heat equation in thin domains. In Equations aux dérivées partielles et applications (Articles dédiés à Jacques-Louis Lions), GauthierVillars, Paris 1998, 787–801. [28] Zuazua, E., Boundary observability for the finite-difference space semi-discretizations of the 2 − d wave equation in the square. J. Math. Pures Appl. 78 (1999), 523–563. [29] Zuazua, E., Optimal and approximate control of finite-diffference schemes for the 1D wave equation. Rend. Mat. Appl. (7) 24 (2) (2004), 201–237. [30] Zuazua, E., Propagation, observation, and control of waves approximated by finite difference methods. SIAM Rev. 47 (2) (2005), 197–243. [31] Zuazua, E., Controllability and Observability of Partial Differential Equations: Some results and open problems. In Handbook of Differential Equations: Evolutionary Differential Equations (ed. by Dafermos, C., Feireisl, E.), vol. 3, Elsevier Science, to appear. Departamento de Matemáticas, Universidad Autónoma, 28049 Madrid, Spain E-mail: [email protected]

Multiscale modeling for epitaxial growth Russel E. Caflisch∗

Abstract. Epitaxy is the growth of a thin film on a substrate in which the crystal properties of the film are inherited from those of the substrate. Because of the wide range of relevant length and time scales, multiscale mathematical models have been developed to describe epitaxial growth. This presentation describes atomistic, island dynamics and continuum models. Island dynamics models are multiscale models that use continuum coarse-graining in the lateral direction, but retain atomistic discreteness in the growth direction. Establishing connections between the various length and time scales in these models is a principal goal of mathematical materials science. Progress towards this goal is described here, including the derivation of surface diffusion, line tension and continuum equations from atomistic, kinetic models. Mathematics Subject Classification (2000). Primary 82D25; Secondary 82C24. Keywords. Epitaxial growth, island dynamics, step edge, step stiffness, Gibbs–Thomson, adatom diffusion, line tension, surface diffusion, renormalization group, kinetic Monte Carlo.

1. Introduction Epitaxy is the growth of a thin film on a substrate in which the crystal properties of the film are inherited from those of the substrate. Since an epitaxial film can (at least in principle) grow as a single crystal without grain boundaries or other defects, this method produces crystals of the highest quality. In spite of its ideal properties, epitaxial growth is still challenging to mathematically model and numerically simulate because of the wide range of length and time scales that it encompasses, from the atomistic scale of Ångstroms and picoseconds to the continuum scale of microns and seconds. The geometry of an epitaxial surface consists of step edges and island boundaries, across which the height of the surface increases by one crystal layer, and adatoms which are weakly bound to the surface. Epitaxial growth involves deposition, diffusion and attachment of adatoms on the surface. Deposition is from an external source, such as a molecular beam. Figure 1 provides a schematic illustration of the processes involved in epitaxial growth. The models that are most often used to describe epitaxial growth include the following: A typical Kinetic Monte Carlo (KMC) method simulates the dynamics of ∗ This work was partially supported by the National Science Foundation through grant DMS-0402276 and by the Army Research Office through grant DAAD19-02-1-0336


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Basic processes in epitaxial growth

(a) deposition (d) attachment (g) diffusion down step (b) diffusion (e) detachment (h) nucleation on top of islands (c) nucleation (f) edge diffusion (i) dimer diffusion

Figure 1. Schematic view of the processes involved in epitaxial growth.

the epitaxial surface through the hopping of adatoms along the surface. The hopping rate has the Arrhenius form e−E/kT in which E is the energy barrier for going from the initial to the final position of the hopping atom. Island dynamics describe the surface through continuum scaling in the lateral directions but atomistic discreteness in the growth direction. Continuum equations approximate the surface using a smooth height function z = h(x, y, t), obtained by coarse graining in all directions. Two other models are used to describe epitaxial growth on a limited time range. Molecular dynamics (MD) consists of Newton’s equations for the motion of atoms on an energy landscape. Because the time scale for MD is femtoseconds (10−15 seconds), this can only be applied to very short time periods. Rate equations describe the surface through a set of bulk variables without spatial dependence. With some exceptions [5], [16], these have been used only for submonolayer growth. The initial theories for epitaxial growth, such as [3], relied on an assumption that the system is close to equilibrium. In many epitaxial systems, however, the system is far from equilibrium so that a kinetic description is required. The emphasis in this article will be on KMC, island dynamics and continuum models for epitaxial systems that are far from equilibrium. A principal goal of mathematical materials science is to analyze the connections between these models. The results presented below, from the work of Margetis [18], Caflisch & Li [7], Margetis & Caflisch [19], Chua et al. [10], and Haselwandter & Vvedensky [14], [15] are for surface diffusion and step stiffness derived from atomistic kinetic models of epitaxy, and general continuum equations from a simplified model. These results are among the first of their kind; e.g., the formula for step stiffness comes from the first derivation of the Gibbs–Thomson formula from an atomistic, kinetic model rather than from a thermodynamic driving force. The results are far from complete. Other effects, such as the nonlinear terms in the continuum equations, have not been derived for a full model of epitaxy. In

Multiscale modeling for epitaxial growth

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addition, the derivations presented here are based on formal asymptotics, rather than rigorous mathematical analysis. Nevertheless, these results are a significant step toward a more complete theory and can serve as a starting point for more rigorous analysis. For simplicity in the presentation, the lattice constant a will be taken to be a = 1, except in a few places where it is useful as a placeholder. Also, all transition rates (with units 1/time) are expressed in terms of equivalent diffusion constants (with units length2 /time); i.e., a rate r is replaced by a diffusion coefficient D = a 2 r.

2. Mathematical models for epitaxial growth In this section, various models for epitaxial growth are described, including atomistic KMC, island dynamics and continuum models, as well as a kinetic model for the structure of a step edge (or island boundary) that is used with island dynamics. 2.1. Atomistic Models. The simplest KMC model is a simple cubic pair-bond solidon-solid (SOS) model [28], [29]. In this model, there is a stack of atoms, without vacancies, above each site on a two-dimensional lattice. New atoms are randomly deposited at a deposition rate F . Any surface atom (i.e., the top atom in the stack of atoms at a lattice point) is allowed to move to its nearest neighbor site at a rate r = D/a 2 in which a is the lattice constant and D is a diffusion coefficient. In the simplest case, D is determined by D = D0 exp{−(ES + nEN )/kB T }.

(1)

In this equation, D0 is a constant prefactor of size 1013 a 2 s −1 , kB is the Boltzmann constant, T is the surface temperature, ES and EN represent the surface and nearest neighbor bond energies, and n is the number of in-plane nearest neighbors. The terrace diffusion coefficient DT for adatoms on a flat terrace and the edge diffusion coefficient DE for adatoms along a step edge (with a single in-plane neighbor) are DT = D0 exp{−ES /kB T }, DE = D0 exp{−(ES + EN )/kB T }.

(2) (3)

Validity of this KMC model for epitaxial growth has been demonstrated by comparison to RHEED measurements from molecular beam epitaxy (MBE) experiments [11]. More complicated models for the diffusion coefficient, subject to the condition of detailed balance, are also used. 2.2. Island dynamics models. Burton, Cabrera and Frank [3] developed the first detailed theoretical description for epitaxial growth. This BCF model is an “island dynamics” model, since it describes an epitaxial surface by the location and evolution of the island boundaries and step edges. It employs a mixture of coarse graining and

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atomistic discreteness, since island boundaries are represented as smooth curves that signify an atomistic change in crystal height. Adatom diffusion on the epitaxial surface is described by a diffusion equation of the form ∂t ρ − DT ∇ 2 ρ = F − 2(dNnuc /dt) (4) in which F is the deposition flux rate and the last term represents loss of adatoms due to nucleation. Desorption from the epitaxial surface has been neglected. The net flux to the step edge from upper and lower terraces is denoted as f+ = f+ (y, t) and f− = f− (y, t), respectively, in which vρ+ + DT n · ∇ρ+ = −f+ , vρ− + DT n · ∇ρ− = f− .

(5) (6)

f = f+ + f− .

(7)

The total flux is Different island dynamics models are distinguished by having different formulas for the diffusive boundary conditions and normal velocity. 1. The island dynamics model with irreversible aggregation: ρ = 0, v = f.

(8)

ρ = ρ∗ , v = f,

(9)

2. The BCF boundary conditions:

in which ρ∗ is the equilibrium adatom density at a step. 3. The island dynamics model with step-edge kinetics: f+ = (DT ρ+ − DE φ) cos θ, f− = (DT ρ− − DE φ) cos θ, v = k w cos θ,

(10)

in which φ and k are the densities of edge-atoms and kinks, and w is the kink velocity, defined in Section 2.3. 4. The island dynamics model with line tension and surface diffusion: f+ = Dd+ (ρ+ − ρ∗ ) − μ+ κ, f− = Dd− (ρ− − ρ∗ ) − μ− κ, v = DT n · [∇ρ] + βρ∗ yy + (μ/DE )κss ,

(11)


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in which κ is curvature and κss is its second derivative along the length of a step edge, ρ∗ is a reference adatom density, Dd± are the attachment/detachment rates, and μ± = (Dd± ρ∗ /kB T )γ˜ in which γ˜ is the step stiffness. This is further discussed in Section 4. For the case of irreversible aggregation, a dimer (consisting of two atoms) is the smallest stable island, and the nucleation rate is dNnuc = Dσ1 ρ 2 , dt

(12)

where · denotes the spatial average of ρ(x, t)2 and σ1 =

4π ln[(1/α)ρD/F ]

(13)

is the adatom capture number. The parameter α reflects the island shape, and α 1 for compact islands. Expression (12) for the nucleation rate implies that the time of a nucleation event is chosen deterministically. The choice of the location of the new island is determined by probabilistic choice with spatial density proportional to the nucleation rate ρ 2 . This probabilistic choice constitutes an atomistic fluctuation that is retained in the island dynamics model [24]. Snapshots of the results from a typical island dynamics simulation are shown in Figure 2. Shown is the island geometry after coverage of 0.25 layers (left) and

Figure 2. Island geometry for island dynamics with irreversible aggregation after deposition of 0.25 layers (left) and 10.25 layers (right).

coverage of 10.25 layers (right). These simulations are for irreversible aggregation with boundary conditions from Eq. (8). Numerical simulation of the island dynamics is performed using a level set method for thin film growth, as described in [4], [8].

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Validation of the island dynamics/level set method has been performed by careful comparison to the results of the atomistic KMC models. Various generalizations and additional physics results are described in [22], [23]. Related work on level set methods for epitaxial growth are found in [9], [25], [26]. The principal dimensionless parameters for epitaxial growth are the ratios of flux and diffusive coefficients, which we refer to as “Péclet numbers” by analogy with fluid mechanics. Let f¯ be a characteristic size for the flux to an edge. Let PT be the terrace Péclet number and PE be the edge Péclet number, defined as PT = F /DT , PE = f/DE ,

(14) (15)

in which DE is the edge diffusion constant. Typical values for PT−1 = DT /F are in the range of 104 to 108 . 2.3. The kinetic edge model. The kinetic edge model of island dynamics was developed in [6]. It involves a statistical description of the crystalline structure of a step edge, including the edge-atom density φ and the kink density k. Edge-atoms are atoms with a single in-plane neighbor along the step; kinks are atoms with two in-plane neighbors. Kinks are of two types – right-facing kinks and left-facing kinks – the densities of which are denoted by kr and k . Figure 3 provides a schematic picture kr

ρ

+

ρ

k

φ

−

Figure 3. The geometry of step edge, showing adatoms with density ρ on the upper and lower terraces + and − , edge-atoms with density φ, and right and left facing kinks with density kr and k .

of the kink density model for a step edge. Related models have been derived by Balykov et al. [1], Balykov & Voigt [2] and Filimonov & Hervieu [13].

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The kinetic edge model consists of a diffusion equation for the edge-atom density φ and a convection equation for the kink density k ∂t φ − DE ∂s2 φ = f+ + f− − f0 , ∂t k + ∂s (w(kr − k )) = 2(g − h).

(16) (17)

In Eq. (16), f± are the net fluxes to the edge from terraces as defined in Eq. (5) and Eq. (6), and f0 is the net loss term due to the attachment of edge-atoms to kinks. In Eq. (17), w is the kink velocity, and g and h represent, respectively, the creation and annihilation of left-right kink pairs. Note that left-facing kinks and right-facing kinks move in opposite directions with velocity w and −w, respectively. The total kink density and the relation between the kink density and the normal angle [3] are kr + k = k, kr − k = tan θ.

(18) (19)

The quantities f+ , f− , f0 , w, g, h, and v are determined by the following constitutive relations (in simplified form): f+ = (DT ρ+ − DE φ) cos θ,

(20)

f− = (DT ρ− − DE φ) cos θ,

(21)

f0 = v (φκ + 1) ,

(22)

w = l1 DE φ + DT (l2 ρ+ + l3 ρ− ) = l123 DE φ + (l2 f+ + l3 f− )/ cos θ,

(23)

g = φ (m1 DE φ + DT (m2 ρ+ + m3 ρ− )) = φ(m123 DE φ + (m2 f+ + m3 f− )/ cos θ),

(24)

h = kr k (n1 DE φ + DT (n2 ρ+ + n3 ρ− )) = kr k (n123 DE φ + (n2 f+ + n3 f− )/ cos θ),

(25)

Xt = v = wk cos θ,

(26)

where DT is the (diffusion) hopping rate of an adatom on a terrace, DE is the (diffusion) hopping rate of an edge-atom along or off an edge, and all li , mi , ni (i = 1, 2, 3) are nonnegative numbers. The geometric parameters li , mi , ni count the number of paths from one state to another, cf. [6] for details. Here, these parameters are generalized to allow a factor relating the macroscopic density ρ or φ to the local density of adatoms or edge atoms at a specific site. For convenience, we have used the notation qij = qi + qj

and

qij k = qi + qj + qk

for q = l, m, or n. For simplicity in this presentation, the constitutive laws (20)–(25) have been simplified by omission of terms that are insignificant for the kinetic steady state solutions of relevance to step-flow growth and by specialization to the case of θ near 0. The terms omitted from (20)–(25) include terms that are important for detailed

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balance, so that they are required for determination of the equilibrium solution for this model. In the more complete analysis of [6], [7], [19], all of the neglected terms are included. There are several significant solutions for the kinetic step edge model Eq. (16)– Eq. (26). First, there is an equilibrium solution that was originally determined in [3] (note that some terms that have been omitted from the presentation here are significant for the equilibrium). Second there is a kinetic (i.e., nonequilibrium) steady state solution, for which the presentation includes all of the significant terms. Suppose that the kink density and edge Peclet number are small (i.e., ak 1 and PE 1) and that the step is symmetric (i.e., ρ+ = ρ− ), then the adatom, edge-adatom and kink densities of the kinetic steady state are approximately ρ = (DE /DT )a −1 ϕ,

(27)

ϕ = (16a/3)k , 1 3 16 k= a −1 . PE 15

(28)

2

(29)

The exponent 1/3 in (29) is related to the critical size for formation of a left-right kink pair. If the critical size were j (i.e., if j + 1 edge-adatoms were required to form a stable kink pair) then the exponent would be j/(j + 2). Figure 4 shows a comparison of this steady state solution (solid line) and computational results from KMC (squares, circles and triangles) for kink density k. The 10

k/L1/3

10

10

1

2

3

4

10

4

10

5

6

10

10

7

10

DE Figure 4. Kink density k, normalized by L1/3 , vs. edge diffusion coefficient DE for kinetic steady state, for various values of terrace width L. Parameter values are flux F = 1 and adatom diffusion DT = 1012 . Results are shown from the kinetic theory (solid line) and KMC computations with L = 25 (squares), L = 50 (), and L = 100 ( ). These are compared to the corresponding equilibrium values for L = 25 (dash-dotted line), L = 50 (dashed line), and L = 100 (dotted line), showing that the steady state and equilibrium differ both qualitatively and quantitatively.


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BCF equilibrium values for k are also plotted (lower three lines) for comparison. In this figure, F = 1 and DT = 1012 , while DE varies between 104 and 107 . The computations are for a periodic step train with straight steps at angle θ = 0 and with distance L = 25, 50, and 100 between the steps. The figure shows excellent agreement between the predictions of the present theory and the results of the KMC simulation, with differences that are less than one standard deviation of the KMC results. The results are significantly different from equilibrium both in size and in dependence on DE . 2.4. Continuum models. Continuum models of epitaxial growth employ coarse graining in all directions. In most cases, they describe the epitaxial surface through a smooth height function z = h(x, y, t). The general equation of this type, as discussed by Haselwandter and Vvedensky [14], [15], is ht = ν2 ∇ 2 h − ν4 ∇ 4 h + λ13 ∇(∇h)3 + λ22 ∇ 2 (∇h)2 + ξ

(30)

in which the ν2 term comes from a height-dependence in the energy, the ν4 term is surface diffusion, the λ terms are nonlinearities, and ξ is a stochastic noise term. This equation generalizes previous models, including the Edwards–Wilkinson (EW) equation with ν4 = λ13 = λ22 = 0, the Mullins–Herring (MH) equation with ν2 = λ13 = λ22 = 0, and the Villain–Lai–Das Sarma (VLDS) equation with ν2 = λ13 = 0. The relations between these models are further discussed in Section 5, through a renormalization group analysis applied to the Edwards–Wilkinson and Wolf–Villain models for epitaxial growth. Derivation of these continuum equations has been mostly through symmetry arguments, thermodynamic driving force or by heuristics. The results reported in Sections 3 and 5 are among the first derivation of these equations from kinetic, atomistic models. Alternative modeling approaches have included additional dependent variables, not just the interface height. For example, Lo & Kohn [17] included adatom density in addition to height. Margetis et al. [20] derive similar results starting from an island dynamics model with a kinetic step edge description, as in Section 2.2 and 2.3.

3. Surface diffusion As first derived by Mullins [21], the surface diffusion equation ht = −ν4 ∇ 4 h

(31)

describes the evolution of a surface through diffusion of the material that comprises the surface. Margetis [18] has given an atomistic, kinetic derivation of surface diffusion for epitaxial growth, and he found the surprising result that surface diffusion is not isotropic. While his derivation is based on detailed asymptotics starting from an island

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dynamics model, the presentation here will be phenomenological but faithful to the spirit of Margetis’s derivation. Consider an epitaxial surface that consists of a series of steps that are nearly parallel. The terrace width between steps is approximately = a/|∇h| in which a is the lattice constant. Assume that the steps are slowly varying but widely spaced, so that a λ in which λ is the length scale for the variation in the steps away from straight. Also, assume that edge diffusion coefficient DE and the edge attachment/detachment rate Dd are much smaller than the terrace diffusion coefficient DT . The analysis is based on the following fundamental property of diffusion: Consider a composite consisting of strips of two material with diffusion coefficients D1 and D2 and with strip widths a1 and a2 . The effective diffusion coefficient D∗ for the composite is the arithmetic average if the diffusion is in the direction along the strips (i.e., a parallel configuration) and it is the harmonic average if the diffusion is in the direction perpendicular to the strips (i.e., a series configuration); i.e., parallel configuration, (a1 D1 + a2 D2 )/(a1 + a2 ) D∗ = (32) −1 −1 −1 series configuration. ((a1 D1 + a2 D2 )/(a1 + a2 )) Define a tangential variable s along the steps and a normal variable n perpendicular to the steps. In the tangential direction, adatoms diffuse at the terrace diffusion rate of DT on the terraces between steps and at the edge diffusion rate DE along the steps. Since the terraces and steps are in parallel in the tangential direction, the corresponding diffusion coefficient Dss = (aDE + DT )/(a + ) ≈ DT .

(33)

Diffusion of adatoms normal the steps is also at rate DT , but it is interrupted by attachment and detachment from the steps at rate Dd . Since the terraces and steps are in a series configuration in the normal direction n, the diffusion coefficient in this direction is Dnn = ((2aDd−1 + DT−1 )/(a + ))− 1 ≈ DT (1 + m|∇h|)−1

(34)

m = 2DT /Dd .

(35)

in which The factor of 2 in the last two formulas is due to the details of the attachment/detachment model used in [18]. Now follow the derivation of diffusion from the thermodynamic driving force (but note that Margetis used a perturbation expansion based on the kinetic equations rather than this near-equilibrium argument). The evolution of the height h is given in terms of the mobility tensor M, current j and chemical potential μ as ht = −∇ · j = ∇ · (M∇μ)

(36)


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since j = −M∇μ and μ = δE/δh = −g1 ∇ · (∇h/|∇h|) − g3 ∇h · (|∇h|∇h). By the argument above, M is the matrix DT ρ∗ 1 0 M= (37) KB T 0 (1 + m|∇h|)−1 in the n, s coordinates.

4. Step stiffness In [7], Caflisch and Li considered the zero limit of the edge Peclet PE number for the kinetic edge model from Section 2.3 for a step that is a slight perturbation of a straight step with θ = 0, i.e., parallel to a crystallographic direction. They used a very specific form for the wavelength and amplitude of the perturbation and their scaling with PE , and they assumed that the solution was close to the kinetic steady state Eq. (29) for θ = 0. They derived the boundary conditions Eq. (11) for the evolution of a step, including the Gibbs–Thomson form of the step stiffness and a term due to edge diffusion of the adatoms that attach to the step. More recently, Margetis and Caflisch [19] performed a more general analysis for a step with variable θ. Under the assumption that PE 1 and that the solution is close to the kinetic steady state, they identified several regimes for the behavior of the solution and the step stiffness coefficient. Since the complete results are complicated, we present the results in an abbreviated form and refer to [19] for a detailed expression. First, suppose that the curvature κ of the step satisfies |κ| < O(PE ) 1. Then the step edge kinetics allow for two regimes for the step stiffness γ˜ : 1/3 (kB T /DT ρ∗ )θ −1 for PE θ 1, γ˜ = 1/3 for 0 < θ PE . (kB T /DT ρ∗ )γ˜0

(38)

(39)

The results for step stiffness imply results for the line tension γ , since γ˜ = γ +γθθ then to leading order 1/3 (kTB /DT ρ∗ )θ log θ for PE θ 1, (40) γ ≈ 1/3 c0 for 0 < θ PE . in which c0 is an undetermined constant. In the outer solution, γθ is nearly infinite for θ small, which predicts a flat facet corresponding to θ = 0. The inner solution provides some curvature to this facet, however. These results are consistent with the recent results of Stasevich et al. [27] for the step stiffness in an Ising model.

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5. Coarse graining In a remarkable series of papers [10], [14], [15], Chua et al. and Haselwandter & Vvedensky performed coarse-graining, followed by a renormalization group analysis, for the Edwards–Wilkinson [12] and Wolf–Villain [30] models of epitaxial growth. In the Wolf–Villain model, particles are randomly deposited on the surface at rate F . Instead of diffusing along the surface, however, each deposited particle makes a single hop to the nearest neighbor site that has the largest number of neighbors (i.e., the highest coordination), or stays where it landed if that hop does not increase the number of neighbors. In two-dimensions (i.e., a one dimensional surface), a particle hops to a position that is no higher than its original position, while in three-dimensions, some particles may increase their coordination by hopping to a higher position. For a general lattice model, Haselwandter and Vvedensky first write the stochastic evolution in terms of a Chapman–Kolmogorov transition probability Tt (H2 |H1 ) for transition between height configuration H1 to H2 in time t Tt+t (H3 |H1 ) = Tt (H3 |H2 )Tt (H2 |H1 ). (41) H2

This can be converted to a Master equation [W ((H − r; r)P (H − r, t) − W (H ; r)P (H, t)] P (H, t) =

(42)

r

in which P (H, t) is the probability for height configuration H at time t, W (H ; r) is the transition rate between H and H + r, and r is the array of jump lengths between configurations. They then apply a Kramers–Moyal–van Kampen expansion with “largeness” parameter , with the lattice size and time between depositions being proportional to −1 . In the limit → ∞, this expansion yields a lattice Langevin equation (1) ∂hij /∂t = Kij + ηij (43) (1)

in which Kij are the first moment of the transition rates and ηij are Gaussian noises with mean zero and covariances given by the second moment of the transition rates. After performing a smoothing and a cutoff Taylor expansion and specializing to the Wolf–Villain (or Edwards–Wilkinson) model, the Langevin equation becomes Eq. (30). Finally they perform a renormalization group (RG) analysis of equation Eq. (30). In the RG “flow”, length and time are scaled at an exponential rate in the flow variable . This analysis shows that the RG fixed points consist of the EW, MH and VLDS equations, as well as three previously unrecognized fixed points. The significance of this result is that as the solution of Eq. (30) evolves, it will linger near the fixed points, so that the solution will approximate a solution of each of these equations. The most important of the equations corresponding to these new fixed points, is their “FP1”.


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Although it is complicated in general, in the two dimensional case it has the form ht = −|ν2 |∇ 2 h − |ν4 |∇ 4 h − |λ13 |∇(∇h)3 + λ22 ∇ 2 (∇h)2 + ξ

(44)

which should be compared to Eq. (30). In two-dimensions this equation corresponds to a stable fixed point, but the corresponding equation in three dimensions corresponds to an unstable fixed point for the RG flow. This coarse graining and RG analysis provides both a derivation of these equations, starting from the EW or WV model, as well as an indication of the regimes of their validity. Acknowledgments. The presentation here benefitted from discussions with Ted Einstein, Christoph Haselwandter, Dionisios Margetis, Tim Stasevich, Axel Voigt, and Dimitri Vvedensky. In particular, Margetis, Haselwandter and Vvedensky provided the author with advance copies of their unpublished work.

References [1] Balykov, L., Kitamura, M., Maksimov, I. L., Nishioka, K., Kinetics of non-equilibrium step structure. Phil. Mag. Lett. 78 (1998), 411–418. [2] Balykov, L., Voigt, A., Kinetic model for step flow growth of [100] steps. Phys. Rev. E 72 (2005), #022601. [3] Burton, W. K., Cabrera, N., Frank, F. C., The growth of crystals and the equilibrium structure of their surfaces. Phil. Trans. Roy. Soc. London Ser. A 243 (1951), 299–358. [4] Caflisch, R. E., Gyure, M. F., Merriman, B., Osher, S., Ratsch, C., Vvedensky, D. D., Zinck, J. J., Island dynamics and the level set method for epitaxial growth. Appl. Math. Lett. 12, 13 (1999), 13–22. [5] Caflisch, R. E., Meyer, D., A reduced order model for epitaxial growth. Contemp. Math. 303 (2002), 9–23. [6] Caflisch, R. E., E, W., Gyure, M. F., Merriman, B., Ratsch, C., Kinetic model for a step edge in epitaxial growth. Phys. Rev. E 59 (1999), 6879–6887. [7] Caflisch, R. E., Li, B., Analysis of island dynamics in epitaxial growth. Multiscale Model. Sim. 1 (2002), 150–171. [8] Chen, S., Kang, M., Merriman, B., Caflisch, R. E., Ratsch, C., Fedkiw, R., Gyure, M. F., Osher, S., Level set method for thin film epitaxial growth. J. Comp. Phys. 167 (2001), 475–500. [9] Chopp, D., A level-set method for simulating island coarsening. J. Comp. Phys. 162 (2000), 104–122. [10] Chua, A. L.-S., Haselwandter, C. A., Baggio, C., Vvedensky, D. D., Langevin equations for fluctuating surfaces. Phys. Rev. E 72 (2005), #051103. [11] Clarke, S., Vvedensky, D. D., Origin of reflection high-energy electron-diffraction intensity oscillations during molecular-beam epitaxy: A computational modeling approach. Phys. Rev. Lett. 58 (1987), 2235–2238.

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[12] Edwards, S. F., Wilkinson, D. R., The surface statistics of a granular aggregate. Proc. Roy. Soc. Ser. A 381 (1982), 17–31. [13] Filimonov, S. N., Hervieu, Y. Y., Terrace-edge-kink model of atomic processes at the permeable steps. Surface Sci. 553 (2004), 133–144. [14] Haselwandter, C. A., Vvedensky, D. D., Multiscale theory of fluctuating interfaces during growth. Preprint, 2005. [15] Haselwandter, C. A., Vvedensky, D. D., Multiscale theory of fluctuating interfaces: From self-affine to unstable growth. Preprint, 2005. [16] Kariotis, R., Lagally, M. G., Rate equation modelling of epitaxial growth. Surface Sci. 216 (1989), 557–578. [17] Lo, T. S., Kohn, R. V., A new approach to the continuum modeling of epitaxial growth: slope selection, coarsening, and the role of the uphill current. Physica D 161 (2002), 237–257. [18] Margetis, D., Unified continuum approach to crystal surface morphological relaxation. Preprint, 2005. [19] Margetis, D., Caflisch, R. E., Modified kinetic model of step edge dynamics: modeling and derivation of step stiffness. Preprint, 2005. [20] Margetis, D., Voigt, A., Caflisch, R. E., private communication, 2005. [21] Mullins, W., Theory of thermal grooving. J. Appl. Phys. 28 (1957), 333–339. [22] Petersen, M., Ratsch, C., Caflisch, R. E., Zangwill, A., Level set approach to reversible epitaxial growth. Phys. Rev. E 64 (2001), #061602. [23] Ratsch, C., Gyure, M., Caflisch, R. E., Gibou, F., Petersen, M., Kang, M., Garcia, J., Vvedensky, D. D., Level-set method for island dynamics in epitaxial growth. Phys. Rev. B 65 (2002), #195403. [24] Ratsch, C., Gyure, M. F., Chen, S., Kang, M., Vvedensky, D. D., Fluctuations and scaling in aggregation phenomena. Phys. Rev. B 61 (2000), 10598–10601. [25] Russo, G., Smereka, P., A level-set method for the evolution of faceted crystals. SIAM J. Sci. Comput. 21 (2000), 2073–2095. [26] Smereka, P., Spiral crystal growth. Physica D 138 (2000), 282–301. [27] Stasevich, T. J., Gebremariam, H., Einstein, T. L., Giesen, M., Steimer, C., Ibach H., Lowtemperature orientation dependence of step stiffness on {111} surfaces. Phys. Rev. B 71 (2005), #245414. [28] Vvedensky, D. D., Atomistic modeling of epitaxial growth: comparisons between lattice models and experiment. Comp. Materials Sci. 6 (1996), 182–187. [29] Weeks, J. D., Gilmer, G. H., Dynamics of crystal growth. Adv. Chem. Phys. 40 (1979), 157–228. [30] Wolf, D. E., Villain, J., Growth with surface diffusion. Europhys. Lett. 13 (1990), 389–394. Mathematics Department, UCLA, Los Angeles, CA 90095-1555, U.S.A. E-mail: [email protected]

Compressive sampling Emmanuel J. Candès ∗

Abstract. Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired resolution of the image, i.e. the number of pixels in the image. This paper surveys an emerging theory which goes by the name of “compressive sampling” or “compressed sensing,” and which says that this conventional wisdom is inaccurate. Perhaps surprisingly, it is possible to reconstruct images or signals of scientific interest accurately and sometimes even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g. the number of pixels in the image. It is believed that compressive sampling has far reaching implications. For example, it suggests the possibility of new data acquisition protocols that translate analog information into digital form with fewer sensors than what was considered necessary. This new sampling theory may come to underlie procedures for sampling and compressing data simultaneously. In this short survey, we provide some of the key mathematical insights underlying this new theory, and explain some of the interactions between compressive sampling and other fields such as statistics, information theory, coding theory, and theoretical computer science. Mathematics Subject Classification (2000). Primary 00A69, 41-02, 68P30; Secondary 62C65. Keywords. Compressive sampling, sparsity, uniform uncertainty principle, underdertermined systems of linear equations, 1 -minimization, linear programming, signal recovery, error correction.

1. Introduction One of the central tenets of signal processing is the Nyquist/Shannon sampling theory: the number of samples needed to reconstruct a signal without error is dictated by its bandwidth – the length of the shortest interval which contains the support of the spectrum of the signal under study. In the last two years or so, an alternative theory of “compressive sampling” has emerged which shows that super-resolved signals and images can be reconstructed from far fewer data/measurements than what is usually considered necessary. The purpose of this paper is to survey and provide some of the key mathematical insights underlying this new theory. An enchanting aspect of compressive sampling it that it has significant interactions and bearings on some fields in the applied sciences and engineering such as statistics, information theory, coding ∗ The author is partially supported by an NSF grant CCF–515362.


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theory, theoretical computer science, and others as well. We will try to explain these connections via a few selected examples. From a general viewpoint, sparsity and, more generally, compressibility has played and continues to play a fundamental role in many fields of science. Sparsity leads to efficient estimations; for example, the quality of estimation by thresholding or shrinkage algorithms depends on the sparsity of the signal we wish to estimate. Sparsity leads to efficient compression; for example, the precision of a transform coder depends on the sparsity of the signal we wish to encode [24]. Sparsity leads to dimensionality reduction and efficient modeling. The novelty here is that sparsity has bearings on the data acquisition process itself, and leads to efficient data acquisition protocols. In fact, compressive sampling suggests ways to economically translate analog data into already compressed digital form [20], [7]. The key word here is “economically.” Everybody knows that because typical signals have some structure, they can be compressed efficiently without much perceptual loss. For instance, modern transform coders such as JPEG2000 exploit the fact that many signals have a sparse representation in a fixed basis, meaning that one can store or transmit only a small number of adaptively chosen transform coefficients rather than all the signal samples. The way this typically works is that one acquires the full signal, computes the complete set of transform coefficients, encode the largest coefficients and discard all the others. This process of massive data acquisition followed by compression is extremely wasteful (one can think about a digital camera which has millions of imaging sensors, the pixels, but eventually encodes the picture on a few hundred kilobytes). This raises a fundamental question: because most signals are compressible, why spend so much effort acquiring all the data when we know that most of it will be discarded? Wouldn’t it be possible to acquire the data in already compressed form so that one does not need to throw away anything? “Compressive sampling” also known as “compressed sensing” [20] shows that this is indeed possible. This paper is by no means an exhaustive survey of the literature on compressive sampling. Rather this is merely an account of the author’s own work and thinking in this area which also includes a fairly large number of references to other people’s work and occasionally discusses connections with these works. We have done our best to organize the ideas into a logical progression starting with the early papers which launched this subject. Before we begin, we would like to invite the interested reader to also check the article [17] by Ronald DeVore – also in these proceedings – for a complementary survey of the field (Section 5).

2. Undersampled measurements Consider the general problem of reconstructing a vector x ∈ RN from linear measurements y about x of the form yk = x, ϕk , k = 1, . . . , K,

or

y = x.

(2.1)

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Compressive sampling

That is, we acquire information about the unknown signal by sensing x against K vectors ϕk ∈ RN . We are interested in the “underdetermined” case K N, where we have many fewer measurements than unknown signal values. Problems of this type arise in a countless number of applications. In radiology and biomedical imaging for instance, one is typically able to collect far fewer measurements about an image of interest than the number of unknown pixels. In wideband radio frequency signal analysis, one may only be able to acquire a signal at a rate which is far lower than the Nyquist rate because of current limitations in Analog-to-Digital Converter technology. Finally, gene expression studies also provide examples of this kind. Here, one would like to infer the gene expression level of thousands of genes – that is, the dimension N of the vector x is in the thousands – from a low number of observations, typically in the tens. At first glance, solving the underdertermined system of equations appears hopeless, as it is easy to make up examples for which it clearly cannot be done. But suppose now that the signal x is compressible, meaning that it essentially depends on a number of degrees of freedom which is smaller than N. For instance, suppose our signal is sparse in the sense that it can be written either exactly or accurately as a superposition of a small number of vectors in some fixed basis. Then this premise radically changes the problem, making the search for solutions feasible. In fact, accurate and sometimes exact recovery is possible by solving a simple convex optimization problem. 2.1. A nonlinear sampling theorem. It might be best to consider a concrete example first. Suppose here that one collects an incomplete set of frequency samples of a discrete signal x of length N. (To ease the exposition, we consider a model problem in one dimension. The theory extends easily to higher dimensions. For instance, we could be equally interested in the reconstruction of 2- or 3-dimensional objects from undersampled Fourier data.) The goal is to reconstruct the full signal f given only K samples in the Fourier domain N −1 1 yk = √ xt e−i2π ωk t/N , (2.2) N t=0 where the ‘visible’frequencies ωk are a subset (of size K) of the set of all frequencies {0, . . . , N − 1}. Sensing an object by measuring selected frequency coefficients is the principle underlying Magnetic Resonance Imaging, and is common in many fields of science, including Astrophysics. In the language of the general problem (2.1), the sensing matrix is obtained by sampling K rows of the N by N discrete Fourier transform matrix. We will say that a vector x is S-sparse if its support {i : xi = 0} is of cardinality less or equal to S. Then Candès, Romberg and Tao [6] showed that one could almost always recover the signal x exactly by solving the convex program1 ( x ˜ 1 := N i=1 |x˜ i |) (P1 )

min x ˜ 1

x∈ ˜ RN

subject to

1 (P ) can even be recast as a linear program [3], [15]. 1

x˜ = y.

(2.3)

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Theorem 2.1 ([6]). Assume that x is S-sparse and that we are given K Fourier coefficients with frequencies selected uniformly at random. Suppose that the number of observations obeys K ≥ C · S · log N. (2.4) Then minimizing 1 reconstructs x exactly with overwhelming probability. In details, if the constant C is of the form 22(δ + 1) in (2.4), then the probability of success exceeds 1 − O(N −δ ). The first conclusion is that one suffers no information loss by measuring just about any set of K frequency coefficients. The second is that the signal x can be exactly recovered by minimizing a convex functional which does not assume any knowledge about the number of nonzero coordinates of x, their locations, and their amplitudes which we assume are all completely unknown a priori. While this seems to be a great feat, one could still ask whether this is optimal, or whether one could do with even fewer samples. The answer is that in general, we cannot reconstruct S-sparse signals with fewer samples. There are examples for which the minimum number of samples needed for exact reconstruction by any method, no matter how intractable, must be about S log N. Hence, the theorem is tight and 1 -minimization succeeds nearly as soon as there is any hope to succeed by any algorithm. The reader is certainly familiar with the Nyquist/Shannon sampling theory and one can reformulate our result to establish simple connections. By reversing the roles of time and frequency in the above example, we can recast Theorem 1 as a new nonlinear sampling theorem. Suppose that a signal x has support in the frequency domain with B = ||. If is a connected set, we can think of B as the bandwidth of x. If in addition the set is known, then the classical Nyquist/Shannon sampling theorem states that x can be reconstructed perfectly from B equally spaced samples in the time domain2 . The reconstruction is simply a linear interpolation with a “sinc” kernel. Now suppose that the set , still of size B, is unknown and not necessarily connected. In this situation, the Nyquist/Shannon theory is unhelpful – we can only assume that the connected frequency support is the entire domain suggesting that all N time-domain samples are needed for exact reconstruction. However, Theorem 2.1 asserts that far fewer samples are necessary. Solving (P1 ) will recover x perfectly from about B log N time samples. What is more, these samples do not have to be carefully chosen; almost any sample set of this size will work. Thus we have a nonlinear analog (described as such since the reconstruction procedure (P1 ) is nonlinear) to Nyquist/Shannon: we can reconstruct a signal with arbitrary and unknown frequency support of size B from about B log N arbitrarily chosen samples in the time domain. Finally, we would like to emphasize that our Fourier sampling theorem is only a special instance of much more general statements. As a matter of fact, the results 2 For the sake of convenience, we make the assumption that the bandwidth B divides the signal length N evenly.

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extend to a variety of other setups and higher dimensions. For instance, [6] shows how one can reconstruct a piecewise constant (one or two-dimensional) object from incomplete frequency samples provided that the number of jumps (discontinuities) obeys the condition above by minimizing other convex functionals such as the total variation. 2.2. Background. Now for some background. In the mid-eighties, Santosa and Symes [44] had suggested the minimization of 1 -norms to recover sparse spike trains, see also [25], [22] for early results. In the last four years or so, a series of papers [26], [27], [28], [29], [33], [30] explained why 1 could recover sparse signals in some special setups. We note though that the results in this body of work are very different than the sampling theorem we just introduced. Finally, we would like to point out important connections with the literature of theoretical computer science. Inspired by [37], Gilbert and her colleagues have shown that one could recover an S-sparse signal with probability exceeding 1 − δ from S · poly(log N, log δ) frequency samples placed on special equispaced grids [32]. The algorithms they use are not based on optimization but rather on ideas from the theory of computer science such as isolation, and group testing. Other points of connection include situations in which the set of spikes are spread out in a somewhat even manner in the time domain [22], [51]. 2.3. Undersampling structured signals. The previous example showed that the structural content of the signal allows a drastic “undersampling” of the Fourier transform while still retaining enough information for exact recovery. In other words, if one wanted to sense a sparse object by taking as few measurements as possible, then one would be well-advised to measure randomly selected frequency coefficients. In truth, this observation triggered a massive literature. To what extent can we recover a compressible signal from just a few measurements. What are good sensing mechanisms? Does all this extend to object that are perhaps not sparse but well-approximated by sparse signals? In the remainder of this paper, we will provide some answers to these fundamental questions.

3. The Mathematics of compressive sampling 3.1. Sparsity and incoherence. In all what follows, we will adopt an abstract and general point of view when discussing the recovery of a vector x ∈ RN . In practical instances, the vector x may be the coefficients of a signal f ∈ RN in an orthonormal basis N f (t) = xi ψi (t), t = 1, . . . , N. (3.1) i=1

For example, we might choose to expand the signal as a superposition of spikes (the canonical basis of RN ), sinusoids, B-splines, wavelets [36], and so on. As a side

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note, it is not important to restrict attention to orthogonal expansions as the theory and practice of compressive sampling accommodates other types of expansions. For example, x might be the coefficients of a digital image in a tight-frame of curvelets [5]. To keep on using convenient matrix notations, one can write the decomposition (3.1) as x = f where is the N by N matrix with the waveforms ψi as rows or equivalently, f = ∗ x. We will say that a signal f is sparse in the -domain if the coefficient sequence is supported on a small set and compressible if the sequence is concentrated near a small set. Suppose we have available undersampled data about f of the same form as before y = f. Expressed in a different way, we collect partial information about x via y = x where = ∗ . In this setup, one would recover f by finding – among all coefficient sequences consistent with the data – the decomposition with minimum 1 -norm min x ˜ 1

such that x˜ = y.

Of course, this is the same problem as (2.3), which justifies our abstract and general treatment. With this in mind, the key concept underlying the theory of compressive sampling is a kind of uncertainty relation, which we explain next. 3.2. Recovery of sparse signals. In [7], Candès and Tao introduced the notion of uniform uncertainty principle (UUP) which they refined in [8]. The UUP essentially states that the K × N sensing matrix obeys a “restricted isometry hypothesis.” Let T , T ⊂ {1, . . . , N } be the K × |T | submatrix obtained by extracting the columns of corresponding to the indices in T ; then [8] defines the S-restricted isometry constant δS of which is the smallest quantity such that (1 − δS ) c 22 ≤ T c 22 ≤ (1 + δS ) c 22

(3.2)

for all subsets T with |T | ≤ S and coefficient sequences (cj )j ∈T . This property essentially requires that every set of columns with cardinality less than S approximately behaves like an orthonormal system. An important result is that if the columns of the sensing matrix are approximately orthogonal, then the exact recovery phenomenon occurs. Theorem 3.1 ([8]). Assume that x is S-sparse and suppose that δ2S + δ3S < 1 or, better, δ2S + θS,2S < 1. Then the solution x to (2.3) is exact, i.e., x = x. In short, if the UUP holds at about the level S, the minimum 1 -norm reconstruction is provably exact. The first thing one should notice when comparing this result with the Fourier sampling theorem is that it is deterministic in the sense that it does not involve any probabilities. It is also universal in that all sufficiently sparse vectors

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are exactly reconstructed from x. In Section 3.4, we shall give concrete examples of sensing matrices obeying the exact reconstruction property for large values of the sparsity level, e.g. for S = O(K/ log(N/K)). Before we do so, however, we would like to comment on the slightly better version δ2S + θS,2S < 1, which is established in [10]. The number θS,S for S + S ≤ N is called the S, S -restricted orthogonality constants and is the smallest quantity such that |T c, T c | ≤ θS,S · c 2 c 2

(3.3)

holds for all disjoint sets T , T ⊆ {1, . . . , N } of cardinality |T | ≤ S and |T | ≤ S . Thus θS,S is the cosine of the smallest angle between the two subspaces spanned by the columns in T and T . Small values of restricted orthogonality constants indicate that disjoint subsets of covariates span nearly orthogonal subspaces. The condition δ2S + θS,2S < 1 is better than δ2S + δ3S < 1 since it is not hard to see that δS+S − δS ≤ θS,S ≤ δS+S for S ≥ S [8, Lemma 1.1]. Finally, now that we have introduced all the quantities needed to state our recovery theorem, we would like to elaborate on the condition δ2S + θS,2S < 1. Suppose that δ2S = 1 which may indicate that there is a matrix T1 ∪T2 with 2S columns (|T1 | = S, |T2 | = S) that is rank-deficient. If this is the case, then there is a pair (x1 , x2 ) of nonvanishing vectors with x1 supported on T1 and x2 supported on T2 obeying (x1 − x2 ) = 0 ⇐⇒ x1 = x2 . In other words, we have two very distinct S-sparse vectors which are indistinguishable. This is why any method whatsoever needs δ2S < 1. For, otherwise, the model is not identifiable to use a terminology borrowed from the statistics literature. With this in mind, one can see that the condition δ2S + θS,2S < 1 is only slightly stronger than this identifiability condition. 3.3. Recovery of compressible signals. In general, signals of practical interest may not be supported in space or in a transform domain on a set of relatively small size. Instead, they may only be concentrated near a sparse set. For example, a commonly discussed model in mathematical image or signal processing assumes that the coefficients of elements taken from a signal class decay rapidly, typically like a power law. Smooth signals, piecewise signals, images with bounded variations or bounded Besov norms are all of this type [24]. A natural question is how well one can recover a signal that is just nearly sparse. For an arbitrary vector x in RN , denote by xS its best S-sparse approximation; that is, xS is the approximation obtained by keeping the S largest entries of x and setting the others to zero. It turns out that if the sensing matrix obeys the uniform uncertainty principle at level S, then the recovery error is not much worse than x − xS 2 .

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Theorem 3.2 ([9]). Assume that x is S-sparse and suppose that δ3S + δ4S < 2. Then the solution x to (2.3) obeys x ∗ − x 2 ≤ C ·

x − xS 1 . √ S

(3.4)

For reasonable values of δ4S , the constant in (3.4) is well behaved; e.g. C ≤ 8.77 for δ4S = 1/5. Suppose further that δS + 2θS,S + θ2S,S < 1, we also have x ∗ − x 1 ≤ C x − xS 1 ,

(3.5)

for some positive constant C. Again, the constant in (3.5) is well behaved. Roughly speaking, the theorem says that minimizing 1 recovers the S-largest entries of an N-dimensional unknown vector x from K measurements only. As a side remark, the 2 -stability result (3.4) appears explicitly in [9] while the ‘1 instance optimality’ (3.5) is implicit in [7] although it is not stated explicitely. For example, it follows from Lemma 2.1 – whose hypothesis holds because of Lemma 2.2. in [8] – in that paper. Indeed, let T be the set where x takes on its S-largest values. Then Lemma 2.1 in [7] gives x ∗ ·1T c 1 ≤ 4 x −xS 1 and, therefore, (x ∗ −x)·1T c 1 ≤ 5 x − xS 1 . We conclude by observing that on T we have √ (x ∗ − x) · 1T 1 ≤ S (x ∗ − x) · 1T 2 ≤ C x − xS 1 , where the last inequality follows from (3.4). For information, a more direct argument yields better constants. To appreciate the content of Theorem 3.2, suppose that x belongs to a weak-p ball of radius R. This says that if we rearrange the entries of x in decreasing order of magnitude |x|(1) ≥ |x|(2) ≥ · · · ≥ |x|(N ) , the ith largest entry obeys |x|(i) ≤ R · i −1/p ,

1 ≤ i ≤ N.

(3.6)

More prosaically, the coefficient sequence decays like a power-law and the parameter p controls the speed of the decay: the smaller p, the faster the decay. Classical calculations then show that the best S-term approximation of an object x ∈ wp (R) obeys x − xS 2 ≤ C2 · R · S 1/2−1/p (3.7) in the 2 norm (for some positive constant C2 ), and x − xS 1 ≤ C1 · R · S 1−1/p in the 1 -norm. For generic elements obeying (3.6), there are no fundamentally better estimates available. Hence, Theorem 3.2 shows that with K measurements only, we can achieve an approximation error which is as good as that one would obtain by knowing everything about the signal and selecting its S-largest entries.

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3.4. Random matrices. Presumably all of this would be interesting if one could design a sensing matrix which would allow us to recover as many entries of x as possible with as few as K measurements. In the language of Theorem 3.1, we would like the condition δ2S + θS,2S < 1 to hold for large values of S, ideally of the order of K. This poses a design problem. How should one design a matrix – that is to say, a collection of N vectors in K dimensions – so that any subset of columns of size about S be about orthogonal? And for what values of S is this possible? While it might be difficult to exhibit a matrix which provably obeys the UUP for very large values of S, we know that trivial randomized constructions will do so with overwhelming probability. We give an example. Sample N vectors on the unit sphere of RK independently and uniformly at random. Then the condition of Theorems 3.1 and 3.2 hold for S = O(K/ log(N/K)) with probability 1 − πN where πN = O(e−γ N ) for some γ > 0. The reason why this holds may be explained by some sort of “blessing of high-dimensionality.” Because the high-dimensional sphere is mostly empty, it is possible to pack many vectors while maintaining approximate orthogonality. • Gaussian measurements. Here we assume that the entries of the K by N sensing matrix are independently sampled from the normal distribution with mean zero and variance 1/K. Then if S ≤ C · K/ log(N/K),

(3.8)

S obeys the condition of Theorems 3.1 and 3.2 with probability 1 − O(e−γ N ) for some γ > 0. The proof uses known concentration results about the singular values of Gaussian matrices [16], [45]. • Binary measurements. Suppose that the entries of the K by N sensing matrix are independently sampled from the symmetric Bernoulli distribution P (ki = √ ±1/ K) = 1/2. Then it is conjectured that the conditions of Theorems 3.1 and 3.2 are satisfied with probability 1 − O(e−γ N ) for some γ > 0 provided that S obeys (3.8). The proof of this fact would probably follow from new concentration results about the smallest singular value of a subgaussian matrix [38]. Note that the exact reconstruction property for S-sparse signals and (3.7) with S obeying (3.8) are known to hold for binary measurements [7]. • Fourier measurements. Suppose now that is a partial Fourier matrix obtained by selecting K rows uniformly at random as before, and renormalizing the columns so that they are unit-normed. Then Candès and Tao [7] showed that Theorem 3.1 holds with overwhelming probability if S ≤ C · K/(log N)6 . Recently, Rudelson and Vershynin [43] improved this result and established S ≤ C · K/(log N)4 . This result is nontrivial and use sophisticated techniques from geometric functional analysis and probability in Banach spaces. It is conjectured that S ≤ C · K/ log N holds.

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• Incoherent measurements. Suppose now that is obtained by selecting K rows uniformly at random from an N by N orthonormal matrix U and renormalizing the columns so that they are unit-normed. As before, we could think of U as the matrix ∗ which maps the object from the to the -domain. Then the arguments used in [7], [43] to prove that the UUP holds for incomplete Fourier matrices extend to this more general situation. In particular, Theorem 3.1 holds with overwhelming probability provided that S≤C·

1 K · , μ2 (log N)4

(3.9)

√ where μ := N maxi,j |Ui,j | (observe that for the Fourier matrix, μ = 1 which gives the result in the special case of the Fourier ensemble above). With U = ∗ , √ μ := N max |ϕi , ψj | (3.10) i,j

which is referred to as the mutual coherence between the measurement basis and the sparsity basis [27], [28]. The greater the incoherence of the measurement/sparsity pair (, ), the smaller the number of measurements needed. In short, one can establish the UUP for a few interesting random ensembles and we expect that in the future, many more results of this type will become available. 3.5. Optimality. Before concluding this section, it is interesting to specialize our recovery theorems to selected measurement ensembles now that we have established the UUP for concrete values of S. Consider the Gaussian measurement ensemble in which the entries of are i.i.d. N(0, 1/K). Our results say that one can recover any S-sparse vector from a random projection of dimension about O(S · log(N/S)), see also [18]. Next, suppose that x is taken from a weak-p ball of radius R for some 0 < p < 1, or from the 1 -ball of radius R for p = 1. Then we have shown that for all x ∈ wp (R) x − x 2 ≤ C · R · (K/ log(N/K))−r ,

r = 1/p − 1/2,

(3.11)

which has also been proven in [20]. An important question is whether this is optimal. In other words, can we find a possibly adaptive set of measurements and a reconstruction algorithm that would yield a better bound than (3.11)? By adaptive, we mean that one could use a sequential measurement procedure where at each stage, one would have the option to decide which linear functional to use next based on the data collected up to that stage. It proves to be the case that one cannot improve on (3.11), and we have thus identified the optimal performance. Fix a class of object F and let EK (F ) be the best reconstruction error from K linear measurements EK (F ) = inf sup f − D(y) 2 , f ∈F

y = f,

(3.12)

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where the infimum is taken over all set of K linear functionals and all reconstruction algorithms D. Then it turns out EK (F ) nearly equals the Gelfand numbers of a class F defined as dK (F ) = inf { sup PV f : codim(V ) < K}, V f ∈F

(3.13)

where PV is the orthonormal projection on the subspace V . Gelfand numbers play an important role in approximation theory, see [40] for more information. If F = −F and F = F +F ≤ cF F , then dK (F ) ≤ EK (F ) ≤ cF dK (F ). Note that cF = 21/p in the case where F is a weak-p ball. The thing is that we know the approximate values of the Gelfand numbers for many classes of interest. Suppose for example that F is the 1 -ball of radius R. A seminal result of Kashin [35] and improved by Garnaev and Gluskin [31] shows that for this ball, the Gelfand numbers obey log(N/K) + 1 log(N/K) + 1 ≤ dk (F ) ≤ C2 · R · , (3.14) C1 · R · K K where C1 , C2 are universal constants. Gelfand numbers are also approximately known for weak-p balls as well; the only difference is that ((log(N/K)+1)/K)r substitutes ((log(N/K) + 1)/K)1/2 . Hence, Kashin, Garnaev and Gluskin assert that with K measurements, the minimal reconstruction error (3.12) one can hope for is bounded below by a constant times (K/ log(N/K))−r . Kashin’s arguments [35] also used probabilistic functionals which establish the existence of recovery procedures for which the reconstruction error is bounded above by the right-hand side of (3.14). Similar types of recovery have also been known to be possible in the literature of theoretical computer science, at least in principle, for certain types of random measurements [1]. In this sense, our results – specialized to Gaussian measurements – are optimal for weak-p norms. The novelty is that the information about the object can be retrieved from random coefficients by minimizing a simple linear program (2.3), and that the decoding algorithm adapts automatically to the weak-p signal class, without knowledge thereof. Minimizing the 1 -norm is adaptive and nearly gives the best possible reconstruction error simultaneously over a wide range of sparse classes of signals; no information about p and the radius R are required.

4. Robust compressive sampling In any realistic application, we cannot expect to measure x without any error, and we now turn our attention to the robustness of compressive sampling vis a vis measurement errors. This is a very important issue because any real-world sensor is subject to at least a small amount of noise. And one thus immediately understands that to be widely applicable, the methodology needs to be stable. Small perturbations in the

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observed data should induce small perturbations in the reconstruction. Fortunately, the recovery procedures may be adapted to be surprisingly stable and robust vis a vis arbitrary perturbations. Suppose our observations are inaccurate and consider the model y = x + e,

(4.1)

where e is a stochastic or deterministic error term with bounded energy e 2 ≤ ε. Because we have inaccurate measurements, we now use a noise-aware variant of (2.3) which relaxes the data fidelity term. We propose a reconstruction program of the form (P2 )

min x ˜ 1

such that

x˜ − y 2 ≤ ε.

(4.2)

The difference with (P1 ) is that we only ask the reconstruction be consistent with the data in the sense that y − x be within the noise level. The program (P2 ) has a unique solution, is again convex, and is a special instance of a second order cone program (SOCP) [4]. Theorem 4.1 ([9]). Suppose that x is an arbitrary vector in RN . Under the hypothesis of Theorem 3.2, the solution x to (P2 ) obeys x − x 2 ≤ C1,S · ε + C2,S ·

x0 − x0,S 1 . √ S

(4.3)

For reasonable values of δ4S the constants in (4.3) are well behaved, see [9]. We would like to offer two comments. The first is that the reconstruction error is finite. This quiet observation is noteworthy because we recall that the matrix is rectangular with many more columns than rows – thus having a fraction of vanishing singular values. Having said that, the mere fact that the severely ill-posed matrix inversion keeps the perturbation from “blowing up” may seem a little unexpected. Next and upon closer inspection, one sees that the reconstruction error is the sum of two terms: the first is simply proportional to the size of the measurement error while the second is the approximation error one would obtain in the noiseless case. In other words, the performance of the reconstruction degrades gracefully as the measurement noise increases. This brings us to our second point. In fact, it is not difficult to see that no recovery method can perform fundamentally better for arbitrary perturbations of size ε [9]. For related results for Gaussian sensing matrices, see [19].

5. Connections with statistical estimation In the remainder of this paper, we shall briefly explore some connections with other fields, and we begin with statistics. Suppose now that the measurement errors in (4.1) are stochastic. More explicitly, suppose that the model is of the form y = x + z,

(5.1)

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where z1 , . . . , zk are i.i.d. with mean zero and variance σ 2 . In this section, we will assume that the zk ’s are Gaussian although nothing in our arguments heavily relies upon this assumption. The problem is again to recover x from y which is a central problem in statistics since this is just the classical multivariate linear regression problem. Because the practical environment has changed dramatically over the last two decades or so, applications have emerged in which the number of observations is small compared to the dimension of the object we wish to estimate – here, K ≤ N. This new paradigm sometimes referred to as “high-dimensional data” is currently receiving much attention and, clearly, the emerging theory of compressive sampling might prove very relevant. The results from the previous sections are directly applicable. Suppose that x is S-sparse to simplify our exposition. Because z 22 is distributed as a chi-squared with K degrees of freedom, the reconstruction (4.2) would obey x ∗ − x 22 ≤ C · Kσ 2

(5.2)

with high probability. While this may seem acceptable to the nonspecialist, modern results in the literature suggest that one might be able to get a better accuracy. In particular, one would like an adaptive error bound which depends upon the complexity of the true unknown parameter vector x ∈ RN . For example, if x only has S significant coefficients, we would desire an error bound of size about Sσ 2 ; the less complex the estimand, the smaller the squared-error loss. This poses an important question: can we design an estimator whose accuracy depends upon the information content of the object we wish to recover? 5.1. Ideal model selection. To get a sense of what is possible, consider regressing the data y onto an arbitrary subset T by the method of least squares. Define x[T ˆ ] to be the least squares estimate whose restriction to the set T is given by xˆT [T ] = (TT T )−1 TT y,

(5.3)

ˆ ] to T and and which vanishes outside T . Above, xˆT [T ] is the restriction of x[T similarly for xT . Since x[T ˆ ] vanishes outside T , we have E x − x[T ˆ ] 2 = xT − xˆT [T ] 2 + |xi |2 , i ∈T /

Consider the first term. We have xT − xˆT [T ] = (TT T )−1 TT (s + z), where s = T c xT c . It follows that E xT − xˆT [T ] 2 = (TT T )−1 TT s 2 + σ 2 Tr((TT T )−1 ).

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However, since all the eigenvalues of TT T belong to the interval [1 − δ|T | , 1 + δ|T | ], we have 1 E xT − xˆT [T ] 2 ≥ · |T | · σ 2 . 1 + δ|T | For each set T with |T | ≤ S and δS < 1, we then have E x − x[T ˆ ] 2 ≥

xi2 +

i∈T c c

1 |T | · σ 2 . 2

We now search for an ideal estimator which selects that estimator x[T ˆ ∗ ] from the family (x[T ˆ ])T ⊂{1,...,N} with minimal Mean-Squared Error (MSE): x[T ˆ ∗ ] = argminT ⊂{1,...,N} E x − x[T ˆ ] 2 . This estimator is ideal because we would of course not know which estimator xˆT is best; that is, to achieve the ideal MSE, one would need an oracle that would tell us which model T to choose. We will consider this ideal estimator nevertheless and take its MSE as a benchmark. The ideal MSE is bounded below by 1 min ( x − x[T ˆ ] 2 + |T | · σ 2 ) 2 T 1 = min(xi2 , σ 2 ). 2

E x − x[T ˆ ∗ ] 2 ≥

(5.4)

i

Letting xS be the best S-sparse approximation to x, another way to express the righthand side (5.4) is in term of the classical trade-off between the approximation error and the number of terms being estimated times the noise level E x − xˆT ∗ 2 ≥

1 inf x − xS 2 + Sσ 2 . 2 S≥0

Our question is of course whether there is a computationally efficient estimator which can mimic the ideal MSE. 5.2. The Dantzig selector. Assume for simplicity that the columns of are normalized (there are straightforward variations to handle the general case). Then the Dantzig selector estimates x by solving the convex program (DS)

˜ 1 min x

x∈ ˜ RN

subject to

sup |(T r)i | ≤ λ · σ

(5.5)

1≤i≤N

for some λ > 0, and where r is the vector of residuals r = y − x. ˜

(5.6)

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The solution to this optimization problem is the minimum 1 -vector which is consistent with the observations. The constraints impose that the residual vector is within the noise level and does not correlate too well with the columns of . For information, there exist related, yet different proposals in the literature, and most notably the lasso introduced by [47], see also [15]. Again, the program (DS) is convex and can be recast as a linear program (LP). The main result in this line of research is that the Dantzig selector is not only computationally tractable, it is also accurate. √ Theorem 5.1 ([10]). Set λ := (1 + t −1 ) 2 log p in (5.5) and suppose that x is S-sparse with δ2S + θS,2S < 1 − t. Then with very large probability, the Dantzig selector xˆ solution to (5.5) obeys min(xi2 , σ 2 ) . (5.7) xˆ − x 2 ≤ O(log p) · σ 2 + i

Our result says that the Dantzig selector achieves a loss within a logarithmic factor of the ideal mean squared error one would achieve with an oracle which would supply perfect information about which coordinates are nonzero, and which were above the noise level. To be complete, it is possible to obtain similar bounds on the MSE. There are extensions of this result to signals which are not sparse but compressible, e.g. for signals which belong to weak-p balls. What is interesting here is that in some instances, even though the number of measurements is much smaller than the dimension of the parameter vector x, the Dantzig selector recovers the minimax rate that one would get if we were able to measure all the coordinates of x directly via y˜ = x + σ z where z is i.i.d. N(0, 1).

6. Connections with error correction Compressive sampling also interacts with the agenda of coding theory. Imagine we wish to transmit a vector x of length M to a remote receiver reliably. A frequently discussed approach consists in encoding the information x with an N by M coding matrix C with N > M. Assume that gross errors occur upon transmission and that a fraction of the entries of Cx are corrupted in a completely arbitrary fashion. We do not know which entries are affected nor do we know how they are affected. Is it possible to recover the information x exactly from the corrupted N-dimensional vector y? To decode, [8] proposes solving the minimum 1 -approximation problem (D1 )

min y − C x ˜ 1 ,

x∈ ˜ RM

(6.1)

which can also be obviously recast as an LP. The result is that if C is carefully chosen, then (6.1) will correctly retrieve the information x with no error provided that the

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fraction ρ of errors is not too large, ρ ≤ ρ ∗ . This phenomenon holds for all x’s and all corruption patterns. To see why this phenomenon occurs, consider a matrix B which annihilates the N × M coding matrix C on the left, i.e. such that BC = 0; B is called a parity-check matrix and is any (N − M) × N matrix whose kernel is the range of C in RN . The transmitted information is of the form y = Cx + e, where e is a sparse vector of possibly gross errors, and apply B on both sides of this equation. This gives y˜ = B(Cx + e) = Be

(6.2)

since BC = 0. Therefore, the decoding problem is reduced to that of recovering the error vector e from the observations Be. Once e is known, Cx is known and, therefore, x is also known since we may just assume that C has full rank. Now the reader knows that we could solve the underdetermined system (6.2) by 1 -minimization. He also knows that if the UUP holds, the recovery is exact. Now (D1 ) and (P1 ) are equivalent programs. Indeed, it follows from the decomposition x˜ = x + h that (D1 ) ⇐⇒ min e − Ch 1 . h∈RM

Now the constraint Bd = Be means that d = e −Ah for some h ∈ RM and, therefore, min d 1 ,

Bd = Be ⇐⇒ minn d 1 , h∈R

d = e − Ah

⇐⇒ minn e − Ah 1 , h∈R

which proves the claim. Hence, if one uses a random coding matrix which is a popular choice, we have the following result, see also [42]: Theorem 6.1 ([8]). Suppose the coding matrix C has i.i.d. N(0, 1) entries. Then with probability exceeding 1 − O(e−γ M ) for some γ > 0, (D1 ) exactly decodes all x ∈ RM provided that the fraction ρ of arbitrary errors obeys ρ ≤ ρ ∗ (M, N). In conclusion, one can correct a constant fraction of errors with arbitrary magnitudes by solving a convenient LP. In [8], the authors reported on numerical results showing that in practice (D1 ) works extremely well and recovers the vector x exactly provided that the fraction of the corrupted entries be less than about 17% in the case where N = 2M and less than about 34% in the case where N = 4M.

7. Further topics Our intention in this short survey was merely to introduce the new compressive sampling concepts. We presented an approach based on the notion of uncertainty principle


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which gives a powerful and unified treatment of some of the main results underlying this theory. As we have seen, the UUP gives conditions for exact, approximate, and stable recoveries which are almost necessary. Another advantage that one can hardly neglect is that this makes the exposition fairly simple. Having said that, the early papers on compressive sampling – e.g. [6], [7], [20] – have spurred a large and fascinating literature in which other approaches and ideas have been proposed. Rudelson and Vershynin have used tools from modern Banach space theory to derive powerful results for Gaussian ensembles [42], [14], [43]. In this area, Pajor and his colleagues have established the existence of abstract reconstruction procedures from subgaussian measurements (including random binary sensing matrices) with powerful reconstruction properties. In a different direction, Donoho and Tanner have leveraged results from polytope geometry to obtain very precise estimates about the minimal number of Gaussian measurements needed to reconstruct S-sparse signals [21], [23], see also [43]. Tropp and Gilbert reported results about the performance of greedy methods for compressive sampling [49]. Haupt and Nowak have quantified the performance of combinatorial optimization procedures for estimating a signal from undersampled random projections in noisy environments [34]. Finally, Rauhut has worked out variations on the Fourier sampling theorem in which a sparse continuoustime trigonometric polynomials is randomly sampled in time [41]. Because of space limitations, we are unfortunately unable to do complete justice to this rapidly growing literature. We would like to emphasize that there are many aspects of compressive sampling that we have not touched. For example, we have not discussed the practical performance of this new theory. In fact, numerical experiments have shown that compressive sampling behaves extremely well in practice. For example, it has been shown that from 3S − 4S nonadaptive measurements, one can reconstruct an approximation of an image in a fixed basis which is more precise than that one would get by measuring all the coefficients of the object in that basis and selecting the S largest [13], [50]. Further, numerical simulations with noisy data show that compressive sampling is very stable and performs well in noisy environments. In practice, the constants appearing in Theorems 4.1 and 5.1 are very small, see [9] and [10] for empirical results. We would like to close this article by returning to the main theme of this paper, which is that compressive sampling invites to rethink sensing mechanisms. Because if one were to collect a comparably small number of general linear measurements rather than the usual pixels, one could in principle reconstruct an image with essentially the same resolution as that one would obtain by measuring all the pixels. Therefore, if one could design incoherent sensors (i.e. measuring incoherent linear functionals), the payoff could be extremely large. Several teams have already reported progress in this direction. For example, a team led by Baraniuk and Kelly have proposed a new camera architecture that employs a digital micromirror array to perform optical calculations of linear projections of an image onto pseudorandom binary patterns [46], [52]. Compressive sampling may also address challenges in the processing of wideband radio frequency signals since high-speed analog-to-digital convertor

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technology indicates that current capabilities fall well short of needs, and that hardware implementations of high precision Shannon-based conversion seem out of sight for decades to come. Finally, compressive sampling has already found applications in wireless sensor networks [2]. Here, compressive sampling allows of energy efficient estimation of sensor data with comparably few sensor nodes. The power of these estimation schemes is that they require no prior information about the sensed data. All these applications are novel and exciting. Others might just be around the corner.

References [1] Alon, N., Matias, Y., Szegedy, B., The space complexity of approximating the frequency moments. J. Comput. System Sci. 58 (1999), 137–147. [2] Bajwa, W. U., Haupt, J., Sayeed, A. M., Nowak, R., Compressive wireless sensing. In Proc. 5th Intl. Conf. on Information Processing in Sensor Networks (IPSN ’06), Nashville, TN, 2006, 134–142. [3] Bloomfield, P., Steiger, W., Least Absolute Deviations: Theory, Applications, and Algorithms. Progr. Probab. Statist. 6, Birkhäuser, Boston, MA, 1983. [4] Boyd, S., Vandenberghe, L., Convex Optimization. Cambridge University Press, Cambridge 2004. [5] Candès, E. J., Donoho, D. L. New tight frames of curvelets and optimal Representations of objects with piecewise C 2 singularities. Comm. Pure Appl. Math. 57 (2004), 219–266. [6] Candès, E. J., Romberg, J., Tao, T., Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Inform. Theory 52 (2006), 489–509. [7] Candès, E. J., Tao, T., Near-optimal signal recovery from random projections and universal encoding strategies. IEEE Trans. Inform. Theory, 2004, submitted. [8] Candès, E. J., Tao, T., Decoding by linear programming. IEEE Trans. Inform. Theory 51 (2005), 4203–4215. [9] Candès, E. J., Romberg, J., Tao, T., Signal recovery from incomplete and inaccurate measurements. Comm. Pure Appl. Math. 59 (8) (2005), 1207–1223. [10] Candès, E. J., Tao, T., The Dantzig selector: statistical estimation when p is much larger than n. Ann. Statist., to appear. [11] Candès, E. J., Romberg, J., The role of sparsity and incoherence for exactly reconstructing a signal from limited measurements. Technical Report, California Institute of Technology, 2004. [12] Candès, E. J., Romberg, J., Quantitative robust uncertainty principles and optimally sparse decompositions. Found. Comput. Math. 6 (2) (2006), 227–254. [13] Candès, E. J., Romberg, J., Practical signal recovery from random projections. In SPIE International Symposium on Electronic Imaging: Computational Imaging III, San Jose, California, January 2005. [14] Candès, E. J., Rudelson, M., Vershynin, R. and Tao, T. Error correction via linear programming. In Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS) (2005), IEEE Comput. Soc. Press, LosAlamitos, CA,295–308.


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[15] Chen, S. S., Donoho, D. L., Saunders, M. A, Atomic decomposition by basis pursuit. SIAM J. Sci. Comput. 20 (1999), 33–61. [16] Davidson, K. R., Szarek, S. J., Local operator theory, random matrices and Banach spaces. In Handbook of the geometry of Banach spaces (ed. by W. B. Johnson, J. Lindenstrauss), Vol. I, North-Holland, Amsterdam 2001, 317–366; Corrigendum, Vol. 2, 2003, 1819–1820. [17] DeVore, R. A., Optimal computation. In Proceedings of the International Congress of Mathematicians (Madrid, 2006), Volume I, EMS Publishing House, Zürich 2006/2007. [18] Donoho, D. L., For most large underdetermined systems of linear equations the minimal 1 -norm solution is also the sparsest Solution. Comm. Pure Appl. Math. 59 (2006), 797–829. [19] Donoho, D. L., For most large underdetermined systems of equations, the minimal 1 -norm near-solution approximates the sparsest near-solution. Comm. Pure Appl. Math. 59 (2006), 907–934. [20] Donoho, D. L., Compressed sensing. Technical Report, Stanford University, 2004. [21] Donoho, D. L., Neighborly polytopes and sparse solutions of underdetermined linear equations. Technical Report, Stanford University, 2005. [22] Donoho, D. L., Logan, B. F., Signal recovery and the large sieve, SIAM J. Appl. Math. 52 (1992), 577–591. [23] Donoho, D. L., Tanner, J., Neighborliness of randomly projected simplices in high dimensions. Proc. Natl. Acad. Sci. USA 102 (2005), 9452–9457. [24] Donoho, D. L., Vetterli, M., DeVore, R. A., Daubechies, I., Data compression and harmonic analysis. IEEE Trans. Inform. Theory 44 (1998), 2435–2476. [25] Donoho, D. L., Stark, P. B., Uncertainty principles and signal recovery. SIAM J. Appl. Math. 49 (1989), 906–931. [26] Donoho, D. L., Huo, X., Uncertainty principles and ideal atomic decomposition. IEEE Trans. Inform. Theory 47 (2001), 2845–2862. [27] Donoho, D. L., Elad, M., Optimally sparse representation in general (nonorthogonal) dictionaries via 1 minimization. Proc. Natl. Acad. Sci. USA 100 (2003), 2197–2202. [28] Elad, M., Bruckstein, A. M., A generalized uncertainty principle and sparse representation in pairs of RN bases. IEEE Trans. Inform. Theory 48 (2002), 2558–2567. [29] Feuer, A., Nemirovski, A., On sparse representation in pairs of bases. IEEE Trans. Inform. Theory 49 (2003), 1579–1581. [30] Fuchs, J. J., On sparse representations in arbitrary redundant bases. IEEE Trans. Inform. Theory 50 (2004), 1341–1344. [31] Garnaev, A., Gluskin, E., The widths of a Euclidean ball. Dokl. Akad. Nauk. USSR 277 (1984), 1048–1052; English transl. Soviet Math. Dokl. 30 (1984), 200–204. [32] Gilbert, A. C., Muthukrishnan, S., Strauss, M., Improved time bounds for near-optimal sparse Fourier representation. In Proceedings of SPIE 5914 (Wavelets XI), ed. by M. Papadakis, A. F. Laine, M. A. Unser, 2005. [33] Gribonval, R., Nielsen, M., Sparse representations in unions of bases. IEEE Trans. Inform. Theory 49 (2003), 3320–3325. [34] Haupt, J., Nowak, R., Signal reconstruction from noisy random projections. IEEE Trans. Inform. Theory, submitted.

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[35] Kashin, B., The widths of certain finite dimensional sets and classes of smooth functions, Izvestia 41 (1977), 334–351. [36] Mallat, S., A Wavelet Tour of Signal Processing. Academic Press, San Diego, CA, 1998. [37] Mansour, Y., Randomized interpolation and approximation of sparse polynomials. SIAM J. Comput. 24 (1995), 357–368. [38] Litvak, A. E., Pajor, A., Rudelson, M., Tomczak-Jaegermann, N., Smallest singular value of random matrices and geometry of random polytopes. Manuscript, 2004. [39] Mendelson, S., Pajor, A., Tomczak-Jaegermann, N., Reconstruction and subgaussian processes. C. R. Math. Acad. Sci. Paris 340 (2005), 885–888. [40] Pinkus, A., N-Widths in Approximation Theory. Ergeb. Math. Grenzgeb. (3) 7, SpringerVerlag, Berlin 1985. [41] Rauhut, H., Random sampling of sparse trigonometric polynomials. Preprint, 2005. [42] Rudelson, M., Vershynin, R., Geometric approach to error-correcting codes and reconstruction of signals. Internat. Math. Res. Notices 2005 (64) (2005), 4019–4041. [43] Rudelson, M., Vershynin, R., Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements. Preprint, 2006. [44] Santosa, F., Symes, W. W., Linear inversion of band-limited reflection seismograms. SIAM J. Sci. Statist. Comput. 7 (1986), 1307–1330. [45] Szarek, S. J., Condition numbers of random matrices. J. Complexity 7 (1991), 131–149. [46] Takhar, D., Laska, J. N., Wakin, M., Duarte, M. F., Baron, D., Sarvotham, S., Kelly, K. F., Baraniuk, R. G., A new compressive imaging camera architecture using optical-domain compression. IS&T/SPIE Computational Imaging IV, San Jose, January 2006. [47] Tibshirani, R., Regression shrinkage and selection via the lasso. J. Roy. Statist. Soc. Ser. B 58 (1996), 267–288. [48] Tropp, J. A., Just relax: convex programming methods for identifying sparse signals in noise. IEEE Trans. Inform. Theory 52 (2006), 1030–1051. [49] Tropp, J. A., Gilbert, A. C., Signal recovery from partial information via orthogonal matching pursuit. Preprint, University of Michigan, 2005. [50] Tsaig, Y., Donoho, D. L., Extensions of compressed sensing. Technical report, Department of Statistics, Stanford University, 2004. [51] Vetterli, M., Marziliano, P., Blu, T., Sampling signals with finite rate of innovation. IEEE Trans. Signal Process. 50 (2002), 1417–1428. [52] Wakin, M., Laska, J. N., Duarte, M. F., Baron, D., Sarvotham, S., Takhar, D., Kelly, K. F., Baraniuk, R. G., Compressive imaging for video representation and coding. Picture Coding Symposium, special session on Next Generation Video Representation, Beijing, April 2006. Applied and Computational Mathematics, California Institute of Technology, Pasadena, CA 91125, U.S.A. E-mail: [email protected]

Total variation based image denoising and restoration Vicent Caselles ∗

Abstract. This paper is devoted to the total variation (TV) based approach to image denoising and restoration. The restored image minimizes total variation in the class of images which satisfy the contraints given by the image acquisition model. We compute some explicit solutions of the denoising model which explain some of the features observed in numerical experiments. We also comment on some alternatives recently proposed by Y. Meyer which lead to u + v image decompositions. Finally we propose a total variation approach to image restoration, i.e., deconvolution and denoising, in which the image acquisition model is incorporated as a set of local constraints. Mathematics Subject Classification (2000). Primary 68U10; Secondary 65K10, 65J20, 94A08.

Keywords. Image restoration, total variation, variational methods.

1. Introduction We assume that the image acquisition system may be modelled by the following image formation model z = h ∗ u + n, (1) where u : R2 → R denotes the ideal undistorted image, h : R2 → R is a blurring kernel, z is the observed image which is represented as a function z : R2 → R, and n is an additive Gaussian white noise with zero mean and standard deviation σ . Let us denote by the interval (0, N]2 . As in most of works, in order to simplify this problem, we shall assume that the functions h and u are periodic of period N in each direction. That amounts to neglecting some boundary effects. Therefore, we shall assume that h, u are functions defined in and, to fix ideas, we assume that h, u ∈ L2 (). Our problem is to recover as much as possible of u, from our knowledge of the blurring kernel h, the statistics of the noise n, and the observed image z. The problem of recovering u from z is ill-posed due to the ill-conditioning of the operator H u = h ∗ u. Several methods have been proposed to recover u. Most of them can be classified as regularization methods which may take into account statistical properties (Wiener filters), information theoretic properties ([19]), a priori ∗ The author acknowledges partial support by the Departament d’Universitats, Recerca i Societat de la Informació de la Generalitat de Catalunya and by PNPGC project, reference BFM2003-02125.


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geometric models ([30]) or the functional analytic behavior of the image given in terms of its wavelet coefficients ([20]). The typical strategy to solve this ill-conditioning is regularization. Probably one of the first examples of regularization method [31] consists in choosing between all possible solutions of (1) the one which minimized the Sobolev (semi) norm of u |Du|2 dx. (2)

Usually, the only information we know about the noise is statistical and limited to an estimate of its mean and its variance. In that case, the model equation (1) is incorporated as a set of constraints for (2): a first constraint corresponding to the assumption that the noise has zero mean, and a second one translating the fact that σ is an upper bound of the standard deviation of n. This formulation was an important step, but the results were not satisfactory, mainly due to the unability of the previous functional to resolve discontinuities (edges) and oscillatory textured patterns. The smoothness required by the Dirichlet integral (2) is too restrictive and information corresponding to high frequencies of z is attenuated by it. Indeed, functions in W 1,2 () (i.e., functions u ∈ L2 () such that Du ∈ L2 ()) cannot have discontinuities along rectifiable curves. These observations motivated the introduction of total variation in image restoration problems by L. Rudin, S. Osher and E. Fatemi in their work [30]. The a priori hypothesis is that functions of bounded variation (the BV model) ([5]) are a reasonable functional model for many problems in image processing, in particular, for restoration problems ([30]). Typically, functions of bounded variation have discontinuities along rectifiable curves, being continuous in some sense (in the measure theoretic sense) away from discontinuities. The discontinuities could be identified with edges. The ability of total variation regularization to recover edges is one of the main features which advocates for the use of this model but its ability to describe textures is less clear, even if some textures can be recovered, up to a certain scale of oscillation. On the basis of the BV model, Rudin–Osher–Fatemi [30] proposed to solve the following constrained minimization problem Minimize |Du| (3) 2 2 |h ∗ u(x) − z(x)| dx ≤ σ ||. subject to

Notice that the image acquisition model (1) is only incorporated through a global constraint. Notice also that, assuming that h ∗ 1 = 1 (energy preservation), the constraint that h ∗ u dx = z(x) is automatically satisfied by its minima [17]. In practice, the above problem is solved via the following unconstrained minimization problem λ Minimize |Du| + |h ∗ u − z|2 dx (4) 2

Total variation based image denoising and restoration

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where the parameter λ is positive. Recall that we may interpret λ−1 as a penalization parameter which controls the trade-off between the goodness of fit of the constraint and the smoothness term given by the total variation. In this formulation, a methodology is required for a correct choice of λ. The connections between (3) and (4) were studied by A. Chambolle and P. L. Lions in [17] where they proved that both problems are equivalent for some positive value of the Lagrange multiplier λ. A particular and important case contained in the above formulation is the denoising problem which corresponds to the case where h = δ, so that equation (1) is written as z = u + n where n is an additive Gaussian white noise of zero mean and variance σ 2 . In this case, the unconstrained variational formulation (5) with h = δ is λ Minimize |Du| + |u − z|2 dx, (5) 2 and it has been the object of much theoretical and numerical research (see [7] for a survey). Even if this model represented a theoretical and practical progress in the denoising problem due to the introduction of BV functions as image models, the experimental analysis readily showed its main drawbacks. Between them, let us mention the staircasing effect (when denoising a smooth ramp plus noise, the staircase is an admissible result), the pixelization of the image at smooth regions and the loose of fine textured regions, to mention some of them. This can be summarized with the simple observation that the residuals z − u, where u represents the solution of (5), do not look like noise. The theoretical analysis of the behavior of solutions of (5) has been the objects of several works [3], [12], [13], [27], [26] and will be developed in Section 2 by exhibiting explicit solutions for specially constructed functions z. In spite of this, a second life in the interest of total variation based regularization was initiated after the proposal of u + v models by Y. Meyer in [26]. The solution u of (5) permits to obtain a decomposition of the data z as a sum of two components u + v where v is supposed to contain the noise and textured parts of the image z, while u contains the geometric sketch of the image z. As Meyer observed, the L2 norm of the residual v := z − u in (5) is not the right one to obtain a decomposition of z in terms of geometry plus texture and he proposed to measure the size of the textured part v in terms of a dual BV norm showing that some models of texture have a small dual BV norm: this will be the object of Section 3. The restoration problem (which corresponds to the case of nontrivial kernel h) has also been the object of much interest due to its applications in many contexts, like satellite, astronomical or video images, to mention a few of them. In Section 4 we shall discuss a total variation based approach to the restoration model in which the image acquisition model is incorporated as a set of local constraints. Indeed, when incorporating (1) as a constraint in (3) we loose the local character of (1) and the restored image does not look satisfactory in textured and smooth regions at the same time. Thus, we propose to incorporate (1) by ensuring that the residuals z−h ∗ u have a variance bounded by σ 2 in a sufficiently large region around each pixel (the sampling process is incorporated in the model), the size of the region has to be sufficient in

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order to estimate the variance of the noise. This gives a constrained formulation of the problem with as many Lagrange multipliers as pixels, and a solution is computed using Uzawa’s method. Finally, in Section 5 we display some experiments on restoration of satellite images which illustrate the results that can be obtained with this method.

2. Explicit solutions of TV based denoising The constrained formulation of the total variation denoising is given by (3) with h = δ. Its unconstrained formulation is given by (5) where λ > 0 is a penalization parameter. Both problems are equivalent for a certain value of λ [17]. Our purpose in this section is to exhibit some qualitative features of total variation denoising by constructing explicit solutions of (5). Those features are well known at the experimental level, and the results give a theoretical justification of these observations. Our solutions will exhibit the possibility to resolve discontinuities, but also the loss of contrast, and the regularization of corners (thus, the image is loosing structure). The staircasing effect was explained in [27]. The construction of explicit solutions of (5) is related to the computation of solutions of the eigenvalue problem for the 1-Laplacian operator. Du = u. (6) −div |Du| We denote by BV(RN ) the space of functions of bounded variation in RN . For definitions concerning bounded variation functions we refer to [5]. The solution of (6) is understood in the following sense ([6], [7], [13]). Definition 2.1. We say that a function u ∈ L2 (RN ) ∩ BV(RN ) is a solution of (6) in RN if there exits a vector field ξ ∈ L∞ (RN ; RN ) with ξ ∞ ≤ 1, such that (ξ, Du) = |Du| and −div ξ = u in D (RN ). If the vector field ξ ∈ L∞ (RN ; RN ) is such that div ξ ∈ L2 (RN ) and u ∈ BV(RN ), the expression (ξ, Du) is a distribution defined by the formula

(ξ, Dw), ϕ := − w ϕ div ξ dx − w ξ · ∇ϕ dx for all ϕ ∈ C0∞ (RN ). RN

RN

Then (ξ, Du) is a Radon measure in RN which coincides with ξ · ∇u when u ∈ L2 (RN ) ∩ W 1,1 (RN ) [11]. The following result is taken from [13] and it explains how can we derive from solutions of (6) data z for which the solution of (5) is explicit. RN ) be such that inf(|ui |, |uj |) = 0, i, j ∈ {1, . . . , m}, Proposition 2.2. Let ui ∈ BV( i = j . Assume that ui and m i=1 ui are solutions of the eigenvalue problem (6),

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i ∈ {1, . . . , m}. Let bi ∈ R, i = 1, . . . , m, z := m i=1 bi ui , and λ > 0. Then the m solution u of the variational problem (5) is u := i=1 sign(bi )(|bi | − λ−1 )+ ui . Assume that m = 1 and u is a solution of (6). If 0 < λ−1 ≤ b, then u := au with a = b − λ−1 is a solution of (5) for the datum z = bu. Indeed, u satisfies the Euler–Lagrange equation of (5) which characterizes its unique solution: Du Du −1 −1 −1 z = bu = au + λ u = au − λ div = u − λ div . |Du| |Du| If λ−1 > b, then u = 0 is the solution of (5). Indeed, in this case λzBV∗ ≤ 1 (the dual norm in BV(RN )∗ ) and there is a vector field ξ ∈ L∞ (RN ; RN ) with ξ ∞ ≤ 1, such that −div ξ = λz. Thus, u = 0 satisfies the Euler–Lagrange equation of (5). The proof when b ≤ 0 is similar and we skip the details. This solution exhibits a loss of contrast of size min(λ−1 , |b|) when the datum is z = bu. Our next theorem gives a family of solutions of (6) and is taken from [12] (see also [4]). 2 Theorem 2.3. Let C1 , . . . , Cm be bounded convex subsets k of R which are disjoint. Let bi > 0, i = 1, . . . , m, k ∈ {1, . . . , m}. Then v := − i=1 bi χ Ci + m i=k+1 bi χ Ci is a solution of (6) if and only if the following conditions holds.

(i) The sets Ci , i = 1, . . . , m, are of class C 1,1 . (ii) bi =

P (Ci ) |Ci |

for any i ∈ {1, . . . , m}.

(iii) The following inequalities hold: ess sup κCi (p) ≤ p∈∂Ci

P (Ci ) |Ci |

for all i = 1, . . . , m.

(iv) If E1 is a solution of the variational problem

min P (E) :

m

Cj ⊆ E ⊆ R \ 2

j =k+1

k

Ci ,

i=1

then we have

m

P (E1 ) =

P (Cj ).

j =k+1

If E2 is a solution of the variational problem k m min P (E) : Ci ⊆ E ⊆ R2 \ Cj , j =k+1

i=1

then we have P (E2 ) =

k

i=1

P (Ci ).

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Moreover, if k = m, then we do not need to assume that the Ci are convex and we can replace condition (i) by the following one: (i ) The sets Ci , i = 1, . . . , m, are convex and of class C 1,1 . This result was essentially proved in [12] (though we only stated the result in its second assertion). Its extension to RN was proved in [4] (replacing the curvature of the boundaries by the sum of principal curvatures) under the assumption that the sets Ci are convex and of class C 1,1 . Let us point out the following corollary for connected sets. Corollary 2.4. Let C ⊂ R2 be a bounded set of finite perimeter, and assume that C is connected. The function v := λχC is a solution of (6) if and only if the following three conditions hold. (i) λ = λC :=

P (C) |C| .

(ii) C is convex and ∂C is of class C 1,1 . (iii) The following inequality holds: ess sup κ∂C (p) ≤ p∈∂C

P (C) . |C|

A convex set C ⊆ R2 such that u := λC χ C is a solution of (6) is called calibrable. The above result gives a characterization of calibrable sets in R2 and was proved in [24], [12]. For convex sets in RN of class C 1,1 the above result is true if we replace the curvature of the boundary by the sum of the principal curvatures [4]. Example 1. Let C ⊂ R2 be the set of Figure 1. It is easy to check that C satisfies the

r

L

Figure 1. A bean-shaped set is calibrable.

assumptions of Corollary 2.4, since C is a convex set with C 1,1 boundary and there holds 1 2π r + 2L P (C) ess sup κ∂C (p) = < = . (7) 2 r πr + 2rL |C| p∈∂C Moreover, since the inequality in (7) is always strict, any convex set C of class C 1,1 close enough to C in the C 1,1 -norm is also calibrable.

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Example 2. Let ⊂ R2 be the union of two disjoint balls B1 and B2 of radius r, whose centers are at distance L (see Figure 2). Then k = 0 and m = 2 in Theorem 2.3 and condition (iv) in it reads as L ≥ π r. Under this condition the set is calibrable. The condition L ≥ π r is nothing else than P (co(B1 ∪ B2 )) ≥ P (B1 ) + P (B2 ) (co denotes the convex envelope) and in this case the solution of the denoising problem with z = χ B1 ∪B2 coincides with the

r L

Figure 2. Two balls as initial datum for the denoising problem.

addition of the solutions obtained with χ B1 and χ B2 . In case that P (co(B1 ∪ B2 )) < P (B1 ) + P (B2 ) there is interaction of the two sets and the solution is not the addition of solutions corresponding to the data χ B1 and χ B2 . These solutions exhibit two features of (5): that discontinuities may be preserved and the loss of contrast. We could expand the above family of solutions by classifying all possible solutions of (6). Along this line, we extended the above results in two directions: in [13] we looked for solutions of (6) which are built up as sums of linear combinations of characteristic functions of convex sets of class C 1,1 (not disjoint, in general), and we considered in [3], [4] the case of general convex sets. Let us illustrate the results in [13] with a simple case. Proposition 2.5. Let K0 , K1 be two bounded open convex sets of R2 with boundary of class C 1,1 such that K1 ⊆ K0 . Let F := K0 \ K1 . Let J :=

P (K0 ) − P (K1 ) > 0. |F |

If ess sup κ∂K0 ≤ J, ∂K0

ess inf κ∂K1 (x) ≥ J, x∈∂K1

then v = λK1 χK1 + J χK0 \K1 is a solution of (6).

ess sup κ∂K1 (x) ≤ λK1 x∈∂K1

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The works [3], [4] describe the denoising of the characteristic function of any convex set of R2 and RN , respectively, and the results in them illustrate the regularization of corners. Even if the more general case of linear combinations of convex sets in R2 and RN is considered, we illustrate the results in [3], [4] with a simple case. Theorem 2.6. Assume that C is a bounded convex set in R2 . Then there is a calibrable set CR ⊆ C such that ∂C \ ∂CR is formed by arcs of circle of radius R such that P (CR ) 1 R = |CR | and for each x ∈ C \CR it passes a unique arc of circle of radius r(x) and λ−1 + those circles fiber C \ CR . Let r(x) = R for x ∈ CR . Then u(x) = 1 − r(x) χC is the solution of (6) for the data z = χ C .

3. Image decomposition models In his work [26], Y. Meyer interpreted the denoising model as a u + v decomposition. Assume that is a bounded connected domain in R2 with Lipschitz boundary. If z ∈ L2 () and u is the solution of (5), then its Euler–Lagrange equation can be written as Du 1 . u + v = z where v = − div λ |Du| This type of decompositions is called a u + v decomposition and u is supposed to be a geometric sketch of the image [26]. As we have shown in the previous section, model (5) does not attain its objective of separating an image into its u + v decomposition. This conclusion was also derived in [26] through complementary arguments. For instance, if z = χ ω where ω is a bounded domain with a C ∞ boundary, then z is not preserved by the Rudin–Osher–Fatemi (ROF) model (contrary to what it should be expected). The v component contains the noise but also part of the image structure and, in particular, part of the texture (depending on the value of λ). On the other hand if z(x) = χ A (x) + p(mx)χ B (x) where A and B are two bounded domains with smooth boundary, m ≥ 1, and p(x), x = (x1 , x2 ), is a smooth 2π-periodic function of the two variables x1 , x2 , then the ROF model does not give u(x) = χ A (x), v(x) = p(mx)χ B (x) [26] (this will be explained after Theorem 3.1). Then to improve the ROF model Meyer proposed a different decomposition [26], which is based in the following variational model inf |Du| dx + λvG , u∈BV(),v∈G(),z=u+v

where λ > 0 and G() denotes the Banach space of distributions f in that may be written f = div ξ where ξ ∈ L∞ (; R2 ). The norm in G is defined by f G := inf{ξ ∞ : ξ ∈ L∞ (; R2 ), f = div ξ }


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where ξ ∞ := ess supx∈ |ξ(x)|. G() is exactly W −1,∞ (), the dual space of W01,1 (). The justification for the introduction of the space G comes from the next result [26]. Theorem 3.1. Let fn be a sequence of functions in L2 () with the following properties (i) There exists a compact set K ⊂ such that the support of fn is contained in K for each n, (ii) There exists q > 2 and C > 0 such that fn q ≤ C, (iii) The sequence fn converges to 0 in a distributional sense. Then fn G converges to 0 as n → ∞. In other words, oscillating textures have a small norm in G(). Now, if z(x) = χ A (x) + p(mx)χ B (x) is as in the first paragraph of this section, then v cannot be p(mx)χ

B(x) for large m [26]. Otherwise we would have p(mx)χ B (x) = 1 Du − λ div |Du| and therefore p(mx)χ B (x)G = λ1 . But we know from Theorem 3.1 that the G-norm of p(mx)χ B (x) is small for large values m (indeed the G-norm of p(mx)χ B (x) is an O(m−1 ) [26]). Theorem 3.1 and other results [26], [25] were the starting point of extensive numerical work on u + v decompositions [32], [28], [10], [9] to explore and compare the relative ability of the G based model versus the ROF model. Meyer’s model was first implemented by Vese–Osher in [32]. A different approach was proposed in [10], [9] where the decomposition is computed by minimizing a convex functional which depends on the two variable u and v, alternatively in each variable. Each minimization is based on the projection algorithm introduced in [16]. The problem to solve is: λ |Du| + |z − u − v|2 dx, (8) inf (u,v)∈BV()×μBG 2 where BG := {v ∈ G : vG ≤ 1}. We refer to [10] for its precise connection with Meyer’s model. Let us mention that other dual Sobolev norms, indeed H −1 , have been explored in [28]. Figure 3 displays the comparison between ROF and model (8) for a simple figure. These images are courtesy of J. F. Aujol and A. Chambolle and have been obtained with the numerical methods developed by the authors in [9], [10]. Figures 3.a and 3.b display the original reference image and the noisy image with an additive Gaussian white noise with σ = 35. Figures 3.c and 3.d display the u and v components obtained with the ROF model with λ chosen so that v = σ . For better visualization, the v component will be always displayed as v + 128. Figures 3.e and 3.f display the u and v components obtained with model (8) with λ = 10 and μ = 55 (for more details on the choice of parameters, see [9], [10]). In this case, for well chosen values of the parameter, the results are quite comparable. But let us point out that model (8) is able

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Figure 3. Comparison of ROF and model (8). From left to right and top to bottom: a) Original reference image. b) Noisy image with σ = 35. c) and d) Result of the ROF model: u and v component (with λ chosen so that v = σ ). For a better visualization, the v component will be displayed as v + 128. e) and f) Result of model (8): u and v component (λ = 10, μ = 55). g) and h) Result of the ROF model: u and v component (in this case σ = 40.8). i) and j) Result of model (8): u and v component (in this case σ = 40.8 and μ = 200). These images are courtesy of J. F. Aujol and A. Chambolle. See the text for more details.

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to separate the horizontal bands from the square for large values of the parameter μ while this does not seem to be possible with the ROF model. This is displayed in the next figures. Figures 3.g and 3.h display the u and v components obtained with the ROF model (the noise corresponds to a value of σ = 40.8) with λ chosen so that v = σ . Figures 3.i and 3.j display the u and v components obtained with model (8) with μ = 200. In any case, the choice of the parameters is open to further analysis and it the separation of the image in two components is related to the different scales present in the image.

4. Image restoration To approach the problem of image restoration from a numerical point of view we shall assume that the image formation model incorporates the sampling process in a regular grid z(i, j ) = h ∗ u(i, j ) + n(i, j ),

(i, j ) ∈ {1, . . . , N }2

(9)

where u : R2 → R denotes the ideal undistorted image, h : R2 → R is a blurring kernel, z is the observed sampled image which is represented as a function z : {1, . . . , N }2 → R, and n(i, j ) is, as usual, a white Gaussian noise with zero mean and standard deviation σ . Let us denote by N the interval (0, N]2 . As we said in the introduction, in order to simplify this problem, we assume that the functions h and u are periodic of period N in each direction. That amounts to neglecting some boundary effects. Therefore, we assume that h, u are functions defined in N . To fix ideas, we assume that h, u ∈ L2 (N ), so that h ∗ u is a continuous function in N (which may be extended to a continuous periodic function in R2 ) and the samples h ∗ u(i, j ), (i, j ) ∈ {1, . . . , N }2 , have sense. Our next purpose is to introduce a restoration model with local constraints and to explain the numerical approach to solve it. For that, let us introduce some notation. We denote by X the Euclidean space RN ×N . Then the image u ∈ X is the vector u = 2 (u(i, j ))N i,j =1 , and the vector field ξ is the map ξ : {1, . . . , N } × {1, . . . , N } → R . If u ∈ X, the discrete gradient is a vector in Y = X × X given by ∇ +,+ u := (∇x+ u, ∇y+ u),

where ∇x+ u(i, j ) =

∇y+ u(i, j )

=

u(i + 1, j ) − u(i, j ) 0

if i < N, if i = N,

u(i, j + 1) − u(i, j ) 0

if j < N, if j = N,

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for i, j ∈ {1, . . . , N }. We denote ∇ +,+ u = (∇x+ u, ∇y+ u). Other choices of the gradient are possible, this one will be convenient for the developments below. Let us define the discrete functional

β Jd (u) = β 2 + |∇ +,+ u(i, j )|2 , β ≥ 0. 1≤i,j ≤N

For any function w ∈ L2 (N ), its Fourier coefficients are (lx+jy) w(x, y)e−2π i N for (l, j ) ∈ Z2 . wˆ l , j = N N

N

Our plan is to compute a band limited approximation to the solution of the restoration problem for (9). For that we define B := u ∈ L2 (N ) : uˆ is supported in {− 21 + N1 , . . . , 21 . We notice that B is a finite dimensional vector space of dimension N 2 which can be identified with X. Both J (u) = N |Du| and Jd0 (u) are norms on the quotient space B/R, hence they are equivalent. With a slight abuse of notation we shall indistinctly write u ∈ B or u ∈ X. 2 We 1shall 1assume 1that the convolution kernel h ∈ L (N ) is such that uˆ is supported ˆ in − 2 + N , . . . , 2 and h(0, 0) = 1. In the discrete framework, the ROF model for restoration is Minimizeu∈X subject to

β

(10)

Jd (u) N

|h ∗ u(i, j ) − z(i, j )|2 ≤ σ 2 N 2 .

(11)

i,j =1

Notice again that the image acquisition model (1) is only incorporated through a global constraint. In practice, the above problem is solved via the following unconstrained formulation β

min max Jd (u) + u∈X λ≥0

N λ 1

2 2 |h ∗ u(i, j ) − z(i, j )| − σ 2 N2

(12)

i,j =1

where λ ≥ 0 is a Lagrange multiplier. The appropriate value of λ can be computed using Uzawa’s algorithm [15], [2] so that the constraint (11) is satisfied. Recall that if we interpret λ−1 as a penalization parameter which controls the importance of the regularization term, and we set this parameter to be small, then homogeneous zones are well denoised while highly textured regions will loose a great part of its structure. On the contrary, if λ−1 is set to be small, texture will be kept but noise will remain in homogeneous regions. On the other hand, as the authors of [15], [2] observed, if we use the constrained formulation (10)-(11) or, equivalently (12), then

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the Lagrange multiplier does not produce satisfactory results since we do not keep textures and denoise flat regions simultaneously, and they proposed to incorporate the image acquisition model as a set of local constraints. Following [2], we propose to replace the constraint (11) by G ∗(h ∗ u − z)(i, j ) ≤ σ 2 ,

for all (i, j ) ∈ {1, . . . , N },

(13)

where G is a discrete convolution kernel such that G(i, j ) > 0 for all (i, j ) ∈ {1, . . . , N }. The effective support of G must permit the statistical estimation of the variance of the noise with (13) (see [2]). Then we shall minimize the functional β Jd (u) on X submitted to the family of constraints (13) (plus eventually the constraint N N i,j =1 h ∗ u(i, j ) = i,j =1 z(i, j )). Thus, we propose to solve the optimization problem: β

min Jd (u) u∈B

subject to G ∗(h ∗ u − z)2 (i, j ) ≤ σ 2

(14) for all (i, j ).

This problem is well-posed, i.e., there exists a solution and is unique if β > 0 and inf c∈R G ∗(z − c)2 > σ 2 . In case that β = 0 and inf c∈R G ∗(z − c)2 > σ 2 , then h ∗ u is unique. Moreover, it can be solved with a gradient descent approach and Uzawa’s method [2]. To guarantee that the assumptions of Uzawa’s method hold we shall use a gradient descent strategy. For that, let v ∈ X and γ > 0. At each step we have to solve a problem like β

min γ |u − v|2X + Jd (u) u∈B

subject to G ∗(h ∗ u − z)2 (i, j ) ≤ σ 2

(15) for all (i, j ).

We solve (15) using the unconstrained formulation min max Lγ (u, {λ}; v), u∈X λ≥0

where λ = (λ(i, j ))N i,j =1 and β

Lγ (u, {λ}; v) = γ |u − v|2X + Jd (u) +

N

λ(i, j )(G ∗(h ∗ u − z)2 (i, j ) − σ 2 ).

i,j =1

Algorithm: TV based restoration algorithm with local constraints 1. Set u0 = 0 or, better, u0 = z. Set n = 0. 2. Use Uzawa’s algorithm to solve the problem min max Lγ (u, {λ}; un ), u∈X λ≥0

that is:

(16)

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(a) Choose any set of values λ0 (i, j ) ≥ 0, (i, j ) ∈ {1, . . . , N }2 , and un0 = un . Iterate from p = 0 until convergence of λp the following steps: (b) With the values of λp solve the problem min Lγ (u, {λp }; un ) u

n be the solution obtained. starting with the initial condition upn . Let up+1 (c) Update λ in the following way:

λp+1 (i, j ) = max(λp (i, j ) + ρ(G ∗(h ∗ upn − z)2 (i, j ) − σ 2 ), 0) for all (i, j ). Let un+1 be the solution of (16). Stop when convergence of un . We notice that, since γ > 0, Uzawa’s algorithm converges if z ∈ h ∗ B. Moreover, if u0 satisfies the constraints, then un tends to a solution u of (14) as n → ∞ [2]. Finally, to solve problem (16) in Step 2.(b) of the algorithm we use either the extension of Chambolle’s algorithm [16] to the restoration case given in [1] if we use β = 0, or the Bermúdez–Moreno algorithm [14] adapted to solve (16) when β > 0 as given in [2]. Being differentiable at when ∇ +,+ u = 0, this second possibility produces slightly smoother solutions in smooth non textured areas. We shall not enter on the comparison of both possibilities here and we shall use β = 0. For more details, we refer to [1], [2]. Let us mention the work [23] where the authors introduce a spatially varying fidelity term which controls the amount of denoising in any region of the image in order to preserve textures and small details. The philosophy is the same as ours but the value of λ(i, j ) is chosen in a different way.

5. Some restoration experiments To simulate our data we use the modulation transfer function corresponding to SPOT 5 HRG satellite with Hipermode sampling (see [29] for more details): ˆ 1 , η2 ) = e h(η

2 2 −4πβ1 |η1 | −4πα η1 +η2

e

sinc(2η1 ) sinc(2η2 ) sinc(η1 ),

(17)

where η1 , η2 ∈ [−1/2, 1/2], sinc(η1 ) = sin(πη1 )/(πη2 ), α = 0.58, and β1 = 0.14. Then we filter the reference image given in Figure 4.a with the filter (17) and we add some Gaussian white noise of zero mean and standard deviation σ (in our case σ = 1, which is a realistic assumption for the case of satellite images [29]) to obtain the image displayed in Figure 4.b. Figure 5.a displays the restoration of the image in Figure 4.b obtained using the algorithm of last section with β = 0. We have used a Gaussian function G of standard deviation σ = 6. The mean value of the constraint is mean((G∗(Ku−z))2 ) = 1.0933 and RMSE = 7.9862. Figure 5.b displays the function λ(i, j ) obtained.


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Figure 4. Reference image and a filtered and noised image. a) Top: reference image. b) Bottom: the data. This image has been generated applying the MTF given in (17) to the top image and adding a Gaussian white noise of zero mean and standard deviation σ = 1.

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Figure 6 displays some details of the results that are obtained using a single global constraint (11) and show its main drawbacks. Figure 6.a corresponds to the result obtained with the Lagrange multiplier λ = 10 (thus, the constraint (11) is satisfied). The result is not satisfactory because it is difficult to denoise smooth regions and keep the textures at the same time. Figure 6.b shows that most textures are lost when using a small value of λ (λ = 2) and Figure 6.c shows that some noise is present if we use a larger value of λ (λ = 1000). This result is to be compared with the same detail of Figure 5.a which is displayed in Figure 6.d. The modulation transfer function for satellite images. We describe here a simple model for the Modulation Transfer Function of a general satellite. More details can be found in [29] where specific examples of MTF for different acquisition systems are shown. The MTF used in our experiments (17) corresponds to a particular case of the general model described below [29]. ˆ is the Fourier transform of the impulse Recall that the MTF, that we denote by h, response of the system. Let (η1 , η2 ) ∈ [−1/2, 1/2] denote the coordinates in the frequency domain. There are different parts in the acquisition system that contribute to the global transfer function: Sensors. Every sensor has a sensitive region where all the photons that arrive are integrated. This region can be approximated by a unit square [−c/2, c/2]2 where c is the distance between consecutive sensors. Its impulse response is then the convolution of two pulses, one in each spatial direction. The corresponding transfer function also includes the effect of the conductivity (diffusion of information) between neighbouring sensors, which is modeled by an exponential decay factor, thus: hˆ S (η1 , η2 ) = sinc(η1 c) sinc(η2 c)e−2πβ1 c|η1 | e−2πβ2 c|η2 | , where sinc(η1 ) = sin(πη1 )/(πη1 ) and β1 , β2 > 0. Optical system. It is considered as an isotropic low-pass filter −2παc hˆ O (η1 , η2 ) = e

η12 +η22

,

α > 0.

Motion. Each sensor counts the number of photons that arrive to its sensitive region during a certain time of acquisition. During the sampling time the system moves a distance τ and so does the sensor; this produces a motion blur effect in the motion direction (d1 , d2 ): hˆ M (η1 , η2 ) = sinc( (η1 , η2 ), (d1 , d2 )τ ). Finally, the global MTF is the product of each of these intermediate transfer functions modeling the different aspects of the satellite: ˆ 1 , η2 ) = hˆ S hˆ O hˆ M . h(η


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Figure 5. Restored image with local Lagrange multipliers. a) Top: the restored image corresponding to the data given in Figure 4.b. The restoration has been obtained using the algorithm of last section We have used a Gaussian function G of standard deviation σ = 6. b) Bottom: the function λ(i, j ) obtained.

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Figure 6. A detail of the restored images with global and local constraints. Top: a), b) and c) display a detail of the results that are obtained using a single global constraint (11) and show its main drawbacks. Figure a) corresponds to the result obtained with the value of λ such that the constraint (11) is satisfied, in our case λ = 10. Figure b) shows that most textures are lost when using a small value of λ (λ = 2)and Figure c) shows that some noise is present if we use a larger value of λ (λ = 1000). Bottom: d) displays the same detail of Figure 5.a which has been obtained using restoration with local constraints.

Acknowledgements. The author is indebted to his coauthors: A. Almansa, F. Alter, F. Andreu, C. Ballester, G. Bellettini, M. Bertalmío, A. Chambolle, J. Faro, G. Haro, J. M. Mazón, M. Novaga, B. Rougé, and A. Solé. I would like to thank also J. F. Aujol and A. Chambolle for providing me with the images in Figure 3. Special thanks and a warm dedication to Andrés Solé who passed away after completing his PhD and started with me his work on restoration.

References [1] Almansa, A., Caselles, V., Haro, G., and Rougé, B., Restoration and zoom of irregularly sampled, blurred and noisy images by accurate total variation minimization with local constraints. Multiscale Model. Simul., to appear. [2] Almansa, A., C. Ballester, C., Caselles, V., Faro, L. J., and Haro, G., A TV based restoration model with local constraints. Preprint, 2006.


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[3] Alter, F., Caselles, V., and Chambolle, A., Evolution of Characteristic Functions of Convex Sets in the Plane by the Minimizing Total Variation Flow. Interfaces Free Bound. 7 (2005), 29–53. [4] Alter, F., Caselles, V., and Chambolle, A., A characterization of Convex Calibrable Sets in RN . Math. Ann. 332 (2005), 329–366. [5] Ambrosio, L., Fusco, N., and Pallara, D., Functions of Bounded Variation and Free Discontinuity Problems. Oxford Math. Monogr., The Clarendon Press, Oxford University Press, New York 2000 [6] Andreu, F., Ballester, C., Caselles, V., and Mazón, J. M., Minimizing total variation flow. Differential Integral Equations 14 (2001), 321–360. [7] Andreu-Vaillo, F., Caselles, V., and Mazón, J. M., Parabolic Quasilinear Equations Minimizing Linear Growth Functionals. Progr. Math. 223, Birkhäuser, Basel 2004 [8] Andrews, H. C., and Hunt, B. R., Digital Image Restoration. Prentice Hall, Englewood Cliffs, NJ, 1977. [9] Aujol, J. F., and Chambolle, A., Dual Norms and Image Decomposition Models. Internat. J. Computer Vision, 63 (2005), 85–104. [10] Aujol, J. F., Aubert, G., Blanc-Feraud, L., and Chambolle, A., Image Decomposition into a bounded variation component and an oscillating component. J. Math. Imaging Vision 22 (2005), 71–88. [11] Anzellotti, G., Pairings between measures and bounded functions and compensated compactness. Ann. Mat. Pura Appl. 135 (1983), 293–318. [12] Bellettini, G., Caselles, V., and Novaga, M., The Total Variation Flow in R N . J. Differential Equations 184 (2002), 475–525. [13] Bellettini, Caselles, V., and Novaga, M., Explicit solutions of the eigenvalue problem

G., Du = u. SIAM J. Math. Anal. 36 (2005), 1095–1129. −div |Du| [14] Bermúdez, A., and Moreno, C., Duality methods for solving variational inequalities. Comput. Math. Appl. 7 (1981), 43–58. [15] Bertalmío, M., Caselles, V., Rougé, B., and Solé, A., TV based image restoration with local constraints. J. Sci. Comput. 19 (2003), 95–122. [16] Chambolle, A., An algorithm for total variation minimization and applications. J. Math. Imaging Vision 20 (2004), 89–97. [17] Chambolle, A., and Lions, P. L., Image recovery via total variation minimization and related problems. Numer. Math. 76 (1997), 167–188. [18] Chan, T. F., Golub, G. H., and Mulet, P., A Nonlinear Primal-Dual Method for Total Variation Based Image Restoration. SIAM J. Sci. Comput. 20 (1999), 1964–1977. [19] Demoment, G., Image reconstruction and restoration: Overview of common estimation structures and problems. IEEE Trans. Acoust. Speech Signal Process. 37 (1989), 2024–2036. [20] Donoho, D., Denoising via soft-threholding. IEEE Trans. Inform. Theory 41 (1995), 613–627. [21] Durand, S., Malgouyres, F., and Rougé, B., Image Deblurring, Spectrum Interpolation and Application to Satellite Imaging. ESAIM Control Optim. Calc. Var. 5 (2000), 445–475.

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[22] Geman, D., and Reynolds, G., Constrained Image Restoration and Recovery of Discontinuities. IEEE Trans. Pattern Anal. Machine Intell. 14 (1992), 367–383. [23] Gilboa, G., Sochen, N., and Zeevi, Y., PDE-based denoising of complex scenes using a spatially-varying fidelity term. In Proc. International Conference on Image Processing 2003, Barcelona, Spain, Vol. 1, 2003, 865–868. [24] Giusti, E., On the equation of surfaces of prescribed mean curvature. Existence and uniqueness without boundary conditions. Invent. Math. 46 (1978), 111–137. [25] Haddad, A., and Meyer, Y., Variational methods in image processing. CAM Reports 04-52, 2004. [26] Meyer, Y., Oscillating patterns in image processing and in some nonlinear evolution equations. The Fifteenth Dean Jacqueline B. Lewis memorial lectures, University Lecture Series 22, Amer. Math. Soc., Providence, RI, 2001. [27] Nikolova, M., Local strong homogeneity of a regularized estimator. SIAM J. Appl. Math 61 (2000), 633–658. [28] Osher, S. J., Sole, A., and Vese, L. A., Image decomposition and restoration using total variation minimization and the H −1 norm. Multiscale Model. Simul. 1 (2003), 349–370. [29] Rougé, B., Théorie de l’echantillonage et satellites d’observation de la terre. In Analyse de Fourier et traitement d’images, Journées X-UPS 1998. [30] Rudin, L., Osher, S., and Fatemi, E., Nonlinear total variation based noise removal algorithms. Physica D 60 (1992), 259–268. [31] Tikhonov, A. N., and Arsenin, V. Y., Solutions of Ill-Posed Problems. Scripta Series in Mathematics, John Wiley & Sons, New York 1977. [32] Vese, L. A., and Osher, S. J., Modeling textures with total variation mkinimization and oscillatikng patterns in image processing. J. Sci. Comput. 19 (2003), 553–572. Departament Tecnologia, Universitat Pompeu Fabra, Passeig de Circumvalació, 8, 08003 Barcelona, Spain E-mail: [email protected]

A wavelet based sparse grid method for the electronic Schrödinger equation Michael Griebel and Jan Hamaekers ∗

Abstract. We present a direct discretization of the electronic Schrödinger equation. It is based on one-dimensional Meyer wavelets from which we build an anisotropic multiresolution analysis for general particle spaces by a tensor product construction. We restrict these spaces to the case of antisymmetric functions. To obtain finite-dimensional subspaces we first discuss semidiscretization with respect to the scale parameter by means of sparse grids which relies on mixed regularity and decay properties of the electronic wave functions. We then propose different techniques for a discretization with respect to the position parameter. Furthermore we present the results of our numerical experiments using this new generalized sparse grid methods for Schrödinger’s equation. Mathematics Subject Classification (2000). 35J10, 65N25, 65N30, 65T40, 65Z05. Keywords. Schrödinger equation, numerical approximation, sparse grid method, antisymmetric sparse grids.

1. Introduction In this article we consider the electronic Schrödinger equation (first without spin for reasons of simplicity) H (x 1 , . . . , x N ) = E(x 1 , . . . , x N )

(1)

with the Hamilton operator 1 p 2 N

H =T +V

where T = −

p=1

and V =−

N N nuc p=1 q=1

Zq 1 + . |xp − R q |2 |x − x | p q 2 q>p N

N

(2)

p=1

∗ The authors were supported in part by the priority program 1145 Modern and universal first-principles methods for many-electron systems in chemistry and physics and the Sonderforschungsbereich 611 Singuläre Phänomene und Skalierung in Mathematischen Modellen of the Deutsche Forschungsgemeinschaft.


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Michael Griebel and Jan Hamaekers

Here, with d = 3, xp := (x1,p , . . . , xd,p ) ∈ Rd denotes the position of the p-th electron, p = 1 . . . , N , and R q ∈ Rd denotes the fixed position of the q-th nucleus, q = 1, . . . , Nnuc . The operator p is the Laplacian acting on the xp -component of , i.e. p = di=1 ∂ 2 /∂(xi,p )2 , Zq is the charge of the q-th nucleus and the norm | . |2 denotes the usual Euclidean distance in Rd . The solution describes the wave function associated to the eigenvalue E. This eigenvalue problem results from the Born–Oppenheimer approximation [51] to the general Schrödinger equation for a system of electrons and nuclei which takes the different masses of electrons and nuclei into account. It is one of the core problems of computational chemistry. Its successful treatment would allow to predict the properties of arbitrary atomic systems and molecules [22]. However, except for very simple cases, there is no analytical solution for (1) available. Also a direct numerical approach is impossible since is a d · N-dimensional function. Any discretization on e.g. uniform grids with O(K) points in each direction would involve O(K d·N ) degrees of freedoms which are impossible to store for d = 3, N > 1. Here, we encounter the curse of dimensionality [8]. Therefore, most approaches resort to an approximation of (1) only. Examples are the classical Hartree–Fock method and its successive refinements like configuration interaction and coupled clusters. Alternative methods are based on density functional theory which result in the Kohn–Sham equations or the reduced density matrix (RDM) [50] and the r12 approach [23] which lead to improved accuracy and open the way to new applications. A survey of these methods can be found in [3], [10], [46]. A major problem with these techniques is that, albeit quite successful in practice, they nevertheless only provide approximations. A systematical improvement is usually not available such that convergence of the model to Schrödinger’s equation is achieved. Instead, we intend to directly discretize the Schrödinger equation without resorting to any model approximation. To this end, we propose a new variant of the so-called sparse grid approach. The sparse grid method is a discretization technique for higher-dimensional problems which promises to circumvent the above-mentioned curse of dimensionality provided that certain smoothness prerequisites are fulfilled. Various sparse grid discretization methods have already been developed in the context of integration problems [27], [28], integral equations [24], [32] and elliptic partial differential equations, see [12] and the references cited therein for an overview. In Fourier space, such methods are also known under the name hyperbolic cross approximation [5], [21], [61]. A first heuristic approach to apply this methodology to the electronic Schrödinger equation was presented in [26]. The sparse grid idea was also used in the fast evaluation of Slater determinants in [33]. Recently Yserentant showed in [67] that the smoothness prerequisite necessary for sparse grids is indeed valid for the solution of the electronic Schrödinger equation. To be more precise, he showed that an antisymmetric solution of the electronic Schrödinger equation with d = 3 1/2,1 1,1 - or Hmix -regularity for the fully antisymmetric and the partially possesses Hmix symmetric case, respectively. This motivated the application of a generalized sparse grid approach in Fourier space to the electronic Schrödinger equation as presented

A wavelet based sparse grid method for the electronic Schrödinger equation

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in [30]. There, sparse grids for general particle problems as well as antisymmetric sparse grids have been developed and were applied to (1) in the periodic setting. Basically, estimates of the type − M H 1 ≤ C(N, d) · M −1/d · H 1,1

mix

could be achieved where M denotes the number of Fourier modes used in the discretization. Here, the norm .H 1,1 involves bounded mixed first derivatives. Thus mix the order of the method with respect to M is asymptotically independent of the dimension of the problem, i.e. the number N of electrons. But, the constants and the 1,1 -norm of the solution nevertheless depend on the number of electrons. While Hmix the dependency of the order constant might be analysed along the lines of [29], the problem remains that the smoothness term H 1,1 grows exponentially with the mix number of electrons. This could be seen from the results of the numerical experiments in [30] and was one reason why in the periodic Fourier setting problems with higher numbers of electrons could not be treated. It was also observed in [69] where a certain scaling was introduced into the definitions of the norms which compensates for this growth factor. In [68], [70] it was suggested to scale the decomposition of the hyperbolic cross into subspaces accordingly and to further approximate each of the subspace contributions by some individually properly truncated Fourier series to cope with this problem. In this article, we present a modified sparse grid/hyperbolic cross discretization for the electronic Schrödinger equation which implements this approach. It uses onedimensional Meyer wavelets as basic building blocks in a tensor product construction to obtain a L2 -orthogonal multiscale basis for the many-electron space. Then a truncation of the associated series expansion results in sparse grids. Here, for the level index we truncate according to the idea of hyperbolic crosses whereas we truncate for the position index according to various patterns which take to some extent the decay of the scaling function coefficients for x → ∞ into account. Note that since we work in an infinite domain this resembles a truncation to a compact domain in which we then consider a local wavelet basis. Here, domain truncation error and scale resolution error should be balanced. Antisymmetry of the resulting discrete wavelet basis is achieved by a restriction of the active indices. The remainder of this article is organized as follows: In Section 2 we present the Meyer wavelet family on R and discuss its properties. In Section 3 we introduce a multiresolution analysis for many particle spaces build by a tensor product construction from the one-dimensional Meyer wavelets and introduce various Sobolev norms. Then we discuss semi-discretization with respect to the scale parameter by means of generalized sparse grids and present a resulting error estimate in Section 4. Section 5 deals with antisymmetric generalized sparse grids. In Section 6 we invoke results on the mixed regularity of electronic wave functions and we discuss rescaling of norms and sparse grid spaces to obtain error bounds which involve the L2 -norm of the solution instead of the mixed Sobolev norm. Then, in Section 7 we comment on the setup

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of the system matrix and on the solution procedure for the discrete eigenvalue problem on general sparse grids and we propose different techniques for the discretization with respect to the position parameter. Furthermore we present the results of our numerical experiments. Finally we give some concluding remarks in Section 8.

2. Orthogonal multilevel bases and the Meyer wavelet family on R We intend to use for the discretization of (1) a L2 -orthogonal basis system.1 This is an important prerequisite from the practical point of view, since it allows to apply the well-known Slater–Condon rules. They reduce the Rd·N - and R2·d·N -dimensional integrals necessary in the Galerkin discretization of the one- and two electron part of the potential function of (1) to short sums of d-dimensional and 2d-dimensional integrals, respectively. Otherwise, due to the structure of the Slater determinants necessary to obtain antisymmetry, these sums would contain exponentially many terms with respect to the number N of electrons present in the system. Let us recall the definition of a multiresolution analysis on R, see also [52]. We consider an infinite sequence · · · ⊂ V−2 ⊂ V−1 ⊂ V0 ⊂ V1 ⊂ V2 ⊂ · · · of nested spaces Vl with l∈Z Vl = 0 and l∈Z Vl = L2 (R). It holds f (x) ∈ Vl ⇔ f (2x) ∈ Vl+1 and f (x) ∈ V0 ⇔ f (x − j ) ∈ V0 , where j ∈ Z. Furthermore, there is a so-called scaling function (or father wavelet) φ ∈ V0 , such that {φ(x − j ) : j ∈ Z} forms an orthonormal basis for V0 . Then l

{φl,j (x) = 2 2 φ(2l x − j ) : j ∈ Z} any u(x) ∈ Vl as u(x) = forms an orthonormal basis of Vl and we can represent ∞ ∗ (x)u(x)dx. With the definition v φ (x) with coefficients v := φ l,j j =−∞ l,j l,j R l,j Wl ⊥ Vl , Vl ⊕ Wl = Vl+1

(3)

we obtain an associated sequence of detail spaces Wl with associated mother wavelet ϕ ∈ W0 , such that {ϕ(x − j ) : j ∈ Z} forms an orthonormal basis for W0 . Thus l

{ϕl,j (x) = 2 2 ϕ(2l x − j ) : j ∈ Z} gives an orthonormal basis for Wl and {ϕl,j : l, j ∈ Z} is an orthonormal basis of L2 (R). Then, we can represent any u(x) in L2 (R) as u(x) =

∞

∞

l=−∞ j =−∞ 1 Note that a bi-orthogonal system would also work here.

ul,j ϕl,j (x)

(4)

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∗ (x)u(x)dx. with the coefficients ul,j := R ϕl,j In the following we focus on the Meyer wavelet family for the choice of φ and ϕ. the definition of the Fourier transform F [f ](ω) = fˆ(ω) = ∞ There, with −iωx dx we set as father and mother wavelet in Fourier space √1 f (x)e −∞ 2π

⎧ ⎪1 1 ⎨ 3 ˆ φ(ω) =√ |ω| − 1)) cos( π2 ν( 2π ⎪ 2π ⎩ 0

ω 1 ϕ(ω) ˆ = √ e−i 2 2π

for |ω| ≤ 23 π, for 2π 3 < |ω| ≤ otherwise,

⎧ π 3 ⎪ ⎨sin( 2 ν( 2π |ω| − 1)) 3 cos( π2 ν( 4π |ω| − 1)) ⎪ ⎩ 0

4π 3 ,

for 23 π ≤ |ω| ≤ 43 π, 8π for 4π 3 < |ω| ≤ 3 , otherwise,

(5)

(6)

where ν : R → R ∈ C r is a parameter function still do be fixed, which has the properties ν(x) = 0 for x ≤ 0, ν(x) = 1 for x > 1 and ν(x) + ν(1 − x) = 1. By dilation and translation we obtain −l ˆ −l ω), F [φl,j ](ω) = φˆ l,j (ω) = 2− 2 e−i2 j ω φ(2 l

F [ϕl,j ](ω) = ϕˆl,j (ω) = 2− 2 e−i2 l

−l j ω

ϕ(2 ˆ −l ω)

where the φˆ l,j and ϕˆl,j denote the dilates and translates of (5) and (6), respectively. This wavelet family can be derived from a partition of unity l χˆ l (ω) = 1 for all ω ∈ R in Fourier space, where for l = 0, 2π φˆ ∗ (ω)φˆ 0,0 (ω) χˆ l (ω) = l 0,0 (7) ∗ 2 π ϕˆl−1,0 (ω)ϕˆl−1,0 (ω) for l > 0, see [4] for details. The function ν basically describes the decay from one to zero of one partition function χˆ l in the overlap with its neighbor. The smoothness of the χˆ l is thus directly determined by the smoothness of ν. The mother wavelets ϕˆl,j and the father wavelets φˆ l,j in Fourier space inherit the smoothness of the χˆ l ’s via the relation (7). There are various choices for ν with different smoothness properties in the literature, see [4], [45], [53], [54]. Examples are the Shannon wavelet and the raised cosine wavelet [63], i.e. (6) with ⎧ ⎪ ⎨0 for x ≤ 0, 0 for x ≤ 21 , ν(x) = ν 0 (x) := and ν(x) = ν 1 (x) x for 0 ≤ x ≤ 1, (8) ⎪ 1 otherwise ⎩ 1 otherwise

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or, on the other hand, ⎧ ⎪ for x ≤ 0 ⎨0 ν˜ (x) ν(x) = ν ∞ (x) := ν˜ (1−x)+˜ for 0<x≤1 ν (x) ⎪ ⎩ 1 otherwise

where ν˜ (x) =

0 e

for x ≤ 0

− x1α

otherwise

(9) with α = 1, 2 [62], respectively. Other types of Meyer wavelets with different smoothness properties can be found in [19], [34], [40], [65]. The resulting mother wavelet functions in real space and Fourier space are given in Figure 1. Note the 0.5

1.5

φ ϕ

1

0.5

0 0

−0.5

φˆ |ϕ| ˆ (ϕ) ˆ (ϕ) ˆ −5

0 ω

−0.5

5

0.5

−1 −8

−6

−4

−2

0 x

2

4

6

1.5

8 φ ϕ

1

0.5

0 0

−0.5

φˆ |ϕ| ˆ (ϕ) ˆ (ϕ) ˆ −5

0 ω

−0.5

5

0.5

−1 −8

−6

−4

−2

0 x

2

4

6

1.5

8 φ ϕ

1

0.5

0 0

−0.5

φˆ |ϕ| ˆ (ϕ) ˆ (ϕ) ˆ −5

0 ω

−0.5

5

−1 −8

−6

−4

−2

0 x

2

4

6

8

Figure 1. Top: (6) with ν 0 from (8) in Fourier space (left) and real space (right). Middle: (6) with ν 1 from (8) in Fourier space (left) and real space (right). Bottom: (6) with ν ∞ from (9) in Fourier space (left) and real space (right).

two symmetric areas of support and the associated two bands with non-zero values of the wavelets in Fourier space which resemble the line of construction due to Wilson,


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Malvar, Coifman and Meyer [17], [20], [39], [49], [64] to circumvent the Balian–Low theorem2 [7], [48]. In real space, these wavelets are C ∞ -functions with global support, in Fourier space, they are piecewise continuous, piecewise continuous differentiable and C ∞ , respectively, and have compact support. Furthermore they possess infinitely many vanishing moments. Finally their envelope in real space decays with |x| → ∞ as |x|−1 for ν 0 , as |x|−2 for ν 1 and faster than any polynomial (subexponentially) for ν ∞ , respectively. To our knowledge, only for the Meyer wavelets with (8) there are analytical formulae in both real and Fourier space available. Certain integrals in a Galerkin discretization of (1) can then be given analytically. For the other types of Meyer wavelets analytical formulae only exist in Fourier space and thus numerical integration is necessary in a Galerkin discretization of (1). For a discretization of (4) with respect to the level-scale l we can restrict the doubly infinite sum to an interval, i.e. l ∈ [L1 , L2 ]. However to obtain the space VL2 we have to complement the sum of detail spaces Wl , l ∈ [L1 , L2 ] by the space VL1 , i.e. we have L2 Wl . VL2 = VL1 ⊕ l=L1

with the associated representation u(x) =

∞

vL1 ,j φL1 ,j (x) +

j =−∞

L2 ∞

ul,j ϕl,j (x).

L1 j =−∞

Note that for the case of R, beside the choice of a finest scale L2 , we here also have a choice of the coarsest scale L1 . This is in contrast to the case of a finite domain where the coarsest scale is usually determined by the size of the domain and is denoted as level zero. Additionally we can scale our spaces and decompositions by a parameter c > 0, c ∈ R. For example, we can set 1

l

Vlc = span{φc,l,j (x) = c 2 2 2 φ(c2l x − j ) : j ∈ Z}. 1 holds. Then we obtain the scaled For c = 2k , k ∈ Z, the obvious identity Vlc = Vl+k decomposition L2 c c VL2 = VL1 ⊕ Wlc l=L1 1

l

with the scaled detail spaces Wlc = span{ϕc,l,j (x) = c 2 2 2 ϕ(c2l x − j ) : j ∈ Z}. 1 holds. For c = 2k , k ∈ Z, the identity Wlc = Wl+k 2 The Balian–Low theorem basically states that the family of functions g

2πimx g(x − n), m, n ∈ Z, m,n (x) = e which are related to the windowed Fourier transform, cannot be an orthonormal basis of L2 (R), if the two 2 dk are both finite. Thus there exists no orthonormal family for a integrals R x 2 |g(x)|2 dx and R k 2 |g(k)| ˆ 2 Gaussian window function g(x) = π −1/4 e−x /2 which is both sufficiently regular and well localized.

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With the choice c = 2−L1 we can get rid of the parameter L1 and may write our wavelet decomposition as L VLc = V0c ⊕ Wlc , (10) l=0

i.e. we can denote the associated coarsest space with level zero and the finest detail space with level L (which now expresses the rescaled parameter L2 ). To simplify notation we will skip the scaling index c in the following. We also introduce with c for l = 0, φl,j ψl,j := (11) c ϕl−1,j for l ≥ 1 for c = 2−L1 a unique notation for both the father wavelets on the coarsest scale and the mother wavelets of the detail spaces. Bear however in mind that in the following the function ψl,j with l = 0 denotes a father wavelet, i.e. a scaling function only, whereas it denotes for l ≥ 1 a true wavelet on scale l − 1. Let us finally consider the wavelet representation of the function e−σ |x−x0 | which is the one-dimensional analogon of the ground state wavefunction of hydrogen centered in x0 = 0. For two types of Meyer wavelets, i.e. with ν 0 from (8) and ν ∞ from (9) with α = 2, Figure 2 gives the isolines to the values 10−3 and 10−4 for both the absolute value of the coefficients vl,j of the representation with respect to the scaling functions and the absolute value of the coefficients ul,j of the representation with respect to the wavelet functions. Here we see the following: For the Meyer wavelet with ν ∞ from (9) where α = 2, the isolines to different values (only 10−3 and 10−4 are shown) are nearly parallel for both the wavelet coefficients ul,j and the scaling coefficients vl,j . For levels larger than −2 the isolines of the wavelet coefficients are even straight lines. Furthermore, on sufficiently coarse levels, the isoline for the wavelet coefficients and the scaling coefficients practically coincide. This is an effect of the C ∞ -property of the underlying mother wavelet. For the Meyer wavelet with ν 0 from (8), i.e. for wavelets which are not C ∞ in both real space and Fourier space, these two observations do not hold. If we compare the isolines of the wavelet coefficients ul,j for the Meyer wavelet with ν ∞ from (9) where α = 2 and that of the Meyer wavelet with ν 0 from (8) we observe that the level on which the bottom kink occurs is exactly the same. However the size of the largest diameter (here roughly on level −2) is substantially bigger for the Shannon wavelet. Note the different scaling of the x-axis of the diagrams on the left and right side. We furthermore observe for the isolines of the scaling coefficients an exponential behavior, i.e. from level l to level l + 1 the associated value for j nearly doubles in a sufficient distance away from point x = 0. With respect to the wavelet coefficients, however, we see that the support shrinks super-exponentially towards the bottom kink with raising level.

1481

A wavelet based sparse grid method for the electronic Schrödinger equation −4 −3

−3 −3

−4−3 −3 −4

−3

−3 0

0

−4

2 4

−4

8 | φ∗l,j (x)u(x) dx| | ϕ∗l,j (x)u(x) dx|

12

200

−60

−40

−20

−100

0 j

−3

−3 −4

8 10 200

12

−4

−4

6

2l | φ∗l,j (x)u(x) dx| 2l | ϕ∗l,j (x)u(x) dx| 100

4

2l | φ∗l,j (x)u(x) dx| 2l | ϕ∗l,j (x)u(x) dx|

−4 −3 −4

−4 −4 −4

−3 −4

8 10

−3

−4 −3 −3 −3−4

−4

−3 −4

l

l

−3

−3

−4

60

−3

−4

−4

−3

2

−4

−3

40

−4

−4 −4

−3 −4 −3

−3

0

−4

20

−3

−3

2

−2

−3 −3

0

−4

−3

−4

−3

−4

−4

−3

−4

−2

−4

−4−4

−4 −3 −4−3 −4

−4

−4

0 j

4 −4− −3 −4 −3 −3 −4

−6

−6

12 −200

100

−4

0 j

−3 −3

−100

−3

12 −200

| φ∗l,j (x)u(x) dx| | ϕ∗l,j (x)u(x) dx|

−4

10

10

−60

−40

−20

0 j

4

8

6

−4

6

6

4

−−34

l

−3

−4

l

−4 −3

−4

−4

−4

−4

4

−4 −3

−4

−4

−4 −3

2

−4

−4

−3 −3

−2

−4

−3

−4

−2

−4

−3

−4 −4 −4 −3 −3

−4

−3

−4

−6

−3

−4 −3

−6

20

40

60

Figure 2. Isolines to the values 10−3 and 10−4 of the absolute value of the coefficients vl,j and ul,j for the Meyer wavelets with ν 0 from (8) (left) and ν ∞ from (9) with α = 2 (right), no scaling (top) and scaling with 2l (bottom).

The relation (3) relates the spaces Vl , Wl and Vl+1 and allows to switch between the scaling coefficients and the wavelet coefficients on level l to the scaling coefficients on level l + 1 and vice versa. This enables us to choose an optimal coarsest level for a prescribed accuracy and we also can read off the pattern of indices (l, j ) which result in a best M-term approximation with respect to the L2 - and H 1 -norm for that prescribed accuracy, respectively. For the Meyer wavelet with ν ∞ from (9) where α = 2, the optimal choice of the coarsest level L1 on which we use scaling functions is just the level where, for a prescribed accuracy, the two absolute values of the wavelet coefficients on one level possess their largest distance, i.e. the associated isoline of the wavelet coefficients shows the largest diameter (here roughly on level −2). The selection of a crossing isoline then corresponds to the fixation of a boundary error by truncation of the further decaying scaling function coefficients on that level which resembles a restriction of R to just a finite domain. From this base a downward pointing triangle then gives the area of indices to be taken into account into the finite sum of best approximation with respect to that error. We observe that the use of the wavelets with ν 0 from (8) would result in a substantially larger area of indices and

1482


thus number of coefficients to be taken into account to obtain the same error level. There, the form of the area is no longer a simple triangle but shows a “butterfly”-like shape where the base of the pattern is substantially larger.

3. MRA and Sobolev spaces for particle spaces In the following we introduce a multiresolution analysis based on Meyer wavelets for particle spaces on (Rd )N and discuss various Sobolev spaces on it. First, let us set up a basis for the one-particle space H s (Rd ) ⊂ L2 (Rd ). Here, we use the d-dimensional product of the one-dimensional system {ψl,j (x), l ∈ N0 , j ∈ Z}. We then define the d-dimensional multi-indices l = (l1 , l2 , . . . , ld ) ∈ Nd0 and j = (j1 , j2 , . . . , jd ) ∈ Zd , the coordinate vector x = (x1 , . . . , xd ) ∈ Rd and the associated d-dimensional basis functions ψl,j (x) :=

d

ψli ,ji (xi ).

(12)

i=1

Note that due to (11) this product may involve both father and mother wavelets depending on the values of the components of the level index l. We furthermore denote d 2 1/2 |l|2 = and |l|∞ = max1≤i≤d |li |. Let us now define isotropic Sobolev i=1 li spaces in d dimensions with help of the wavelet series expansion, i.e. we classify functions via the decay of their wavelet coefficients. To this end, we set λ(l) := |2l |2 = |(2l1 , . . . , 2ld )|2

(13)

and define

H s (Rd ) = u(x) = l∈Nd , ul,j ψl,i (x) : 0 j ∈Zd

u2H s (Rd ) =

l∈Nd 0, j ∈Zd

(14) λ(l)2s · |ul,j |2 ≤ c2 < ∞ ,

∗ (x)u(x)d x and c is a constant which depends on d. where ul,j = Rd ψl,j Based on the given one-particle basis (12) we now define a basis for many-particle spaces on Rd·N . We then have the d · N-dimensional coordinates x := (x 1 , . . . , x N ) where x i ∈ Rd . To this end, we first employ a tensor product construction and define the multi-indices l = (l 1 , ..., l N ) ∈ Nd·N and the associated multivariate wavelets 0 ψl,j ( x ) :=

N

p=1

ψlp ,j p (xp ) =

N

ψlp ,j p (x 1 , . . . , x N ).

(15)

p=1

Note again that this product may involve both father and mother wavelets depending on the values of the components of the level index l. The wavelets ψl,j span the


1483

subspaces Wl,j := span{ψl,j } whose union forms3 the space V =

(16)

Wl,j .

l∈NdN 0 j ∈ZdN

We then can uniquely represent any function u from V as u( x) = ul,j ψl,j ( x)

(17)

l∈NdN 0 j ∈ZdN

with coefficients ul,j =

RdN

ψ∗ ( x )u( x )d x . l,j

Now, starting from the one-particle space H s (Rd ) we build Sobolev spaces for many particles. Obviously there are many possibilities to generalize the concept of Sobolev spaces [1] from the one-particle case to higher dimensions. Two simple possibilities are the additive or multiplicative combination i.e. an arithmetic or geometric averaging of the scales for the different particles. We use the following definition that combines both possibilities. We denote N

λmix (l) :=

λ(lp )

and

λiso (l) :=

p=1

N

λ(lp ).

(18)

p=1

Now, for −∞ < t, r < ∞, set t,r ((Rd )N ) Hmix = u( x ) = l∈NdN ul,j ψl,j ( x) : 0 j ∈ZdN

u2H t,r ((Rd )N ) = mix

l∈NdN 0

(19)

λmix (l)2t · λiso (l)2r ·

j ∈ZdN

|ul,j |2 ≤ c2 < ∞

with a constant c which depends on d and N. The standard isotropic Sobolev spaces [1] as well as the Sobolev spaces of dominating mixed smoothness [58], both generalized to the N-particle case, are included here. They can be written as the special cases 0,s H s ((Rd )N ) = Hmix ((Rd )N )

and

t,0 t Hmix ((Rd )N ) = Hmix ((Rd )N ),

respectively. Hence, the parameter r from (19) governs the isotropic smoothness, t,r whereas t governs the mixed smoothness. Thus, the spaces Hmix give us a quite flexible framework for the study of problems in Sobolev spaces. Note that the relations t/N t/N t t Hmix ⊂ H t ⊂ Hmix for t ≥ 0 and Hmix ⊂ H t ⊂ Hmix for t ≤ 0 hold. See [58] t and [36] for more information on the spaces Hmix . 3 Except for the completion with respect to a chosen Sobolev norm, V is just the associated Sobolev space.

1484


4. Semidiscrete general sparse grid spaces We now consider truncation of the series expansion (17) with respect to the level parameter l but keep the part of the full series expansion with respect to the position parameter j . To this end, we introduce, besides the parameter L (after proper scaling with c) which indicates the truncation of the scale with respect to the one-particle space, an additional parameter T ∈ (−∞, 1] which regulates the truncation pattern for the interaction between particles. We define the generalized sparse grid space

VL,T :=

l∈L,T

where Wl = span{ψl,j , j ∈ ZdN }

Wl

(20)

with associated generalized hyperbolic cross with respect to the scale-parameter l : λmix (l) · λiso (l)−T ≤ (2L )1−T }. L,T := {l ∈ Nd·N 0

(21)

128

128

96

96

96

96

64

32

0 0

64

32

32

64 l1

96

128

0 0

l2

128

l2

128

l2

l2

The parameter T allows us to switch from the full grid case T = −∞ to the conventional sparse grid case T = 0, compare [12], [31], [42], and also allows to create with T ∈ (0, 1] subspaces of the hyperbolic cross/conventional sparse grid space. Obviously, the inclusions VL,T1 ⊂ VL,T2 for T1 ≤ T2 hold. Figure 3 displays the index sets for various choices of T for the case d = 1, N = 2 and L = 128.

64

32

32

64 l1

96

128

64

32

0 0

32

64 l1

96

128

0 0

32

64 l1

96

128

Figure 3. 128,T for T = 0.5, 0, −2, −10 (from left to right), d = 1, N = 2; the conventional sparse grid/hyperbolic cross corresponds to T = 0. For T = −∞ we get a completely black square.

We then can uniquely represent any function u from VK,T as u( x) =

l∈L,T ,j ∈Zd·N

ul,j ψl,j ( x ).

Such a projection into VK,T introduces an error. Here we have the following error estimate: t,r Lemma 1. Let s < r + t, t ≥ 0, u ∈ Hmix ((Rd )N ). Let u˜ L,T be the best approximation in VL,T with respect to the H s -norm and let uL,T be the interpolant of u in

1485


VL,T , i.e. uL,T =

l∈L,T

j ∈ZdN

ul,j ψl,j ( x ). Then, there holds

inf u − vH s = u − u˜ L,T H s ≤ u − uL,T H s ⎧ N−1 ⎨O((2L )s−r−t+(T t−s+r) N−T ) · u t,r for T ≥ Hmix ≤ ⎩O((2L )s−r−t ) · u t,r for T ≤ H

VL,T

mix

s−r t ,

(22)

s−r t .

For a proof, compare the arguments in [31], [42], [43], [30]. This type of estimate was already given for the case of a dyadically refined wavelet basis with d = 1 for the periodic case on a finite domain in [31], [42], [43]. It is a generalization of the energy-norm based sparse grid approach of [11], [12], [29] where the case s = 1, t = 2, r = 0 was considered using a hierarchical piecewise linear basis. 0,r Let us discuss some cases. For the standard Sobolev space Hmix (i.e. t = 0, r = 2) and the spaces VL,T with T ≥ −∞ the resulting order is dependent of T and dependent on the number of particles N. In particular the order even deteriorates t,0 with larger T . For the standard Sobolev spaces of bounded mixed derivatives Hmix s (i.e. t = 2, r = 0) and the spaces VL,T with T > 2 the resulting order is dependent of T and dependent on the number of particles N whereas for T ≤ 2s the resulting order is independent of T and N. If we restrict the class of functions for example 1,1 (i.e. t = 1, r = 1) and measure the error in the H 1 -norm (i.e. s = 1) the to Hmix approximation order is dependent on N for all T > 0 and independent on N and T for all T ≤ 0. Note that in all cases the constants in the O-notation depend on N and d.

5. Antisymmetric semidiscrete general sparse grid spaces Let us now come back to the Schrödinger equation (1). Note that in general an electronic wave function depends in addition to the positions x i of the electrons also on their associated spin coordinates σi ∈ − 21 , 21 . Thus electronic wave functions are N → R : ( x , σ ) → ( x , σ ) with spin coordinates defined as : (Rd )N × − 21 , 21 σ = (σ1 , . . . , σN ). Furthermore, physically relevant eigenfunctions obey the following two assumptions: First, elementary particles are indistinguishable from each other (fundamental principle of quantum mechanics). Second, no two electrons may occupy the same quantum state simultaneously (Pauli exclusion principle). Thus, we consider only wave functions which are antisymmetric with respect to an arbitrary simultaneous permutation P ∈ SN , of the electron positions and spin variables, i.e. which fulfil (P x , P σ ) = (−1)|P | (P x , P σ ). Here SN is the symmetric group. The permutation P is a mapping P : {1, . . . , N } → {1, . . . , N } which translates to a permutation of the corresponding numbering of

1486


electrons and thus to a permutation of indices, i.e. we have P (x 1 , . . . , x N )T := (x P (1) , . . . , x P (N ) )T and P (σ1 , . . . , σN )T := (σP (1) , . . . , σP (N) )T . In particular, the symmetric group is of size |SN | = N! and the expression (−1)|P | is equal to the determinant det P of the associated permutation matrix. N we define the associated spatial Now, to a given spin vector σ ∈ − 21 , 21 d component of the full wave function by σ : (R )N → R : x → ( x , σ ). Then, since there are 2N possible different spin distributions σ , the full Schrödinger equation, i.e. the eigenvalue problem H = E, decouples into 2N eigenvalue problems for the 2N associated spatial components σ . Here, the spatial part σ to a given σ obeys the condition σ (P x ) = (−1)|P | σ (P x )

for all P ∈ Sσ := {P ∈ SN : P σ = σ }.

(23)

In particular, the minimal eigenvalue of all eigenvalue problems for the spatial components is equal to the minimal eigenvalue of the full eigenvalue problem. Moreover, the eigenfunctions of the full system can be composed by the eigenfunctions of the eigenvalue problems for the spatial parts. Although there are 2N possible different spin distributions σ , the bilinear form (P ·)|H |(P ·) is invariant under all permutations P ∈ SN of the position coordinates x . Thus it is sufficient to consider the eigenvalue problems which are associated (N,S) (N,S) to the spin vectors σ (N,S) = (σ1 , . . . , σN ) where the first S electrons possess 1 spin − 2 and the remaining N − S electrons possess spin 21 , i.e. (N,S)

σj

=

⎧ ⎨− 1

for j ≤ S,

⎩

for j > S.

2

1 2

In particular, it is enough to solve only the N/2 eigenvalue problems which correspond to the spin vectors σ (N,S) with S ≤ N/2. For further details see [66]. Therefore, we consider in the following without loss of generality only spin distribu(N,S) (N,S) tions σ (N,S) = (σ1 , . . . , σN ). We set S(N,S) := Sσ (N,S) . Note that there holds |S(N,S) | = S!(N − S)!. Now we define spaces of antisymmetric functions and their semi-discrete sparse grid counterparts. The functions of the N-particle space V from (16) which obey (N,S) the anti-symmetry condition (23) for a given σ (N,S) form a linear subspace V A of V . We define the projection into this subspace, i.e. the antisymmetrization operator (N,S) A(N,S) : V → V A by A(N,S) u( x ) :=

1 (−1)|P | u(P x ). S!(N − S)! P ∈SN,S

(24)

1487


For any basis function ψl,j of our general N-particle space V we then have ) = A(N,S) A(N,S) ψl,j x

S

N ψlp ,j p (x 1 , . . . , x S ) ψlp ,j p (x S+1 , . . . x N )

p=1

p=S+1

S N ψlp ,j p (x 1 , . . . , x S ) A(N −S,N−S) ψlp ,j p ( x S+1 , . . . , x N ) = A(S,S) p=1

= =

S 1

S!

p=S+1

ψlp ,j p x 1 , . . . , x S

p=1

1 S!(N − S)!

P ∈SN,S

1 (N − S)!

(−1)|P | ψl,j (P x ) =

ψlp ,j p x S+1 , . . . , x N

N p=S+1

1 (−1)|P | ψP l,P j ( x ). S!(N − S)! P ∈SN,S

In other words, the classical product x ) := ψl,j (

N

ψlp ,j p (xp ) =

p=1

N

ψlp ,j p (x 1 , . . . , x N )

p=1

gets replaced by the product of two outer products S 1 ψlp ,j p (x 1 , . . . , x S ) S!

and

p=1

N 1 ψlp ,j p (x S+1 , . . . , x N ) (N − S)! p=S+1

that correspond to the two sets of coordinates and one-particle bases which are associated to the two spin values − 21 and 21 . The outer product involves just the so-called slater determinant [55], i.e. ψl ,j (x 1 ) . . . ψl ,j (x N ) N 1 1 1 1 .. .. .. ψlp ,j p (x 1 , . . . , x N ) = . . . . p=1 ψl ,j (x 1 ) . . . ψl ,j (x N ) N

N

N

N

Note here again that due to (11) both father wavelet functions and mother wavelet functions may be involved in the respective products. The sequence A(N,S) ψl,j l∈NdN ,j ∈ZdN only forms a generating system of the 0

and no basis since many functions A(N,S) ψl,j are antisymmetric subspace V A identical (up to the sign). But we can gain a basis for the antisymmetric subspace (N,S) VA if we restrict the sequence A(N,S) ψl,j l∈NdN ,j ∈ZdN properly. This can be (N,S)

0

done in many different ways. A possible orthonormal basis B (N,S) for V A given with help of (N,S) ( x) l,j

:= √

(N,S)

is

S N 1 · ψlp ,j p (x 1 , . . . , x S ) · ψlp ,j p (x S+1 , . . . , x N ) S!(N − S)! p=1 p=S+1

(25)

1488


as follows:

(N,S) d·N B (N,S) := : l ∈ Nd·N , (l 1 , j 1 ) < · · · 0,

4π 2 (3x+2π)2

for |k| ≤ for

2π 3

2π 3 ,

≤ |k| ≤

for |k| ≥

4π 3 ,

4π 3 .

This choice results in just a representation with respect to the Meyer wavelet series with ν ∞ , i.e. (9) with α = 2, compare also (7). The Fourier series expansion of each ˆ R,K,l (k) then introduces just the j -scale, while the k-scale of the Fourier space relates to the l-scale of the Meyer wavelets. All we now need is a good decomposition Y of HR,K into subdomains, a choice of smooth χˆl ’s and a proper truncation of the ˆ Fourier series expansion of each of the R,K,l ’s. This corresponds to a truncation of d·N with respect to both the l- and the j -scale. the Meyer wavelet expansion of in R Presently, however, it is not completely clear what choice of decomposition and what kind of truncation of the expansion within each subband l is most favourable with respect to both the resulting number M of degrees of freedom and the corresponding accuracy of approximation for varying number N of electrons. Anyway, with the choice K = 2L the set of indices in l, j -wavelet space which is associated to (29) reads A HY

(N,S)

R,2L

(N,S) := (l, j ) ∈ A :

S i=1

˜ i ) 2 ˜ 2 λ(l 1 + λ(lRi ) + N ≤ 22L , i=S+1 1 + R

2

2

where for l ∈ Nd0 we define λ˜ (l) :=

min

k∈supp(χˆ l )

{|k|2 }.

Note that this involves a kind of octavization due to the size of the support of the χˆ l . For example, we obtain for the Shannon wavelet λ˜ (l) = |(λ˜ ν 0 (l1 ), . . . , λ˜ ν 0 (ld ))|2 with 0 for l = 0, λ˜ ν 0 (l) = l−1 otherwise. cπ2

1494


7. Numerical experiments We now consider the assembly of the discrete system matrix which is associated (N,S) to a generalized antisymmetric sparse grid space VA with corresponding finite(N,S) (N,S) (N,S) A(N,S) A ⊂ and basis functions { : (l, j ) ∈ A } dimensional set (N,S) with { } from (25) in a Galerkin discretization of (1). l,j

l,j

To this end, we fix N > 0

and 0 ≤ S ≤ N and omit for reasons of simplicity the indices S and N in the following. (N,S) To each pair of indices (l, j ), (l , j ), each from A , and associated functions (N,S) (N,S) , we obtain one entry in the stiffness matrix, i.e. l,j l ,j (N,S) (N,S) (N,S)∗ (N,S) A(l,j ),(l ,j ) := |H | = ( x )H ( x ) d x. (33) l,j

l ,j

l,j

l ,j

Since we use L2 -orthogonal one-dimensional Meyer wavelets as basic building blocks in our construction, also the one-particle basis functions are L2 -orthogonal and we furthermore have L2 -orthogonality of the antisymmetric many-particle basis func(N,S) tions (x). We then can take advantage of the well-known Slater–Condon rules l,j

[18], [55], [60]. Consequently, quite a few entries of the system matrix are zero and the remaining non-zero entries can be put together from the values of certain d- and 2d-dimensional integrals. These integrals can be written in terms of the Fourier transformation of the Meyer wavelets. In case of the kinetic energy operator we obtain for l α , l β ∈ Nd0 and j α , j β ∈ Zd 1 1 ψl α ,j α | − |ψl β ,j β = ∇ψl∗α ,j α (x) · ∇ψl β ,j β (x)dx 2 2 Rd d d

1 kμ2 ψˆ l∗α,(μ) ,j α,(μ) (kμ )ψˆ l β,(μ) ,j β,(μ) (kμ ) dkμ δl α,(ν) ,l β,(ν) δj α,(ν) ,j β,(ν) = 2 R ν=μ

μ=1

and for the integrals related to the d-dimensional Coulomb operator v(x) = 1/|x|2 we can write ψl∗α ,j α (x)v(x)ψl β ,j β (x)dx ψl α ,j α |v|ψl β ,j β = d R = v(k)( ˆ ψˆ l α ,j α ∗ ψˆ l β ,j β )(k) dk. Rd

For l α , l β , l α , l β ∈ Nd0 and j α , j β , j α , j β ∈ Zd we obtain the integrals related to the electron-electron operator v(x − y) = 1/|x − y|2 in the form ψl∗α ,j α (x)ψl∗ ,j (y)v(x − y)ψl β ,j β (x)ψl β ,j β (y)dx dy α α d d R R d v(k)( ˆ ψˆ l α ,j α ∗ ψˆ l β ,j β )(k)(ψˆ l α ,j α ∗ ψˆ l β ,j β )(k) dk. = (2π) 2 Rd


1495

d Here, f ∗ g denotes the Fourier convolution, namely (2π )− 2 Rd f (x − y)g(y) dy. Note that, in the case of the Meyer wavelet tensor-product basis, the d-dimensional Fourier convolution can be written in terms of the one-dimensional Fourier convolution d

ˆ ˆ (ψˆ l α,(μ) ,j α,(μ) ∗ ψˆ l β,(μ) ,j β,(μ) )(kμ ). (ψl α ,j α ∗ ψl β ,j β )(k) =

μ=1

Thus the d-dimensional and 2d-dimensional integrals in real space which are associated to the Coulomb operator and the electron-electron operator can be written in form of d-dimensional integrals of terms involving one-dimensional convolution integrals. For the solution of the resulting discrete eigenvalue problem we invoke a parallelized conventional Lanczos method taken from the software package SLEPc [35] which is based on the parallel software package PETSc [6]. Note that here also other solution approaches are possible with improved complexities, like multigridtype methods [13], [15], [44], [47] which however still need to be carried over to the setting of our generalized antisymmetric sparse grids. Note that an estimate for the accuracy of an eigenfunction relates to an analogous estimate for the eigenvalue by means of the relation |E − E app | ≤ 4 · − app 2L2 where E and denote the exact minimal eigenvalue and associated eigenfunction of H , respectively, and E app and app denote finite-dimensional Galerkin approximations in arbitrary subspaces, see also [66]. Then, with Lemma 1, we would obtain for the case d = 3 with s = 0 and, for example, r = 1, t = 1 and S = 0 the estimate (N,0) 2 N−1 ) E − E A(N,0) ≤ 4 · − A(N,0) 2 2 ≤ O((2L )2·(−2+(T +1) N−T 1,1 ) · A L,T L,T L H

mix

and we see that the eigenvalues are in general much better approximated than the eigenfunctions. For example, for T = 0, this would result in a (squared) rate of the order −4 + 2(N − 1)/N which is about −4 for small numbers of N but gets −2 for N → ∞. Let us now describe our heuristic approach for a finite-dimensional subspace (N,S) choice in wavelet space which hopefully gives us efficient a-priori patterns A (N,S) and associated subspaces VA . We use a model function of the Hylleraas-type [16], [41], [57]4 N N

e−αp |xp |2 (34) h( x) = e−βp,q |xp −x q |2 p=1

q>p

which reflects the decay properties, the nucleus cusp and the electron-electron cusps of an atom in real space with nucleus fixed in the origin as guidance to a-priori derive a pattern of active wavelet indices in space and scale similar to the simple 4 Note that we omitted here any prefactors for reasons of simplicity.

1496


one-dimensional example of Figure 2. The localization peak of a Meyer wavelet ψl,j in real space (e.g. after proper scaling with some c analogously to (10)) is given by j for l = 0, θ (l, j ) = ι(l, j )2−l where ι(l, j ) = 1 + 2j otherwise, which leads in the multidimensional case to θl,j = (θ(l1 , j1 ), . . . , θ(ld , jd )) ∈ Rd θl,j = (θ(l 1 , j 1 ), . . . , θ(l N , j N )) ∈ (Rd )N We now are in the position to describe different discretizations with respect to both the l-scale and the j -scale. We focus with respect to the j -scale on three cases: First, we restrict the whole real space to a finite domain and take the associated wavelets on all incorporated levels into account. Note that in this case the number of wavelets grows from level to level by a factor of 2. Second, we use on each level the same prescribed fixed number of wavelets. And third we let the number of wavelets decay from level to level by a certain factor which results in a multivariate analog to the triangular subspace of Figure 2 (right). With respect to the l-scale we rely in all cases on the regular sparse grid with T = 0. These three different discretization approaches are illustrated in the Figures 5–8 for d = 1, N = 1 and d = 1, N = 2, respectively. 0

1

1

2

2

2

3

3

3

l

l

0

1

l

0

4

4

4

5

5

5

6

6

7 −300

−200

−100

0 j

100

200

300

7 −4

6 −2

0 j

2

4

7 −4

−2

0 j

2

4

A A A Figure 5. From left to right: Index sets full (L,J,R) , rec (L,J,R) and tri (L,J,R) with d = 1, N = 1, L = 8, J = 4, R = 1 and α1 = 1. (N,S)

(N,S)

(N,S)

1

2

2

2

3

3

3 l

0

1

l

0

1

l

0

4

4

4

5

5

5

6

6

7 −4

−2

0 x

2

4

7 −4

6 −2

0 x

2

4

7 −4

−2

0 x

2

4

Figure 6. From left to right: Localization peaks of basis functions in real space corresponding (N,S) (N,S) (N,S) to index sets A , A and A with d = 1, N = 1, L = 8, J = 4, rec (L,J,R) full (L,J,R) tri (L,J,R) R = 1 and α1 = 1.

1497

4

4

3

3

2

2

1

1 x2

x2


0

0

−1

−1

−2

−2

−3 −4 −4

−3 −2

0 x1

2

−4 −4

4

−2

0 x1

2

4

4

4

3

3

2

2

1

1 x2

x2

Figure 7. From left to right: Localization peaks of basis functions in real space corresponding to (N,S) A(N,S) A(N,S) the index sets A , rec (L,J,R) and tri (L,J,R) with d = 2, N = 1, L = 8, J = 4, full (L,J,R) R = 1 and α1 = 1.

0

0

−1

−1

−2

−2

−3 −4 −4

−3 −2

0 x1

2

−4 −4

4

−2

0 x1

2

4

Figure 8. From left to right: Localization peaks of basis functions in real space corresponding to (N,S) A(N,S) A(N,S) the index sets A , rec (L,J,R) and tri (L,J,R) with d = 1, N = 2, L = 8, J = 4, full (L,J,R) R = 1, S = 1 and α1 = α2 = β1,2 = 21 .

To this end, we define with the parameter J ∈ N+ the pattern for the finite domain with full wavelet resolution, i.e. the full space (with respect to j -scale after a finite domain is fixed), as (N,S) (N,S) A := (l, j ) ∈ A : h(θ(l, j )) > e−J full (L,J,R) HY R,2L

= (l, j ) ∈ +

A(N,S) HY L R,2

:

N p=1

!

αp |θ(lp , j p )|2

! β |θ(l , j ) − θ(l , j )| < J p,q p q 2 p q q>p

N

with prescribed αp , βp,q . Note here the equivalence of the sum to ln(h(θ(l, j ))). To describe the other two cases we set with a general function which still has to be fixed (N,S) (N,S) := (l, j ) ∈ A : A (L,J,R) full (L,J,R) ! −1 N N θ l p , j p − θ l q , j q 2 < J . p=1 αp lp , j p 2 + q>p βp,q θ

1498


Note that θ −1 denotes the inverse mapping to θ . It holds ι˜−1 (l, ι(l, j ) − ι(l , j )2l−l ) −1 θ θ (l, j ) − θ l , j = −1 ι˜ (l , ι(l, j )2l −l − ι(l , j ))

for l ≥ l , for l ≥ l

where ι˜(l, j ) = (l, ι(l, j )). We now define the rectangular index set A rec (L,J,R) via d d the following choice of : For l ∈ N0 and j ∈ Z we set (N,S)

rec (l, j ) := (rec (l1 , j1 ), . . . , rec (ld , jd )) and for l ∈ N0 and j ∈ Z we set rec (l, j ) :=

|j |, | 21 + j |

for l = 0, otherwise.

Finally we define the triangle space A with help of tri (L,J,R) (N,S)

tri (l, j ) := (tri (l1 , j1 ), . . . , tri (ld , jd )) where for l ∈ N0 and j ∈ Z we set ⎧ |j | ⎪ l ⎪ ⎪ ⎨ 1− 1Lmax +1 | 2 +j | tri (l, j ) := l ⎪ 1− ⎪ ⎪ Lmax +1 ⎩ ∞

for l = 0, for 0 < l ≤ Lmax , otherwise

with Lmax as the maximum level for the respective triangle. Let us now discuss the results of our first, very preliminary numerical experiments with these new sparse grid methods for Schrödinger’s equation. To this end, we restrict ourselves for complexity reasons to the case of one-dimensional particles only. The general three-dimensional case will be the subject of a forthcoming paper. We use in the following in (1) the potential V =−

N N nuc

Zq v(xp − R q ) +

p=1 q=1

with

D − |r|2 v(r) = 0

N N

v(xp − x q )

(35)

p=1 q>p

for |r|2 ≤ D, otherwise

which is truncated at radius D and shifted by D. Note that lim|r|2 →∞ v(r) = 0. Up to truncation and the shift with D, |r|2 is just the one-dimensional analogue to the

1499


Coulomb potential. The Fourier transform reads ⎧√ ⎨ √ 2 1 2 (1 − cos(D|k|2 )) π |k|2 v(k) ˆ = ⎩ √D 2 2π

for |k|2 = 0, otherwise.

Note that vˆ is continuous. We study for varying numbers N of particles the behavior of the discrete energy E, i.e. the smallest eigenvalue of the associated system matrix A, as L and J increase. (N,S) (N,S) , A Here, we use the generalized antisymmetric sparse grids A rec (L,J,R) full (L,J,R) and A and focus on the two cases S = 0 or S = N/2. We employ the tri (L,J,R) (N,S)

Meyer wavelets with (9) where ν ∞ , α = 2, and the Shannon wavelet with ν 0 from (8). Tables 1 and 2 give the obtained results. Here, M denotes the number of degrees of freedom and #A denotes the number of the non-zero matrix entries. Furthermore, E denotes the difference of the obtained values of E and ε denotes the quotient of the values of E for two successive rows in the table. Thus, ε indicates the convergence rate of the discretization error. Table 1. d = 1, N = 1, c = 1, R = 1, α1 = 1, Lmax = L, D = 8. (N,S) Afull (L,J,R)

J 2 4 8 16 32 64 16 16 16 16 16

L 1 1 1 1 1 1 1 2 3 4 5

M 5 9 17 33 65 129 33 65 129 257 513

ν∞ #A

25 81 289 1089 4225 16641 1089 4225 16641 66049 263169

E −7.187310 −7.189322 −7.189334 −7.189335 −7.189335 −7.189335 −7.189335 −7.191345 −7.191376

(N,S) rec (L,J,R)

A J 16 16 16 16 16 16

L 1 2 3 4 5 6

M 33 65 97 129 161 193

#A 1089 4225 9409 16641 25921 37249

E −7.189335 −7.191345 −7.191376 −7.191377 −7.191377 −7.191377

(N,S)

M 33 55 73 91 107 125

2.01e−03 1.14e−05 1.08e−06 4.60e−07 1.36e−09

175.1 10.6 2.3 336.4

2.00e−03 3.19e−05

62.9

E −7.186261 −7.188615 −7.188674 −7.188683 −7.188684 −7.188685 −7.188683 −7.190920 −7.190958 −7.190959 −7.190959

#A 1089 3025 5329 8281 11449 15625

E −7.189335 −7.191314 −7.191357 −7.191366 −7.191366 −7.191371

E

ε

2.35e−03 5.92e−05 9.25e−06 1.29e−06 1.73e−07

39.7 6.4 7.1 7.4

2.23e−03 3.80e−05 1.00e−06 3.04e−08

58.7 37.6 33.1

ν0

E

ε

2.00e−03 3.19e−05 8.37e−07 2.51e−08 7.53e−10

62.9 38.0 33.3 33.4

E −7.188683 −7.190920 −7.190956 −7.190957 −7.190957 −7.190957

ν∞

tri (L,J,R)

L 1 2 3 4 5 6

ν0 ε

ν∞

A J 16 16 16 16 16 16

E

E

2.23e−03 3.61e−05 9.52e−07 2.85e−08 8.84e−10

ε

61.8 37.9 33.3 32.2

ν0

E

ε

1.97e−03 4.25e−05 9.28e−06 2.13e−07 5.07e−06

46.5 4.5 43.4 0.042

E −7.189335 −7.190747 −7.190825 −7.190865 −7.190866 −7.190900

E

ε

1.41e−03 7.80e−05 4.04e−05 5.42e−07 3.39e−05

18.0 1.9 74.5 0.016

1500


In Table 1, with just one particle, i.e. N = 1, we see that the minimal eigenvalues for the Shannon wavelet are slightly, i.e. by 10−3 − 10−2 , worse than the minimal eigenvalues for the Meyer wavelet with ν ∞ . Furthermore, from the first part of the table where we fix L = 1 and vary J and alternatively fix J = 16 and vary L it gets clear that it necessary to increase both J and L to obtain convergence. While just an increase of J with fixed L = 1 does not improve the result at all (with D fixed), the increase of L for a fixed J at least gives a convergence to the solution on a bounded domain whose size is associated to the respective value of D and J . In the second (N,S) (N,S) and A for the part of the table we compare the behavior for A rec (L,J,R) full (L,J,R)

wavelets with ν ∞ and ν 0 . While we see relatively stable monotone rates of around 33 (N,S) (N,S) and better in case of A , the convergence behavior for A is more rec (L,J,R) full (L,J,R) erratic. Nevertheless, when we compare the achieved results for the same amount of matrix entries #A we see not much difference. For example, with ν ∞ , we get for J = 16, L = 6 with 125 degrees of freedom and 15625 matrix entries a value of (N,S) (N,S) whereas we get for A with J = 16, L = 4 −7.191371 for A rec (L,J,R) full (L,J,R) with about the same degrees of freedom and matrix entries nearly the same value −7.191377.

Let us now consider the results for N > 1 given in Table 2. Here we restricted (N,S) due to complexity reasons. We see that the ourself to the sparse grid A tri (L,J,R) computed minimal eigenvalues in the case S = N2 are higher than that in the case S = 0, as to be expected. Furthermore, our results suggest convergence for rising L. 3 If we compare the cases R = 1 and R = 2 2 for N = 2, we see that both the number of degrees of freedom and the minimal eigenvalues are for R = 1 approximately the 3 same as for R = 2 2 on the next coarser level. An analogous observation holds in the case N = 4. Note furthermore that the sparse grid effect acts only on the fully antisymmetric subspaces of the total space. This is the reason for the quite large number of degrees of freedom for the case N = 4, S = 2. Note finally that our present simple numerical quadrature procedure is relatively expensive. To achieve results for higher numbers of particles with sufficiently large L and J , the numerical integration scheme has to be improved. Moreover, to deal in the future with the case of three-dimensional particles using the classical potential (2) and the Meyer wavelets with ν ∞ , an efficient and accurate numerical quadrature still has to be derived.5

5 Such a numerical quadrature scheme must be able to cope with oscillatory functions and also must resolve the singularity in the Coulomb operator.


1501

Table 2. d = 1, c = 1, D = 8, A , ν0. tri (L,J,R) (N,S)

J 8 8 8 8

L 4 5 6 7

J 8 8 8 8

L 3 4 5 6

J 8 8 8 8

L 4 5 6 7

Z = 2, N M 1037 1401 1623 1943

= 2 , S = 1, R #A 1029529 1788305 2324081 3240369

= 1, α1 = α2 = β1,2 = 21 E E −28.818529 −28.819933 1.40e−03 −28.819954 2.07e−05 −28.819963 8.81e−06

ε

67.55 2.35

3

Z = 2, N = 2, S = 1, R = 2 2 , α1 = α2 = β1,2 = 21 M #A E E 1067 1092649 −28.818529 1425 1856129 −28.819933 1.40e−03 1637 2369721 −28.819954 2.07e−05 1957 3829849 −28.819963 8.81e−06 Z = 2, N = 2, S = 0, R = 1, α1 = α2 = β1,2 = 21 M #A E E 383 138589 −27.134075 501 234965 −27.134725 6.49e−04 614 341696 −27.134725 3.98e−07 731 470569 −27.134725 3.77e−07

ε

67.55 2.35 ε

1630.99 1.05

3

J 8 8 8 8

L 3 4 5 6

J 8 8

L 4 5

J 8

L 1

J 8 8 8

L 4 5 6

J 8 8 8

L 1 2 3

Z = 2, N = 2, S = 0, R = 2 2 , α1 = α2 = β1,2 = 21 M #A E E 475 215125 −27.134075 622 353684 −27.134725 6.49e−04 714 455420 −27.134725 3.98e−07 852 624244 −27.134725 3.77e−07 Z = 4, N = 4, S = 2, R = 1, αp = βp,q = 41 M #A E E 24514 17003256 −106.755154 39104 32716440 −106.756364 1.20e−03 Z = 4, N = 4, S = 2, R = 8, αp = βp,q = 41 M #A E E 31592 22864800 −106.755154 Z = 4, N = 4, S = 0, R = 1, αp = βp,q = 41 M #A E E 1903 313963 −102.659381 2842 647688 −102.659503 1.22e−04 4039 1063101 −102.660489 9.86e−04 Z = 4, N = 4, S = 0, R = 8, αp = βp,q = 41 M #A E E 3527 761851 −102.659381 6029 1558219 −102.659503 1.22e−04 8098 2343162 −102.660489 9.85e−04

ε

1631.10 1.05 ε

ε

ε

0.12 ε

0.12

8. Concluding remarks In this article we proposed to use Meyer’s wavelets in a sparse grid approach for a direct discretization of the electronic Schrödinger equation. The sparse grid constructions promises to break the curse of dimensionality to some extent and may allow a numerical treatment of the Schrödinger equation without resorting to any model approximation. We discussed the Meyer wavelet family and their properties and built on them an anisotropic multiresolution analysis for general particle spaces. Furthermore

1502


we studied a semidiscretization with respect to the level and introduced generalized semidiscrete sparse grid spaces. We then restricted these spaces to the case of antisymmetric functions with additional spin. Using regularity and decay properties of the eigenfunctions of the Schrödinger operator we discussed rescaled semidiscrete sparse grid spaces due to Yserentant. They allow to get rid of the terms that involve 1/2,1 1,1 the Hmix - and Hmix -norm of the eigenfunction which may grow exponentially with the number of electrons present in the system. Thus a direct estimation of the approximation error can be achieved that only involves the L2 -norm of the eigenfunction. We also showed that a Fourier series approximation of a splitting of the eigenfunctions living on a scaled hyperbolic cross in Fourier space essentially just results in Meyer wavelets. Therefore, we directly tried to discretize Schrödinger’s equation in properly chosen wavelet subspaces. We only presented preliminary numerical results with one-dimensional particles and a shifted and truncated potential. For the Meyer wavelets with ν ∞ and for the classical, not truncated Coulomb potential, substantially improved quadrature routines have to be developed in the future to achieved reasonable run times for the set up of the stiffness matrix. Furthermore, the interplay and the optimal choice of the coarsest scale, i.e. of c, the scaling parameter R, the domain truncation parameter J , the scale truncation parameter L and the parameters Lmax , αp , βp,q is not clear at all and needs further investigation. Finally more experiments are necessary with other types of sparse grid subspaces beyond the ones derived from the Hylleraas-type function (34) to complete our search for an accurate and cost effective approximation scheme for higher numbers N of electrons. Probably not the best strategy for subspace selection was yet used and substantially improved schemes can be found in the future. This may be done along the lines of best M-term approximation which, from a theoretical point of view, would however involve a new, not yet existing Besov regularity theory for high-dimensional spaces in an anisotropic setting. Or, from a practical point of view, this would involve new adaptive sparse grid schemes using tensor product Meyer wavelets which need proper error estimators and refinement strategies for both the boundary truncation error and, balanced with it, the scale truncation error. The sparse grid approach is based on a tensor product construction which allows to treat the nucleus–electron cusps properly which are aligned to the particlecoordinate axes of the system but which does not fit to the “diagonal” directions of the electron–electron cusps. Here, proper a-priori refinement or general adaptivity must be used which however involves for d = 3 at least the quite costly resolution of three-dimensional manifolds in six-dimensional space which limits the approach. To this end, new features have to brought into the approximation like for example wavelets which allow additionally for multivariate rotations in the spirit of curvelets [14]. Also an approach in the spirit of wave-ray multigrid methods [9] may be envisioned. Alternatively an embedding in still higher-dimensional formulations which allows to express the electron-electron pairs as new coordinate directions might be explored. This, however, is future work.


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[41] Klopper, W., r12-dependent wavefunctions. In The Encyclopedia of Computational Chemistry (ed. by P. von Ragué Schleyer, N. L. Allinger, T. Clark, J. Gasteiger, P. A. Kollman, H. F. Schaefer, and P. R. Schreiner), John Wiley and Sons, Chichester 1998, 2351–2375. [42] Knapek, S., Approximation und Kompression mit Tensorprodukt-Multiskalenräumen. Dissertation, Universität Bonn, 2000. [43] Knapek, S., Hyperbolic cross approximation of integral operators with smooth kernel. Tech. Report 665, SFB 256, Universität Bonn, 2000. [44] Knyazev, A., and Neymeyr, K., Efficient solution of symmetric eigenvalue problem using multigrid preconditioners in the locally optimal block conjugate gradient method. Electron. Trans. Numer. Anal. 15 (2003), 38–55. [45] Lemarié, P., and Meyer, Y., Ondelettes et bases hilbertiennes. Rev. Mat. Iberoamericana 2 (1–2) (1986), 1–18. [46] Levine, I., Quantum chemistry. 5th ed., Prentice-Hall, 2000. [47] Livne, O., and Brandt, A., O(N log N) multilevel calculation of N eigenfunctions. In Multiscale Computational Methods in Chemistry and Physics (ed. by A. Brandt, J. Bernholc, and K. Binder), NATO Science Series III: Computer and Systems Sciences 177, IOS Press, 2001, 112–136. [48] Low, F., Complete sets of wave packets. In A Passion for Physics—Essays in Honor of Geoffrey Chew, World Scientific, Singapore 1985, 17–22. [49] Malvar, H., Lapped transform for efficient transform/subband coding. IEEE Trans. Acoust. Speech Signal Process. 38 (1990), 969–978. [50] Mazziotti, D., Variational two-electron reduced density matrix theory for many-electron atoms and molecules: Implementation of the spin- and symmetry-adapted T-2 condition through first-order semidefinite programming. Phys. Rev. A 72 (3) (2005), 032510. [51] Messiah, A., Quantum mechanics. Vol. 1, 2, North-Holland, Amsterdam 1961/62. [52] Meyer, Y., Wavelets and operators. Cambridge Stud. Adv. Math. 37, Cambridge University Press, Cambridge 1992. [53] Meyer, Y., Wavelets, vibrations and scalings. CRM Monograph Ser.9, Amer. Math. Soc., Providence, RI, 1998. [54] Pan, G., Wavelets in electromagnetics and device modeling. Wiley–IEEE Press, 2003. [55] Parr, R., andYang, W., Density functional theory of atoms and molecules. Oxford University Press, New York 1989. [56] Persson, A., Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator. Math. Scand. 8, (1960), 143–153. [57] Rodriguez, K., and Gasaneo, G., Accurate Hylleraas-like functions for the He atom with correct cusp conditions. J. Phys. B: At. Mol. Opt. Phys. 38 (2005), L259–L267. [58] Schmeisser, H., and Triebel, H., Fourier analysis and functions spaces. John Wiley, Chichester 1987. [59] Simon, B., Schrödinger operators in the twentieth century. J. Math. Phys. 41 (2000), 3523–3555. [60] Slater, J., The theory of complex spectra. Phys. Rev. 34 (10) (1929), 1293–1322. [61] Smolyak, S., Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl. Akad. Nauk SSSR 148 (1963), 1042–1045; English. transl. Soviet Math. Dokl. 4 (1963), 240–243.

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[62] Walnut, D., An introduction to wavelet analysis. Appl. Numer. Harmon. Anal., Birkhäuser, Boston 2002. [63] Walter, G., and Zhang, J., Orthonormal wavelets with simple closed-form expressions. IEEE Trans. Signal Process. 46 (8) (1998), 2248–2251. [64] Wilson, K., Renormalization group and critical phenomena. II. Phase-space cell analysis of critical behavior. Phys. Rev. B 4 (1971), 3184–3205. [65] Yamada, M., and Ohkitani, K., An identification of energy cascade in turbulence by orthonormal wavelet analysis. Prog. Theor. Phys. 836 (4) (1991), 799–815. [66] Yserentant, H., On the electronic Schrödinger equation. Report 191, SFB 382, Universität Tübingen, 2003. [67] Yserentant, H., On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98 (2004), 731–759. [68] Yserentant, H., Sparse grid spaces for the numerical solution of the electronic Schrödinger equation. Numer. Math. 101 (2005), 381–389. [69] Yserentant, H., The hyperbolic cross space approximation of electronic wavefunctions. Numer. Math. submitted. [70] Yserentant, H., The hyperbolic cross space approximation of electronic wavefunctions. Talk at IHP-EU Network Workshop/Winter School Breaking Complexity: Nonlinear/Adaptive Approximation in High Dimensions, Physikzentrum Bad Honnef, Germany, 15th December 2005. Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany E-mail: [email protected] Institut für Numerische Simulation, Universität Bonn, Wegelerstr. 6, 53115 Bonn, Germany E-mail: [email protected]

Mathematical and numerical analysis for molecular simulation: accomplishments and challenges Claude Le Bris

Abstract. Molecular simulation is explored from the mathematical viewpoint. The field comprises computational chemistry and molecular dynamics. A variety of mathematical and numerical questions raised is reviewed. Placing the models and the techniques employed for simulation on a firm mathematical ground is a difficult task, which has begun decades ago. The time is right for assessing the field, and the issues and challenges ahead. Mathematics Subject Classification (2000). 35Bxx, 35Jxx, 35Pxx, 49Kxx, 81Q05, 81Q10, 82Bxx. Keywords. Computational chemistry, molecular simulation, molecular dynamics, Schrödinger equations, Hartree–Fock theory, Thomas–Fermi theory, Density Functional Theory, nonlinear eigenvalue problems, spectral theory, spectral methods, elliptic partial differential equations, optimization, reduced basis methods, sparse grids, Hamiltonian dynamics, symplectic methods, geometric integration, stochastic differential equations, Markov chains, Monte-Carlo techniques.

1. Introduction Molecular simulation is an increasingly important field of scientific computing. It comprises computational chemistry, focused on the calculations of electronic structures and the related properties, and molecular dynamics, devoted to the simulation of molecular evolutions, evaluations of ensemble averages and thermodynamic quantities. Examples of reference treatises are [68], [70] and [1], [39], respectively. We also refer to [14, Chapter 1] for a mathematically-oriented introductory text. 1.1. Ubiquity of molecular simulation. The field has several intimate connections with many other fields. Indeed, molecular simulation is above all important because many macroscopic properties of matter originate from phenomena at the microscopic scale. Instances are: electrical conductivities, colors, chemical reactivities, mechanical behaviour, aging. Accurate calculations on representative microscopic systems allow for the evaluation of such properties. Additionally, even the macroscopic phenomena that proceed from bulk effects, and which thus necessitate the consideration of large size microscopic systems, may now be studied by advanced techniques in molecular simulation. Recent record calculations simulate the dynamics of billions Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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of atoms over a microsecond. Molecular biology, chemistry and physics are thus inseparable today from molecular simulation. An easy observation sheds some light on this. Roughly one publication out of ten in chemistry journals presents some numerical simulations performed on theoretical models. This is an impressive ratio for a field so much experimentally oriented. Computations are first seen as complements to experiments. For instance, all the information about the electronic properties is contained in the wave function; the latter cannot be measured but it can be computed. Computations are also seen as an alternative to experiment. It is possible to simulate molecular systems that have not been synthesized yet, or phenomena inaccessible to experiments (huge temperature or pressure, time scales smaller than the femtosecond, evolutions on decades or more). Additionally, computations can serve for the laser control of molecular systems ([6]), and other emerging fields of high energy physics. Other, apparently distant, fields also make an extensive use of molecular simulation. Rheology of complex fluids and more generally materials science were once focused at the macroscopic scale and based on purely macroscopic descriptions. They used to be far from molecular concerns. However, the accuracy needed in the quantitative evaluation of many properties (think e.g. of constitutive laws or slip boundary conditions) requires models to be more and more precise, involving the finest possible scales in the simulation. This eventually includes the molecular scale. The last application field that we shall mention, besides the fields using the macroscopic impact of molecular simulation, regards the emerging field of nanotechnology. Nanosystems are indeed accessible today to a direct molecular simulation. Overall, major technological challenges for the years to come may, or more properly stated must, be addressed by molecular simulation techniques. Examples are the detailed simulation of protein folding, and the description of the long time radiation damage of materials in nuclear power plants. To appreciate this ubiquity of molecular simulation, it is sufficient to consider the enormous proportion of computational time devoted to molecular simulation in the largest centers of computational resources worldwide. 1.2. Relation to mathematics. On the other hand, the interface of molecular simulation with mathematics is not yet comparable to the practical importance of the field. Molecular simulation, and more precisely computational quantum chemistry, were born in the 1950s for molecular systems consisting of a few electrons. Contemporary methods and techniques now allow for the simulation of molecules of hundreds of electrons, modelled by very precise quantum models, up to samples of billions of particles modelled by molecular dynamics. This is an enormous success. The calculations are often surprisingly accurate, but also sometimes desperately inaccurate. Experts in chemistry have constantly improved the models and the methods. They have turned the field into an almighty tool. However, in many respects, molecular simulation is still an art. It relies upon a delicate mix of physical intuition, prag-

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matic cleverness, and practical know-how. Mathematics has already provided with significant contributions to the theoretical understanding. Also, its companion fields, numerical analysis and scientific computing, have definitely improved the efficiency of the techniques. Yet, they all need to irrigate more molecular simulation. To state it otherwise, there is an enormous gap between the sophistication of the models and the success of the numerical approaches used in practice and, on the other hand, the state of the art of their rigorous understanding. We are witnessing an evolution that is due to two different reasons. First, the mathematical knowledge on the models is rather satisfactory. Efforts were initiated as early as the 1970s by pioneers such as E. H. Lieb, B. Simon, W. Thirring, Ch. Feffermann, focusing on fundamental theoretical issues. Questions were addressed about the well-posedness of the models, and the relation between the various models, in various asymptotic regimes. Researchers such as R. Benguria, J. P. Solovej, V. Bach, G. Friesecke, to only name a few, continued the effort over the years. Those were later joined by contributors following the impulsion given by P-L. Lions: M. J. Esteban, I. Catto, E. Séré, X. Blanc, M. Lewin, and the author. A number of researchers, experts in analysis, spectral theory, partial differential equations, evolution equations, now become involved in the field. The enclosed bibliography cites several of them. Second, and as a natural follow-up to mathematical analysis, numerical analysis has indeed come into the picture. The numerical analysis of computational chemistry methods was a completely unexplored subject until the mid 1990s. Boosted by the state of the mathematical analysis, it is now a quickly developing topic. The work in this field was pioneered by E. Cancès. Researchers such as Y. Maday, M. Griebel, W. Hackbush, Ch. Lubich, W. E, well known for their contributions in various other fields of the engineering sciences, now get involved, along with their collaborators (G. Turinici, …) in electronic structure calculations or in molecular dynamics. It is therefore a good time for assessing the field, and the issues and challenges ahead. Doing so might help to boost the research in the area. The present contribution rapidly reviews some commonly used models and their mathematical nature, indicating the progress achieved over the last decades in their mathematical understanding. Questions of numerical analysis are also addressed. Important unsolved issues are emphasized. Owing to the evident space limitation, this review is not meant to be exhaustive: see [54], [55] for more comprehensive reviews, and [56] for a recent collection of various contributions. This is rather an invitation for mathematicians to get involved in the endeavour of placing the field on a firm mathematical ground.

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2. Mathematical overview of the models of computational chemistry 2.1. The Schrödinger equation. For most applications of molecular simulation, the matter is described by an assembly of nuclei, which are point particles treated classically, equipped with electrons, which are light particles modelled by quantum mechanics. For systems of limited size, called molecular systems, there are M nuclei, of charge zk , located at xk , and N electrons of unit charge. The finest models are called ab initio models since they only involve universal constants and no experimentally determined parameters. Assuming the molecular system nonrelativistic, placing it at zero temperature, and, for clarity of exposition, omitting the spin variable, the state of the electrons is modelled by the N-body Hamiltonian Hex 1 ,...,x M = −

N 1

2

i=1

xi −

M N i=1 k=1

zk + |xi − x k |

1≤i<j ≤N

1 , |xi − xj |

(1)

where the terms respectively model the kinetic energy, the attraction between nuclei and electrons, the repulsion between electrons. Notice that the positions xk of the nuclei are parameters of this operator. The electronic ground-state is by definition the minimizer of the energy: zk zl W (x1 , x2 , . . . , xM ) = inf ψ, Hex 1 ,...,x M ψ, ψ ∈ WN + . (2) |xk − xl | 1≤k c/b. (1) Kin selection theory has been tested in numerous experimental studies. Many cooperative acts among animals occur between close kin (Frank 1998, Hamilton 1998). The exact relationship between kin selection and other mechanisms such as group selection and spatial reciprocity, however, remains unclear. A recent study even suggests that much of cooperation in social insects is due to group selection rather than kin selection (Wilson & Hölldobler 2005). Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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2. Direct reciprocity In 1971, Robert Trivers published a landmark paper entitled ‘The evolution of reciprocal altruism’ (Trivers 1971). Trivers analyzed the question of how natural selection could lead to cooperation between unrelated individuals. He discusses three biological examples: cleaning symbiosis in fish, warning calls in birds and human interactions. Trivers cites Luce & Raiffa (1957) and Rapoport & Chammah (1965) for the Prisoner’s Dilemma, which is a game where two players have the option to cooperate or to defect. If both cooperate they receive the ‘reward’, R. If both defect they receive the ‘punishment’, P . If one cooperates and the other defects, then the cooperator receives the ‘sucker’s payoff’, S, while the defector receives the ‘temptation’, T . The PD is defined by the ranking T > R > P > S. Would you cooperate or defect? Assuming the other person will cooperate it is better to defect, because T > R. Assuming the other person will defect it is also better to defect, because P > S. Hence, no matter what the other person will do it is best to defect. If both players analyze the game in this ‘rational’ way then they will end up defecting. The dilemma is that they both could have received a higher payoff if they had chosen to cooperate. But cooperation is ‘irrational’. We can also imagine a population of cooperators and defectors and assume that the payoff for each player is determined by many random interactions with others. Let x denote the frequency of cooperators and 1−x the frequency of defectors. The expected payoff for a cooperator is fC = Rx + S(1 − x). The expected payoff for a defector is fD = T x + P (1 − x). Therefore, for any x, defectors have a higher payoff than cooperators. In evolutionary game theory, payoff is interpreted as fitness. Successful strategies reproduce faster and outcompete less successful ones. Reproduction can be cultural or genetic. In the non- repeated PD, in a well mixed population, defectors will outcompete cooperators. Natural selection favors defectors. Cooperation becomes an option if the game is repeated. Suppose there are m rounds. Let us compare two strategies, ‘always defect’ (ALLD), and GRIM, which cooperates on the first move, then cooperates as long as the opponent cooperates, but permanently switches to defection if the opponent defects once. The expected payoff for GRIM versus GRIM is nR. The expected payoff for ALLD versus GRIM is T + (m − 1)P . If nR > T + (m − 1)P then ALLD cannot spread in a GRIM population when rare. This is an argument of evolutionary stability. Interestingly, Trivers (1971) quotes ‘Hamilton (pers. commun.)’ for this idea. A small problem of the above analysis is that given a known number of rounds it is best to defect in the last round and by backwards induction it is also best to defect in the penultimate round and so on. Therefore, it is more natural to consider a repeated game with a probability w of having another round. In this case, the expected number of rounds is 1/(1 − w), and GRIM is stable against invasion by ALLD provided w > (T − R)/(T − P ). We can also formulate the PD as follows. The cooperator helps at a cost, c, and the other individual receives a benefit b. Defectors do not help. Therefore we have

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Evolutionary dynamics of cooperation

T = b, R = b − c, P = 0 and S = −c. The family of games that is described by the parameters b and c is a subset of all possible Prisoner’s Dilemma games as long as b > c. For the repeated PD, we find that ALLD cannot invade GRIM if w > c/b.

(2)

The probability of having another round must exceed the cost to benefit ratio of the altruistic act (Axelrod & Hamilton 1981, Axelrod 1984). Thus, the repeated PD allows cooperation, but the question arises – what is a good strategy for playing this game? This question was posed by the political scientist, Robert Axelrod. In 1979, he decided to conduct a tournament of computer programs playing the repeated PD. He received 14 entries, from which the surprise winner was tit-for-tat (TFT), the simplest of all strategies that were submitted. TFT cooperates in the first move, and then does whatever the opponent did in the previous round. TFT cooperates if you cooperate, TFT defects if you defect. It was submitted by the game theorist Anatol Rapoport (who is also the co-author of the book Rapoport & Chammah,1965). Axelrod analyzed the events of the tournament, published a detailed account and invited people to submit strategies for a second championship. This time he received 63 entries. John Maynard Smith submitted ‘tit-for-two-tats’, a variant of TFT which defects only after the opponent has defected twice in a row. Only one person, Rapoport, submitted TFT, and it won again. At this time, TFT was considered to be the undisputed world champion in the heroic world of the repeated PD. But one weakness became apparent very soon (Molander 1985, May 1987). TFT cannot correct mistakes. The tournaments were conducted without strategic noise. In the real world, ‘trembling hands’ and ‘fuzzy minds’ cause erroneous moves. If two TFT players interact with each other, a single mistake leads to a long sequence of alternating defection and cooperation. In the long run two TFT players get the same low payoff as two players who flip coins for every move in order to decide whether to cooperate or to defect. Errors destroy TFT. In 1989, we began to conduct ‘evolutionary tournaments’ (Nowak & Sigmund 1992). Instead of inviting experts to submit programs, we asked mutation and selection to explore (some portion of) the strategy space of the repeated PD in the presence of noise. The initial random ensemble of strategies was quickly dominated by ALLD. If the opposition is nonsensical, it is best to defect. A large portion of the population began to adopt the ALLD strategy and everything seemed lost. But after some time, a small cluster of players adopted a strategy very close to TFT. If this cluster is sufficiently large, then it can increase in abundance, and the entire population swings from ALLD to TFT. Reciprocity (and therefore cooperation) has emerged. We can show that TFT is the best catalyst for the emergence of cooperation. But TFT’s moment of glory was brief and fleeting. In all cases, TFT was rapidly replaced by another strategy. On close inspection, this strategy turned out to be ‘generoustit-for-tat’ (GTFT) which always cooperates if the opponent has cooperated on the previous move, but sometimes (probabilistically) even cooperates when the opponent has defected. Natural selection had discovered ‘forgiveness’.

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After many generations, however, GTFT is undermined by unconditional cooperators, ALLC. In a society, where everybody is nice (using GTFT), there is almost no need to remember how to retaliate against a defection. A biological trait which is not used is likely to be lost by random drift. Birds that escape to islands without predators lose the ability to fly. Similarly, a GTFT population is softened and turns into an ALLC population. Once most people play ALLC, there is an open invitation for ALLD to seize power. This is precisely what happens. The evolutionary dynamics run in cycles: from ALLD to TFT to GTFT to ALLC and back to ALLD. These oscillations of cooperative and defecting societies are a fundamental part of all our observations regarding the evolution of cooperation. Most models of cooperation show such oscillations. Cooperation is never a final state of evolutionary dynamics. Instead it is always lost to defection after some time and has to be re-established. These oscillations are also reminiscent of alternating episodes of war and peace in human history (Figure 1).

Tit-for-tat

Always defect

Generous Tit-for-tat

Always cooperate

Win-stay, lose-shift Figure 1. Evolutionary cycles of cooperation and defection. A small cluster of tit-for-tat (TFT) players or even a lineage starting from a single TFT player in a finite population can invade ‘always defect’ (ALLD). In fact, TFT is the most efficient catalyst for the first emergence of cooperation in an ALLD population. But in a world of ‘fuzzy minds’ and ‘trembling hands’, TFT is soon replaced by generous-tit-for-tat (GTFT), which can re-establish cooperation after occasional mistakes. If everybody uses GTFT, then ‘always cooperate’ (ALLC) is a neutral variant. Random drift leads to ALLC. An ALLC population invites invasion by ALLD. But ALLC is also dominated by ‘win-stay, lose-shift’ (WSLS), which leads to more stable cooperation than tit-for-tat-like strategies.

A subsequent set of simulations, exploring a larger strategy space, led to a surprise (Nowak & Sigmund 1993). The fundamental oscillations were interrupted by another strategy which seems to be able to hold its ground for a very long period of time. Most surprisingly, this strategy is based on the extremely simple principle of winstay, lose-shift (WSLS). If my payoff is R or T then I will continue with the same move next round. If I have cooperated then I will cooperate again, if I have defected then I will defect again. If my payoff is only S or P then I will switch to the other move next round. If I have cooperated then I will defect, if I have defected then I will cooperate (Figure 2). If two WSLS strategists play each other, they cooperate

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Win-stay C (3) …. C C

D (5) …. D C

Lose-shift C (0) …. D D

D (1) …. C (probabilistic) D

Figure 2. ‘Win-stay, lose-shift’ (WSLS) embodies a very simple principle. If you do well then continue with what you are doing. If you are not doing well, then try something else. Here we consider the Prisoner’s Dilemma (PD) payoff values R = 3, T = 5, P = 1 and S = 0. If both players cooperate, you receive 3 points, and you continue to cooperate. If you defect against a cooperator, you receive 5 points, and you continue to defect. But if you cooperate with a defector, you receive 0 points, and therefore you will switch from cooperation to defection. If, on the other hand, you defect against a defector, you receive 1 point, and you will switch to cooperation. Your aspiration level is 3 points. If you get at least 3 points then you consider it a ‘win’ and you will ‘stay’ with your current choice. If you get less than 3 points, you consider it a ‘loss’ and you will ‘shift’ to another move. If R > (T + P )/2 (or b/c > 2) then WSLS is stable against invasion by ALLD. If this inequality does not hold, then our evolutionary simulations lead to a stochastic variant of WSLS, which cooperates after a DD move only with a certain probability. This stochastic variant of WSLS is then stable against invasion by ALLD.

most of the time. If a defection occurs accidentally, then in the next move both will defect. Hereafter both will cooperate again. WSLS is a simple deterministic machine to correct stochastic noise. While TFT cannot correct mistakes, both GTFT and WSLS can correct mistakes. But WSLS has an additional ace in its hand. When WSLS plays ALLC it will discover after some time that ALLC does not retaliate. After an accidental defection, WSLS will switch to permanent defection. Therefore, a population of WSLS players does not drift to ALLC. Cooperation based on WSLS is more stable than cooperation based on tit-for-tat-like strategies. The repeated PD is mostly known as a story of tit-for-tat, but win-stay, lose-shift is a superior strategy in an evolutionary scenario with errors, mutation and many generations (Fudenberg & Maskin 1990, Nowak & Sigmund 1993). Incidentally, WSLS is stable against invasion by ALLD if b/c > 2. If instead 1 < b/c < 2 then a stochastic variant of WSLS dominates the scene; this strategy cooperates after a mutual defection only with a certain probability. Of course, all strategies of direct reciprocity, such as TFT, GTFT or WSLS can only lead to the evolution of cooperation if the fundamental inequality (2) is fulfilled.

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3. Indirect reciprocity While direct reciprocity embodies the idea ‘You scratch my back and I scratch yours’, indirect reciprocity suggests ‘You scratch my back and I scratch someone else’s’. Why should this work? Presumably I will not get scratched if it becomes known that I scratch nobody. Indirect reciprocity, in this view, is based on reputation (Nowak & Sigmund 1998a, b, 2005). But why should you care about what I do to a third person? The main reason why economists and social scientists are interested in indirect reciprocity is because one-shot interactions between anonymous partners in a global market become increasingly common and tend to replace the traditional long-lasting associations and long-term interactions between relatives, neighbors, or members of the same village. A substantial part of our life is spent in the ‘company of strangers’, and many transactions are no longer face-to-face. The growth of e-auctions and other forms of e-commerce is based, to a considerable degree, on reputation and trust. The potential to exploit such trust raises what economists call moral hazards. How effective is reputation, especially if information is only partial? Evolutionary biologists, on the other hand, are interested in the emergence of human societies, which constitutes the last (up to now) of the major transitions in evolution. In contrast to other eusocial species, such as bees, ants or termites, humans display a high degree of cooperation between non-relatives (Fehr & Fischbacher 2003). A considerable part of human cooperation is based on moralistic emotions, such as anger directed towards cheaters or the ‘warm inner glow’ felt after performing an altruistic action. Intriguingly, humans not only feel strongly about interactions which involve them directly, they also judge actions between third parties as evidenced by the contents of gossip. There are numerous experimental studies of indirect reciprocity based on reputation (Wedekind & Milinski 2000, Milinski et al. 2002, Wedekind & Braithwaite 2002, Seinen & Schramm 2006). A simple model of indirect reciprocity (Nowak & Sigmund 1998a, b) assumes that, within a well-mixed population, individuals meet randomly, one in the role of the potential donor, the other as the potential recipient. Each individual experiences several rounds of this interaction in each role, but never with the same partner twice. A player can follow either an unconditional strategy, such as always cooperate or always defect, or a conditional strategy, which discriminates among the potential recipients according to their past interactions. In a simple example, a discriminating donor helps a recipient if her score exceeds a certain threshold. A player’s score is 0 at birth, increases whenever that player helps and decreases whenever the player withholds help. Individual-based simulations and direct calculations show that cooperation based on indirect reciprocity can evolve provided the probability, p, of knowing the social score of another person exceeds the cost-to-benefit ratio of the altruistic act, p > c/b.

(3)

The role of genetic relatedness that is crucial for kin selection is replaced by social acquaintanceship. In a fluid population, where most interactions are anonymous and

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people have no possibility of monitoring the social score of others, indirect reciprocity has no chance. But in a socially viscous population, where people know one another’s reputation, cooperation by indirect reciprocity can thrive (Nowak & Sigmund 1998a). In a world of binary moral judgments (Nowak & Sigmund 1998b, Leimar & Hammerstein 2001, Panchanathan & Boyd 2003, Fishman 2003, Brandt & Sigmund 2004, 2005), there are four ways of assessing donors in terms of ‘first-order assessment’: always consider them as good, always consider them as bad, consider them as good if they refuse to give, or consider them as good if they give. Only this last option makes sense. Second-order assessment also depends on the score of the receiver; for example, it can be deemed good to refuse help to a bad person. There are 16 second-order rules. Third-order assessment also depends on the score of the donor; for example, a good person refusing to help a bad person may remain good, but a bad person refusing to help a bad person remains bad. There are 256 third-order assessment rules. We display three of them in Figure 3. Using the Scoring assessment rule,

Action of donor

Three assessment rules Reputation of donor and recipient GG GB BG BB C G G G G Scoring D B B B B C D

G B

G G

G B

G Standing B

C D

G B

B G

G B

B B

Judging

Reputation of donor after the action

Figure 3. Assessment rules specify how an observer judges an interaction between a potential donor and a recipient. Here we show three examples of assessment rules in a world of binary reputation, good (G) and bad (B). For ‘Scoring’, cooperation (C) earns a good reputation and defection (D) earns a bad reputation. ‘Standing’ is very similar to Scoring, the only difference is that a ‘good’ donor can defect against a ‘bad’ recipient without losing his ‘good’ reputation. Note that Scoring is associated with costly punishment (Fehr & Gaechter 2002, Sigmund et al. 2001), whereas for Standing punishment of ‘bad’ recipients is cost-free. For ‘Judging’ it is ‘bad’ to help a ‘bad’ recipient.

cooperation, C, always leads to a good reputation, G, whereas defection, D, always leads to a bad reputation, B. Standing (Sugden 1986) is like Scoring, but it is not bad if a good donor defects against a bad recipient . With Judging, in addition, it is bad to cooperate with a bad recipient. An action rule for indirect reciprocity prescribes giving or not giving, depending on the scores of both donor and recipient. For example, you may decide to help if the recipient’s score is good or your own score is bad. Such an action might increase your

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own score and therefore increase the chance of receiving help in the future. There are 16 action rules. If we view a strategy as the combination of an action rule and an assessment rule, we obtain 4096 strategies. In a remarkable calculation, Ohtsuki & Iwasa (2004, 2005) analyzed all 4096 strategies and proved that only eight of them are evolutionarily stable under certain conditions and lead to cooperation (Figure 4). Both Standing

Ohtsuki & Iwasa’s ‘Leading eight’

C D

GG

GB

BG

BB

G B

* G

G B

* *

C

D

C

Assessment

C/D Action

If a ‘good’ donor meets a ‘bad’ recipient, the donor must defect, and this action does Not reduce his reputation. The three *s can be set G or B. G If a column in the assessment module is B then the action must be C otherwise D.

Figure 4. Ohtsuki & Iwasa (2004, 2005) analyzed the combination of 28 = 256 assessment modules with 24 = 16 action modules. This is a total of 4096 strategies. They found that 8 of these strategies can be evolutionarily stable and lead to cooperation, provided that everybody agrees on each other’s reputation. (In general, uncertainty and incomplete information might lead to private lists of the reputation of others.) The three asterisks in the assessment module indicate a free choice between G and B. There are therefore 23 = 8 different assessment rules which make up the ‘leading eight’. The action module is built as follows: if the column in the assessment module is G and B, then the corresponding action is C, otherwise the action is D. Note that Standing and Judging are members of the leading eight, but neither Scoring nor Shunning is included.

and Judging belong to the leading eight, but not Scoring. We expect, however, that Scoring has a similar role in indirect reciprocity to that of tit-for-tat in direct reciprocity. Neither strategy is evolutionarily stable, but their ability to catalyze cooperation in adverse situations and their simplicity constitute their strength. In extended versions of indirect reciprocity in which donors can sometimes deceive others about the reputation of the recipient, Scoring is the ‘foolproof’ concept of ‘I believe what I see’. Scoring judges the action and ignores the stories. There is also experimental evidence that in certain situations humans follow scoring rather than standing (Milinski et al. 2001). In human evolution, there must have been a tendency to move from the simple cooperation promoted by kin or group selection to the strategic subtleties of direct and indirect reciprocity. Direct reciprocity requires precise recognition of individual


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people, a memory of the various interactions one had with them in the past, and enough brain power to conduct multiple repeated games simultaneously. Indirect reciprocity, in addition, requires the individual to monitor interactions among other people, possibly judge the intentions that occur in such interactions, and keep up with the ever changing social network of the group. Reputation of players may not only be determined by their own actions, but also by their associations with others. We expect that indirect reciprocity has coevolved with human language. On one hand, it is helpful to have names for other people and to receive information about how a person is perceived by others. On the other hand, a complex language is especially necessary if there are intricate social interactions. The possibilities for games of manipulation, deceit, cooperation and defection are limitless. It is likely that indirect reciprocity has provided the very selective scenario that led to cerebral expansion in human evolution.

4. Graph selection (or network reciprocity) Game theory was invented by von Neumann and Morgenstern (1944) as a mathematical approach to understanding the strategic and economic decisions of humans. Hamilton (1967), Trivers (1971) and Maynard Smith & Price (1973) brought game theory to biology. Instead of analyzing the interaction between two rational players, evolutionary game theory explores the dynamics of a population of players under the influence of natural selection (Maynard Smith 1982). In the classical setting of the replicator equation, the population size is infinite and interactions are equally likely between any two individuals (Taylor & Jonker 1978, Hofbauer et al. 1979, Zeeman 1980). Each individual obtains an average payoff which is interpreted as biological fitness: strategies reproduce according to their payoff. Successful strategies spread and eliminate less successful ones. The payoff depends on the frequency of strategies in the population. Hence, natural selection is frequency dependent. The replicator equation is deeply connected to the concept of an evolutionarily stable strategy (ESS) or Nash equilibrium. In the framework of the replicator equation, an ESS cannot be invaded by any mutant strategy (Hofbauer & Sigmund 1998). For recent books on game theory and evolutionary game theory we refer to Fudenberg & Tirole 1991, Binmore 1994, Weibull 1995, Samuelson 1997, Fudenberg & Levine 1998, Hofbauer & Sigmund 1998, Gintis 2000, Cressman 2003. Recent reviews of evolutionary game dynamics are Hofbauer & Sigmund (2003) and Nowak & Sigmund (2004). The traditional model of evolutionary game dynamics assumes that populations are well-mixed, which means that interactions between any two players are equally likely. More realistically, however, the interactions between individuals are governed by spatial effects or social networks. Let us therefore assume that the individuals of a population occupy the vertices of a graph (Nakamaru et al. 1997, 1998, Skyrms & Pemantle 2000, Abramson & Kuperman 2001, Ebel & Bornholdt 2002, Lieberman et al. 2005, Nakamaru & Iwasa 2005, Santos et al. 2005, Santos & Pacheco 2005).

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The edges of the graph determine who interacts with whom (Figure 5). Consider a population of N individuals consisting of cooperators and defectors. A cooperator b

b

D

D

D C

b

C

D 2b-5c

C 2b-2c

D

C

C 2b-3c

Figure 5. Games on graphs. The members of a population occupy the vertices of a graph (or social network). The edges denote who interacts with whom. Here we consider the specific example of cooperators, C, competing with defectors, D. A cooperator pays a cost, c, for every link. Each neighbor of a cooperator receives a benefit, b. The payoffs of some individuals are indicated in the figure. The fitness of each individual is a constant, denoting the ‘baseline fitness’, plus the payoff of the game. For evolutionary dynamics, we assume that in each round a random player is chosen to die, and the neighbors compete for the empty site in a mode proportional to their fitness. A simple rule emerges: if b/c > k then selection favors cooperators over defectors. Here k is the average number of neighbors per individual.

helps all individuals to whom it is connected. If a cooperator is connected to k other individuals and i of those are cooperators, then its payoff is bi − ck. A defector does not provide any help, and therefore has no costs, but it can receive the benefit from neighboring cooperators. If a defector is connected to k other individuals and j of those are cooperators, then its payoff is bj . Evolutionary dynamics are described by an extremely simple stochastic process: at each time step, a random individual adopts the strategy of one of its neighbors proportional to their fitness. We note that stochastic evolutionary game dynamics in finite populations is sensitive to the intensity of selection. In general, the reproductive success (fitness) of an individual is given by a constant, denoting the baseline fitness, plus the payoff that arises from the game under consideration. Strong selection means that the payoff is large compared to the baseline fitness; weak selection means the payoff is small compared to the baseline fitness. It turns out that many interesting results can be proven for weak selection, which is an observation also well known in population genetics. The traditional, well-mixed population of evolutionary game theory is represented by the complete graph, where all vertices are connected, which means that all individuals interact equally often. In this special situation, cooperators are always opposed by natural selection. This is the fundamental intuition of classical evolutionary game theory. But what happens on other graphs? We need to calculate the probability, ρC , that a single cooperator, starting in a random position, turns the whole population from defectors into cooperators. If selection neither favors nor opposes cooperation, then this probability is 1/N, which


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is the fixation probability of a neutral mutant. If the fixation probability ρC is greater than 1/N, then selection favors the emergence of cooperation. Similarly, we can calculate the fixation probability of defectors, ρD . A surprisingly simple rule determines whether selection on graphs favors cooperation. If b/c > k, (4) then cooperators have a fixation probability greater than 1/N and defectors have a fixation probability less than 1/N. Thus, for graph selection to favor cooperation, the benefit-to-cost ratio of the altruistic act must exceed the average degree, k, which is given by the average number of links per individual . This relationship can be shown with the method of pair-approximation for regular graphs, where all individuals have exactly the same number of neighbors (Ohtsuki et al. 2006). Regular graphs include cycles, all kinds of spatial lattices and random regular graphs. Moreover, computer simulations suggest that the rule b/c > k also holds for non-regular graphs such as random graphs and scale free networks. The rule holds in the limit of weak selection and k N. For the complete graph, k = N, we always have ρD > 1/N > ρC . The basic idea is that natural selection on graphs (in structured populations) can favor unconditional cooperation without any need of strategic complexity, reputation or kin selection. Games on graphs grew out of the earlier tradition of spatial evolutionary game theory (Nowak & May 1992, Herz 1994, Killingback & Doebeli 1996, Mitteldorf & Wilson 2000, Hauert et al. 2002, Le Galliard et al 2003, Hauert & Doebeli 2004, Szabo & Vukov 2004) and investigations of spatial models in ecology (Durrett & Levin 1994a, b, Hassell et al. 1994, Tilman & Kareiva 1997, Neuhauser 2001) and spatial models in population genetics (Wright 1931, Fisher & Ford 1950, Maruyama 1970, Slatkin 1981, Barton 1993, Pulliam 1988, Whitlock 2003).

5. Group selection The enthusiastic approach of early group selectionists to explain the evolution of cooperation entirely from this one perspective (Wynne-Edwards 1962) has met with vigorous criticism (Williams 1966) and has led to a denial of group selection for decades. Only an embattled minority of scientists defended the approach (Eshel 1972, Wilson 1975, Matessi & Jayakar 1976, Wade 1976, Uyenoyama & Feldman 1980, Slatkin 1981, Leigh 1983, Szathmary & Demeter 1987). Nowadays, however, it seems clear that group selection acts as a powerful mechanism for the promotion of cooperation (Sober & Wilson 1998, Keller 1999, Michod 1999, Swenson et al. 2000, Kerr & Godfrey-Smith 2002, Paulsson 2002, Boyd & Richerson 2002, Bowles & Gintis 2004, Traulsen et al. 2005). We only have to make sure that its basic requirements are fulfilled in a particular situation (Maynard Smith 1976). We would like to illustrate exactly what these requirements are through the use of a simple model (Traulsen & Nowak 2006).

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Imagine a population of individuals subdivided into groups. For simplicity, we assume the number of groups is constant and given by m. Each group contains between one and n individuals. The total population size can fluctuate between the bounds m and nm. Again, there are two types of individuals, cooperators and defectors. Individuals interact with others in their group and thereby receive a payoff. At each time step a random individual from the entire population is chosen proportional to payoff in order to reproduce. The offspring is added to the same group. If the group size is less than or equal to n nothing else happens. If the group size, however, exceeds n then with probability q the group splits into two. In this case, a random group is eliminated (in order to maintain a constant number of groups). With probability 1 − q, the group does not divide, but instead a random individual from that group is eliminated (Figure 6). This minimalist model of multi-level selection has some interesting fea-

D D D C C D

C

D

C D

C CC

Figure 6. A simple model of group selection. A population consists of m groups of maximum size n. Individuals interact with others in their group in the context of an evolutionary game. Here we consider the game between cooperators, C, and defectors, D. For reproduction, individuals are chosen from the entire population with a probability proportional to their payoff. The offspring is added to the same group. If a group reaches the maximum size, n, then it either splits in two or a random individual from that group is eliminated. If a group splits, then a random group dies, in order to keep the total population size constant. This meta-population structure leads to the emergence of two levels of selection, although only individuals reproduce.

tures. Note that the evolutionary dynamics are entirely driven by individual fitness. Only individuals are assigned payoff values. Only individuals reproduce. Groups can stay together or split (divide) when reaching a certain size. Groups that contain fitter individuals reach the critical size faster and therefore split more often. This concept leads to selection among groups, even though only individuals reproduce. The higher level selection emerges from lower level reproduction. Remarkably, the two levels of selection can oppose each other. As before, we can compute the fixation probabilities, ρC and ρD , of cooperators and defectors in order to check whether selection favors one or the other. If we add a single cooperator to a population of defectors, then this cooperator must first take over a group. Subsequently the group of cooperators must take over the entire population. The first step is opposed by selection, the second step is favored by selection. Hence,


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we need to find out if the overall fixation probability is greater to or less than what we would obtain for a neutral mutant. An analytic calculation is possible in the limit q 1 where individuals reproduce much more rapidly than groups divide. In this case, most of the groups are at their maximum size and hence the total population size is almost constant and given by N = nm. We find that selection favors cooperators and opposes defectors, ρC > 1/N > ρD , if b n >1+ . c m−2

(5a)

This result holds for weak selection. Smaller group sizes and larger numbers of competing groups favor cooperation. We also notice that the number of groups, m, must exceed two. There is an intuitive reason for this threshold. Consider the case of m = 2 groups with n = 2 individuals. In a mixed group, the cooperator has payoff −c and the defector has payoff b. In a homogeneous group, two cooperators have payoff b − c, while two defectors have payoff 0. Thus the disadvantage for cooperators in mixed groups cannot be compensated for by the advantage they have in homogeneous groups. Interestingly, however, for larger splitting probabilities, q, we find that cooperators can be favored even for m = 2 groups. The reason is the following: for very small q, the initial cooperator must reach fixation in a mixed group; but for larger q, a homogeneous cooperator group can also emerge if a mixed group splits giving rise to a daughter group that has only cooperators. Thus, larger splitting probabilities make it easier for cooperation to emerge. Let us also consider the effect of migration between groups. The average number of migrants accepted by a group during its life-time is denoted by z. We find that selection favors cooperation provided n b >1+z+ . c m

(5b)

In order, to derive this condition we have assumed weak selection and q 1, as before, but also that both the numbers of groups, m, and the maximum group size, n, are much large than one. Group selection (or multi-level selection) is a powerful mechanism for the evolution of cooperation if there are a large number of relatively small groups and migration between groups is not too frequent.

6. Conclusion I have presented five simple (Equations 1–5) rules that determine whether particular mechanisms can promote the evolution of cooperation. In all five theories, b is the benefit for the recipient and c the cost for the donor of an altruistic act. The comparison of the five rules enables us to understand the crucial quantities that are responsible

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for the natural selection of cooperation by the various mechanisms that have been proposed. 1. Kin selection leads to cooperation if b/c > 1/r, where r is the coefficient of genetic relatedness between donor and recipient (Hamilton 1964a). 2. Direct reciprocity leads to cooperation if b/c > 1/w, where w is the probability of playing another round in the repeated Prisoner’s Dilemma (Axelrod & Hamilton 1981). 3. Indirect reciprocity leads to cooperation if b/c > 1/q, where q is the probability to know the reputation of a recipient (Nowak & Sigmund 1998a). 4. Graph selection (or ‘network reciprocity’) leads to cooperation if b/c > k, where k is the degree of the graph, that is the average number of neighbors (Ohtsuki et al. 2006). 5. Group selection leads to cooperation if b/c > 1 + z + n/m, where z is the number of migrants accepted by a group during its life-time, n is the group size and m is the number of groups (Traulsen & Nowak 2006).

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Fractional Brownian motion: stochastic calculus and applications David Nualart

Abstract. Fractional Brownian motion (fBm) is a centered self-similar Gaussian process with stationary increments, which depends on a parameter H ∈ (0, 1) called the Hurst index. In this note we will survey some facts about the stochastic calculus with respect to fBm using a pathwise approach and the techniques of the Malliavin calculus. Some applications in turbulence and finance will be discussed. Mathematics Subject Classification (2000). Primary 60H30; Secondary 60G18. Keywords. Fractional Brownian motion, stochastic integrals, Malliavin calculus, Black– Scholes formula, stochastic volatility models.

1. Introduction A real-valued stochastic process X = {Xt , t ≥ 0} is a family of random variables Xt : → R defined on a probability space (, F , P ). The process X is called Gaussian if for all 0 ≤ t1 < t2 < · · · < tn the probability distribution of the random vector (Xt1 , . . . , Xtn ) on Rn is normal or Gaussian. From the properties of the normal distribution it follows that the probability distribution of a Gaussian process is entirely determined by the mean function E(Xt ) and the covariance function Cov(Xt , Xs ) = E((Xt − E(Xt ))(Xs − E(Xs ))), where E denotes the mathematical expectation or integral with respect to the probability measure P . One of the most important stochastic processes used in a variety of applications is the Brownian motion or Wiener process W = {Wt , t ≥ 0}, which is a Gaussian process with zero mean and covariance function min(s, t). The process W has independent t increments and its formal derivative dW dt is used as input noise in dynamical systems, giving rise to stochastic differential equations. The stochastic calculus with respect to the Brownian motion, developed from the works of Itô in the forties, permits to formulate and solve stochastic differential equations. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Motivated from some applications in hydrology, telecommunications, queueing theory and mathematical finance, there has been a recent interest in input noises without independent increments and possessing long-range dependence and self-similarity properties. Long-range dependence in a stationary time series occurs when the covariances tend to zero like a power function and so slowly that their sums diverge. The self-similarity property means invariance in distribution under a suitable change of scale. One of the simplest stochastic processes which is Gaussian, self-similar and it has stationary increments is fractional Brownian motion, which is a generalization of the classical Brownian motion. As we shall see later, the fractional Brownian motion possesses long-range dependence when its Hurst parameter is larger than 1/2. In this note we survey some properties of the fractional Brownian motion, and describe different methods to construct a stochastic calculus with respect to this process. We will also discuss some applications in mathematical finance and in turbulence.

2. Fractional Brownian motion A Gaussian process B H = {BtH , t ≥ 0} is called fractional Brownian motion (fBm) of Hurst parameter H ∈ (0, 1) if it has mean zero and the covariance function E(BtH BsH ) = RH (t, s) =

1 2H s + t 2H − |t − s|2H . 2

(2.1)

This process was introduced by Kolmogorov [25] and studied by Mandelbrot and Van Ness in [30], where a stochastic integral representation in terms of a standard Brownian motion was established. The parameter H is called Hurst index from the statistical analysis, developed by the climatologist Hurst [24], of the yearly water run-offs of Nile river. The fractional Brownian motion has the following properties. H , t ≥ 0} and 1. Self-similarity: For any constant a > 0, the processes {a −H Bat {BtH , t ≥ 0} have the same probability distribution. This property is an immediate consequence of the fact that the covariance function (2.1) is homogeneous of order 2H , and it can be considered as a “fractal property” in probability.

2. Stationary increments: From (2.1) it follows that the increment of the process in an interval [s, t] has a normal distribution with zero mean and variance E (BtH − BsH )2 = |t − s|2H . (2.2) Hence, for any integer k ≥ 1 we have (2k)! E (BtH − BsH )2k = |t − s|2H k . k!2k

(2.3)

Fractional Brownian motion: stochastic calculus and applications

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Choosing k such that 2H k > 1, Kolmogorov’s continuity criterion and (2.3) imply that there exists a version of the fBm with continuous trajectories. Moreover, using Garsia–Rodemich–Rumsey lemma [19], we can deduce the following modulus of continuity for the trajectories of fBm: For all ε > 0 and T > 0, there exists a nonnegative random variable Gε,T such that E(|Gε,T |p ) < ∞ for all p ≥ 1, and, almost surely, H B − B H ≤ Gε,T |t − s|H −ε , t s for all s, t ∈ [0, T ]. In other words, the parameter H controls the regularity of the trajectories, which are Hölder continuous of order H − ε, for any ε > 0. For H = 1/2, the covariance can be written as R1/2 (t, s) = min(s, t), and the process B 1/2 is an ordinary Brownian motion. In this case the increments of the process in disjoint intervals are independent. However, for H = 1/2, the increments are not independent. H , n ≥ 1. Then {X , n ≥ 1} is a Gaussian stationary sequence Set Xn = BnH −Bn−1 n with unit variance and covariance function 1 ρH (n) = (n + 1)2H + (n − 1)2H − 2n2H 2 ≈ H (2H − 1)n2H −2 → 0, as n tends to infinity. Therefore, if H > 21 , ρH (n) > 0 for n large enough and ∞ n=1 ρH (n) = ∞. We say that the sequence {Xn , n ≥ 1} has long-range dependence. Moreover, this sequence presents an aggregation behavior which can be used to describe cluster phenomena. For H < 21 , ρH (n) < 0 for n large enough and ∞ n=1 |ρH (n)| < ∞. In this case, {Xn , n ≥ 1} can be used to model sequences with intermittency. 2.1. Construction of the fBm. In order to show the existence of the fBm we should check that the symmetric function RH (t, s) defined in (2.1) is nonnegative definite, that is, n

ai aj RH (ti , tj ) ≥ 0

(2.4)

i,j =1

for any sequence of real numbers ai , i = 1, . . . , n and for any sequence ti ≥ 0. Property (2.4) follows from the integral representation 1 1 1 BtH = ((t − s)+ )H − 2 − ((−s)+ )H − 2 dWs , (2.5) C1 (H ) R where {W (A), A Borel subset of R} is a Brownian measure on R and

∞ 1 1 2 H − 21 H − 21 2 C1 (H ) = (1 + s) −s ds + , 2H 0

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obtained by Mandelbrot and Van Ness in [30]. The stochastic integral (2.5) is well 1 1 defined, because the function ft (s) = ((t − s)+ )H − 2 − ((−s)+ )H − 2 , s ∈ R, t ≥ 0 satisfies R ft (s)2 ds < ∞. On the other hand, the right-hand side of (2.5) defines a zero mean Gaussian process such that E (BtH )2 = t 2H and E (BtH − BsH )2 = (t − s)2H , which implies that B H is an fBm with Hurst parameter H . 2.2. p-variation of the fBm. Suppose that X = {Xt , t ≥ 0} is a stochastic process with continuous trajectories. Fix p > 0. We define the p-variation of X on an interval [0, T ] as the following limit in probability: lim

n→∞

n X j T − X (j −1)T p . n

j =1

n

If the p-variation exists and it is nonzero a.s., then for any q > p the q-variation is zero and for any q < p the q-variation is infinite. For example, the 2-variation (or quadratic variation) of the Brownian motion is equal to the length of the interval T . Rogers has proved in [40] that the fBm B H has finite 1/H -variation equals to cp T , where cp = E(|B1H |p ). In fact, the self-similarity property implies that the sequence n H 1/H B j T − B H (j −1)T j =1

n

n

has the same distribution as n 1/H T H Bj − BjH−1 , n j =1

and by the Ergodic Theorem this converges in L1 () and almost surely to E(|B1H |p )T . As a consequence, the fBm with Hurst parameter H = 1/2 is not a semimartingale. Semimartingales are the natural class of processes for which a stochastic calculus can be developed, and they can be expressed as the sum of a bounded variation process and a local martingale which has finite quadratic variation. The fBm cannot be a semimartingale except in the case H = 1/2 because if H < 1/2, the quadratic variation is infinite, and if H > 1/2 the quadratic variation is zero and the 1-variation is infinite.


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Let us mention the following surprising result proved by Cheridito in [8]. Suppose that {BtH , t ≥ 0} is an fBm with Hurst parameter H ∈ (0, 1), and {Wt , t ≥ 0} is an ordinary Brownian motion. Assume they are independent and set Mt = BtH + Wt . Then {Mt , t ≥ 0} is not a semimartingale if H ∈ 0, 21 ∪ 21 , 43 , and it is a semimartingale, in law to a Brownian motion on any finite time interval [0, T ], equivalent if H ∈ 43 , 1 . The 1/H -variation of Wick stochastic integrals with respect to the fractional Brownian motion with parameter H > 1/2 has been computed by Guerra and Nualart in [20].

3. Stochastic calculus with respect to the fBm The aim of the stochastic calculus is to define stochastic integrals of the form

T

ut dBtH ,

0

(3.1)

where u = {ut , t ∈ [0, T ]} is some stochastic process. If u is a deterministic function there is a general procedure to define the stochastic integral of u with respect to a Gaussian process using the convergence in L2 (). We will first review this general approach in the particular case of the fBm. 3.1. Integration of deterministic processes. Consider an fBm B H = {BtH , t ≥ 0} with Hurst parameter H ∈ (0, 1). Fix a time interval [0, T ] and denote by E the set of step functions on [0, T ]. The integral of a step function of the form ϕt =

m

aj 1(tj −1 ,tj ] (t)

j =1

is defined in a natural way by

T 0

ϕt dBtH

=

m j =1

aj (BtH − BtH ). j j −1

We would like to extend this integral to a more general class of functions, using the convergence in L2 (). To do this we introduce the Hilbert space H defined as the closure of E with respect to the scalar product 1[0,t] , 1[0,s] H = RH (t, s).

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T Then the mapping ϕ −→ 0 ϕt dBtH can be extended to a linear isometry between H and the Gaussian subspace HT (B H ) of L2 (, F , P ) spanned by the random variables {BtH , t ∈ [0, T ]}. We will denote this isometry by ϕ −→ B H (ϕ). We would like to interpret B H (ϕ) as the stochastic integral of ϕ ∈ HT (B H ) with T respect to B H and to write B H (ϕ) = 0 ϕt dBtH . However, we do not know whether the elements of H can be considered as real-valued functions. This turns out to be true for H < 21 , but is false when H > 21 (see Pipiras and Taqqu [38], [39]). We state without proof the following results about the space H. 3.1.1. Case H > function

1 2.

In this case the second partial derivative of the covariance ∂ 2 RH = αH |t − s|2H −2 , ∂t∂s

where αH = H (2H − 1), is integrable, and we can write t s |r − u|2H −2 dudr. RH (t, s) = αH 0

(3.2)

0

Formula (3.2) implies that the scalar product in the Hilbert space H can be written as T T ϕ, ψ H = αH |r − u|2H −2 ϕr ψu dudr (3.3) 0

0

for any pair of step functions ϕ and ψ in E . As a consequence, we can exhibit a linear space of functions contained in H in the following way. Let |H| be the Banach space of measurable functions ϕ : [0, T ] → R such that T T 2 ϕ|H| = αH |r − u|2H −2 |ϕr ||ϕu | dudr < ∞. 0

0

It has been shown in [39] that the space |H| equipped with the inner product ·, · H is not complete and it is isometric to a subspace of H. The following estimate has been proved in [31] using Hölder and Hardy–Littlewood inequalities. Lemma 3.1. Let H >

1 2

1

and ϕ ∈ L H ([0, T ]). Then ϕ|H| ≤ bH ϕ

1

L H ([0,T ])

,

for some constant bH . Thus we have the embeddings 1

L2 ([0, T ]) ⊂ L H ([0, T ]) ⊂ |H| ⊂ H,

(3.4)

1547


T and Wiener-type integral 0 ϕt dBt can be defined for functions ϕ in the Banach space |H|. Notice that we can integrate more functions that in the case of the Brownian motion, and the isometry property of the Itô stochastic integral is replaced here by the formula

T 2 T T H E ϕt dBt |r − u|2H −2 ϕr ϕu dudr = ϕ2H . = αH 0

3.1.2.

Case H
H − 21 . The first inclusion is a direct consequence of formula (3.6), and the second one follows from Hardy–Littlewood inequality. Roughly speaking, in this case the fractional Brownian motion is more irregular than the classical Brownian motion, and some Hölder continuity is required for a function to be integrable. Moreover the computation of the variance of an integral using formula (3.5) is more involved. 3.2. Integration of random processes. Different approaches have been used in the literature in order to define stochastic integrals with respect to the fBm. Lin [26] T and Dai and Heyde [13] have defined a stochastic integral 0 ut dBtH as limit in L2 of Riemann sums in the case H > 21 . The techniques of Malliavin calculus have been used to develop the stochastic calculus for the fBm starting from the pioneering

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David Nualart

work of Decreusefond and Üstünel [14]. We refer to the works of Carmona and Coutin in [7], Alòs, Mazet and Nualart [1], [2], Alòs and Nualart [3], and the recent monograph by Hu [21], among others. We will first describe a path-wise approach based on Young integrals. T 3.2.1. Path-wise approach. We can define 0 ut dBtH using path-wise Riemann– Stieltjes integrals taking into account the results of Young in [43]. In fact, Young T proved that the Riemann–Stieltjes integral 0 ft dgt exists, provided that f, g : [0, T ] → R are Hölder continuous functions of orders α and β with α + β > 1. Therefore, if u = {ut , t ∈ [0, T ]} is a stochastic process with γ -Hölder continuous T trajectories, where γ > 1 − H , then the Riemann–Stieltjes integral 0 ut dBtH exists T path-wise. That is for any element ω ∈ , the integral 0 ut (ω) dBtH (ω) exists as the point-wise limit of Riemann sums. In particular, if H > 1/2, the path-wise Riemann– T Stieltjes integral 0 F (BtH ) dBtH exists if F is a continuously differentiable function. Moreover the following change of variables formula holds: t H F (BsH ) dBsH (3.7)

(Bt ) = (0) + 0

if

= F . In the case of the form

1 4

1/2. Then for any ξ ∈ Rd we have 2

T 1 i ξ,BtH

H E e dBt (3.20) ≤ C(1 ∧ |ξ | H −2 ), 0

where zC =

d

i=1 z

izi

C

and C is a constant depending on T , d and H .

Proof. From (3.18) we get T T H H ei ξ,Bt dBtH = ei ξ,Bt 3 dBtH + H 0

0

T

iξ ei ξ,Bt t 2H −1 dt. H

(3.21)

0

We denote by πξ (x) = x − |ξξ|2 ξ, x the projection operator on the orthogonal subspace of ξ . Clearly

T T iξ H i ξ,BtH

H i ξ,BtH

H e dBt = πξ e dBt + 2 (ei ξ,BT − 1), (3.22) |ξ | 0 0 and, as a consequence, it suffices to show the estimate (3.20) for the first summand in the right-hand side of (3.22). From (3.21) it follows that

T

T i ξ,BtH

H i ξ,BtH

H Z := πξ e dBt e 3 dBt . = πξ 0

0

1553


Then we need to compute the expectation of the square norm of the C3 -valued random variable Z. This is done using the duality relationship (3.11) and the commutation formula (3.16). The composition of the projection operator πξ and the derivative H operator D vanishes on a random variable of the form ei ξ,Bt . Hence, only the H first term in the commutation formula (3.16) applied to ut = ei ξ,Bt will contribute to E(Z2C ) and we obtain d E(Z2C ) = E(Z j Z j ) j =1

(ξ j )2 H H E e−i ξ,B· , e−i ξ,B· H 2 |ξ | j =1 T T H H = (d − 1)αH E ei ξ,Bs −Br |s − r|2H −2 dsdr 0 0 T T |s−r|2H 2 e− 2 |ξ | |s − r|2H −2 dsdr, = (d − 1)αH =

d

1−

0

0

which leads to the desired estimate.

1

2

1

Proposition 3.3 also holds for H ∈ 4 , 2 if we replace the path-wise integral in the right-hand side of (3.18) by the Stratonovich integral defined as the limit in probability of symmetric sums T n H 1 H H F (BtH ) dBtH = lim F B (i−1)T + F (B H iT ) B iT − B (i−1)T . n→∞ 2 n n n n 0 i=1

For H = 1/2 the Wick integral appearing in Equation (3.18) is the classical Itô integral and it is the limit of forward Wick or ordinary Riemann sums: 0

T

n 1/2 1/2 1/2 1/2 1/2 F Bt 3 dBt = lim F B (i−1)T 3 B iT − B (i−1)T n→∞

= lim

n→∞

i=1 n

n

n

n

1/2 1/2 1/2 F B (i−1)T B iT − B (i−1)T .

i=1

n

n

n

Nevertheless, for H < 1/2 the forward Riemann sums do not converge in general. For example, in the simplest case F (x) = x, we have, with the notation ti = iTn n n

1 2H H H H 2H E ti − ti−1 (Bti−1 (Bti − Bti−1 )) = − (ti − ti−1 )2H 2 i=1

i=1

1 = T 2H (1 − n1−2H ) → −∞, 2 as n tends to infinity.

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The convergence 1 1of the forward Wick Riemann sums to the forward Wick integral in the case H ∈ 4 , 2 has been recently established in [36] and [5]. More precisely, the following theorem has been proved in [36]. Theorem 3.5. Suppose H ∈ 41 , 21 and let F be a function of class C 7 such that F together with its derivatives satisfy the growth condition (3.17). Then, the forward Wick integral 0

T

F (BtH ) 3 dBtH = lim

n→∞

n H H F BH (i−1)T 3 B iT − B (i−1)T n

n

i=1

n

exists and the Wick–Itô formula (3.19) holds. More generally, we can replace the fractional Brownian motion B H by an arbitrary Gaussian process {Xt , t ≥ 0} with zero mean and continuous covariance function R(s, t) = E(Xs Xt ). Suppose that the variance function Vt = E(Xt2 ) has bounded variation on any finite interval and the following conditions hold for any T > 0: n 2 E X iT − X (i−1)T X j T − X (j −1)T → 0, lim

n→∞

i,j =1

lim

n→∞

n

n

n

sup

n

2 E X iT − X (i−1)T Xt → 0.

i=1 0≤t≤T

n

(3.23)

n

(3.24)

n

T Then it is proved in [36] that the forward Wick integral 0 F (Xt ) 3 dXt exists and the following version of the Wick–Itô formula holds: T 1 T

F (XT ) = F (X0 ) + F (Xt ) 3 dXt + F (Xt ) dVt . 2 0 0

4. Application of fBm in turbulence The observations of three-dimensional turbulent fluids indicate that the vorticity field of the fluid is concentrated along thin structures called vortex filaments. In his book Chorin [10] suggests probabilistic descriptions of vortex filaments by trajectories of self-avoiding walks on a lattice. Flandoli [17] introduced a model of vortex filaments based on a three-dimensional Brownian motion. A basic problem in these models is the computation of the kynetic energy of a given configuration. Denote by u(x) the velocity field of the fluid at point x ∈ R3 , and let ξ = curlu be the associated vorticity field. The kynetic energy of the field will be 1 1 ξ(x) · ξ(y) 2 H= |u(x)| dx = dxdy. (4.1) 2 R3 8π R3 R3 |x − y|


1555

We will assume that the vorticity field is concentrated along a thin tube centered in a curve γ = {γt , 0 ≤ t ≤ T }. Moreover, we will choose a random model and consider this curve as the trajectory of a three-dimensional fractional Brownian motion B H = {BtH , 0 ≤ t ≤ T } with Hurst parameter H . That is, the components of the process B H are independent fractional Brownian motions. This modelization is justified by the fact that the trajectories of the fractional Brownian motion are Hölder continuous of any order H ∈ (0, 1). For technical reasons we are going to consider only the case H > 21 . Then the vorticity field can be formally expressed as T · ξ(x) = δ(x − y − BsH )B H ds ρ (dy), (4.2) s R3

0

where is a parameter called the circuitation, and ρ is a probability measure on R3 with compact support. Substituting (4.2) into (4.1) we derive the following formal expression for the kinetic energy: H= Hxy ρ(dx)ρ(dy), (4.3) R3

R3

where the so-called interaction energy Hxy is given by the double integral Hxy =

3 2 T T 1 dBsH,i dBtH,i . H H| 8π |x + B − y − B 0 t s i=1 0

(4.4)

We are interested in the following problems: Is H a well defined random variable? Does it have moments of all orders and even exponential moments? In order to give a rigorous meaning to the double integral (4.4) we introduce the regularization of the function | · |−1 : σn = | · |−1 ∗ p1/n ,

(4.5)

where p1/n is the Gaussian kernel with variance n1 . Then the smoothed interaction energy Hnxy

T 3 2 T H H H,i = σn (x + Bt − y − Bs ) dBs dBtH,i 8π 0 0

(4.6)

i=1

is well defined, where the integrals are path-wise Riemann–Stieltjes integrals. Set n H = Hnxy ρ(dx)ρ(dy). (4.7) R3

R3

The following result has been proved in [35].

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Theorem 4.1. Suppose that the measure ρ satisfies 1 |x − y|1− H ρ(dx)ρ(dy) < ∞. R3

R3

(4.8)

Let Hnxy be the smoothed interaction energy defined by (4.5). Then Hn defined in (4.7) converges, for all k ≥ 1, in Lk () to a random variable H ≥ 0 that we call the energy associated with the vorticity field (4.2). If H = 21 , the fBm B H is a classical three-dimensional Brownian motion. In this case condition (4.8) would be R3 R3 |x − y|−1 ρ(dx)ρ(dy) < ∞, which is the assumption made by Flandoli [17] and Flandoli and Gubinelli [18]. In this last paper, using Fourier approach and Itô’s stochastic calculus, the authors show that E(e−β H ) < ∞ for sufficiently small negative β. The proof of Theorem 4.1 is based on the stochastic calculus with respect to fBm and the application of Fourier transform. Using Fourier transform we can write 1 e−i ξ,z

= (2π)3 dξ |z| |ξ |2 R3 and

σn (x) =

R3

|ξ |−2 ei ξ,x −|ξ |

2 /2n

dξ.

(4.9)

Substituting (4.9) into in (4.6), we obtain the following formula for the smoothed interaction energy: Hnxy

3 −|ξ |2 /2n 2 T T H,j H,j i ξ,x+Bt −y−Bs e = e dBs dBt 2 8π |ξ | 0 R3 j =1 0 2 2 = |ξ |−2 ei ξ,x−y −|ξ | /2n Yξ 2C dξ, (4.10) 8π R3

where

T

Yξ = 0

ei ξ,Bt dBtH . H

Integrating with respect to ρ yields 2 2 n H = Yξ 2C |ξ |−2 | ρ (ξ )|2 e−|ξ | /2n dξ ≥ 0. 8π R3 From Fourier analysis and condition (4.8) we know that 1 1− H1 |x − y| ρ(dx)ρ(dy) = CH | ρ (ξ )|2 |ξ | H −4 dξ < ∞. R3

R3

R3

(4.11)

(4.12)


1557

Then, taking into account (4.12) and (4.11), in order to show the convergence in Lk () of Hn to a random variable H ≥ 0 it suffices to check that k( H1 −2) . (4.13) E(Yξ 2k C ) ≤ Ck 1 ∧ |ξ | For k = 2 this has been proved in Proposition 3.4. The general case k ≥ 2 follows by similar arguments making use of the local nondeterminism property of fBm (see Berman [4]):

Var (BtHi − BsHi ) ≥ kH (ti − si )2H . i

i

5. Application to financial mathematics Fractional Brownian motion has been used to describe the behavior to prices of assets and volatilities in stock markets. The long-range dependence self-similarity properties make this process a suitable model to describe these quantities. We refer to Shiryaev [41] for a general description of the applications of fractional Brownian motion to model financial quantities. We will briefly present in this section two different uses of fBm in mathematical finance. 5.1. Fractional Black and Scholes model. It has been proposed by several authors to replace the classical Black and Scholes model which has no memory and is based on the geometric Brownian motion by the so-called fractional Black and Scholes model. In this model the market stock price of the risky asset is given by

σ 2 2H St = S0 exp μt + σ BtH − t , (5.1) 2 where B H is an fBm with Hurst parameter H , μ is the mean rate of return and σ > 0 is the volatility. The price of the non-risky assets at time t is ert , where r is the interest rate. Consider an investor who starts with some initial endowment V0 ≥ 0 and invests in the assets described above. Let αt be the number of non-risky assets and let βt the number of stocks owned by the investor at time t. The couple (αt , βt ), t ∈ [0, T ] is called a portfolio and we assume that αt and βt are stochastic processes. Then the investor’s wealth or value of the portfolio at time t is Vt = αt ert + βt St . We say that the portfolio is self-financing if t t rs Vt = V0 + r αs e ds + βs dSs . 0

0

(5.2)

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This means that there is no fresh investment and there is no consumption. We see here that the self-financing condition requires the definition of a stochastic integral with respect to the fBm, and there are two possibilities: path-wise integrals and Wick-type integrals. The use of path-wise integrals leads to the existence of arbitrage opportunities, which is one of the main drawbacks of the model (5.1). Different authors have proved the existence of arbitrages for the fractional Black and Scholes model (see Rogers [40], Shiryaev [41], and Cheridito [9]). By definition, an arbitrage is a selffinancing portfolio which satisfies V0 = 0, VT ≥ 0 and P (VT > 0) > 0. In the case H > 21 , one can construct an arbitrage in the following simple way. Suppose, to simplify, that μ = r = 0. Consider the self-financing portfolio defined by βt = St − S0 , t αt = βs dSs − βt St . 0

This portfolio satisfies V0 = 0 and Vt = (St − S0 )2 > 0 for all t > 0, and hence it is an arbitrage. In the classical Black and Scholes model (case H = 21 ), there exists an equivalent probability measure Q under which μ = r and the discounted price process St = −rt e St is a martingale. Then, the discounted value of a self-financing adapted portfolio T Ss2 ds < ∞ is a martingale on the time interval [0, T ] given by satisfying EQ 0 βs2 the Itô stochastic integral t t = V0 + V βs d Ss . 0

As a consequence, Vt = e−r(T −t) EQ (VT |Ft ), and the price of an European option with payoff G at the maturity time T is given by e−r(T −t) EQ (G|Ft ). The probability Q is called the martingale measure. In the case H = 21 , there exist an equivalent 2

probability Q under which μ = r and St = S0 exp(σ BtH − σ2 t 2H ) has constant expectation. However, e−rt St is not a martingale under Q. The existence of arbitrages can be avoided using forward Wick integrals to define the self-financing property (5.2). In fact, using the Wick–Itô formula in (5.1) yields dSt = μSt dt + σ St 3 dBtH , and then the self-financing condition (5.2) could be written as t t rs (rαs e + μβs Ss )ds + σ βs Ss 3 dBsH . Vt = V0 + 0

0

Applying the stochastic calculus with respect to the Wick integral, Hu and Øksendal in [22], and Elliott and Hoek in [16] have derived the following formula for the value

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of the call option with payoff (ST − K)+ at time t ∈ [0, T ]: where

y± =

ln

C(t, St ) = St (y+ ) − Ke−r(T −t) (y− ),

(5.3)

σ 2 (T 2H − t 2H ) 2H St + r(T − t) ± σ T − t 2H . K 2

(5.4)

In [6] Björk and Hult argue that the definition of a self-financing portfolio using the Wick product is quite restrictive and in [37] Nualart and Taqqu explain the fact that in formula (5.4) only the increment of the variance of the process in the interval [t, T ] appears, and extend this formula to price models driven by a general Gaussian process. 5.2. Stochastic volatility models. It has been observed that in the classical Black imp and Scholes model the implied volatility σt,T obtained from formula (5.3) for different options written on the same asset is not constant and heavily depends on the time t, the time to maturity T − t and the strike price St . The U -shaped pattern of implied volatilites across different strike prices is called “smile”, and it is believed that this and other features as the volatility clustering can be explained by stochastic volatility models. Hull and White have proposed in [23] an option pricing model in which the volatility of the asset price is of the form exp(Yt ), where Yt is an Ornstein–Uhlenbeck process. Consider the following stochastic volatility model based on the fractional Ornstein– Uhlenbeck process. The price of the asset St is given by dSt = μSt dt + σt St dWt , where σt = f (Yt ) and Yt is a fractional Ornstein–Uhlenbeck process: dYt = α(m − Yt )dt + βt dBtH . The process Wt is an ordinary Brownian motion and BtH is a fractional Brownian motion with Hurst parameter H > 21 , independent of W . Examples of functions f are f (x) = ex and f (x) = |x|. Comte and Renault studied in [11] this type of stochastic volatility model which introduces long memory and mean reverting in the Hull and White setting. The long-memory property allows this model to capture the well-documented evidence of persistence of the stochastic feature of Black and Scholes implied volatilities, when time to maturity increases. Hu has proved in [21] the following properties of this model. 1) The market is incomplete and martingale measures are not unique. 2) Set γt = (r − μ)/σt and dQ = exp dP

0

T

1 γt dWt − 2

T

|γt | dt . 2

0

Then Q is the minimal martingale measure associated with P .

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3) The risk minimizing-hedging price at time t = 0 of an European call option with payoff (ST − K)+ is given by C0 = e−rT EQ [(ST − K)+ ].

(5.5)

As a consequence of (5.5), if Gt denotes the filtration generated by fBm, we obtain C0 = e−rT EQ [EQ ((ST − K)+ |GT )] = e−rT EQ [CBS (σ )]. Here σ =

T 0

σs2 ds and CBS (σ ) is the Black and Scholes price function given by CBS = S0 (y+ ) − Ke−rT (y− ),

where ln SK0 + (r ± y± = √ σ T

σ2 2 )T

.

References [1] Alòs, E., Mazet, O., Nualart, D., Stochastic calculus with respect to fractional Brownian motion with Hurst parameter lesser than 21 . Stoch. Proc. Appl. 86 (1999), 121–139. [2] Alòs, E., Mazet, O., Nualart, D., Stochastic calculus with respect to Gaussian processes. Ann. Probab. 29 (2001), 766–801. [3] Alòs, E., Nualart, D., Stochastic integration with respect to the fractional Brownian motion. Stoch. Stoch. Rep. 75 (2003), 129–152. [4] Berman, S., Local nondeterminism and local times of Gaussian processes. Indiana Univ. Math. J. 23 (1973), 69–94. [5] Biagini, F., Øksendal, B., Forward integrals and an Itô formula for fractional Brownian motion. Preprint, 2005. [6] Björk, R., Hult, H., A note on Wick products and the fractional Black-Scholes model. Preprint, 2005. [7] Carmona, P. , Coutin, L., Stochastic integration with respect to fractional Brownian motion. Ann. Inst. Henri Poincaré 39 (2003), 27–68. [8] Cheridito, P., Mixed fractional Brownian motion. Bernoulli 7 (2001), 913–934. [9] Cheridito, P., Regularizing Fractional Brownian Motion with a View towards Stock Price Modelling. PhD Dissertation, ETH, Zürich, 2001. [10] Chorin, A., Vorticity and Turbulence. Appl. Math. Sci. 103, Springer-Verlag, New York 1994. [11] Comte, F., Renault, E., Long memory in continuous-time stochastic volatility models. Math. Finance 8 (1998), 291–323


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[12] Coutin, L., Qian, Z., Stochastic analysis, rough paths analysis and fractional Brownian motions. Probab. Theory Related Fields 122 (2002), 108–140. [13] Dai, W., Heyde, C. C., Itô’s formula with respect to fractional Brownian motion and its application. J. Appl. Math. Stochastic Anal. 9 (1996), 439–448. [14] Decreusefond, L., Üstünel, A. S., Stochastic analysis of the fractional Brownian motion. Potential Anal. 10 (1998), 177–214. [15] Duncan, T. E., Hu, Y., Pasik-Duncan, B., Stochastic calculus for fractional Brownian motion I. Theory. SIAM J. Control Optim. 38 (2000), 582–612. [16] Elliott, R. J., van der Hoek, J., A general fractional white noise theory and applications to finance. Math. Finance 13 (2003), 301–330. [17] Flandoli, F., On a probabilistic description of small scale structures in 3D fluids. Ann. Inst. Henri Poincaré 38 (2002), 207–228. [18] Flandoli, F., Gubinelli, M., The Gibbs ensemble of a vortex filament. Probab. Theory Related Fields 122 (2001), 317–340. [19] Garsia, A. M., Rodemich, E., Rumsey, H., Jr., A real variable lemma and the continuity of paths of some Gaussian processes. Indiana Univ. Math. J. 20 (1970/1971), 565–578. [20] Guerra, J., Nualart, D., The 1/H -variation of the divergence integral with respect to the fractional Brownian motion for H > 1/2 and fractional Bessel processes. Stoch. Process. Appl. 115 (2005), 91–115. [21] Hu, Y., Integral transformations and anticipative calculus for fractional Brownian motions. Mem. Amer. Math. Soc. 175 (2005). [22] Hu, Y. , Øksendal, B., Fractional white noise calculus and applications to finance. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 6 (2003), 1–32. [23] Hull, J., White, A., The pricing of options on assets with stochastic volatilities. J. Finance 3 (1987), 281–300. [24] Hurst, H., E. Long-term storage capacity in reservoirs. Trans. Amer. Soc. Civil Eng. 116 (1951), 400–410. [25] Kolmogorov, A. N., Wienersche Spiralen und einige andere interessante Kurven im Hilbertschen Raum. C. R. (Doklady) Acad. URSS (N.S.) 26 (1940), 115–118. [26] Lin, S. J., Stochastic analysis of fractional Brownian motions. Stoch. Stoch. Rep. 55 (1995), 121–140. [27] Lyons, T., Differential equations driven by rough signals. Rev. Mat. Iberoamericana 14 (1998), 215–310. [28] Lyons, T., Qian, Z., System control and rough paths. Oxford Math. Monogr., Oxford University Press, Oxford 2002. [29] Malliavin, P. , Stochastic calculus of variations and hypoelliptic operators. In Proceedings of the International Symposium on Stochastic Differential Equations (Kyoto, 1976), Wiley, New York, Chichester, Brisbane 1978, 195–263. [30] Mandelbrot, B. B., Van Ness, J. W., Fractional Brownian motions, fractional noises and applications. SIAM Review 10 (1968), 422–437. [31] Memin, J., Mishura, Y., Valkeila, E., Inequalities for the moments of Wiener integrals with respecto to fractional Brownian motions. Statist. Prob. Letters 55 (2001), 421–430.

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[32] Nualart, D., The Malliavin calculus and related topics. 2nd edition, Probab. Appl., Springer Verlag, New York 2005. [33] Nualart, D., Stochastic integration with respect to fractional Brownian motion and applications. Contemp. Math. 336 (2003), 3–39. [34] Nualart, D., Pardoux, E., Stochastic calculus with anticipating integrands. Probab. Theory Related Fields 78 (1988), 535–581. [35] Nualart, D., Rovira, C., Tindel, S., Probabilistic models for vortex filaments based on fractional Brownian motion. Ann. Probab. 31 (2003), 1862–1899. [36] Nualart, D., Taqqu, M. S., Wick-Itô formula for Gaussian processes. Stoch. Anal. Appl., to appear. [37] Nualart, D., Taqqu, M. S., Some issues concerning Wick integrals and the Black and Scholes formula. Preprint. [38] Pipiras, V., Taqqu, M. S., Integration questions related to fractional Brownian motion. Probab. Theory Related Fields 118 (2000), 121–291. [39] Pipiras, V., Taqqu, M. S., Are classes of deterministic integrands for fractional Brownian motion on a interval complete? Bernoulli 7 (2001), 873–897. [40] Rogers, L. C. G., Arbitrage with fractional Brownian motion. Math. Finance 7 (1997), 95–105. [41] Shiryaev, A. N., Essentials of Stochastic Finance: Facts, Models and Theory. Adv. Ser. Stat. Sci. Appl. Probab. 3, World Scientific, Singapore 1999. [42] Skorohod, A. V., On a generalization of a stochastic integral. Theory Probab. Appl. 20 (1975), 219–233. [43] Young, L. C., An inequality of the Hölder type connected with Stieltjes integration. Acta Math. 67 (1936), 251–282. Department of Mathematics, University of Kansas, Lawrence, Kansas 66045, U.S.A. E-mail: [email protected]

Atomistic and continuum models for phase change dynamics Anders Szepessy

Abstract. The dynamics of dendritic growth of a crystal in an undercooled melt is determined by macroscopic diffusion-convection of heat and capillary forces acting on length scales compared to the nanometer width of the solid-liquid interface. Its modeling is useful for instance in processing techniques based on casting. The phase field method is widely used to study evolution of such microstructures of phase transformations on a continuum level; it couples the energy equation to a phenomenological Allen–Cahn/Ginzburg–Landau equation modeling the dynamics of an order parameter determining the solid and liquid phases, including also stochastic fluctuations to obtain the qualitative correct result of dendritic side branching. This lecture presents some ideas to derive stochastic phase field models from atomistic formulations by coarse-graining molecular dynamics and kinetic Monte Carlo methods. Mathematics Subject Classification (2000). 82C31, 65C30. Keywords. Phase transformation, phase-field, coarse-grained, molecular dynamics, Brownian dynamics, Langevin equation, Smoluchowski equation, kinetic Monte Carlo.

1. Introduction to phase-field models The phase field model for modeling a liquid solid phase transformation is an Allen– Cahn/Ginzburg–Landau equation coupled to the energy equation ∂t φ = div(k1 ∇φ) − k0 (f (φ) + g (φ)k4 T ) + noise, ∂t (cv T + k2 g(φ)) = div(k3 ∇T )

(1.1)

with a double-well potential f having local minima at ±1, smoothed step function g, temperature T and specific heat cv , cf. [3]. The phase field variable φ : Rd ×[0, ∞) → [−1, 1] interprets the solid and liquid phases as the domains {x ∈ Rd : φ(x) > 0} and {x ∈ Rd : φ(x) < 0} respectively. To have such an implicit definition of the phases, as in the level set method, is a computational advantage compared to a sharp interface model, where the necessary direct tracking of the interface introduce computational drawbacks. This phenomenological phase-field model, with free energy potentials motived by thermodynamics, has therefore become a popular and effective computational method to solve problems with complicated microstructures of dendrite and eutectic growth, cf. [1], [3]. The phase-field model has mathematical wellposedness and convergence to sharp interface results [34]. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Assuming that the reaction term in the Allen–Cahn equation takes a given form, e.g. a standard choice is f (φ) := (1 − φ 2 )2 , 1 5 2 3 1 g(φ) := 15 16 5 φ − 3 φ + φ + 2 , then the parameters k0 , k1 , k2 , k3 , k4 in the phase-field model can be determined from atomistic molecular simulations [19]; an alternative in [1] uses a steeper step function g to easy derive consistency with sharp interface models. The evolution of the phase interface depends on the orientation of the solid crystal; this is modeled by an anisotropic matrix k1 . Added noise to system (1.1) is also important, e.g. to obtain sidebranching dendrites [22] explained in Section 5.4. Phase changes can be modeled on an atomistic level by molecular dynamics or kinetic Monte Carlo methods. This lecture first presents some ideas and questions to derive a stochastic phase field model by coarse-graining molecular dynamics, to determine the reaction term (i.e. f and g) and the noise. This is made in three steps in Sections 2 to 4: to give a precise quantitative atomistic definition of the phase-field variable, to introduce an atomistic molecular dynamics model based on Brownian dynamics, and to derive the dynamics for the coarse-grained phase-field. Section 5 derives stochastic hydrodynamical limits of solutions to an Ising model with long range interaction, i.e. coarse-graining a kinetic Monte Carlo method following [24]. Section 5.4 presents a simple kinetic Monte Carlo method for dendrite dynamics.

2. Quantitative atomistic definition of the phase-field variable The aim is to give a unique definition of the phase-field variable, so that it can be determined precisely from atomistic simulations. The usual interpretation is to measure interatomic distances and use structure functions (or similar methods) to measure where the phase is solid and where it is liquid, which then implicitly defines the phase-field variable [3]. Here we instead use the energy equation for a quantitative and explicit definition of the phase-field variable. The macroscopic energy equation with a phase transformation and heat conduction is ∂t (cv T + m) = div(k∇T )

(2.1)

where m corresponds to the latent heat release. In (1.1) the latent heat determines the parameter k2 , since φ is defined to jump from 1 to −1 in the phase transformation. We will instead use this latent heat to directly define the phase field function, and not only the parameter k2 . The total energy, cv T + m, can be defined from molecular dynamics of N particles with position Xi , velocity vi and mass μ in a potential V , see [20], [18], N |vi |2 cv T + m = + V (X1 , . . . , XN ). μ (2.2) 2 i=1

Atomistic and continuum models for phase change dynamics

1565

Assume that the potential can be defined from pair interactions 1 (Xi − Xj ), V (X) = 2 N

(2.3)

i=1 j =i

where : R3 → R is a molecular dynamics pair potential, e.g. a Lennard–Jones potential σ 12 σ 6 (x) = z1 − z2 . |x| |x| In the macroscopic setting the jump of m in a phase change is called the latent heat, which depends on the thermodynamic variables kept constant: with constant N, T and volume it is called the internal energy andwith constant pressure instead of volume it is called enthalpy. The kinetic energy i μ|vi |2 /2 is related to the temperature. It is therefore natural to let the phase field variable be determined by the potential energy V (X). In a pointwise setting the potential energy can be represented by the distribution N 1 (Xi − Xj )δ(x − Xi ) 2 i=1 j =i

where δ is the point mass at the origin [20]. We seek an averaged variant and we will study a microscopic phase change model where the interface is almost planar in the microscopic scale with normal in the x1 direction. Therefore we take a smooth average and define the phase-field variable by 1 (Xi − Xj ) η(x − Xi ) 2 i=1 j =i

N

m(X, x) :=

(2.4)

mi (X)

where η : R3 → (0, ∞) is a smooth approximation of the point (delta) mass, with scale εi > 0 in the xi direction, η(x) :=

2 2 3 e−|xi | /(2εi )

i=1

(2π εi2 )1/2

.

(2.5)

Smooth averages have been used in molecular dynamics for fluid dynamics, cf. [18] and for the vortex blob method and the smoothed particle hydrodynamics approximation of moving particles in fluid dynamics, cf. [29], [2]. Sections 3–4 present a molecular dynamics model for the potential energy (2.4) and Section 5.4 formulates a kinetic Monte Carlo model. Question 2.1. How accurate is it to say that the (macroscopic) latent heat is equal to a jump in V ?

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3. An atomistic Brownian dynamics model The standard method to simulate molecular dynamics is to write Newton’s laws for the particles, cf. [10], [32]. We will instead use Brownian dynamics with the Ito differential equations

(3.1) dXit = −∂Xi V (Xt )dt + 2γ dWit where Wi are independent Brownian motions and the notation Xit := Xi (t) is the position of the i’th particle at time t. This equation, called the Smoluchowski equation, is the zero relaxation time limit (i.e. τ → 0+) of Langevin’s equation ( cf. [25], [30], [32], [21]) d Xˆ is = pi /μds dpis

ps = −∂Xi V (X )ds − i ds + τ ˆs

2γ μ ˆ s d Wi , τ

(3.2)

in the faster time scale s = μt/τ , where μ is the mass and Wˆ i are independent Brownian motions. The zero relaxation time limit is explained more in Remark 3.2. The simplified Brownian dynamics has the same invariant measure with density proportional to e−V (X)/γ as in Monte-Carlo molecular dynamics simulations of equilibrium problems with γ = kB T , where T is the absolute temperature and kB is the Boltzmann constant. In this sense, the parameter γ /kB in the Brownian dynamics is the local temperature T . In contrast to the standard Monte-Carlo method, the model (3.1) includes the time variable. Our microscopic model of a phase change is then the Brownian dynamics model (3.1) for the phase-field (latent heat) variable m in (2.4) coupled to the macroscopic energy equation (2.1). The Brownian dynamics uses γ := kB T , where the temperature varies on the macroscopic scale, due to the energy equation, so that T is almost constant on the microscopic scale of a molecular dynamics simulation and makes its Gibbs equilibrium density proportional to e−V (X)/(kB T (x)) reasonable. We have two reasons to use Brownian dynamics instead of standard deterministic Newton dynamics (τ = ∞ in (3.2)): the most important reason is to have a formulation that separates the noise and the mean drift, which is a much harder issue in deterministic many particle dynamics, in fact so far the only derivation of the Euler equations of conservation laws from particle dynamics use a weak noise perturbation of a Hamiltonian system in [31]; and the second reason is to try to simulate molecular dynamics longer time. Question 3.1. Is Brownian dynamics a reasonable alternative to standard molecular dynamics here? Remark 3.2 (Smoluchowski limit of the Langevin equation). The Smoluchowski high friction limit of the Langevin equation has been computed with different methods using strong [30] and weak convergence [25]. Strong convergence has the drawback to

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yield error estimates of order eKt τ , due to a Gronwall estimate and Lipschitz bound K of the forces; in contrast, error estimates of probabilities using weak convergence can show good accuracy for long time. The proof that the Langevin solution Xˆ μt/τ converges weakly (i.e. in law) to the Smoluchowski solution Xt as τ → 0+ in [25], [28] uses a Chapman–Enskog expansion of the Kolmogorov backward equation, for the Langevin dynamics in the diffusion time scale t, combined with a general convergence result for such diffusion processes in [26]. Dissipative particle dynamics [15] has dissipation-fluctuation perturbations of a Hamiltonian system where the momentum is conserved, in contrast to the analogous Langevin dynamics. The work [25] also shows that a Smoluchowski type limit seems more subtly for dissipative particle dynamics.

4. Coarse-grained phase-field dynamics We want to determine a mean drift function a(m) and a diffusion function b(m) so that the coarse-grained approximation mt , solving the coarse-grained equation dmt = a(mt )dt +

M

bk (mt )d W˜ kt ,

k=1

is an optimal approximation to the phase field m(X t , · ) defined in (2.4), where X t solves the Brownian dynamics (3.1). Here W˜ k , k = 1, . . . , M are all independent Brownian motions, also independent of all Wi . For this purpose we seek to minimize the error of the expected value at any time T E[g(m(XT , · ))] − E[g(mT )] for any given function g with the same initial value m0 = m(X 0 , · ). Here the expected value of a stochastic variable w, with set of outcomes and probability measure P , is defined by E[w] :=

w dP .

The first idea, in Section 4.1, is that Ito’s formula and the Brownian dynamics (3.1) determine functions α and β, depending on the microscopic state X, so that dm(Xt , · ) = α(Xt )dt +

N

βj (Xt )dWjt .

(4.1)

j =1

The next step, in Section 4.2, is to estimate the error, using the Kolmogorov equations for m and (4.1) similar to [35], [24], which leads to T N M T T E[g(m(X , · ))−g(m )] = E u , α−a +u , βj ⊗βj − bk ⊗bk dt , 0

j =1

k=1

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where u , · is the L2 (R) scalar product corresponding to the variable x with u , which is the Gateaux derivative (i.e. functional derivative) of the functional E[g(mT ) | mt = n] with respect to n; and similarly u , · is the L2 (R × R) scalar product with the second Gateaux derivative u of the functional E[g(mT ) | mt = n] with respect to n. The notation bk ⊗ bk (x, x ) := bk (x)bk (x ) is the tensor product. The final step, in Section 4.3, is to use molecular dynamics simulations for a planar two phase problem and take averages in cross sections parallel to the interface, where , u , a, u k bk ⊗ bk are constant, to evaluate approximations to the functions a and b ⊗ b by k k k T 1 a= E α dt , T 0 T N 1 bk ⊗ bk = E βj ⊗ βj dt . T 0 j =1

k

4.1. The Ito formula for the phase-field. The Ito formula (cf. [13]) implies N N (−∂Xj m ∂Xj V + γ ∂Xj Xj m) dt + 2γ ∂Xj m dWj . dm(X , x) = j =1 j =1 βj (Xt )

t

(4.2)

α(Xt )

The definition in (2.4), m(Xt , x) =

mi (X)η(x − Xit ),

i

yields ∂Xj m =

∂Xj mi η(x − Xi ) + mj ∂Xj η(x − Xj ).

i

In (4.2) we will use (2.5) to evaluate the last derivative as ∂Xj η(x − Xj ) = −∂x η(x − Xj )

∂Xj η(x − Xj ) = −η(x − Xj )

in dt terms,

(x − Xj )1 (x − Xj )2 (x − Xj )3 , , ε12 ε22 ε32

in dWj terms,

in order to avoid spatial derivatives on the diffusion coefficient, while including them in the drift. Since 1 mi = (Xi − Xk ) 2 k =i

and V (X) =

1 (Xi − Xj ) 2 i

j =i

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there holds 1 1 (Xi − Xk )δij − (Xi − Xj )(1 − δij ), 2 2 k =i ∂Xj V (X) = (Xj − Xk ),

∂Xj mi =

k =j

where δij :=

1 0

i = j, i = j

is the Kronecker symbol. The second derivatives are ∂Xj Xj mi η(x − Xi ) − 2∂Xj mj ∂x η(x − Xj ) + mj ∂xx η(x − Xj ), ∂Xj Xj m = i

with

1 1 (Xi − Xk )δij + (Xi − Xj )(1 − δij ) 2 2

∂Xj Xj mi =

k =i

and all terms in (4.2) are now expressed in terms of , its gradient and Hessian . We note that the drift, α, has the form ∂x

N

N n2i (Xt )η(x − Xit ) + n1i (Xt )η(x − Xit )

i=1

i=1

of conservative and non conservative reaction terms. Similarly the diffusion, βj , takes the form N n3i (Xt )η(x − Xit )(x − Xit ). i=1

4.2. The error representation. The conditioned expected value u(n, t) := E[g(mT ) | mt = n]

(4.3)

satisfies the Kolmogorov equation (cf. [35], [24]) M bk ⊗ bk = 0 ∂t u + u , a + u ,

(4.4)

k=1

u( ·, T ) = g Let mt := m(X t , · ). The final condition in (4.4) and the definition (4.3) show that T T T T 0 t E[g(m(X , · )) − g(m )] = E[u(m , T )] − u(m , 0) = E du(m , t) . 0

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Use the Ito formula and (4.2) to evaluate du(mt , t) and Kolmogorov’s equation (4.4) to replace ∂t u in this right hand side to obtain the error representation E[g(m(XT , · )) − g(mT )] T N =E u , α + u , βj ⊗ βj + ∂t u dt 0

=E

j =1 T

u , α − a + u ,

0

N j =1

βj ⊗ βj −

M

bk ⊗ bk dt .

k=1

4.3. Computation of averages in cross sections. The optimal choice of the funcT tion a is to minimize E 0 u , α − a dt , which seems hard to determine exactly since u (m(Xt , · ), t) depends on X t . However, the function u (m(Xt , · ), t) depends only mildly on the coarse-grained m(X t , · ) and not directly on Xt . Therefore a reasonable approximation of this optimum is to think of an expansion of u in α − a and T determine a by the leading order condition E 0 α − a dt = 0, which means that the drift a(x) := a(m( ·, x)) is T 1 a(x) = E α(x) dt , T 0 and similarly for the diffusion matrix 1 d(x, x ) = E T

0

T

N

βj (x) ⊗ βj (x ) dt .

j =1

We expect the spatial averages of the microscopic variables to vary on a much smaller scale in the x1 direction normal to the phase front than in its orthogonal directions. Consequently we use an average function η in (2.4) with higher resolution in the x1 direction, so that 0 < ε1 ε2 = ε3 . In a microscopic simulation the molecular dynamics (3.1) has a small spatial volume, so that ε2 is much larger than the size of the simulation box. Consequently we may first think of α and β depending only on the x1 coordinate. In practice, the drift a and diffusion d can only be determined for a discrete set of points {(x1 (1), x2 (1), x3 (1)), . . . , (x1 (M/3), x2 (M/3)), x3 (M/3))} =: XM and XM × XM , respectively, related to the scales εi . The diffusion coefficient b, as a function of x, can then be obtained from Choleski factorization of the M × M matrix d M bk (x)bk (x ) = d(x, x ). k=1

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T We expect that x1 → T −1 E 0 mt dt is monotone, for fixed (x2 , x3 ), so that its inverse function, denoted by m−1 , is well defined. Then the coarse-grained drift and the diffusion can be obtained as function of m by a(m) := a(m−1 (m)), and similarly for bj . Question 4.1. Will the computed a and b be reasonable? Question 4.2. Can the phase-field method be coupled to the molecular dynamics method for improved localized resolution? Question 4.3. Note that the approximation error E[g(m(XT , · )) − g(mT )] becomes proportional to the variances T E α − a, α − a dt , 0

E 0

T

N j =1

βj ⊗ βj −

M k=1

bk ⊗ bk ,

N

βj ⊗ βj −

j =1

M

bk ⊗ bk dt .

k=1

The first variations ∂u (m(Xt , · ), t)/∂α and ∂u (m(Xt , · ), t)/∂βj determine the factors of proportionality. Can this be used to adaptively determine the resolution scale ε? Remark 4.4. If we integrate the noise term over all x1 , i.e. take ε1 very large, and let g(m) = m2 , then the error E[g(m(XT , · )) − g(mT )] we are studying is the usual fluctuation of energy E[V 2 −E[V ]2 ] (proportional to the specific heat [21]), provided we set m = E[V ].

5. An atomistic kinetic Monte Carlo method Kinetic Monte Carlo methods can also be used to simulate solid-liquid phase changes on an atomistic level, cf. [14]. Here the reaction states and rates are given a priori, which makes it possible to simulate crystal growth on larger time scales than in molecular dynamics. The reaction rates and states can in principle be determined from a molecular dynamics simulations on smaller systems, cf. [37]; however often several reactions are involved making this a demanding modeling task. This section is a short version of [24] and derives stochastic hydrodynamical limits of the Ising model with long range interaction, which is the simplest model of this kind of an stochastic interacting particle system on a square lattice with two possible states in each lattice point, cf. [21]. Define a periodic lattice L := γ Z d ∩[0, 1]d , with neighboring sites on distance γ , and consider spin configurations σ : L × [0, T ] → {−1, 1} defined on this lattice.

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Introduce a stochastic spin system where the spin σt (x), at site x ∈ L and time t, will flip to −σt (x) with the rate c(x, σt ( · ))dt, in the time interval (t, t + dt), independent of possible flips at other sites, cf. [27]. Let σ x denote the configuration of spins after a flip at x of state σ , i.e. σ (y) y = x, x σ (y) = −σ (x) y = x, the probability density P (σ, t) of finding the spin system in configuration σ ∈ {−1, 1}L at time t then solves the master equation dP (σ, t) (c(x, σ x )P (σ x , t) − c(x, σ )P (σ, t)), (5.1) = dt x∈L where the gain term x c(x, σ x )P (σ x , t) is the probability of jumping to state σ at time t and the loss term x c(x, σ )P (σ, t) is the probability to leave state σ . Similar master equations are used for microscopic models of chemical reactions and phase transformations, cf. [36], [14], where lattice sites are occupied by different species of particles. For instance with two species the state space could be {0, 1} × {0, 1} instead of {−1, 1} for the classical spin model above. We want a spin system that has statistical mechanics relevance, which can achieved e.g. by choosing the rate function c as follows. Consider the Hamiltonian 1 J (x − y)σ (x)σ (y) − h(x)σ (x) H (σ ) = − 2 x∈L y=x

J = γ d J0 ,

x∈L

J0 (x) = 0

for |x| ≥ 1, C 2 (Rd ),

is compactly supported and where the long range interaction potential, J0 ∈ the function h ∈ C 2 (Rd ) is a given external field. Define the Glauber Markov process on {−1, 1}L with generator d c(x, σ )(f (σ x ) − f (σ )) (5.2) E[f (σt )|σ ] = Lf (σ ) = dt x∈L

for f : {−1, 1}L → R and the flip rate 1 e−βU (x)σ (x) = 1 − σ (x) tanh(βU (x)) , −βU (x) βU (x) 2 e +e J (x − y)σ (y) =: h(x) + J ∗ σ (x) − J (0)σ (x), U (x) = h(x) +

c(x, σ ) =

(5.3)

y=x

where β > 0 is the inverse temperature. This flip rate has built in invariance of the Gibbs density, e−βH (σ ) / σ e−βH (σ ) , since it satisfies the detailed balance c(x, σ )e−βH (σ ) = c(x, σ x )e−βH (σ ) , x

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which implies that this Gibbs density is a time independent (invariant) solution to (5.1). Having this invariant Gibbs measure implies that the model has statistical mechanics relevance, see [12], [4], [5], [6], [11]. For example in a neighborhood of x ∈ L, where h and J ∗ (1, . . . , 1) are positive, the construction of the flip rate c makes the system favor phases with spins mostly equal to 1 as compared to phases with spins mostly equal to −1. We will study localized projection averages of σ on scale ε. In particular we will find approximations to expected values of such averages. The error analysis uses consistency with the backward equation ∂t u˜ + Lu˜ = 0 for t < T ,

u( ˜ ·,T ) = g

corresponding to the master equation (5.1) for expected values u(ξ, ˜ t) := E[g(σT )| σt = ξ ]. 5.1. A coarse-grained kinetic Monte Carlo method. Define the coarse periodic lattice L := qγ Z d ∩ [0, 1]d with neighboring sites on distance qγ =: ε, where q is an even positive integer and q d is the number of fine sites projected to a coarse site: the lattice points y ∈ L define the coarse cells Cy = {x ∈ L : −qγ /2 ≤ xi − yi < qγ /2}, of q d neighboring points in the fine lattice and the averaging operator 1/q d if x and z are in the same coarse cell Cy , Aε (z, x) = 0 if x and z are in different coarse cells. We will study the behavior of the localized projection averages X(z) :=

(5.4)

Aε (z, x)σ (x),

x∈L

for z ∈ L. The coarse-grained average X can be interpreted as a function on the coarse lattice since the restriction of X to each coarse cell Cz is constant, i.e. X = d x∈C· σ (x)/q . The work [23] derives a coarse-grained kinetic Monte Carlo equation approximating the average X. The next section shows as in [24] that the average spin, X, can be approximated by the solution, X : L × [0, T ] × → R, to the Ito stochastic differential equation dXt (x) = a(Xt )(x)dt + b(Xt )(x)dW x ,

X0 = X0 ,

(5.5)

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Anders Szepessy ¯

¯

¯

¯

with the drift, a : RL → RL , and diffusion, b : RL → RL , coefficients given by a(X) = −X + tanh(β(J ∗ X + h − J (0)X)), γ d/2 b(X)(x) = |1 − X tanh(β(J ∗ X + h − J (0)X))(x)| η(X(x)), ε 1 for x ∈ [−1, 1], η(r) = 0 for x ∈ (−∞, −ˆr ) ∪ (ˆr, ∞),

(5.6)

rˆ := min(1 + e−2β(2|J |1 +hL∞ ) , 3/2) and a Wiener process W : L×[0, T ]× → R on a probability space ( , P , {Ft }Tt=0 ), with the set of outcomes , probability measure P and sigma algebra Ft of events up to time t. Here W x are independent one dimensional standard Brownian motions for x ∈ L, so that formally E[dWtx ] = 0, y

E[dWsx dWt ] = 0 for s = t, y

E[dWtx dWt ] = 0 for x = y, and E[dWtx dWtx ] = dt. The C ∞ cut-off function η : R → [0, 1], with compact support, is introduced to handle the complication that |X(x)| may be larger than 1, although |X(x)| is not, so that 1 − X tanh(β(J ∗ X + h − J (0)X))(x) may be close to zero causing large values on derivatives of

|1 − X tanh(β(J ∗ X + h − J (0)X))(x)| , note that we have |X(x)| ≤ 1 and consequently the cut-off η improves the approximation by switching off the noise before 1 − X tanh(β(J ∗ X + h − J (0)X))(x) becomes zero making b a C ∞ function. ¯ The approximation uses that the high dimensional value function u : RL ×[0, T ] → R defined by u(ξ, t) = E[g(XT )|Xt = ξ ] solves a corresponding Kolmogorov backward equation, where the drift and diffusion coefficients in (5.6) are chosen to minimize the error E[g(XT )] − E[g(XT )]. To define the Kolmogorov backward equation introduce the weighted scalar products wy vy εd for w, v ∈ 2 (L), w · v := y∈L

w · v :=

wxy vxy ε2d

2

for w, v ∈ 2 (L ),

x,y∈L

w · v :=

x,y,z∈L

wxyz vxyz ε3d

3

for w, v ∈ 2 (L ).

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Then u satisfies the Kolmogorov backward equation ∂t u + a · u + D · u = 0

for t < T ,

u( ·, T ) = g,

where Dxy =

(1 − X tanh(β(J ∗ X + h))(x))η2 (X(x)) 0

y = x, y = x,

and u (ξ, t) = ∂ξ u(ξ, t) and u (ξ, t) are the first and second order Gateaux derivatives 2

of u in 2 (L) and 2 (L ), respectively. 5.2. Stochastic hydrodynamical limit of the particle system. The main result in [24] is Theorem 5.1. The average spin, X, can be approximated by the solution, X, to the Ito stochastic differential equation (5.5) with error E[g(XT )] − E[g(XT )] = O T ε + T (γ /ε)2d (5.7) provided that the Gateaux derivatives u (Xt , t), u (Xt , t) and u (Xt , t) on the path X are bounded in the weighted norms 1 (Li ) up to time T . Note that a = 0 gives O(1) error, while b = 0 gives O((γ /ε)d ) error so that b defined by (5.6) is justified for γ ε γ 2d/(2d+1) , with T fixed. The stochastic differential equation (5.5) has C ∞ coefficients, where perturbations of solutions may grove exponentially in time. The work [24] verifies that mean square estimates of X and its variations up to order three give bounds on the weighted 1 -norm of the derivatives of u that depend exponentially on time, i.e. eCT . Proof of the theorem. The definitions of u, the generator (5.2) and the average (5.4) imply E[g(XT )] − E[g(XT )] = E[u(XT , T )] − E[u(X0 , 0)] T =E du(Xt , t) 0

T

=

0

= = 0

T

E E[Lu − a · u − D · u |Xt ] dt

0

T

(5.8) E E c(x, σ ) u(X(σ x )) − u(X(σ )) − a · u − D · u |Xt dt

T

E E c(x, σ ) u(X(σ ) − 2Aε (x, · )σ (x)) − u(X(σ ))

0

E[Lu + ∂t u] dt =

x∈L

x∈L

− a · u − D · u |Xt

dt.

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Anders Szepessy

The first step to estimate this error is to write the differences in u in terms of its Gateaux derivatives by Taylor expansion, for some s ∈ [0, 1], u(X(σ ) − 2Aε (x, · )σ (x)) − u(X(σ )) = −2u (X) · Aε (x, · )σ (x) + 2u (X) · Aε (x, · )Aε (x, · )σ 2 (x) 4 − u (X − 2sAε (x, · )σ (x)) · Aε (x, · )Aε (x, · )Aε (x, · )σ 3 (x), 3 so that the error representation (5.8) becomes E[g(XT )] − E[g(XT )] T = E E u (X) · (−2c(x, σ )Aε (x, · )σ (x) − a) 0

(5.9)

(5.10)

x∈L

+ u (X) · (2c(x, σ )Aε (x, · )Aε (x, · )σ 2 (x) − D) 4 − u (X − 2sAε (x, · )σ (x)) · c(x, σ )Aε (x, · )Aε (x, · )Aε (x, · )σ 3 (x) | Xt dt. 3 The next step is to determine the optimal a and b which minimize the error (5.10). For this purpose we shall in the flipping rate approximate the coupling J ∗ σ and J (0)σ = O(γ d ) with J ∗ X and J (0)X, using the long range O(1) interaction distance of J . The definition of the average (5.4) implies J ∗X = J ( · − y)Aε (y, z)σ (z) z,y∈L

and consequently the coupling has the uniform error estimate J ∗ σ − J ∗ X∞ ≤ J ( · − z) − J ( · − y)Aε (y, z) 1 σ ∞ = O(ε). (5.11) y∈L

This error estimate, the flip rate (5.3) and J (0) = O(γ d ) imply 2c(x, σ )Aε (x, · )σ (x) − x∈L

(5.12)

= −X + Aε · tanh(β(J ∗ σ + h − J (0)σ )) = −X + tanh(β(J ∗ X + h − J (0)X)) + O(ε + γ d ), and

2c(x, σ )Aε (x, · )Aε (x, · )σ 2 (x)

x∈L

=

γ d

1 − X tanh(β(J ∗ X + h − J (0)X)) ε + O (γ /ε)2d + ε + γ 2d .

(5.13)

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We have

Aε (x, · )Aε (x, · ) x A (x − · )A (x − · )A (x − · ) ε ε ε

∞

∞

x

= (γ /ε)d , = (γ /ε)2d ,

(5.14)

which together with the expansions (5.10), (5.12) and (5.13) proves the theorem.

2

We also have Lemma 5.2. Suppose that the Gateaux derivatives u (Xt , t) and u (Xt , t) on the path X are bounded in the weighted norms 1 (Li ) up to time T and that the initial spin σ0 has expected value m, where σ0 (x) − mx are i.i.d. with bounded variance and second order difference quotients |d 2 m/dx 2 | = O(1). Then the deterministic mean ¯ field solution, Xˆ : RL × [0, T ] → R, ˆ ˆ d X/dt = −Xˆ + tanh(β(J ∗ Xˆ + h − J (0)X)),

Xˆ 0 = E[X0 ],

depends on ε only through the initial data and satisfies E[g(XT )] − E[g(Xˆ T )] = O(ε + (γ /ε)d ) provided the drift a is defined by (5.6). Proof. Think of Xˆ as an X with b = 0 and apply the corresponding expansion (5.8), (5.9) and (5.14). Then it remains to verify that the initial data satisfy E[u(X0 , 0) − u(Xˆ 0 , 0)] = O (γ /ε)d , but this is a direct consequence of the central limit theorem and the initial σ0 − E[σ0 ] being i.i.d. with bounded variance. 2 5.3. Alternative invariant measure diffusion for mean exit times. Not all expected values E[g(XT )] can be approximated using the stochastic differential equation (5.5) with Einstein diffusion, due to the required bounds on the derivatives of u; such an example is to determine the expected first exit time τ (Y ) = inf{t : Yt ∈ A} from a neighborhood A of an equilibrium point y ∈ A, where a(y ) = 0 and Y0 ∈ A. Then the expected exit time is exponentially large, i.e. d d limγ /ε→0+ γε log E[τ (X)] and limγ /ε→0+ γε log E[τ (X)] (5.15) are both strictly positive. These expected values are related to transition rates k and E[τ ] = 1/k in simple cases, see [17], [9]. Hanggi et al. [16] have proposed a remedy by approximating the master equation by a different stochastic differential equation with the same asymptotic drift

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but a modified diffusion, to leading order, chosen so that the SDE invariant density d Z −1 e−U/(γ /ε) is asymptotically the same as for the master equation. One perspective on the two different SDEs with Einstein diffusion or invariant measure diffusion is that the two limits, coarse-graining and time tending to infinity, do not commute. Because of (5.15) the theory of large deviations for rare events is relevant for exit times, cf. [9]. Let γ1 := γ /ε. Consider an SDE d/2 ˜

dXt (x) = (a(Xt ) + γ1d c(Xt ))(x)dt + γ1

b(Xt )(x)dWtx ,

with the generator Lf = (a + γ1d c) · f + γ1d D˜ · f ,

Dij = b˜i b˜j δij ;

the idea in [16] is to find c and D so that the corresponding SDE asymptotically has the d same invariant density e−U/γ1 /Z as the master equation. Hanggi et al. [16] determine the diagonal diffusion matrix D˜ and the small contribution to the drift γ1d c by D˜ ii = −ai /Ui , ci = −∂xi D˜ ii ;

(5.16)

note that since a and U have the same zeros, the constructed function D˜ ii is positive in general. The equation (5.16) can be obtained by the WKB expansion d 0 L∗ e−U/γ1 = γ1−d (ai Ui + D˜ ii Ui Ui ) + γ10 (∂i ai + 2Ui ∂i D˜ ii + Uii D˜ ii + ci Ui ) d + γ1d (∂i c − ∂ii D˜ ii ) e−U/γ1 together with the two leading order conditions that the terms of order γ1−d and γ10 vanish; here L∗ is the Chapman–Enskog operator adjoint to L. Consequently the ˜ d choice (5.16) will in general generate an SDE with an invariant density e−U/γ1 /Z, where |U˜ − U | = O(γ12d ). Let us indicate why good approximation of the invariant measure implies that also the expected values, E[τ ], for exit problems related to rare events with large deviations, are accurately computed: the work [9] shows that lim γ1d log E[τ (X)] = inf U (y) − U (y ),

γ1 →0+

y∈∂A

(5.17)

for one stable attracting equilibrium point y ∈ A. The work [24] shows that the exit time (5.17) with SDE’s and invariant measure diffusion is asymptotically the same as for the master equation for the 1D Curie–Weiss model: lim γ1d log E[τ (X)] − log E[τ (X)] = 0 , (5.18) γ1 →0+


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where E[τ (X)] and E[τ (X)] denote the mean exit time for the Hanggi SDE and the Curie–Weiss master equation, respectively. The Curie–Weiss model is a simple adsorption/desorption Ising model with constant interaction potential, cf. Section 5.4. The technique to establish this asymptotic agreement is to use logarithmic (Hopf– Cole) transformations of the two mean exit times, as functions of the initial location, which transforms the corresponding two linear Kolmogorov backward equations to two nonlinear Hamilton–Jacobi equations, cf. [8]. The two processes give rise to two different asymptotic Hamilton–Jacobi equations, however the key observation is that they have the same viscosity solution since they are both convex and have the same set of zeros. 5.4. Dendrites with Einstein diffusion. We see by Theorem 5.1 and Lemma 5.2 that the mean field differential equation solution is also an accurate approximation to the spin dynamics, provided the derivatives of the value function are bounded; this indicates that the stochastic differential equation (5.5) then only offers a small quantitative improvement. However, if the derivatives of the value function are large the mean field solution may give a qualitatively wrong answer, with O(1) error as γ /ε → 0+, while the stochastic differential equation still yields an asymptotically correct limit; such an example is dendrite formation in phase transformations, cf. [22], [19], [3], [14]. Let us try to motivate why the noise in Theorem 5.1 seems applicable to dendrite formation. Dendrite dynamics can be formulated by the phase field method with an Allen–Cahn/Ginzburg–Landau equation coupled to a diffusion equation for the energy, as in (1.1), and by master equations coupled to the energy equation, cf. [14]. Mean field equations related to such a phase field system have been derived from a spin system coupled to a diffusion equation, see [7]. A master equation variant of the molecular dynamics model in Sections 2–4 is to let the coarse-grained potential energy be defined by 1 m(σ, z) := J (x − y)σ (y) − h σ (x)Aε (x, z), 2 x y=x

where A is the average in (5.4), and replace the Glauber dynamics with Arrhenius dynamics. That is, the microscopic dynamics is given by independent spins σ (x) ∈ {0, 1}, for each lattice point x ∈ L, flipping with adsorption rate ca (x) = d0 1 − σ (x) from states 0 to 1, and with desorption rate 1 cd (x) = d0 σ (x) exp − J (x − y)σ (y) − h kB T y =x

from states 1 to 0, where h is a surface binding energy or an external field and d0 is a given rate, cf. [23]. Arrhenius dynamics also satisfies detailed balance with the same

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Gibbs density

e

x

y =x

J (x−y)σ (x)σ (y)/2− x hσ (x) /(kB T )

as for Glauber dynamics. The dynamics for the potential energy variable can then be coupled to the energy equation (2.1) ∂t (cv T + m) = div(k∇T ) by letting the temperature T vary on the coarse-grained scale. The dendrite grows with a positive non vanishing speed. Without noise in the model there is no side branching, while the side branching is present with added noise to the phase field model, cf. [3], or to the mean field model derived in [14]. This noise induced side branching is explained by the high sensitivity with respect to small perturbations at the dendrite tip, cf. [22]. Therefore the derivatives of an appropriate value function are large. Here the value function, u, could for instance measure the total dendrite surface at a fixed time. The inconsistent approximation of the mean field solution could by Lemma 5.2 be explained by having (γ /ε)d u 1 = O(1).

(5.19)

The smallest scale in the problem is the dendrite tip radius ρ; with a bounded value function its derivatives could then be u 1 = O(1/ρ), u 1 = O(1/ρ 2 ), u 1 = O(1/ρ 3 ). Consequently (5.19) yields (γ /ε)d/2 = ρ, so that the noise error for the stochastic differential equation with the Einstein diffusion of Theorem 5.1 would be bounded by (γ /ε)2d u 1 = O((γ /ε)d/2 ), which tends to zero as γ /ε → 0+. Therefore, this adsorption/desorption kinetic Monte Carlo model with long range interaction generates an approximating stochastic differential equation, which could be applicable also to coupling with the energy equation if the derivation remains valid with slowly varying temperature. An essential and maybe more difficult question is to find accurate kinetic Monte Carlo methods for real systems with dendrite dynamics, e.g. using ideas from the molecular dynamics coarse-graining in Sections 2–4.

References [1] Amberg, G., Semi sharp phase-field method for quantitative phase change simulations. Phys. Rev. Lett. 91 (2003), 265505–265509. [2] Beale, J. T., Majda, A. J., Vortex methods. I. Convergence in three dimensions. Math. Comp. 39 (1982), 1–27.


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[3] Boettinger, W. J., Warren, J. A., Beckermann, C., Karma, A., Phase field simulation of solidification. Ann. Rev. Mater. Res. 32 (2002), 163–194. [4] De Masi, A., Orlandi, E., Presutti, E., Triolo, L., Glauber evolution with the Kac potentials. I. Mesoscopic and macroscopic limits, interface dynamics. Nonlinearity 7 (1994), 633–696. [5] De Masi, A., Orlandi, E., Presutti, E., Triolo, L., Glauber evolution with Kac potentials. II. Fluctuations. Nonlinearity 9 (1996), 27–51. [6] De Masi, A., Orlandi, E., Presutti, E., Triolo, L., Glauber evolution with Kac potentials. III. Spinodal decomposition. Nonlinearity 9 (1996), 53–114. [7] Dirr, N., Luckhaus, S., Mesoscopic limit for non-isothermal phase transition. Markov Processes and Related Fields 7 (2001), 355–381. [8] Fleming, W. H., Soner, H. M., Controlled Markov Processes and Viscosity Solutions. Appl. Math. (New York) 25, Springer-Verlag, New York 1993. [9] Freidlin, M. I., Wentzell, A. D., Random Perturbations of Dynamical Systems. Grundlehren Math. Wiss. 260, Springer-Verlag, New York 1984. [10] Frenkel, D., Smit, B., Understanding Molecular Simulation. Comput. Sci. Ser. 1, Academic Press, Orlando, FL, 2002. [11] Giacomin, G., Lebowitz, J. L., Presutti, E., Deterministic and stochastic hydrodynamic equations arising from simple microscopic model systems. In Stochastic partial differential equations: six perspectives, Math. Surveys Monogr. 64, Amer. Math. Soc., Providence, RI, 1999, 107–152. [12] Glauber, R. J., Time-dependent statistics of the Ising model. J. Math. Phys. 4 (1963), 294–307. [13] Goodman, J., Moon, K.-S., Szepessy, A., Tempone, R., Zouraris, G., Stochastic and partial differential equations with adapted numerics. http://www.math.kth.se/~szepessy/ sdepde.pdf [14] Gouyet, J. F., Plapp, M., Dietrich, W., Maass, P., Description of far-from-equilibrium process by mean-field lattice gas models. Adv. in Phys. 52 (2003), 523–638. [15] Groot, R. D., Warren, P. B., Dissipative particle dynamics: bridging the gap between atomistic and mesoscopic simulation. J. Chem. Phys. 107 (1997), 4423–4435. [16] Hanggi, P., Grabert, H., Talkner, P., Thomas, H., Bistable systems: master equation versus Fokker-Planck modeling. Phys. Rev. A 3 (1984), 371–378. [17] Hanggi, P., Talkner, P., Borkovec, M., Reaction-rate theory - 50 years after Kramers. Rev. Modern Phys. 62 (1990), 251–341. [18] Hardy R. J., Formulas for determining local properties in molecular dynamics: shock waves. J. Chem. Phys. 76 (1982), 622-628. [19] Hoyt, J. J., Asta, M., Karma, A., Atomistic and continuum modeling of dendritic solidification, Materials Science Engineering R-Reports 41 (2003), 121–163. [20] Irving, J. H., Kirkwood, J. G., The statistical mechanics of transport processes: IV the equations of hydrodynamics. J. Chem. Phys. 18 (1950), 817–829. [21] Kadanoff, L. P., Statistical physics : statics, dynamics and renormalization. World Scientific, Singapore 2000. [22] Karma, A., Rappel, W. J., Phase-field model of dendritic side branching with thermal noise. Phys Rev. E. 60 (1999), 3614–3625.

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[23] Katsoulakis, M. A., Majda, A. J., Vlachos, D. G., Coarse-grained stochastic processes and Monte Carlo simulations in lattice systems. J. Comput. Phys. 186 (2003), 250–278. [24] Katsoulakis, M., Szepessy, A., Stochastic hydrodynamical limits of particle systems. Preprint, 2005; http://www.nada.kth.se/∼szepessy/ [25] Kramer, P. R., Majda, A. J., Stochastic mode reduction for particle-based simulation methods for complex microfluid systems. SIAM J. Appl. Math. 64 (2003), 401–422. [26] Kurtz, T. G.,A limit theorem for perturbed operator semigroups with applications to random evolutions. J. Funct. Anal. 12 (1973), 55–67. [27] Liggett, T. M., Interacting particle systems. Classics Math., Springer-Verlag, Berlin 2005. [28] Majda, A. J., Timofeyev, I., Vanden Eijnden, E., A mathematical framework for stochastic climate models. Comm. Pure Appl. Math. 54 (2001), 891–974. [29] Mas-Gallic, S., Raviart, P.-A., A particle method for first-order symmetric systems. Numer. Math. 51 (1987), 323–352. [30] Nelson, E., Dynamical Theories of Brownian Motion. Princeton University Press, Princeton, NJ, 1967. [31] Olla, S., Varadhan, S. R. S., Yau, H.-T., Hydrodynamical limit for a Hamiltonian system with weak noise. Comm. Math. Phys. 155 (1993), 523–560. [32] Schlick, T., Molecular modeling and simulation. Interdiscip. Appl. Math. 21, SpringerVerlag, New York 2002. [33] Shardlow, T., Splitting for dissipative particle dynamics. SIAM J. Sci. Comput. 24 (2003), 1267–1282. [34] Soner, H. M., Convergence of the phase-field equations to the Mullins-Sekerka Problem with kinetic undercooling. Arch. Rat. Mech. Anal. 31 (1995), 139–197. [35] Szepessy, A., Tempone, R., Zouraris, G., Adaptive weak approximation of stochastic differential equations. Comm. Pure Appl. Math. 54 (2001), 1169-1214. [36] van Kampen, N. G., Stochastic Processes in Physics and Chemistry. North-Holland, Amsterdam 1981. [37] Yip, S. (ed.), Handbook of Materials Modeling. Springer-Verlag, Berlin 2005. Institutionen för Matematik, Kungl. Tekniska Högskolan, 10044 Stockholm, Sweden E-mail: [email protected]

Competitions and mathematics education Petar S. Kenderov∗

Abstract. Mathematics competitions, together with the people and organizations engaged with them, form an immense and vibrant global network today. This network has many roles. Competitions help identify students with higher abilities in mathematics. They motivate these students to develop their talents and to seek professional realization in science. Competitions have positive impact on education and on educational institutions. Last but not least, a significant part of the classical mathematical heritage known as “Elementary Mathematics” is preserved, kept alive and developed through the network of competitions and competition-related activities. Nevertheless, competitions need to evolve in order to meet the demands of the new century. These and many other items are outlined and discussed in the paper. Mathematics Subject Classification (2000). Primary 97U40; Secondary 97C60. Keywords. Mathematics competitions, olympiads, higher ability students.

1. Introduction Competition is essential and intrinsic to life. Every day, living things in nature and economic subjects in society compete for resources, for better living conditions, and for higher efficiency. The desire to compete in overcoming a challenge is deeply rooted in human nature and has been employed for centuries to help people sharpen their skills and improve their performance in various activities. Competitions, however hotly debated, praised, or condemned, remain central and inherent in education. Both the traditional marking (grading) of students in school and the more innovative measuring of their basic scholastic abilities (implemented by methods such as PISA, TIMSS, or SAT) inevitably create, directly or indirectly, competition among students, among teachers, among schools, and even among whole countries. Heated debates aside, few would deny the positive influence such competitions bring to the process of teaching and learning, and to the overall performance of the educational system. The interaction between competition and education is more complex, however. It is not only that competitions enhance education. Education itself can be viewed as ∗ The author is grateful to the Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, and to his colleagues for their support of (and the involvement with) the Bulgarian system of competitions in mathematics, informatics, and mathematical linguistics. Special thanks to E. Belogay, participant in IMO 1979, for carefully reading and editing the early drafts of this paper.


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preparation of individuals (or groups of individuals, even whole nations) for future competitions. In what follows, we give a brief history of contemporary math competitions and present the state of the art in this area. Then we outline how competitions help identify, motivate, and develop higher-ability and talented students. Next we focus on the impact of competitions on education, on educational institutions and on mathematics as a science. Finally, we pose challenges and identify venues for improvement.

2. Brief history of mathematics competitions It is difficult to trace precisely the origins of mathematics competitions for school students; after all, in-class testing (which often resembles small-scale competitions) has accompanied the school system from its very beginning. In fact, the archetype of some competitions can be found outside school, in the society. Newspapers and recreational journals frequently offer prizes for solving crosswords, puzzles, and problems of a deeper mathematical nature. This practice is widely used today by many mathematical journals that publish problems and give awards to school students who provide good solutions. V. Berinde [2] reports that a primary school math competition with 70 participants was held in Bucharest, Romania, as early as 1885. There were eleven prizes awarded to 2 girls and 9 boys. It cannot be excluded that other competitions were held elsewhere before or after that date too. Nevertheless, the 1894 Eötvös competition in Hungary is widely credited as the forerunner of contemporary mathematics (and physics) competitions for secondary school students. The competitors were given four hours to solve three problems individually (no interaction with other students or teachers was allowed). The problems in the Eötvös competition were specially designed to challenge and check creativity and mathematical thinking, not just acquired technical skills; the students were often asked to prove a statement. As an illustration, here are the three problems given in the very first Eötvös competition in 1894 (the entire collection of problems and their solutions is maintained by John Scholes at www.kalva.demon.co.uk/eotvos.html): P1. Show that {(m, n) : 17 divides 2m + 3n} = {(m, n) : 17 divides 9m + 5n}. P2. Given a circle C, and two points A, B inside it, construct a right-angled triangle P QR with vertices on C and hypotenuse QR such that A lies on the side P Q and B lies on the side P R. For which A, B is this not possible? P3. A triangle has sides length a, a + d, a + 2d and area S. Find its sides and angles in terms of d and S. Give numerical answers for d = 1, S = 6. The Eötvös competition model still dominates the competition scene.

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The year 1894 is notable also for the birth of the famous mathematics journal KöMaL (an acronym of the Hungarian name of the journal, which translates to High School Mathematics and Physics Journal). Founded by Dániel Arany, a high school teacher in Györ, Hungary, the journal was essential to the preparation of students and teachers for competitions (about one third of each issue was devoted to problems and problem solving and readers were asked to send solutions). As noted by G. Berzsenyi in the preface of [3], about 120–150 problems were published in KöMaL each year; about 2500–3000 solutions were received. The best solutions and the names of their authors were published in following issues. This type of year-round competition helped many young people discover and develop their mathematical abilities; many of them later became world-famous scientists. (For more information, see the journal web site, komal.elte.hu.) About the same time, similar development occurred in Hungary’s neighbor, Romania. The first issue of the monthly Gazeta Matematicˇa, an important journal for Romanian mathematics, was published in September 1895. The journal organized a competition for school students, which improved in format over the years and eventually gave birth to The National Mathematical Olympiad in Romania. For legal reasons, the journal was transformed to Society Gazeta Matematicˇa in August 1909. The following year, the Romanian Parliament approved the legal status of the new society and this is considered to be the birthday of the Romanian Mathematical Society [2]. What happened in Hungary and Romania in the late 1800’s was not something isolated and special to these two countries only; most likely, it reflected a much broader trend. Indeed, international collaboration and solidarity were rising steadily and many national math societies were founded around the same time. The Olympic Games were revived in 1896. The First International Congress of Mathematicians took place in Zürich in 1897. Within several decades, other countries started to organize mathematics competitions. In 1934, a Mathematical Olympiad was organized in Leningrad, USSR (now St. Petersburg, Russia).

3. Mathematics competitions today Today the world of mathematics competitions encompasses millions of students, teachers, research mathematicians, educational authorities, and parents, who organize and take part in hundreds of competitions and competition-like events with national, regional, and international importance every year. Even greater is the number of books, journals, and other printed and electronic resources that help students and their mentors prepare for the various types of competitions. 3.1. International Mathematical Olympiad (IMO). Of course, the most important and most prestigious math competition is the International Mathematical Olympiad (IMO) – an annual competition for high school students. Directly or indirectly, all other competition activities in mathematics and sciences are related to the IMO.

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The idea to organize an international mathematics competition crystallized during the Fourth Congress of Romanian Mathematicians in 1956. Paul Jainta [4] points out that “IMO, the pinnacle of competitions among individuals, was the brainchild of Romania’s Tiberiu Roman, an educator of monumental vision.” The first IMO took place in Romania (1959) with participants from seven countries: Bulgaria, Czechoslovakia, German Democratic Republic, Hungary, Poland, Romania, and the Soviet Union (USSR). The second IMO (1960) was organized by Romania as well, but since then it is hosted by a different country every year (except 1980, when no IMO was held). Over the years, the participation grew dramatically: the 2005 IMO in Mexico gathered 513 competitors from 93 countries! Strict formalized rules govern every aspect of the IMO, such as participation, problem selection, assessment of solutions, and distribution of medals (for a description of the IMO, browse erdos.fciencias.unam.mx). Each country sends a team of up to eight (four in 1982; since 1983, six) high-school students, chaperoned by a team leader and a deputy team leader. The competition itself is held on two consecutive days; each day, the students have four and a half hours to solve three problems. Each year, just before the competition, the six problems are selected by an international jury formed by the national team leaders and representatives of the host country. Even though confined to secondary school mathematics, the problems are rather difficult and solving them requires a significant degree of inventive ingenuity and creativity. Each problem is worth seven points, so the perfect score is 42 points. Formally, like the Olympic Games, the IMO is a competition for individuals; participants are ranked according to their score and (multiple) individual medals are awarded. Nevertheless, again as in the Olympic Games, the medals and points obtained by the participants from each country are totaled and the countries are unofficially ranked, providing grounds for comparison between countries. The two days of heavy problem-solving are followed by a social program for all the participants. Students get to know each other, discuss alternative solutions to the competition problems, and make plans for their future, while the team leaders share their experiences and best practices in creating new problems and preparing their students for the competition. With its high standards, the IMO prompts the participating countries to constantly improve their educational systems and their methods for selecting and preparing the students. This yielded a great variety of competitions and mathematical enrichment activities around the world which resists any classification. There are “Inclusive” (open for all) competitions which are intended for students of average abilities, while “exclusive” (by invitation only) events target talented students (a prime example of the second type is the IMO and the national olympiad rounds beyond the first). There are “Multiple-choice” competitions where each problem is supplied with several answers, from which the competitor has to find (or guess, as no justification is required) the correct one. In contrast, “classical style” competitions (like the IMO) require the students to present arguments (proofs) in written form. In “correspondence” com-


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petitions, such as those organized by KöMaL and Gazeta Matematicˇa, the students do not necessarily meet each other, while in “presence” competitions (which form the majority of math competitions) the participants are gathered together, which is believed to provide “equal rights" to all students. There are even mixed-style competitions, with a presence-style first stage and correspondence-style subsequent stages. (We will present some newer styles in more detail later.) Another indication of the importance of the IMO is the fact that other sciences, such as physics, chemistry, and biology, soon followed suit and started international olympiads of their own. Bulgaria organized the first international olympiads in informatics/computer science (1989) and in mathematical linguistics (2003). 3.2. Mathematics competition networks. Like any event with positive social impact, each math competition creates and maintains its network of dedicated people. Numerous math competition journals complement these networks, connecting editorial staff, authors, and readers. Good examples in this direction are Kvant (Russia), Crux Mathematicorum (Canada), Mathematics Magazine and Mathematical Spectrum (UK). The math competition networks range in size from regional to international networks that are associated with large and famous competitions, such as the IMO, Le Kangourou Sans Frontiéres [www.mathkang.org], the Australian Mathematics Competition [www.amt.canberra.edu.au], the International Mathematics Tournament of Towns [www.amt.canberra.edu.au/imtot.html], the Ibero-American Mathematics Olympiad [www.campus-oei.org/oim/], and the Asian-Pacific Mathematics Olympiad [www.cms.math.ca/Competitions/APMO/] – the list is far too short to enumerate all networks that deserve to be mentioned. The different competition networks are not isolated, as many people naturally belong to more than one network. A different and more formal tie is provided by the World Federation of National Mathematics Competitions (WFNMC). The WFNMC was founded in 1984, during the Fifth International Congress of Mathematical Education (ICME5) in Adelaide, Australia. Since then it has a “reserved slot" in the programs of every ICME. Every second year after ICME the WFNMC organizes its own Conference. It has an award, named after Paul Erdös, which is given to people with outstanding contributions to mathematics competitions. The Federation publishes also its journal Mathematics Competitions [www.amt.canberra.edu.au/wfnmc.html] which is another powerful tool for networking people engaged with competitions. In 1994 the WFNMC became an Affiliated Study Group of the International Commission on Mathematical Instruction (ICMI), which, in turn, is a commission of the International Mathematical Union [1]. In this way the competitions networks are incorporated into the global mathematical community. Taken together, these networks form a large global network in the field of mathematics competitions and, more generally, in the classical area known under the (somewhat misleading) name Elementary Mathematics. Like in any other area of science, this network operates and lives through its journals, conferences, and workshops, but the periodical regularity of its math competitions adds to its strength and

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vitality since the people meet more often. In addition, this global network facilitates the dissemination of best practices in curriculum development and in the work with talented youngsters. New problem solving techniques, new classes of problems, and new ideas about organizing competitions spread quickly around the world. We should not forget also that, through this global network, the Elementary Mathematics (which constitutes an important part of our mathematical heritage) is preserved, kept alive and further developed.

4. Why are the competitions needed? Here is a short and incomplete list of reasons on which we expand later on: 1. higher abilities and talent are identified, motivated and developed; 2. what happens before and after the competition is good for education; 3. talented students are steered to careers in science; 4. competitions raise the reputation of an educational institution. 4.1. Finding higher abilities and talent. The educational systems in most countries target mainly students of average mathematical abilities (who form the majority in schools). Additional care is often provided for lower-ability students, so that they could cover the educational standards. The standard curriculum and syllabus requirements pose no significant challenge however to students with higher abilities. They do not feel the need to work hard and, as a result, their mathematical abilities and talent remain undiscovered and undeveloped. This is a pity, of course, since these higher-ability youngsters are a very important resource for the development of society, provided they are properly educated, motivated, and supported. Unlike other natural resources, such as mineral deposits, which remain preserved for the future generations, if undiscovered and unused, the talent of a young person is lost forever, if it is not identified, cultivated, and employed properly. Competitions and other enrichment activities are obvious remedies for this shortcoming, as they allow students to exhibit their abilities and talent. Moreover, competitions motivate the participants to work hard while preparing for them and, as a result, further develop their abilities and talent. 4.2. Before and after competitions. Some opponents to competitions complain that there is no apparent direct connection between the competitions and the mathematics as taught in the classroom. This, in our mind, is a rather narrow approach to the issue. Classroom is only one of the many homes of the educational process. One should take into account the integral impact of competitions and competition-related activities on education. What frequently escapes public attention, which often focuses on a rather small group of happy winners, is the fact that the other, “non-winner” participants,


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also gain a lot. While preparing for the competition, and trying to solve the problems during the competition itself, all participants increase their knowledge significantly. Taking into account that in some competitions hundreds of thousands of students are taking part, the integral impact on the learning of mathematics becomes significant for the overall development of the contemporary society. From this point of view the contribution of the International Competition “European Kangaroo" with more than 3 millions of participants is difficult to overestimate. We should not neglect also what happens in the corridors of the school (or outside the school) after the competition is over. The students are sharing their experiences (successes, failures, new ideas generated, etc.). This has a tremendous educational effect which however is not always given proper attention. The competitions and mathematics enrichment activities can be viewed as events that provide impetus for subsequent discussions among the students (as well as among their friends, parents, etc.). From the viewpoint of acquiring new mathematical knowledge (facts and techniques) these after competition discussions might be as important as the preparation for and the competition itself. Many of us owe a significant part of our knowledge to just such “corridor mathematics". From this point of view the social program after IMO gains additional importance. All this could (and should) have some practical implications for the ways the competitions and other enrichment activities are planned and organized. One should deliberately incorporate possibilities (the more the better) for “after event" discussions, reflections and interactions. There is an unexhausted potential for introduction and sharing new practices in this area. Finally, while preparing their students for competitions the teachers gain experience how to teach mathematical topics that are currently not in the curriculum. This may become important at later stages, if some of these topics become a part of the school program. 4.3. Steering talented students to careers in science. The health and longevity of any social sector depends on how many talented young people are attracted to it. The role of math competitions in identifying talented young people and in attracting them to science should be obvious. Indeed, the fact that a significant number of successful participants in math competitions later become famous scientists was recognized rather early. On 17 July 1929, John von Neumann, who was born in Hungary and was influenced by math competitions, wrote in his letter from Berlin to Professor Lipót Fejér in Budapest ([5]): Dear highly honored Professor, I had the opportunity several times to speak to Leo Szilard about the student competitions of the Eötvös Mathematical and Physical Society, also about the fact that the winners of these competitions, so to say, overlap with the set of mathematicians and physicists who later became well-respected world-figures. Taking the general bad reputations of examinations world-wide into account it is to be considered as a great achievement if the selection works with a 50

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percent probability of hitting the talent. Szilard is very interested in whether this procedure can be applied in the German context and this has been the subject of much discussion between us. However, since we would like above all to learn what the reliable statistical details are, we are approaching you with the following request. We would like: 1. to have a list of names of the winners and runners-ups of the student competitions, 2. to see marked on the list those who were adopted on a scientific basis and those adopted for other work, 3. to know your opinion about the extent to which the prizewinner and the talented are the same people and, for example, what proportion of the former would be worthy of financial support from the State in order to make their studies possible. Very often the future professional realization of a young person is often predetermined by the “first success.” The first area where positive results are achieved often becomes the preferred area in which a person invests time and efforts, which in turn brings more success, stronger motivation, and higher professionalism. Math competitions provide such opportunity for early success and thus help attract good young minds to mathematical and scientific careers. In this way competitions contribute to the development and progress of mathematics and other sciences. 4.4. Raising the reputation of an educational institution. The academic reputation of a university depends primarily on the merit of the intellectual achievements of its academic staff. “The higher the reputation of the professors, the higher the reputation of the university” is the essence of this widely accepted belief. What is often overlooked, though, is that the level of the students also has a significant impact on the outcome of the educational process and, in the long run, on the reputation of the institution. While higher-ability students still have the chance of becoming good professionals if trained by ordinary professors, even outstanding professors can fail to produce high-level specialists from mediocre and unmotivated students. Teachers know well that a few good students in class not only motivate the other students and make them work harder, but also place higher demands on the preparation of the teachers themselves. This two-way challenge influences positively the educational process and improves, directly or indirectly, the reputation of the entire educational institution. It is no wonder that many universities try hard to attract good students. One of the best ways to achieve this is to organize competitions for secondary school students and to offer incentives, such as stipends or entrance exam waivers, to the winners. Such policies usually yield the expected results, as a special type of relationship develops between organizers and the winners during the preparation for the competition, the competition itself, and the post-competition period, which encourages the winners to


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consider seriously (sometimes as the first option) enrolling in the university where the competition (and/or the preparation for it) takes place. In addition to the obvious advantages, enrolling competition winners has a delayed “value-added” effect to the reputation of a university. After graduation, math competition winners, as people with good problem-solving skills, are more likely to get rapid professional recognition, because they are likely to find solutions to difficult and complex real-life problems easier and faster than others. Once their success is noticed and registered by the working environment, the recognition of the problem-solvers’ alma mater increases immediately and almost automatically. As a success story, consider the University of Waterloo, Canada, and the breathtaking rise of its reputation during the seventies and eighties of the last century. Alongside other plausible explanations, such as good management and excellent academic staff, its success can also be attributed to the fact that the University of Waterloo was the host of the Canadian Mathematics Competition [www.cemc.uwaterloo.ca], which attracted a good portion of the best young minds in Canada. The William Lowell Putnam Competition, widely known as the “Putnam Exam” and administered by the Mathematical Association of America, is the flagship of annual competitions for university students in North America. While enrolled at the University of Waterloo, the former winners in school competitions performed consistently well in the Putnam Exam, securing a prominent presence of Waterloo in the top five teams in North America. This also was contributing to the reputation of the institution. It is no wonder that, within less than 20 years, the University of Waterloo became one of the leading centers for mathematics and computer sciences in the world. There is another success story related to the University of Waterloo and the Canadian Mathematics Competition, which shows how a new implementation of an inspiring idea at a new place can yield fantastic results. The Australian mathematician Peter O’Halloran (1931–1994) spent a part of his 1972–73 sabbatical leave from the Canberra College of Advanced Education (now University of Canberra) at the University of Waterloo. There he gained, as Peter Taylor (Executive Director of the Australian Mathematics Trust) recalls ([6]), ... the idea of a broadly based mathematics competition for high school students. On his return he often enthused to his colleagues about the potential value of such a competition in Australia. In 1976, while President of the Canberra Mathematical Association, he established a committee to run a mathematics competition in Canberra. This was so successful that the competition became national by 1978 as the Australian Mathematics Competition, sponsored by the Bank of New South Wales (now Westpac Banking Corporation). It is now well known that this competition has grown to over 500,000 entries annually, and is probably the biggest mass-participation event in the country. The success of Peter O’Halloran was encouraging for others. André Deledicq started in 1991 the Kangaroo Competition in France (the name reveals the Australian

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influence). The Kangaroo Competition is now truly international (albeit with focus on Europe), enjoying more than 3 million participants each year. It is an appropriate place here to pay tribute to Peter O’Halloran, who had the vision for the future of mathematics competitions and knew the strategies how to achieve the goals. He understood the role of international collaboration in this field and was the major force behind the inception of WFNMC and its association with ICMI as an Affiliated Study Group.

5. Competitions and science Before we go any further, we need to consider a natural question: Why are math competitions so good in revealing higher mathematical abilities and inclination to doing research? The simplest and obvious answer seems to be: Because both higher abilities and inclination to doing research are necessary to be successful in a math competition. Necessary, but not sufficient. To be successful in a competition, a student often needs not only a good mind, but a very quick one. Most competitions are limited in time to just 3–4 hours, imposing a significant stress on the nervous system of their participants. Not only do students have to solve the problems correctly, they have to do so quickly and in the presence of their direct competitors. Yet, there are many highly creative students, who do not perform well under pressure. Such “slow thinkers” often come up with new and valuable ideas a mere day (or even just five minutes) after the end of the competition, yet receive no reward or incentive. Traditional competitions disadvantage such students, even though some of them are highly creative and could become good inventors or scientists. Indeed, what matters in science is rarely the speed of solving difficult problems posed by other people. More often, what matters is the ability to formulate questions and pose problems, to generate, evaluate, and reject conjectures, to come up with new and nonstandard ideas. All these activities require ample thinking time, access to information resources in libraries or the Internet, communication with peers and experts working on similar problems, none of which are allowed in traditional competitions. Obviously, other types of competitions are needed to identify, encourage, and develop such special “slower” minds. The competitions should reflect the true nature of research, containing a research-like phase, along with an opportunity to present results to peers – precisely as it is in real science. As a matter of fact, such competitions, designed to identify students with an inclination to scientific (not only mathematical) research, already exist. Below we present three of them.


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5.1. Germany/Switzerland. Jugend Forscht (Youth Quests) celebrated its 40-th anniversary in 2005. It is a German annual competition for students under the age of 21, who work, alone or in teams, on projects of their own. The projects are presented at special sessions, where the winners are awarded [www.jugend-forscht.de]. Switzerland has a similar competition, which is organized by the Schweizer Jugend Forscht (SwissYouth Quests) foundation, established in 1970. The competition, which covers all scientific directions, including social sciences and humanities, has existed since 1967 [www.sjf.ch]. A Google search for the phrase “Jugend Forscht” produced 25 000 hits in 2002; the same search produced half a million hits in 2005! This 20-fold increase speaks for itself, especially since only German language area is included. 5.2. USA. Many such programs exist in the USA. As a matter of fact, Jugend Forscht was originally shaped after the many “Science Fairs” in USA. We mention only one such program here, because it emphasizes mathematics and because it was used as a model for similar programs in other countries. The Virginia-based Center for Excellence in Education (CEE) was founded by Admiral H. G. Rickover in 1983. It has the following goals [www.cee.org]: The Center for Excellence in Education nurtures careers of excellence and leadership in science and technology for academically talented high school and college students. CEE is as well dedicated to encouraging international understanding among future leaders of the world. CEE’s programs challenge students and assist them on a long-term basis to become creators, inventors, scientists and leaders of the 21st century. The major CEE event, sponsored jointly with the Massachusetts Institute of Technology, is the Research Science Institute (RSI) [www.cee.org/rsi/]: Each summer approximately 75 high school students gather for six of the most stimulating weeks of their young lives. Selected from the United States and other nations, these students participate in a rigorous academic program which emphasizes advanced theory and research in mathematics, the sciences, and engineering. Students attend college-level classes taught by distinguished professors. Nationally recognized teachers conduct classes designed to sharpen research skills. In addition, students complete hands-on research with top mentors at corporations, universities, and research organizations. Only outstanding, carefully selected students are admitted to the program. RSI starts with a series of professional lectures in mathematics, biology, physics, and chemistry. The students are paired with experienced scientists and mentors, who introduce them to interesting research topics and share with them the joy and excitement of exploring new territories. The RSI days are filled with research, evening lectures,

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ultimate Frisbee, sport events, etc. At the end of the program, the students present their own research, both in written and oral form, and awards are given to the best performers. The RSI is an international program: almost a third of its students come from other countries. It provides a unique environment for talented students from different parts of the world to meet, live and work together for a relatively long period of time (six weeks seems to be optimal – it is neither too long to become boring nor too short to put unbearable stress). Again, one should not neglect the importance of the networking and friendships fostered by the RSI program for the future development of the participants. The fact that they know each other will make their future collaboration more fruitful. Year after year, the Bulgarian participants in RSI emphasize the social character of the event and the unique atmosphere created during the RSI. 5.3. Bulgaria. Before the 1989 political changes, Bulgaria had a venue for talented young people, very similar to the above-mentioned Jugend Forscht and RSI. It was called Movement for Technical and Scientific Creativity of the Youth (abbreviated in Bulgarian to TNTM). Students worked on individual scientific projects and presented their work on special sessions, where winners were awarded. Like almost everything else related with the youth, the TNTM movement was under the umbrella of theYoung Communists’ League (Komsomol). After the democratization of the Bulgarian society, the Communist League disappeared, along with everything related to it, including TNTM. A decade later it became absolutely clear that actions were needed to revive those activities at the level of contemporary challenges and requirements. The RSI model was adapted to the conditions in Bulgaria and, as one of the “Year of Mathematics” initiatives, the new High School Students’ Institute of Mathematics and Informatics (HSSIMI ) was founded in 2000. Throughout one academic year, the involved high school students (grade 8–12) work on freely chosen topics (projects) in mathematics and/or informatics (computer science). They work individually or in teams and are supervised by a teacher, a university student, a relative, or just any specialist in the field, willing to help. In fact, some recent HSSIMI projects were successfully supervised by former HSSMI participants, who are now university students. Warmly accepted by the mathematical community in Bulgaria, the HSSIMI organizes three major events: two competition-like sessions and a Research Summer School. The sessions are held at the stand-alone Students Conference for High School students in January and at the School Section of the Annual Spring Conference of the Union of Bulgarian Mathematicians(UBM) in April. The latter section is actually the most visited section at the Spring Conference of UBM, attended by university professors, researchers, teachers, parents, and school peers. To participate in the HSSIMI sessions, students submit a written paper with the results of their work. Specialists referee the papers, assess the projects, and suggest improvements. Students present their research at the sessions and winners are


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awarded. As special award, two of the winners are sent to USA in order to participate in RSI. The authors of the best projects are invited to a three-week Research Summer School. During the first two weeks, eminent specialists from universities, research institutes, and software companies give lectures and practical courses in mathematics and informatics. As in similar programs, the main goal of this preliminary training is to expand the students’ knowledge in topics of their interest and to offer new problems for possible projects. During the third week, students hold a High School Students Workshop, where they briefly present their ideas for new projects. For the short period of its existence, the HSSIMI became a valuable addition to the established (and rather densely populated) system of traditional competitions in Bulgaria. As was planned and expected, the HSSIMI attracted students who were not regulars in the traditional competitions. Similar initiatives can be found in other countries. There are positive signs of networking between them as well. Good examples in this direction are the Tournament of Towns Summer Conference and the annual International Mathematics Projects Competition (IMPC) in Kazakhstan. Reflecting more closely the nature and spirit of research process, these kinds of activities also attract excellent minds to mathematics and definitely deserve better recognition and support by the professional mathematical communities around the world.

6. What to do next? In addition to enhancing the traditional math competitions and developing the nontraditional initiatives discussed above, there are other venues for future improvements, such as implementing the current science trends into competitions, targeting other audiences, and supporting and developing the human resources standing behind competitions and other related activities. 6.1. Algorithms in mathematics. The nature of mathematical research has changed significantly since considerable computing power came to the desk of almost every researcher and student. Mathematicians today can conduct complicated numerical experiments, use software for complex algebraic and analytic transformations, find patterns in huge data sets. Like the experiments in other sciences, this could help reject some conjecture or formulate a new one. Thus, research in mathematics became similar to research in the other sciences. All this is based on mathematical algorithms. Algorithmic thinking is getting higher importance and successfully complements the “axiomatic” approach and thinking in mathematics. This change should be duly reflected in the creation and selection of competition problems. Perhaps more problems should be offered at various competitions where algorithms and their properties are focused in order to cultivate algorithmic thinking.

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Otherwise, we will become witnesses of a “brain-drain” and the best young minds will be driven to competitions in informatics. 6.2. Teamwork. Working in teams is a well-established trend in modern science. For centuries, research in mathematics has been a solitary endeavor. Today, we see more and more teamwork in mathematics and, especially, in its applications. This reveals yet another similarity between modern mathematics and the other sciences (where teamwork has traditionally deeper roots). The ability to work in a team is valuable skill that could and should be cultivated early on. Mathematics team competitions could contribute a lot in this direction. There are many such competitions around the world; it only makes sense to make them more popular. 6.3. Competitions for university students. Even though they are not the focus of this paper, mathematics competitions for university students, among other virtues, help attract talented young people to academic careers in mathematics. Some of these university-level competitions are highly respected and have existed for many years. The above-mentioned Putnam Exam is more than 65 years old. The International Mathematics Competition for University Students began in 1994 [www.imc-math.org]. Of course, there are many other such competitions, but their number is still much smaller than the number of competitions for secondary school students, providing plenty of opportunities for new initiatives and international collaboration in this area. 6.4. Teachers and the competitions. In many countries, year after year, some schools consistently “produce” more competition winners than other schools. What is the reason behind this phenomenon? Why are some schools more successful than others? The reasons may be numerous and fairly different in nature. Very often, however, the prominent success of a particular school can be attributed to the dedicated efforts of a single teacher or a small group of teachers. For these excellent teachers, teaching is a vocation, a mission, and not just means to make both ends meet. Such special teachers are real assets for the school and for the whole country. They possess both the necessary scientific ability and the extraordinary personality needed to identify and motivate for hard work the future winners in competitions. Such teachers need special care, though. Their higher scientific ability is acquired very slowly, at the expense of great personal efforts. It is no secret that the success of these teachers depends very strongly on their working environment and on the appreciation by their colleagues and administration. Very often however the actual working conditions in the schools do not support the work and the development of these dedicated teachers. There is a lot that can (and have to) be done in order to improve the situation. For instance, the materials available to the teachers should not include problems and


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solutions only, but also provide didactic instructions for the teachers how to use these materials in their work with higher ability students and what type of reactions and difficulties to expect on the side of students. For this to happen a special research is needed, conducted with the help of professional math educators. Many organizations which are involved with competitions are also organizing seminars and workshops for teachers. There is a valuable experience in many countries in the work with such teachers. The positive results and the problems could be discussed and evaluated with the aim of disseminating the good practices. Teachers are the major human resource for the development of competitions and related activities. Another problem is that often competition-like activities are not “at home” (and therefore not appreciated) both in Mathematics Departments (because “they concern Elementary Mathematics”) and in Mathematics Education Departments (because they are “too mathematical and refer to the relatively small group of talented students”). It is time for both communities (research mathematicians and mathematics educators) to understand their joint interest in supporting competitions and competitions – related activities.

7. Summary Competitions have influenced positively mathematics education and its institutions in different ways for more than a century. Engaging millions of students and educators, math competitions have a distinguished way to identify, motivate, and develop young talent, steering it to careers in science. Mathematics competitions have matured and formed an immense and vibrant global network which contributes significantly to the preservation and the maintenance of mathematical heritage. The flagship IMO not only serves as the “golden standard” for numerous other competitions in mathematics and the sciences (especially with its often-overlooked social program), but it also provides a constant stimulus for improvement of school systems around the world. Traditional competitions are complemented by more inclusive and less known events that emulate more closely real research and engage even broader student audience. Nevertheless, stronger consolidation and collaboration of teachers, schools, universities, and educational authorities is needed in order to meet the challenges of the new century.

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References [1] Bass, H., Hodgson, B. R., The International Commission on Mathematical Instruction. Notices Amer. Math. Soc. 51 (2004), 639–644. [2] Berinde, V., Romania – The Native Country of International Mathematical Olympiads. A brief history of Romanian Mathematical Society. CUB PRESS 22, Baia Mare 2004. [3] Century 2 of KöMaL. V. Oláh (editor), G. Berzsenyi, E. Fried and K. Fried (assoc. editors), KoMaL, Janos Bolyai Mathematical Society/Roland Eotvos Physical Society, Budapest 1999. [4] Jainta, P., Problems Corner: Contests from Romania, EMS Newsletterr 35 (2000), 20–24; www.emis.de/newsletter/newsletter35.pdf. [5] von Neumann, J., Letter to Professor Lipót Fejér in Budapest (1929). In Neumann Archive, OMIKK, Budapest; www.kfki.hu/fszemle/archivum/fsz9905/papp.html. [6] Taylor, P., Obituary: Peter Joseph O’Halloran (1931-1994), Mathematics Competitions, Journal of WFNMC 7 (2) (1978), 12–17; www.amt.canberra.edu.au/obitpoh.html. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Akad. G. Bonchev Street, Block 8, 1113 Sofia, Bulgaria E-mail: [email protected]

Understanding and misunderstanding the Third International Mathematics and Science Study: what is at stake and why K-12 education studies matter Alan Siegel∗

Abstract. The technical portion of this paper concerns a videotape classroom study of eighth grade mathematics lessons in Japan, and how methodological design errors led to conclusions that are refuted by the actual video data. We document these errors, and trace their distillation into one- and two-sentence education policy recommendations articulated in U.S. government position papers, implemented in classrooms across the U.S. and imported by countries around the world. We also present the historical context needed to understand the misrepresentations cited in support of questionable education policy. Mathematics Subject Classification (2000). Primary 97D40; Secondary 97-02. Keywords. Third International Mathematics and Science Study, TIMSS.

1. Introduction The outstanding results for the top-performing countries in the Third International Mathematics and Science Study (TIMSS) have generated widespread interest in best teaching practices around the world. In the TIMSS Videotape Classroom Study by James Stigler et al. [31], the teaching styles in Germany, Japan, and the U.S. were compared in an effort to discover what makes some programs so successful. The conclusions from this comparison are striking and have been widely cited, but often in a highly trivialized and even inaccurate manner. Moreover, this particular study, as we will show, is marred by design errors that raise serious doubt about some of its most influential conclusions. Indeed, it is these very findings that have been cited and accidentally distorted in support of the latest reform programs and education policies – both in the U.S. and elsewhere. For example, it is widely acknowledged (cf. [31, p. 134]) that Japanese lessons often use very challenging problems as motivational focal points for the content being taught. According to the Glenn Commission1 Report [10, p. 16], ∗ The author is grateful to the ICM, NSF and AMS for their support of this presentation. Disclaimer: although the assessments and statements in this paper have been made in good faith by the author, they should not necessarily be viewed as representative of or endorsed by the ICM, NSF or AMS. 1 The commission’s proper name is the National Commission on Mathematics and Science Teaching for the 21st Century. It was chaired by former U.S. Senator and astronaut John Glenn. The year-long Commission was


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“In Japan, . . . closely supervised, collaborative work among students is the norm. Teachers begin by presenting students with a mathematics problem employing principles they have not yet learned. They then work alone or in small groups to devise a solution. After a few minutes, students are called on to present their answers; the whole class works through the problems and solutions, uncovering the related mathematical concepts and reasoning.” We revisit the TIMSS Videotape Classroom Study to resolve the one crucial classroom question that both the Glenn Commission and the TIMSS Classroom Study group failed to address: How can Japanese eighth graders, with just a few minutes of thought, solve difficult problems employing principles they have not yet learned? We will see that the technique required to solve the challenge problem of the day will have already been taught, and that the lesson begins with a review of the fundamental method needed to solve the problem. Students begin working on these problems individually – not in groups. Sometimes group-work is allowed for second efforts on a given assignment, but only after individual seat-work. These lessons include student-presented solutions, but the presentations are closely supervised by the teacher, and the time allocated for this activity is limited so that students will be able to work on a second challenge exercise of the same type, and the teacher will have enough time to show how to apply a fundamental technique as many as ten times – all in a single lesson. Stigler’s videotapes reveal master teaching of substantial content hidden within a warm and inviting teaching style. Students do indeed participate, but in moderation, and subject to the vigilant oversight of instructors who ensure that no one wanders off course. It is also worth noting that the Videotape Classroom Study identified some of the significant differences between the current reform positions and Japanese teaching practices. For example, it pointed out that students did not use calculators in the Japanese classes, and that Japanese teaching has a far higher concentration of proofs and derivations than both reform and traditional programs in the U.S. The Videotape Study also found that Japanese teachers spend more time lecturing than even traditional U.S. teachers. These distinctions not withstanding, the notion that Japanese teaching might be implementing U.S. reforms is given far greater emphasis in a major Government report, which flatly declares: “Japanese teachers widely practice what the U.S. mathematics reform recommends, while U.S. teachers do so infrequently [25, p. 9].”

mandated to develop a strategy to raise the quality of mathematics and science teaching in all of the nation’s classrooms. Unfortunately, the cited quote was, quite possibly, the most substantive paragraph in their report to the nation. The preliminary Glenn Commission report cited Stigler and his TIMSS Videotape Classroom Study as the source of this finding, although the final version omitted the specific citation.

Understanding and misunderstanding TIMSS

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This report on best teaching practices worldwide makes no mention of any differences between the U.S. reforms and Japanese teaching styles. Evidently, its perspective (see also [25, pp. 40–43]) differs from that of its source of primary information, which is the more cautiously worded TIMSS Videotape Study [31]. Moreover, the differences identified in the Videotape Study – which concern direct instruction, calculators, and teacher-managed demonstrations – are all matters of contention in the U.S. debate over classroom reform. Finally, we note that studies of individual classroom lessons – no matter how comprehensive – are necessarily incomplete. They cannot detect how coherent a curriculum might be day-by-day, much less over the course of years, and are illequipped to assess the completeness of a given math curriculum.

2. Background The need for sound – and indeed first-rate – K-12 mathematics programs is well understood. In the U.S., many reform programs have been implemented over the last fifty years, but the evidence shows that on balance, we have made very modest progress toward this goal of world-class math education. The majority of our past reform efforts can be characterized as a tug of war between traditional and student-centric education movements. Just one of these programs was sufficiently different to deserve special mention: the so-called New Math that originated in the 1950s, and which was widely implemented in the ’60s. This reform was pioneered by mathematicians, and was the only program ever to attempt to teach elementary mathematics from an informal set-theoretic perspective. It failed, in part, because its implementations did not provide safeguards to ensure that mainstream American students – and teachers – could handle the material, which is an error that the current reformers have been very careful to avoid. Finally, the program has historical importance because its failure led to a fairly sharp separation between those concerned with K-12 math education and those interested in mathematics research and college teaching. In the mid-1980s, a new version of student-centric learning and teaching began taking hold in the mathematics education community, and it is fair to say that these ideas have swept the American schools of education, and are likewise well represented by advocates in many other parts of the world. In 1989, these ideas were codified into teaching policy when “educators . . . carefully articulated a new vision of mathematics learning and curriculum in the National Council of Teachers of Mathematics’ (NCTM’s) Curriculum and Evaluation Standards for School Mathematics [6].” The 1989 Curriculum Standards [20], together with the follow-up 1991 Teaching Standards [21] and the 1995 Assessment Standards [22] called for a redirection of focus from what to teach grade by grade to new ideas about how to teach and how to assess student progress. And with the publication of these documents, the NCTM completed its transformation from an organization that

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began in the 1920s with ties to the Mathematical Association of America, and that had been led by content-oriented math teachers who endorsed the revolutionary New Math of the ’60s, to an organization led by professors of mathematics education who endorsed a new type of revolutionary math program2 in the ’90s. Loosely put, the theoretical core of this new vision of education is called constructivism. Like most complex social theories, constructivism is founded on a few main principles, has many interpretations and derived consequences, and a bewildering variety of implementations. A thumbnail (and necessarily incomplete) sketch of the main principles of constructivism is as follows. The philosophical basis of constructivism is that everyone learns differently, and that we learn best by integrating new knowledge into our own core understandings and thought processes. Therefore, education is most effective when it engages the learner to become the main agent in the learning process. That is, learning should be engaging in every sense of the word. Since we learn by discovering and by doing, learning is a quintessentially social process wherein through mutual interaction, we organize, communicate, share, and thereby develop deepened understanding. Moreover, content should be based on real-world problems to reach each learner’s core knowledge base, and to maximize the purposefulness of each lesson. As stated, these objectives have merit – especially for teaching younger learners. Indeed, the author believes that the debate over abstract constructivism misses the point. However, the teaching reforms advocated by the NCTM include, in addition to abstract principles, very applied recommendations that have significant impact on curricula, pedagogy and the opportunity for students to learn mathematics. Thus, the real questions concern the content and training provided by the reform program implementations, as well as the consequences of the derivative theories of learning and testing that are put forward as logical consequences of constructivist principles. And it is this debate about what kinds of education programs work that defines the context for the TIMSS Videotape Classroom Study and the classification of Japanese pedagogy. The impact of reform principles on classroom structure and course content. The applied education theories advanced by contemporary reformers must be sketched out if the various assertions about Japanese teaching and the latest reform recommendations are to make sense. The principle of discovery-based learning aims to have the students themselves discover mathematical principles and techniques. According to Cobb et al. [5, p. 28], 2 The NCTM reform program was also endorsed by the federal department of Education and Human Resources, which provided funds to create reform-compliant textbooks, to support their use, and to support studies designed to prove that the new programs were effective. To date, more than $75 million has been allocated to produce these new mathematics textbooks, and about $1 billion has been spent on programs to foster their use. The Educational Systemic Reform programs, for example, ran for nine years with an annual budget of about $100 million, and related programs for K-12 math and science education received comparable funding. More about the history of these programs can be found in [32].


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“It is possible for students to construct for themselves the mathematical practices that, historically, took several thousand years to evolve.” In the 1999 Yearbook of the National Council of Teachers of Mathematics, the article “Teaching Fractions: Fostering Children’s Own Reasoning” by Kamii and Warrington [15] advises: “1. Do not tell children how to compute by using numerical algorithms. . . . 2. Do not tell children that an answer is right or wrong. . . . 3. Encourage children to use their own reasoning instead of providing them with ready-made representations or ‘embodiments.’ 4. Ask children to estimate solutions to problems first because estimation is an effective way to build strong number sense.” To be fair to the authors, it should be pointed out that they provide alternatives to prohibitions 1, and 2. For example, they recommend that the issue of correctness be resolved by the entire class through cooperative discussion. These discovery-based policies are often implemented via the workshop model of teaching where students are seated in clusters of four desks facing each other with no central lecture place in the classroom. This organization is designed to foster collaborative learning and to reinforce the teacher’s role as a “guide on the side” as opposed to the “sage on the stage.” In some programs, the purpose of the teacher is to introduce the exercise of the day. The students then work in groups of four to discover what they can about the problem. In the next phase, the students present their findings to the class, and an active discussion typically ensues. The teacher might have a role that is confined to being a moderator to maintain order in the discussions. Likewise, some of the programs feature unsupervised group-work with the teacher serving mainly as a passive observer. In the higher grades, the U.S. discovery-based programs feature markedly diminished content depth, and the project-based texts exhibit poor coherence in their management of topics and offering of reinforcement exercises. To date, some reform programs simply omit material that does not fit within this model. Moreover, this style of teaching, absent sufficient guidance from the teacher, is typically very time consuming, and the slow pace cannot help but limit the curriculum. For example, on page 315 of a tenth grade reform geometry textbook [37], exercise 24 asks the student to draw an equilateral, an isosceles, and a scalene triangle, and to draw the medians and observe the outcome in each case. The assignment also asks the students to measure the lengths of the medians and the distance from the vertices of each triangle to its centroid. The problem finishes by asking, “What do you conclude?” No proofs are offered or requested, and for good reason. The study of similar triangles begins in Chapter 13 on page 737, where the final two chapters of the book present content that is less observation-based. In 2001, I was invited to observe some of these workshop model classes at a magnet high school in lower Manhattan. In one of the ninth grade classes, the lesson

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problem of the day was (in mathematical terms) to determine the equation of a line through the origin that does not intersect any additional points on the integer Cartesian lattice in R2 . The students began the exercise working unsupervised in groups of four. Then the class convened as a whole to discuss their findings with the teacher serving as moderator. The tenth (or so) student to speak observed that if the line were to intersect another lattice point, then it would have a rational slope. The teacher then called on another student, and this key observation was soon lost. The discussion devolved into an unsuccessful effort to understand the difference between rational numbers with finite decimal representations and those with repeating decimal expansions, and the math period ended with no solution to either question. In a televised eleventh grade lesson [24] from a reform textbook series [7], students seated in groups of four were given the following problem. The teacher displayed boards of different lengths, widths, and thicknesses suspended between pairs of bricks. A karate expert, he explained, can deliver the tremendous energy necessary to break a strong board. For the first part of the lesson, the students were asked to determine a formula for the energy necessary to break a board as a function of its length and thickness. The students discussed the question with great enthusiasm. There was no evidence of any physical modeling, and it was not clear if the class knew Hooke’s law or not. In the second portion of the lesson, a representative from each group presented the group’s thoughts to the class. The first to speak was able to intuit that a longer thinner board would be easier to break, but nevertheless went on to opine that the formula for the energy E, as a function of the length L and thickness T , should be E = L + T . Another group thought that the formula should be E = kLT , where k is a constant that depends on the physical properties of the wood. In the next portion of the lesson, students were given strands of dried spaghetti to form a bridge between two tables, pennies to use as weights, and a paper cup plus paper clip to suspend on the strand(s) of spaghetti. They then conducted tests with different lengths and strand counts to see how many pennies were necessary to break the spaghetti – thus measuring the breaking force, which was misrepresented as energy. The use of multiple strands served to emulate different thicknesses (albeit incorrectly). Data was gathered for 1 to 5 strands, and distances of 2 to 5 inches. Then the students used their graphing calculators under the supervision of the teacher to determine the best fit for the data, which was E = 10 TL , where E is measured in pennies, T in spaghetti strands, and L in inches. The TV program closed by noting that with the introduction of this new curriculum, grades were higher, and more students were electing to take three and four years of math classes. Of course, the stacking of spaghetti strands to model thicker pasta constitutes a fundamental conceptual error. It is no accident that plywood is manufactured with bonded layers, and as straightforward mathematical modeling shows, strength, in a simple model of deformation, is proportional to the square of a beam’s thickness. Likewise, the confusion between force and energy ill serves the students, as does the lesson’s implication that mathematics might be an experimental science.

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The reforms seeking to maximize engagement include mandates to avoid drill and – by extension – the kind of practice necessary to instill knowledge transfer to long-term memory. In concrete terms, the reform programs do not teach the multiplication table in elementary school. Ocken reviewed all of the printed materials produced by one of the elementary school reform programs [33] for grades K-5, and found fewer than 30 problems asking students to multiply two whole numbers, both of which contain a digit greater than five[23]. This program implements the reduced emphasis on pencil and paper calculations as recommended in the 1989 NCTM Standards, and, as recommended, supports student work with calculators even in the earliest grades. Likewise, the standard place-based rules for multiplication of multidigit integers are no longer taught as essential material. Opponents of these reforms see the structure of place-based multiplication as precursor knowledge that helps the learner internalize the more abstract operations of polynomial arithmetic. In one textbook series [16], the division of fractions was simply omitted from the curriculum. And long division is long gone from these programs. To maximize engagement, reformers recommend that problems and content be situated, which means that exercises, derivations and even theorems should be presented in an applied context whenever possible. More generally, abstraction and symbolic methods are eschewed. Of course, the foregoing comments about abstraction and symbolic methods are just words. In order to understand them, we again take a few quick peeks into the reform textbooks to see how these theories and recommendations are turned into practice. For example, one ninth grade reform book [8] has, scattered among its 515 pages, only 25 pages that even contain an equal sign. Of these, only pages 435 and 436 actually concern algebra. The totality of the information about algebra is on page 436, and is as follows. “Some such equations are easier to solve than others. Sometimes the particular numbers involved suggest tricks or shortcuts that make them easy to solve. In each of the equations below, the letter x stands for an unknown number. Use any method you like to find the number x stands for, but write down exactly how you do it. Be sure to check your answers and write down in detail how you find them. x =7 5 4 x+1 = 3 6

x 72 = 6 24 5 19 = 13 x

x 11 = 8 4 2 =6 x

x 5 = 7 3 9 x = ” x 16

The preference for encouraging ad hoc “tricks” and “shortcuts” instead of teaching systematic methods is evident. Indeed, the text does not present any methods for solving these problems. The passage also illustrates how the these new programs encourage students to write expository explanations and avoid teaching students to

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develop and record logical solution strategies based on correct operations, problem decomposition, and the layered application of systematic methods. On page 416 of this ninth grade text, problem 3 reads as follows. a.

b.

c.

d.

3. Consider the following pairs of figures. In each case, state whether you consider the shape to be the same or not, and why.

The chapter goes on to explore some of the most elementary properties of similarity, but the development is probably closer to the level of sixth grade than ninth, and the overall content of the textbook is far weaker than, say, the standard sixth grade books used in Singapore [13]. The comparison with the Singapore books is worthy of elaboration. In anAmerican Educator article [1], the mathematician Ron Aharoni writes about what he learned using the Singapore math program to teach first grade in Israel. He points out that these lessons encourage students to describe problems in words, and feature more discussion than is common in traditional programs. These characteristics are consistent with some of the constructivist principles. There are, however, fundamental differences between this teaching style and the applied recommendations and prohibitions that characterize – and indeed define – the latest reform practices. Aharoni describes how he actively teaches insights based on his mathematical knowledge – even in first grade. And he also points out that significant reinforcement is necessary to help first graders integrate this first grade content into their own thinking. Interestingly, the fifth and sixth grade Singapore texts [13] exhibit a transition from this verbal/expository approach of reasoned problem representation to an informal but precise prealgebra. The books present – with many detailed examples – a kind of pictorial algebra, where a physical segment might be used to represent an unknown length. The modeling allows graphical unknowns to be added, subtracted, and multiplied and divided by integers in physical representations of equations. Students solve many carefully constructed word problems with this modeling process and its physical representation of variables. This representation is used to strengthen intuition and understanding as preparation for variables and algebra. By the sixth grade, the students are using the method to solve sophisticated word problems that would challenge U.S. high schoolers. No U.S. reform text presents such a coherent curriculum, and none provides a systematic increase of content and problem depth chapter-by-chapter and over the course of years to build deepening layers of understanding on behalf of the learner. In terms of pedagogy, Aharoni emphasizes the importance of deep content knowledge and a deep understanding of what is being taught as prerequisites for deciding how to teach a particular topic [1, p. 13]. He says that the understanding of fundamental mathematical principles can be taught, but this instruction requires active teaching by a very knowledgeable teacher.


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The current reform programs, by way of contrast, aim to teach less, not more. In a ninth grade reform algebra text, for example, students receive enough training to solve for x in the equation y = 3x + 2, but there is just one equation in the book that uses variable coefficients. This one exception, which is on page 748 reads [9]: “Show how to derive the quadratic formula by applying completing the square to the general quadratic equation, ax 2 + bx + c = 0.” This question requires a tremendous leap in skill given the text’s limited use of equations with variable coefficients. Moreover, the presentation on completing the square is so weak that it is inconceivable how any but the most exceptional student could learn enough to solve this problem. The totality of the exposition reads: “Here’s an example of how to use completing the square to solve the quadratic equation x 2 + 6x − 2 = 5. Since − 2 doesn’t make x 2 + 6x a perfect square, it is in the way. Move it to the other side: x 2 + 6x = 7. Add 9 to both sides to make the left side a perfect square: x 2 +6x +9 = 16. Write the left side as a perfect square: (x + 3)2 = 16.” There is no attempt to teach a systematic approach for completing the square, or to explain how the magical 9 was selected for use in this particular case. The avoidance of abstraction and symbolic coefficients, and the recommendations against teaching systematic methods have undermined the quality of the textbook. This instance of teaching by one explicit example cannot instill wide-spread understanding. And the inclusion of the exercise to derive the quadratic formula (which is just about the last problem in a very long book) would appear to be based less on it being an appropriate exercise than on the need to include the topic in the curriculum.3 Ralston recommends the outright abandonment of pencil and paper calculations in favor of mental arithmetic supplemented by calculators [26]. Non-reformers disagree, and suggest that proficiency in arithmetic is not taught for its own sake but rather to strengthen the learner’s core knowledge and intuition as a prerequisite for understanding fractions. Arithmetic fluency is even more important for a mastery of and fluency in algebra, where the rules of arithmetic are revisited at an abstract level with the introduction of variables and exponents. Many teachers report that those who lack a grounding in the concrete operations of arithmetic experience great difficulty with algebra and its manipulation of symbols. Other non-reformers argue that the written record of pencil and paper problem solving documents a student’s approach to a problem, which can be reviewed by the student and the teacher for conceptual errors as well as computational mistakes. Non-reformers also argue that it is the use of the written record that allows learners to combine fundamental steps into more 3 It is also fair to say that some of the most project-based reform texts are designed around sequences of typically unrelated projects, which result in a disorganized and incomplete curriculum with very few review and reinforcement exercises (cf. [33], [8], [7], [9]).

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complex solutions that are too detailed to retain as mental calculations. In addition, it is argued that the written representations of algebra bring a precision of expression, of computation and of modeling that surpasses the written word in accuracy, clarity, and simplicity. The purpose of this inside review of American mathematics education was to identify the controversies arising from the latest reforms in concrete (i.e. situated) – as opposed to abstract – terms. It is time to explain why Japanese pedagogy has become a topic of worldwide interest, and to investigate how well it aligns with the latest reform principles. The Third International Mathematics and Science Study. TIMSS is an enormous umbrella project that seeks to measure academic achievement around the world, and which includes many subsidiary studies that analyze a host of related issues in an effort to determine how best to improve math and science education. TIMSS began in 1994–95 with the testing of 400,000 students worldwide at grades four, eight, and twelve. It has grown into a quadrennial program that conducted additional testings and data acquisitions in 1999 and 2003, and has already begun to lay the groundwork for the next round in 2007. The program now includes nearly fifty countries, and the studies cover a large number of independent projects with publications in the many thousands of pages. Although there have been some fluctuations in the TIMSS rankings over the last decade, and the participating countries have varied to some degree over time, the overall results have been much more consistent than not. This fact is probably a testament to the meticulous effort to maintain balanced student samples from the participating countries, and the care that is exercised in the testing protocols and data analyses. The project also deserves very high marks for adhering to a wonderfully high standard of scholarship. The research projects produce not only reports of findings but also detailed documentation of the data acquisition and analysis procedures and indeed every aspect of project methodology. When feasible, these studies even publish enough raw data for independent researchers to review every step of the research effort for independent assessment. Despite the wealth of information provided by the TIMSS publications, it is fair to say that two specific TIMSS findings have captured the majority of the headlines, and have had the greatest influence on classroom practice and education policy. The most eye-opening results come from the achievement scores of students around the world. For example, in the little multicultural, multilingual, top-performing country of Singapore, some 46% of the eighth graders scored in the top 10% of the world. And 75% of their students placed among the top 25% of all eighth graders worldwide. Just 1% of their students placed among the bottom 25% of all eighth graders around the world. This is a stunning achievement. Singapore has indeed shown what it really means to have an education system where no child is left behind. Moreover, these performance results have held up with remarkable consistency in each of the TIMSS testing rounds. Just a notch down from Singapore, the next group


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of top performers have been Korea, Hong Kong, Chinese Taipei (formerly known as Taiwan) and Japan (mostly in this order) with Flemish Belgium trailing somewhat behind, but consistently next in line. The U.S. scores are also worth mentioning. Roughly put, American fourth graders and eighth graders scored somewhat above the international average. But at the twelfth grade, the U.S. scored at the bottom of the industrialized world, and only significantly out-performed two countries: South Africa and Cyprus. No other country fell so far so fast. There was also a more sophisticated twelfth grade test that was reserved for twelfth graders in advanced math programs in the participating countries. On that test, the U.S. was next-to-last; even Cyprus performed significantly better. For completeness, it should be noted that the twelfth grade testing has not been repeated since 1995 and the U.S. plummet just described. This is unfortunate because the lack of follow-up testing forces us to infer whether the American mathematics programs have recovered from the results documented in 1995. Moreover, the real purpose of a K-8 program is to prepare students for subsequent study as opposed to an eighth grade TIMSS test. So our understanding of mathematics education around the world would be greatly enhanced by a schedule of testing that includes grade twelve as well as grades four and eight.4 In view of the absence of follow-up twelfth grade testing, one could speculate that the American TIMSS scores might show that the newest programs are beginning to make a difference. After all, the latest math reforms are often introduced at the earlier grades first, and then extended by one grade level per year. Could it be that U.S. high school students are performing better now because more of them are participating in reform math programs? The answer seems to be a clear no. A variety of studies5 have documented very little progress in high school math achievement over the last decade. To date, the NAEP scores, for example, have been most notable for their lack of improvement. In short, TIMSS testing shows that the US, and indeed most of the world have K-12 mathematics programs that are nowhere near the quality of the best programs worldwide. These results constitute a compelling argument for continued testing on an international scale. Simply stated, TIMSS is one of our best mechanisms for identifying unforeseen weaknesses in national programs, and for discovering exemplary programs that can be investigated in an effort to improve domestic teaching. The other finding that has generated enormous impact can be traced to “TIMSS Videotape Classroom Study: Methods and Findings from an Exploratory Research Project on Eighth-Grade Mathematics Instruction in Germany, Japan, and the United States” [31]. For convenience, we condense the TIMSS Videotape Classroom Study’s name to TVCS. 4 For countries such as Singapore, which do not have a twelfth grade, the testing might well be given at the

completion of the secondary education system. 5 See, for example, Too Little Too Late: American High Schools in an International Context by William H. Schmidt. In Brookings Education Policy papers 2003 (ed. by Diane Ravitch), pp. 253–277.

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The Videotape Classroom Study documentation. During 1994–95, the TVCS team recorded 231 eighth-grade mathematics lessons in Germany, Japan and the U.S. The TVCS project report by Stigler et al. [31] contains an extensive analysis of these tapes and a description of the data acquisition and analysis methodologies. Stigler and James Hiebert subsequently conducted a joint study of Japanese training in pedagogy, which has strong cultural traditions that are surprisingly different from the programs of teacher development in the U.S. [30]. In 1999, Hiebert and Stigler began a second TIMSS videotape classroom study [11] that covered a broader selection of higher performing countries. These videotape study projects produced a variety of supporting documentation [34], [35], [36], [14], [12], but the follow-up study did not record a new series of Japanese lessons and instead relied on the earlier tapings. We cover the main findings from the second study and the differences in its methodology and conclusions (which may well have resulted from criticisms of the earlier project), but will focus primarily on the 1995 TVCS, which remains the far more influential of the two publications. The 1995 project produced a publicly available videotape [34] that begins with Stigler presenting an overview of the Japanese lessons that is very similar to the description already quoted from the Glenn Commission Report. It then shows carefully selected representative excerpts of the geometry and algebra lessons recorded in Germany, Japan, and the U.S. The German and American lesson samples were produced in addition to the original 231 recordings, which are not in the public domain due to confidentiality agreements. The Japanese excerpts were selected from the original 50 tapings recorded in Japan, and disclosure permissions were obtained after the fact. The TIMSS videotape kit includes a guide to the excerpts [36] and a CD ROM [35] is available with the same excerpts, but without Stigler’s introduction.

3. What the Japanese video excerpts show Geometry. The tape shows the Japanese geometry lesson beginning with the teacher asking what was studied the previous day. After working to extract a somewhat meaningful answer from the class, he himself gives a summary: Any two triangles with a common base (such as AB in Figure 1) and with opposing vertices on a line parallel to the base (such as the line through D, C and P ) have the same area because the lengths of their bases are equal, and their altitudes are equal. The teacher states this principle and uses his computer graphics system to demonstrate its potential application by moving vertex Figure 1 P along the line CD. The demonstration

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shows how to deform triangle ABP in a way that preserves its area. Next, he explains that this principle or method is to be the “foundation [36, p. 136]” for the forthcoming problem, which he then presents. It is the following. Eda and Azusa each own a piece of land that lies between the same pair of lines. Their common boundary is formed by a bent line segment as shown. The problem is to change the bent line into a straight line segment that still divides the region into two pieces, each with the same area as before. Despite the previous review, the probEda lem is still going to be a challenge for Azusa eighth graders, and it is fair to infer that the teacher understands this very well. In geometry, one of the most difficult chalFigure 2 lenges in a construction or proof is determining where to put the auxiliary lines. These lines are needed to construct the angles, parallel lines, triangle(s), etc. that must be present before a geometry theorem or principle can be applied to solve the problem. For the exercise in Figure 2, the key step is to draw two crucial auxiliary lines. One defines the base of a triangle that must be transformed in a way that preserves its area. The other is parallel to this base, and runs through its opposing vertex. So what should a master instructor do? The answer is on the tape. After explaining the problem, the teacher asks the students to estimate where the solution line should go, and playfully places his pointer in various positions that begin in obviously incorrect locations and progress toward more plausible replacements for the bent line. Now here is the point. With the exception of two positions held for about one second (which come shortly after the frame shown in Figure 4), none of his trial placements approximate either of the two answers that are the only solutions any student will find.

Figure 3

Figure 4

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Rather, they are all suggestive of the orientation for the auxiliary lines that must be drawn before the basic method can be applied. He is giving subtle hints, and calling the students’ attention to the very geometric features that must be noticed if the problem is to be solved. It is surely no accident that the teacher pauses with his pointer placed in two particular locations far longer than anywhere else. One of the locations is shown in Figure 4. The other is parallel to this placement, but located at the opposing vertex, which forms the bend in the boundary between Eda and Azusa. Only after this telling warm-up – the heads-up review of the solution technique necessary to get the answer, and the casual discussion loaded with visual cues about what must be done – are the children allowed to tackle the problem. But this is not the end of the lesson, and the students only get an announced and enforced three minutes to work individually in search of a solution. As the children work, the teacher circulates among the students and gives hints, typically in the form of leading questions such as: “Would you make this the base? [The question is] that somewhere there are parallel lines, okay [36, p. 140]?” He then allocates an additional 3 minutes where those who have figured out the solution discuss it with the other teacher. Weaker students are allowed to work in groups or to use previously prepared hint cards. The excerpt does not show what happens next. The TIMSS documentation [36] reports that students prepare explanations on the board (9 minutes). Then a student presents his solution. The construction is clearly correct, and he starts out with a correct explanation. But when the time comes to demonstrate the solution, he gets lost and cannot see how to apply the area preserving transformation that solves the problem. The teacher then tells him to use “the red triangle” as the target destination. The advice turns out to be insufficient, and the teacher steps in to redraw the triangle that solves the problem, and calls the student’s attention to it with the words, “over here, over here.” The student seems to understand and begins the explanation afresh. But he soon winds up saying, “Well I don’t know what I am saying, but . . . ” He then regains his confidence, and the presentation comes to an end without additional explanation. A number of students say that they do not understand. Figure 5

Then another student explains her answer, but the presentation is omitted from the tape. According to the Moderator’s Guide [36, pp. 139–41], these two student presentations take altogether less than three minutes. Next, the teacher explains how to solve the problem. There are two equivalent answers that correspond to moving the middle vertex in Figure 1 to the left or right. Both directions solve the problem, and he shows this. For completeness, we also show the two ways that the triangle transformation technique can be used to solve the problem. In order to make the connection between

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Figure 6

Azusa Eda

Eda

Azusa

the review material and the challenge problem absolutely clear, the problem and its two answers have been rotated to present the same perspective as the triangle transformations in Figure 1, which began the day’s lesson. Evidently, no one devised an alternative solution method.

Figure 7

In his discussion of the solution, the teacher points out [36, p. 141] that this line straightening technique eliminates one of the two corners at the base of the triangle in Figures 6 and 7. This observation exposes a subtlety in that the corner that is eliminated is not the apex of the triangle, which is the point being moved to straighten out the line. The lesson then continues with the teacher posing a new problem that can be solved with the same technique. This time the figure is a quadrilateral, and the exercise is to transform it into a triangle with the same area. At this point, the basic solution method should be within a student’s reach, although the problem still requires a sound understanding of the basic method. There is also added difficulty due to the need to recognize that two Figure 8 consecutive sides of the quadrilateral should be viewed as representing the bent line of Figure 2, and that the other two sides should be extended as auxiliary lines to recast this new problem into a version of the Eda–Azusa exercise. The basic line straightening method can be applied so that any one of the four vertices can serve as the point where the line bends, and this designated vertex can be shifted in either of two directions to merge one of its two connecting sides with one of the auxiliary lines. The students again work individually for three minutes, and then are allowed to work in groups, use hint cards or ask the teacher. Figure 9 The TIMSS documentation indicates that this joint phase lasts for 20 minutes, and includes student presentations of their answers. There are apparently eight such presentations, which were selected to illustrate all eight ways the basic method can be applied: there are four vertices that can each be moved two ways. Then the teacher analyzes these eight ways in greater depth,

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and explains how they all use the same idea. All students remain seated during this portion of the lesson, and he controls the discussion very carefully and does almost all of the speaking. For homework, the teacher asks the students to transform a five-sided polygon6 into a triangle with the same area. An analysis of the teaching and its content. This lesson is nothing less than a masterpiece of teaching, and the management of classroom time is remarkable. Although many students did not solve the first problem of the day, the assignment certainly succeeded in engaging the attention of everyone. The second problem was no giveaway, but it gave students the chance to walk in the teacher’s footsteps by applying the same ideas to turn a quadrilateral into a triangle. The teacher-led study of all possible solutions masked direct instruction and reinforcement practice in an interesting and enlightening problem space. Evidently, no student ever developed a new mathematical method or principle that differed from the technique introduced at the beginning of the lesson. Altogether, the teacher showed how to apply the method 10 times. Yet the lesson is an excellent example of how to teach problem solving, because each successive problem required a complete understanding of the basic proof technique. The homework assignment is yet another application of the same method, and gives everyone a chance to revisit the lesson of the day once more. It also hints at the use of induction. It is also worth pointing out that this geometry lesson, which is a specific application of measure preserving transformations, has additional uses. It appears, for example, in Euclid’s proof of the Pythagorean Theorem (cf. Book I Prop 47 of Euclid’s Elements).7 More advanced exercises of this type appear on national middle school mathematics competitions in China and regional high school entrance examinations in Japan. And it is not much of a stretch to suggest that measure preserving transformations lie at the heart of those mysterious changes of variables in the study of integration. All in all, the lesson is a wonderful example of the importance of a deep understanding of fundamental mathematics. Algebra. The Japanese algebra lesson begins with student-presented answers for each of the previous day’s six homework problems [36, p. 114]. These activities, along with the accompanying classroom discussion are omitted from the excerpts. Then the teacher presents a more challenging problem that uses the same basic calculation method that the students have been studying, but needs one common-sense extension. The problem is this. 6 The problem probably should be restricted to convex figures; otherwise it includes irregular cases that are difficult to formalize. On the other hand, this concern is just a minor technicality that has no effect on the pedagogical value of the problem. 7 In fact, the technique is central to Euclid’s development of area in general, which is based on transforming any polygon into a square with the same area. And the natural extension of this problem became a question for the ages: how to square the circle.


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There are two kinds of cakes for sale. They must be bought in integer multiples; you cannot buy a fraction of a cake. The most delicious cake costs 230 yen, and a less tasty one is available for 200 yen. You wish to purchase 10 cakes but only have 2,100 yen. The problem is to buy 10 cakes and have as many of the expensive cakes as possible while spending no more than 2,100 yen. The reproduction of the six homework exercises as shown in the TIMSS Moderator’s Guide [36, p. 114] confirms that the class was already experienced with the technical mechanics necessary to solve problems with inequalities. Evidently, prior lessons had also covered word problems and the translation of word problems into equations and inequalities. Indeed, the teacher introduces the problem with the remarks, “Today will be the final part of the sentence problems [36, p. 159].” Thus, it is fair to infer that the only difference between the cake problem and the material they had just reviewed is the requirement that the solution use whole numbers of cakes. After making sure that the students understand the problem, he asks them to devise a way to solve it. They get an announced and enforced three minutes. Next, the teacher solicits solution approaches from the students. A student volunteers that she tried all possibilities. Her approach was to try 10 cheap cakes, then 9 cheap ones and 1 expensive one, etc., until she had the best answer. However, she was unable to finish in the three minutes that the teacher allocated for the problem. The teacher emphasizes the point, and it will soon become clear that part of the lesson is to show that this unstructured approach is unsound. He then briefly discusses another way to solve the problem. The approach, which is quite inventive, uses a notion of marginal cost. If we buy 10 of the most expensive cakes, we exceed our budget by 200 yen. Trading in an expensive cake for a cheaper cake gives a net savings of 30 yen. Evidently, seven cakes have to be traded in, which shows that the answer is three expensive cakes and seven cheaper ones. As the teacher expected [36, p. 164], no student solved the problem this way. Then he calls on another student, who explains how she set up the problem as an inequality, solved it as an equality, and then rounded the number of expensive cakes down to the nearest lesser integer. As she explains the equation, he writes it on the board. Only a few students understand the explanation, and he asks for another explanation of the same process. In subsequent activities that are only summarized on the tape and in the Moderator’s Guide, the teacher then passes out a worksheet and works through a detailed analysis of the solution for the class. After the detailed presentation, another problem of the same type was assigned, but with larger numbers. The teacher’s words are telling: “If you count one by one, you will be in an incredibly terrible situation. In the same way that we just did the cake situation, set up an inequality equation by yourself and find out . . . [the answer]. Because finding the answers one by one is hard, I wonder if you see the numerous good points of setting up inequality equations . . . ”

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The students worked on the problem individually. After 11 minutes, the teacher went over the problem with the class. The class ended with the teacher summarizing the solution technique that constituted the lesson of the day. The video excerpts contain no group-based problem solving in this algebra lesson, and the Moderator’s Guide confirms that none of the class time included problem solving in groups. An analysis of the teaching and its content. Students never developed new solution methods. In the algebra class, the students were given the opportunity to learn first-hand why ad hoc trial-and-error approaches (which are encouraged by some of the latest reform recommendations) do not work. Although the tape does not explicitly show how many students were able to solve the original cake problem in the allotted time, the student responses suggest that no more than five could have possibly succeeded. But the three minutes of struggle might well have served to make the lesson more purposeful. From a mathematical perspective, the cake problem was designed to require a deep understanding of inequality problems and their solutions. Mathematicians would say that when we solve a problem, we find all of the answers. If the cake problem had allowed fractional purchases, and had simply required that altogether any mix of ten cakes be purchased for at most 2100 yen, then the algebraic formulation would read, 230x + 200(10 − x) ≤ 2100, where x is the number of expensive cakes purchased, and 10 − x is the number of the inexpensive ones. The problem would also require that x be non-negative, since you cannot buy negative quantities of cake. A little manipulation gives: 0≤x≤

10 . 3

Now, the point is that every x in this interval is a solution to the simplified problem, and every solution answer, to the problem is in this interval. So if we want a special 10 . If we want is the place to look. If we want the largest x, it is the interval 0, 10 3 3 the largest integer x, it is 3. And if we wanted the largest even integer, for example, we would look nowhere else but into 0, 10 3 to conclude that this answer is x = 2. Incidentally, a complete answer must also observe that the number of inexpensive items must be non-negative. This problem variant is more than a matter of common sense; it exposes students to a deep understanding of solutions to inequalities and the implications of real world constraints. Moreover, the problem illustrates the idea of decomposing a complex exercise into a more basic problem whose solution can then be adapted to achieve the original objective. Evidently, the video excerpts feature challenge problems that cover fundamental principles, techniques, and methods of systematic thought that lie at the heart of mathematics and problem solving. As such, they ought to provide experiences that


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build a powerful foundation of intuition and understanding for more advanced material yet to come. As a derivative benefit, these problems are so rich they can be readily transformed into follow-up exercises for use as reinforcement problems in class and as homework. Both lesson excerpts exemplify a multi-round teaching and reinforcement pedagogy that begins with review of the fundamental (and systematic) principle that is the key to solving the challenge problem. The review is followed by two or three rounds (when homework is counted) that feature equivalent problems, often with additional educational content. Between each round, the teacher guides the students through the solution process to open the eyes of each learner to the basic idea, and to give the students yet another chance to apply the technique by themselves and to integrate the material into their own understanding – all in an engaging style without rote or tedium.

4. What can be deduced about Japanese teaching? Many publications claim that the Japanese lessons teach students to invent solutions, develop methods and discover new principles. For example, this view is expressed in the Glenn Commission report [10, p. 4], and is clearly stated in TVCS as well: “[In Japan, the] problem . . . comes first [and] . . . the student has . . . to invent his or her own solutions [31, p. vi].” In fact, TVCS reports that the 50 Japanese lessons averaged 1.7 student-presented alternative solution methods per class [31, Figure 22, p. 55]. Yet the excerpts exhibit no signs of such activity. They contain just one student-devised solution alternative, and it failed to produce an answer. These differences are fundamental, and they should be reconciled. Part of the problem is that students are unlikely to devise their own solutions when the time is limited, the problems are so difficult that hints are needed, and the exercises are (clearly) designed to teach the value and use of specific techniques. Students would presumably have a better chance of finding alternative solution methods for less challenging exercises. And they would have an even better chance with problems that can be solved by a variety of methods that have already been taught. Examples might include geometry problems where different basic theorems can be used, and studies of auxiliary lines where the exercises are designed so that different auxiliary lines build different structures that have already been studied. TVCS illustrates alternative solution methods with the U.S. assignment to solve x 2 + 43x − 43 = 0 by completing the square and by applying the quadratic formula [31, p. 97]. Of course, this problem directed students to use different methods they already knew. The example contains no hint of any discovery. So the question remains: where are the alternative solution methods, and when do they demonstrate signs of student-discovery? The answers are in TVCS. It presents the actual examples that were used to train the data analysts who counted the “Student Generated Alternative Solution Methods”

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(SGSM1, SGSM2, . . . ) in each lesson. The training lessons, it turns out, were the Japanese excerpts that we have just analyzed. The two student presentations for the Eda–Azusa problem are coded as SGSM1 and SGSM2 [31, p. 26–27]. Similarly, the second problem, where each of four vertices could be moved in two directions, has the codings SGSM1–SGSM8. Altogether, this lesson is counted as having 10 student-generated alternative solution methods, even though it contains no studentdiscovered methods whatsoever. And the failed try-all-possibilities approach in the Japanese algebra excerpt is counted as yet another student-discovered solution method. (See also “Teacher and Students Presenting Alternative Solution Methods [36, pp. 161–163].”) TVCS also contains a partial explanation for the source of these judgments. It reports that the data coding and interpretation procedures were developed by four doctoral students – none of whom were in mathematics programs [31, p. 24]. Moreover, TVCS states that the project’s supporting mathematicians only saw coder-generated lesson tables, and were denied access to the actual tapes [31, p. 31]. It is reasonable to infer, therefore, that they did not participate in the design of these coding practices. As for the question of invention, TVCS explains: “When seat-work is followed by students sharing alternative solution methods, this generally indicates that students were to invent their own solutions to the problem [31, p. 100].” Altogether, there appears to have been a sequence of misinterpretations that counted student presentations as alternative solution methods, which became student-generated, and then invented and which ultimately evolved into invented discoveries that might even depend on new principles the students had not yet learned ([31], [25], [10]). On the other hand, the contributions by the Japanese teachers received much less generous recognition. Yet in the defining examples of student discovery, the teachers – not the students – manage the ideas and lead the education process. Additional statistics from the TIMSS projects. It is worth reiterating that in the sample Japanese lessons, students began working individually – and not in groups – on each of the four representative exercises. Similarly, the Stigler–Hiebert analysis [30, p. 79] states that “Students rarely work in small groups to solve problems until they have worked first by themselves.” TVCS contains no comparable statement, and even implies otherwise: “[After the problem is posed, the Japanese] students are then asked to work on the problem . . . sometimes individually and sometimes in groups [31, p. 134].” However, not one of the 86 figures and bar charts documents instances where problems began with students working in groups. Chart 41 [31, p. 78] indicates that of the seat-work time spent on problem solving, 67.2% of the time comprised individual effort and 32.8% of the time was spent in group-work. Another TIMSS study addressed this issue in the statistics it gathered for a carefully balanced sampling of 3750 or so eighth graders from each participating country. One of its questionnaires asked teachers about their classroom organization and whether most of their lessons included students working in small groups, individually, as a class, etc. The results, which were weighted by the number of students

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in each responding teacher’s class, are reproduced below for the U.S. and Japan [3, pp. 154–155]. Percent of Students Whose Teachers Report Using Each Organizational Approach "Most or Every Lesson"

Country

Work Together as Work Together as Work in Pairs or Work in Pairs or Work Individually Work Individually Small Groups with a Class with Small Groups a Class with Students without Assistance Teacher Teaching with Assistance without Assistance Assistance from from Teacher from Teacher Responding to One the Whole Class Teacher from Teacher Another

Japan United States

22 r

22

78 r

49

27 r

50

15 r

19

7 r

26

1 r

12

An "r" indicates teacher response data available for 70−84% of students.

Figure 10

The table shows that Japanese lessons do not have significant numbers of small-group activities. In fact, American classes evidently contain about 4 times as many such lessons. Of course, it should be noted that the data is based on questionnaires and depends, therefore, on the judgment of each respondent. The meaning of “most or every lesson” might have cultural biases, as might the definitions of “small groups” and “teacher assistance.” Still, these TIMSS statistics support the notion that the Japanese style of teaching is substantially different from many of the U.S. reform practices. Placing Japanese teaching in the context of U.S. reform. The video excerpts show Japanese lessons with a far richer content than the corresponding offerings from the U.S. and Germany. TVCS reports that the eighth-grade lessons recorded in Japan, Germany, and the U.S. covered material at the respective grade levels 9.1, 8.7, and 7.4 by international standards [31, p. 44]. We suspect that the interactive nature of the teaching style, the coherent, concept-based exercises with disguised reinforcement problems, the motivated direct instruction, and the deep understanding of the teachers all contribute to the quality of the Japanese curriculum. Additional analysis shows that 53% of the Japanese lessons used proof-based reasoning, whereas the comparable statistic for the US lessons – which included both traditional and reform programs – stood at zero [31, p. vii]. And comparisons evaluating the development of concepts – including their depth and applicability – and the overall coherence of the material likewise judged the Japanese programs to be vastly superior [30, p. 59]. By all evidence, the use of proof-based reasoning as reported in Japan is not at all representative of the reform programs in the U.S., and the use of such remarkably challenging problems is beyond the scope of any American program past or present. When comparing U.S. reform practices and Japanese teaching methods, TVCS offers somewhat guarded conclusions that are sometimes difficult to interpret: “Japanese teachers, in certain respects, come closer to implementing the spirit of current ideas advanced by U.S. reformers than do U.S. teachers. For example, Japanese lessons include high-level mathematics, a clear

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focus on thinking and problem solving, and an emphasis on students deriving alternative solution methods and explaining their thinking. In other respects, though, Japanese lessons do not follow such reform guidelines. They include more lecturing and demonstration than even the more traditional U.S. lessons [a practice frowned upon by reformers], and [contrary to specific recommendations made in the NCTM Professional Standards for Teaching Mathematics] 8 we never observed calculators being used in a Japanese classroom [31, p. vii].” Subsequent elaboration on the similarities between U.S. reform and Japanese pedagogy recapitulates these ideas in the context of various reform goals, but again offers no statistical evidence to compare with the data accumulated from the analysis of Japanese teaching practices [31, pp. 122–124]. Consequently, it is difficult – absent additional context – to compare these reform notions in terms of mathematical coherence, depth, international grade level, or the preparation of students for more advanced studies and challenging problems. And no matter what “the spirit of current reform ideas” may mean, it is clear that Japanese and U.S. reform pedagogies differ in their management of classroom time, their use of proof-based reasoning, their tradeoffs between student-discovery and the use of direct instruction, as well as their use of individual and small group activities. For completeness, we note that TVCS makes a distinction between the idealized goals as prescribed in the NCTM Professional Standards for Teaching Mathematics, and as embodied in actual classroom practices of some reform programs. In particular, TVCS discusses two reform-style lessons. One involved students playing a game that was purported by the teacher as being NCTM compliant, but happens to have very little mathematics content: “It is clear to us that the features this teacher uses to define high quality instruction can occur in the absence of deep mathematical engagement on the part of the students [31, p. 129].” The other lesson was deemed compliant with the spirit of NCTM reforms. It began with the teacher whirling an airplane around on a string. The eighth graders then spent the period working in supervised groups to determine the speed of the plane, and came to realize that the key issues were the number of revolutions per second, and the circumference of the plane’s circular trajectory. The problem also required a realization that units conversions would be needed to state the speed in miles per hour. The problem engaged the class, and a variant to compute the speed of a bird sitting on the midpoint of the string was evidently a challenge. The homework for this math class was a writing assignment: the students were asked to describe the problem, to summarize their group’s approach, and to write about the role they played in the group’s work [31, p. 127]. TVCS did not evaluate this lesson or the homework in terms of international grade level or its coherence within a curriculum. 8 The bracketed additions are elaborations from page 123 of TVCS, where the discussion of calculator usage is reworded and thereby avoids the slight grammatical misconstruction we have caused with the unedited in-place insertion.


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Other characterizations of Japanese classroom practices. Studies that use human interaction as a primary source of data must rely on large numbers of interpretations to transform raw, complex, occasionally ambiguous, and even seemingly inconsistent behavior into meaningful evidence. Given the complexity of the lessons, it is not surprising that different interpretations should arise. TVCS – to its credit – documents an overview of these decision-making procedures, although the actual applications were far too numerous to publish. Moreover, TVCS actually contains widely diverse observations, ideas, and conclusions that sometimes get just occasional mention, and that are necessarily excluded from the Executive Summary. Understandably, this commentary is also missing – along with any supporting context – from the onesentence to one-paragraph condensations in derivative policy papers (cf. [25], [10]). Perhaps the seventh and eighth words in the opening line of the TVCS Executive Summary explain this issue as succinctly as possible: “preliminary findings [31, p. v].” It is now appropriate to explore these larger-picture observations and to place them within the context of actual lessons. TVCS even offers some support for our own observations: “[Japanese] students are given support and direction through the class discussion of the problem when it is posed (figure 50), through the summary explanations by the teacher (figure 47) after methods have been presented, through comments by the teacher that connect the current task with what students have studied in previous lessons or earlier in the same lesson (figure 80), and through the availability of a variety of mathematical materials and tools (figure 53) [31, p. 134].” Unfortunately, these insights are located far from the referenced figures and the explanations that accompany them. The words are effectively lost among the suggestions to the contrary that dominate the report. It is also fair to suggest that the wording is too vague to offer any inkling of how powerful the “support and direction through class discussion” really was. Similarly, the value of the connections to previous lessons is left unexplored. This discussion does not even reveal whether these connections were made before students began working on the challenge problems, or after. For these questions, the video excerpts provide resounding answers: the students received masterful instruction. The Math Content Group analyzed a representative collection of 30 classroom lesson tables. Their assessments, as sampled in TVCS, agree with our overall observations, apart from the use of hints, which were mostly omitted from the tables. These analyses are highly stylized with abstract representations for use in statistical processing and were, presumably, not intended to be a reference for the actual teaching.9 9 For example, the analysis of the excerpted geometry lesson consists of a directed graph with three nodes, two links and nine attributes. The first node represents the basic principle (attribute PPD) illustrated in Figure 1. The node’s link has the attributes NR (Necessary Result) and C+ (Increased Complexity). It points to a node representing the first challenge exercise. The representations were used to get a statistical sense of various

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Another sentence in TVCS begins with teachers helping students, but ends with students inventing methods. “The teacher takes an active role in posing problems and helping students examine the advantages of different solution methods [however, rather than elaborating on how this takes place, the sentence changes direction with the words], but the students are expected to struggle with the mathematical problems and invent their own methods. [31, p. 136].” This interpretation of student work as inventive discovery appears throughout TVCS. In its analysis of the excerpted Japanese geometry lesson, TVCS categorizes the teacher’s review of the basic solution method (shown in Figure 1) as “APPLYING CONCEPTS IN NEW SITUATION [31, Figure 63, p. 101],” but inexplicably switches tracks to count the student applications as invented student-generated alternative solution methods. Another such instance reads, “students will struggle because they have not already acquired a procedure to solve the problem [31, p. 135].” Similarly, TVCS never explains how teachers participate in the problem solving by teaching the use of methods and by supplying hints. Its only discussion about hinting is to acknowledge the offer of previously prepared hint cards [31, pp. 26–30]. And by the time the Glenn Commission finished its brief encapsulation of student progress, even the struggle had disappeared along with proper mention of extensive teacher-based assistance.

5. The matter of pedagogy Having sequenced through the Japanese lesson excerpts to determine exactly what took place in the classrooms, we now compare these applied teaching practices with current reform principles.10 One of the most important differences between these two approaches to teaching concerns discovery-based learning. As with any idealized theory, the real issue is how well it works in practice. Discovery-based lessons can make sense – in moderation – provided suitable safeguards are in place. In particular: • Judgments must resolve how much time is needed for students to discover the mathematics, and the necessary tradeoffs among time for guided discovery, time for additional (or deeper) lessons, and time for practice. • There must be detection/correction mechanisms for incomplete “discoveries”. • There must be allowances for the fact that in even the best of circumstances, only a few students will succeed in discovering non-trivial mathematical principles. The lesson excerpts reveal a teaching style that is surprising and very different from the U.S reforms – in theory and practice. In the Japanese classes, the time allotted for the first round of grappling with problems is remarkably modest. Consequently, the broad-brush characteristics of the lessons [31, pp. 58–69]. 10 See [2] for an enlightening albeit jargon heavy exposition on the differences between the theories of learning advanced by educators and by cognitive psychologists.


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remaining time is sufficient for teacher-assisted student presentations to help identify conceptual weaknesses, and for direct instruction to present new insights, as well as for follow-up problems designed to solidify understanding. Due to the time limitations and the difficulty of the more challenging problems, many students will be learning via a model of “grappling and telling.” That is, most students will struggle with a tough problem in class, but not find a solution. They will then learn by being told how to solve it, and will benefit by contrasting their unsuccessful approaches against methods that work [27]. There is no question that preliminarily grappling with a problem is both motivational and educational (cf. [4, p. 11] and [27]). And discussions to understand why some approaches fail, to understand why a solution might be incomplete, and to explore alternative problem solving techniques are all sound investments of class time. However, the use of grappling and telling raises the implementation question: Who should do the telling? In some teaching practices, the theory of discovery-based learning is extended to include the notion of cooperative learning, which holds that the students should teach one another because they “understand” each other. In contrast, the TIMSS videotape and the data in Figure 10 show that Japanese teaching is by no means purely or principally based on cooperative learning. Although students get to explain their solutions, the video excerpts show that Japanese teachers are by no means passive participants. Student explanations frequently need – and get – supervision, and students can be remarkably incoherent (cf. Figure 5) even when their solutions are absolutely perfect. When all is said and done, the teachers do the teaching – and the most important telling – but in an interactive style that is highly engaging and remarkably skillful. Stigler and Hiebert report that the lessons do not adhere to a fixed organization. Some lessons feature more direct instruction or extended demonstrations, whereas others demand that the students memorize basic facts [30, pp.48–51]. Students might even be asked to memorize a mandate to think logically [30, p. 49]. Aharoni’s article on experimental math programs in Israel deserves mention in this context. In the late 1970s, Israel developed a unique and nearly unrecognizable adaptation of the 1960s New Math, which is still in use to this day. The curriculum has been controversial; Israel had placed first on the original 1964 precursor to the TIMSS exams, and had fallen to 28th place on TIMSS 1999. Of course, this small country has experienced demographic shifts and many other sources of instability, so this drop in rank is by no means proof that the curriculum has failed, but there were other concerns about the program, and the TIMSS results gave little reason to believe that all was well. Israel was just months away from adopting the latest U.S. reform standards when circumstances led to a reconsideration and the decision to test a program based on translations of the Singapore textbooks (from English). Aharoni is participating in this experiment, and writes about his experiences with these textbooks. He argues that teachers must have a deep knowledge of fundamental mathematics if they are to instill a sound understanding of elementary arithmetic. His first-grade

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teaching uses deep insights to provide a purposeful understanding of the most basic arithmetic operations. For example, he guides first-graders through story problems designed to open their eyes to the many different ways that a single operation – such as subtraction – can be used in the modeling of problems so that all students will enter the higher grades with the intuition and core knowledge necessary to master the translation of word problems into the native language of algebra. Only time will tell if the program is successful, but if so, his observations would have implications about best practices and teacher training. This perspective places high demands on teachers and – by extension – on schools of education. Currently, most education programs allocate modest resources for courses on mathematics content, and very few programs are prepared to offer the kind of deep applied understanding that Aharoni describes. Instead, schools of education typically emphasize courses on developmental psychology, learning theory, and related topics such as authentic assessment, which is a grading practice based on portfolios of student work such as a study of how ancient Greek geometry was used 2000 years ago, or on real-life applications of periodicity – as opposed to exams. Similarly, very few mathematics departments feature course offerings on deep knowledge for K-12 instruction. This problem is further compounded by the certainty that most education majors would not have attended K-12 programs where such deep understanding would have been taught. A small, but highly respected and widely cited comparative study by Liping Ma gives additional insight into this problem. In her study, American and Chinese elementary school teachers were asked to compute 1 43 ÷ 21 , and to give a physically meaningful problem where the answer is determined by this computation. In the U.S., only 43% of those questioned performed the calculation correctly, and just one of the 23 teachers provided a conceptually correct story problem. In China, all 76 teachers performed the calculation correctly, and 80% came up with correct story problems [17]. In contrast, Hiebert and Stigler came to very different conclusions about how best to foster world-class teaching. They began with the TVCS tapes and findings, and conducted new investigations into Japanese teaching traditions. Their findings are published in The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom [30]. According to the authors, “differences” such as “teaching techniques, . . . and [teaching] basic skills [versus teaching for] conceptual understanding . . . paled” in comparison to the differences they observed in the culture of teaching. In their view, the Japanese tradition of life-long reflection on how to teach, and the culture of teachers sharing these ideas among each other in a continuing process of professional development was more significant than any of these other issues, which comprise the entirety of the debate over education reform in the U.S. and elsewhere. That is, they opined that the Japanese practices of ongoing collaborative- and selfimprovement were even more important than the current state of the Japanese art of teaching as well as the curriculum differences reported in their book. However, in a follow-up videotape classroom study of teaching in Australia, the


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Czech Republic, Hong Kong, Japan, the Netherlands, Switzerland, and the United States, Stigler and Hiebert came to different conclusions [11]. For this study, new data coding schemes were developed to replace those used in the 1995 TVCS. Two of the findings are particularly noteworthy. First, the new study does not mention student-invented or student-discovered solution methods, and instead of reporting an average of 1.7 student-presented solution alternatives per Japanese lesson, the new study reports that 17% of the Japanese problems featured presentations of alternative methods [11, p. 94], and that students had a choice of methods in 31% of the lessons. Second, the study found no unifying theme to explain why the stronger countries perform so well. According to the authors: “A broad conclusion that can be drawn from these results is that no single method of teaching eighth-grade mathematics was observed in all the relatively higher achieving countries participating in this study [12, p. 11].” “It was tempting for some people who were familiar with the 1995 study to draw the conclusion that the method of teaching mathematics seen in the Japanese videotapes was necessary for high achievement [11, p. 119].” Evidently, this positional retreat (see also [11, p. 1]) must include Stigler, Hiebert, and the Glenn Commission, among others. And the fact that the follow-up videotape study did not report student-discovered mathematics suggests that the earlier finding of student discoveries was inaccurate. These changes in understanding notwithstanding, the earlier TVCS and the followup The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom will almost certainly outlive the more recent Hiebert–Stigler classroom study. These earlier publications continue to make must-read lists on education, and continue to inspire calls for reforms based on their findings. For example, on November 21, 2005, a New York Times editorial titled “Why the United States Should Look to Japan for Better Schools” cited the Teaching Gap book, and issued a call to reconsider “how teachers are trained and how they teach what they teach” (emphasis added). Not one word was spent on the importance of what content is taught, and what a teacher should know in depth [29].

6. Conclusions Mathematicians often ask what they can do to help preserve the integrity of K-12 math programs. In 1999, a letter protesting the new textbooks was signed by more than 200 leading American mathematicians and scientists and was published in the Washington Post. It had some positive results, but failed to stop the latest reforms. A similar protest in Israel was successful – but just barely. In California, protests

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supported by grassroots parents organizations, mathematicians, scientists, concerned journalists, and politicians were able to secure a sound revision of the State K-12 math standards in 1997 – after more than five years of struggle. In many countries, mathematics societies will probably be most effective by lobbying as a group and by seeking a role in the textbook adoptions and in overseeing the assessment programs. In the U.S., reform curricula have often been introduced in conjunction with new testing programs designed and even managed by the publishers of the newly adopted textbooks. This practice eliminates the opportunity to compare pre- and post-reform student achievement. And publishers seldom provide in-depth testing on the weakest aspects of their own programs. It is also worth pointing out that program validation tests should cover an entire curriculum. Whereas achievement tests should concentrate on the most important material that can be covered in the allotted time, the testing of education programs should use sampling to achieve comprehensive coverage at a nominal marginal cost in the overall testing process. Needless to say, the oversight required for these assessment programs should be of the highest caliber. Some tests use closely guarded questions. The secrecy allows the same questions to be used year after year to maintain consistency in the scoring. For example, one of the more widely cited validation studies relied mainly on a test that to the best of my knowledge has had only three of its questions appear in the literature. This achievement test was devised to align with the new math reforms, but is also reported to assess basic computational skills. It is given over a period of three days with the teachers retaining custody of the materials after school. So its questions are not really secret, and the administrative procedures lack safeguards to protect the integrity of the assessment program. Sometimes students were even allowed to rework questions from the previous day. Moreover, the test manufacturer does not require the test to be given with time limits, which are optional even for the testing of basic skills. The validation project reported year-by-year improvement of fourth-grade scores with the new reform program, but this progress was not matched by the scores for the more securely administered state testing of fifth graders. In the U.S., the government-mandated No Child Left Behind (NCLB) testing (with state-determined tests) shows good progress for the majority of our states year by year, whereas the National Assessment of Educational Progress (NAEP) math testing shows that the net achievement of our twelfth graders has been unchanged nationwide for more than a decade. Something does not quite add up. The NAEP uses a mix of new and secret questions but is designed to be free of the biases that result from test-specific instruction and cramming. It is given to randomly selected schools, and the performance results are reported at the state level with additional results for subcategories based on gender and socio-economic status. Each student is given a randomly selected subset of test questions, and no performance results are released for students, schools, school districts, or education programs. Consequently, there is little incentive to teach to the test. The majority of the California achievement test questions are released and retired each year, and state law forbids the use of these

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materials in classroom preparation for forthcoming tests. There are programs in place to detect cheating, but it is not possible to know how effective they are, and students can always use these questions for practice independently of their school assignments. In New York, there are no such prohibitions, and many New York City schools use old tests routinely in required after-school preparation sessions held during the six weeks prior to the State and City testing. But although the NAEP may be our most uncompromised testing program, it is far from perfect. The test is consensus-based, with an oversight committee that has limited authority and where only about 10% of its members are mathematicians. The web-released sample questions suggest that the twelfth grade test is probably at a sixth grade level, on average. A representative question on fractions might be to compute two-thirds of 12 marbles. Evidently, the NAEP Governing Board (NAGB) has not reached a consensus about the benefits of knowing if an American high school education enables seniors to evaluate, say, 1/6 − 1/9, much less 2 19 − 4 16 . To date, just one of the released algebra problems is categorized as solving a system of equations. This twelfth grade multiple choice question reads: What number, if placed in each box below, makes both equations true? 4×

=

and 3 ×

=

:

A) 0

B) 1

C) 2

D) 3

E) 4

A) 3

B) 4

C) 8

D) 9

E) 12

A “hard” problem reads: For what value of x is 8

12

x

= 16 ?

Only 34% of our high school seniors found the correct answer even though calculators were available for use on this problem. The NAEP testing also asked students if they used a calculator for this question, but this data, unfortunately, does not appear to have been released on the web. Needless to say, the TIMSS test questions and testing procedures, unlike many U.S. practices, stand out as a beacon of hope. But we must take care to ensure that all of the TIMSS analyses are well documented, are open to external review, and are as accurate as possible. And with so many challenges in the search for sound education reform, we may all be able to contribute somewhere in this complex of vital activities. We close with the following summary assessments. 1. The undisciplined appeal to constructivist ideas has produced American programs that are more a betrayal of true constructivism than an advance of its principles. The result is an unprecedented reduction in the transmission of mathematical content. 2. The reform books and classroom curricula focus on examples, tricks, and experiments rather than fundamental mathematical principles, systematic methods, and deep understanding. 3. The justification for these “reforms” is based on mostly inaccurate interpretations of the best teaching practices in other countries. In particular, paradigmatic classroom examples from Japan have been misconstrued by researchers to suggest that students discover mathematical principles. In fact, the teacher conveys these

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principles quite explicitly, albeit engagingly and through examples. 4. As a consequence of these misinterpretations, “exemplary” math lessons in the U.S. convey little content, take too much time, and can even lead to false “discoveries” of mathematical principles. 5. A proper understanding of best practices suggests that i. teachers must be trained to understand, at a deep level, the mathematics they are teaching ii. teachers should encourage individual work, but must ensure that important principles are conveyed in an orderly and cumulative manner. 6. Mathematicians, guided by proven programs such as those in Singapore, should be involved in determining the principles that are taught, the examples that help convey them, and the exercises that reinforce the net learning. 7. Mathematicians must play an active role in overseeing the quality of achievement tests in an effort to determine where our education programs are succeeding and where they are not. Acknowledgments. The author is greatly indebted to Professor Michiko Kosaka for her tutoring on Japanese culture, and her verification of translated and mistranslated passages (cf. Figure 1) in the TIMSS Videotape Study documentation. It is also a pleasure to thank Clilly Castiglia and Kevin Feeley of the NYU Center for Advanced Technology and the Media Research Lab, who graciously provided the VHS frame processing. Portions of this paper are adapted from Testing Student Learning, Evaluating Teaching Effectiveness, edited by W. M. Evers and H. J. Walberg [28].

References [1] Aharoni, R., What I Learned in Elementary School. Amer. Edctr. 29 (3) (2005), 8–13. [2] Anderson, J., Reder, L., Simon, H., Applications and Misapplications of Cognitive Psychology to Mathematics Education. Texas Education Review (Summer, 2000). http://actr.psy.cmu.edu/people/ja/misapplied.html [3] Beaton, A. E., et al., Mathematics Achievement in the Middle School Years: IEA’s Third International Mathematics and Science Study. International Association for the Evaluation of Educational Achievement (IEA), Chestnut Hill, MA, 1996. [4] Bransford, J. D., et al., How People Learn: Brain, Mind, Experience and School. National Research Council, National Academy Press, Wash., DC, 2000. [5] Cobb, P., Yackel, E., Wood, T., A constructivist alternative to the representational view of mind in mathematics education. JMRE 23 (1992), 2–33. [6] Cook, C., The Significance of the NCTM Standards to the Pathways Critical Issues in Mathematics. North Central Regional Educational Laboratory, 1995. www.ncrel.org/ sdrs/areas/issues/content/ctnareas/math/ma0/htm


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[7] Coxford, A. F., et al., Contemporary mathematics in context: a unified approach. Core-Plus, Janson Publications, Dedham, MA, 1997. [8] Fendel, D., et al., Interactive Mathematics Program: Integrated High School Mathematics 1,2 3, 4. Key Curriculum Press, Emeryville, CA, 1997. [9] Garfunkel, S., et al., Mathematics Modeling Our World, 1,2,3,4. COMAP, Inc., W.H. Freeman and Co., New York, NY, 1998. [10] Glenn, J. et al., Before It’s Too Late, A report to the Nation from the National Commission on Mathematics and Science Teaching for the 21st Century. 2000. www.ed.gov/ inits/Math/glenn/report.pdf [11] Hiebert, J., et al., Teaching Mathematics in Seven Countries: Results From the TIMSS 1999 Video Study. National Center for Education Statistics (NCES), Wash., DC, 2003. [12] Hiebert, J., et al., Highlights From the TIMSS 1999 Video Study of Eighth-Grade Mathematics Teaching. NCES, NCES 2003-011, Wash., DC, 2003. [13] Hong, K. T., et al., Primary Mathematics 5A, 5B, 6A, 6B. Curriculum Planning & Development Division, Ministry of Education, Federal Publications, Singapore, 2000. [14] Jacobs, J., et al., Third International Mathematics and Science Study 1999 Video Study Technical Report Volume 1: Mathematics. NCES, Wash., DC, 2003. [15] Kamii, C., Warrington, M. A., Teaching Fractions: Fostering Children’s Own Reasoning, Developing Mathematical Reasoning in Grades K-12. In 1999 Yearbook of the NCTM (ed. by L. Stiff and F. Curcio). NCTM, Reston, VA 1999, 82–92. [16] Lappan, G., et al., Connected Mathematics 6, 7, 8. Connected Mathematics Project, Prentice Hall, Upper Saddle River, NJ, 1999. [17] Ma, L., Knowing and Teaching Elementary Mathematics: Teachers’Understanding of Fundamental Mathematics in China and the United States. Lawrence Erlbaum Associates, Mahwah, NJ, 1999. [18] Martin, M. O., et al., School Contexts for Learning and Instruction IEA’s Third International Mathematics and Science Study. IEA, Chestnut Hill, MA, 1999. [19] Mullis, I. V. S., et al., TIMSS 1999 International Mathematics Report Findings from the IEA’s Repeat of the Third International Mathematics and Science Study at the Eighth Grade. IEA, Boston College, Chestnut Hill, MA, 2000. [20] NCTM, Curriculum and evaluation standards for school mathematics. NCTM, Reston, VA, 1989. [21] NCTM, Professional Standards for Teaching Mathematics. NCTM, Reston, VA, 1991. [22] NCTM, Assessment Standards for School Mathematics. NCTM, Reston, VA, 1995. [23] Ocken, S., personal communication. [24] Public Broadcasting Service, Stressed to the Breaking Point. High School Math Project – Focus on Algebra, TV show, MathLine series, J. Peters, Producer, 1997. See also the lesson plan at www.pbs.org/teachersource/mathline/lessonplans/pdf/hsmp/ stressedtobreaking.pdf. [25] Peak, L., et al., Pursuing Excellence: A Study of U.S. Eighth-Grade Mathematics and Science Teaching, Learning, Curriculum, and Achievement in International Context. NCES, NCES 97-198, Wash., DC, 1996. [26] Ralston, A., Let’s Abolish Pencil-and-Paper Arithmetic. Journal of Computers in Mathematics and Science Teaching 18 (2) (1999), 173–194.

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[27] Schwartz, D. L., Bransford, J. D., A time for telling. Cognition and Instruction 16 (4) (1998), 475–522. [28] Siegel, A., Telling Lessons from the TIMSS Videotape. In Testing Student Learning, Evaluating Teaching Effectiveness (ed. by W. M. Evers and H. J. Walberg), Hoover Press, Stanford, CA, 2004, 161–194. [29] Staples, B., Why the United States Should Look to Japan for Better Schools, Editorial Observer, New York Times, Section A, Page 22, Column 1, Nov. 21, 2005. [30] Stigler, J. W., Hiebert, J., The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom. Free Press, New York, NY, 1999. [31] Stigler, J. W., et al., The TIMSS Videotape Classroom Study: Methods and Findings from an Exploratory Research Project on Eighth-Grade Mathematics Instruction in Germany, Japan, and the United States. NCES, NCES 99-074, Wash., DC, 1999. [32] Stotsky, S., ed., What’s at Stake in the K-12 Standards Wars A Primer for Educational Policy Makers. Peter Lang, New York, NY, 2000. [33] TERC, Investigations in Number, Data, and Space. K-5 book series, Dale Seymour Publications, Parsippany, NJ, 1998. [34] U.S. Dept. of Education, Eighth-Grade Mathematics Lessons: United States, Japan, and Germany. Video Tape, U.S. Dept. of Education, NCES, Wash., DC, 1997. [35] U.S. Dept. of Education, Eighth-Grade Mathematics Lessons: United States, Japan, and Germany. CD ROM, U.S. Dept. of Education, NCES, Wash., DC, 1998. [36] U.S. Dept. of Education, Moderator’s Guide to Eighth-Grade Mathematics Lessons: United States, Japan, and Germany. U.S. Dept. of Education, Wash., DC, 1997. [37] Usiskin, Z., et al., Geometry. University of Chicago School mathematics Project, Prentice Hall, Upper Saddle River, NJ, 1999. Courant Institute, NYU, 251 Mercer St., New York, NY 10012, U.S.A. E-mail: [email protected]

Mathematics, the media, and the public Ian Stewart

Abstract. It is becoming increasingly necessary, and important, for mathematicians to engage with the general public. Our subject is widely misunderstood, and its vital role in today’s society goes mostly unobserved. Most people are unaware that any mathematics exists beyond what they did at school. So our prime objective must be to make people aware that new mathematics is constantly being invented, and that the applications of mathematics are essential in a technological world. The mass media can play a significant role in encouraging such understanding, but the world of the media is very different from the academic world. I describe what it is like to engage with the media, concentrating on my own experiences of the past 40 years. Mathematics Subject Classification (2000). Primary 00A06; Secondary 00A08. Keywords. Popularisation of mathematics.

1. Introduction For most of the 20th Century, mathematicians were free to pursue their subject essentially independently of the rest of human society and culture. In his celebrated book A Mathematician’s Apology (Hardy [3]) the analyst G. H. Hardy wrote: ‘It is a melancholy experience for a professional mathematician to find himself writing about mathematics.’ In Hardy’s view, writing about existing mathematics paled into insignificance when compared to creating new mathematics. In many ways he was, and still is, right. But the two activities are not mutually exclusive. Moreover, as the 20th Century has given way to the 21st, it has become increasingly vital for mathematicians to take steps to increase public awareness of their motives, activities, concerns, and contributions. Such awareness has direct benefits for the mathematical enterprise, even if that is viewed entirely selfishly: ultimately, the public purse funds our private obsessions, and will cease to do so unless the guardians of that purse are assured that the money would not be better spent elsewhere. Public awareness of mathematics (within the broader context of the ‘public understanding of science’) also benefits the populace at large, because we live in an increasingly technological world that cannot function effectively without substantial input from mathematics and mathematicians. However, the role of mathematics in maintaining society is seldom appreciated – mostly because it takes place behind the scenes. The computer industry has made sure that it takes the credit (and sometimes the blame) for anything even vaguely related to its machines, but we mathematicians have failed completely to make it known Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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that without our contributions, such as algorithms (and of course much else that has nothing to do with us) those machines would be unable to add 1 and 1 and make 10. So we have a lot of work to do if we want to demonstrate that mathematics is not – as many imagine – a subject that has been rendered obsolete by the computer, but a vital part of what makes computers work. And almost everything else. To quote the preface of my recent book [9]: No longer do mathematicians believe that they owe the world an apology. And many are now convinced that writing about mathematics is at least as valuable as writing mathematics – by which Hardy meant new mathematics, new research, new theorems. In fact, many of us feel that it is pointless for mathematicians to invent new theorems unless the public gets to hear of them. Not the details, of course, but the general nature of the enterprise. In particular, that new mathematics is constantly being created, and what it is used for. At the end of the 19th Century, it was not unusual for the leading mathematicians of the day to engage with the public. Felix Klein and Henri Poincaré both wrote popular books. David Hilbert gave a radio broadcast on the future of mathematics. But within a few decades, the attitude typified by Hardy seems to have taken over. Fortunately, we are now reverting to the attitudes of the late 19th Century. Distaste for mere vulgarisation gave way to grudging acceptance of its occasional necessity, and this in turn has given way to active encouragement and approval. Even today, the role of populariser is not all sweetness and light, but the days when (as happened to a colleague at another institution) a senior member of the administration would castigate a member of his academic staff for daring to write a column in a major daily newspaper are long gone. If anything, we are now more likely to be castigated for not not writing a column in a major daily newspaper. Since my first appointment at the University of Warwick in 1969, indeed even before that, I have been involved in many different forms of mathematical popularisation – mainly books, magazines, newspapers, radio, and television. I generally feel much more comfortable doing popularisation than talking about it – in fact the main advice I give to people who are interested in becoming a populariser is to get on with it – so my intention here is to describe what it is like to be engaged in such activities, with specific reference to my own experiences. I hope that this may prove useful for others who may wish to play the role of media mathematician, and informative for those who prefer to watch from the sidelines but would like to understand the nature of the game better.

2. What is popularisation? I have given many talks that popularised mathematics, but I once gave a talk about popularising mathematics, which is not the same thing. One example I mentioned

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was a description of the Galois group of the quintic equation in comic book form [8]. Here a character in the story juggled five turnips (the ‘roots’ of the equation) in a blur, showing that they were indistinguishable – in short, the Galois group of the general quintic equation is the symmetric group S5 . A mathematics teacher in the audience objected that this was not popularisation. Just as Monsieur Jourdain, in Moliere’s The Bourgeois Gentleman, was astonished to discover that he had been speaking prose all his life, I was astonished to discover that I had not been speaking popularisation. The teacher then explained that popularising mathematics meant making it accessible to children and getting them excited about it. No: that’s education. Not, perhaps, education in the sense currently envisaged in the UK (and increasingly everywhere else), which is a sterile process in which boxes are ticked to indicate that the child has temporarily mastered some small item of knowledge or technique, regardless of context, but education in the sense it used to mean, which was teaching things to children. Explaining things in a comprehensible manner, and enthusing children about the topic, are essential features of education at school level – and, indeed, in adult education too. It was particularly clear that the teacher’s view of what constituted ‘mathematics’ differed from mine. She was referring to the nuts and bolts of the school syllabus; my main concern was, and always will be, the frontiers of past of present mathematical research. The two are about as similar as do-re-mi and Wagner’s Ring Cycle. There is, of course, common ground. It is possible to popularise school mathematics among children without trying to teach it to them. But one of the biggest misconceptions among otherwise intelligent adults is that the ‘mathematics’ they did at school is all there is. One of the most important aspects of popularisation is to make it clear to both children and adults that this presumption is wrong. By ‘popularisation’ I mean attempts to convey significant ideas from or about mathematics to intelligent, mostly sympathetic non-specialists, in a manner that avoids scaring them silly and exploits whatever interests them. I say ‘attempts’ because success can be elusive. The level of exposition can range from humorous short puzzles to books on hot research topics.

3. The public The phrase ‘public understanding of science’ is widely used but seldom clarified. Which public? What are they supposed to understand? Why don’t they understand it already? The schoolteacher mentioned above had a very different idea of what the words ‘public’ and ‘understand’ meant, compared to what I meant. Many scientists consider the public to be anyone who is not a scientist, and view their alleged lack of understanding as a deficiency to be remedied by supplying the required information. Thus members of the public who are concerned about possible effects of genetically modified organisms are directed, by such scientists, towards research that demonstrates the

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(alleged) safety of GMOs as food; people concerned about the safety of nuclear power are directed to statistical analyses of the probability of accidents, and so on. In this view, the public – whoever they may be – are considered ignorant, and the objective of the ‘public understanding of science’ is to remedy this deplorable deficiency. I don’t find this view helpful. Even when correct, it is patronising and selfdefeating. But mostly it is not correct. Often the public, for all their ignorance of technical details, have a much clearer grasp of overall issues than specialist scientists. A major problem with GMOs, for instance, is not their safety as food, but potential damage done to the ecosystem by introducing alien species. You don’t need to know any genetics to observe that numerous confident pronouncements about GMOs made by scientific experts have turned out to be wrong, and badly so. Not long ago people in the UK were assured that genetically modified DNA could not be transferred more than a few metres by pollen. It quickly transpired that such transfer routinely occurred over distances of several kilometres. It is not necessary to prove that such transfer is harmful to notice that the experts did not have a clue what they were talking about, or that their alleged expertise had led them to wildly inaccurate conclusions. On many issues of public concern, reassurance by scientists serves only to educate the public in the limitations of reductionism and the narrow-mindedness of many scientific experts. Some scientists even seem to think that it is possible to draw up some list of basic scientific ‘facts’that members of the public should know, and then teach them. So they should know that the Earth goes round the Sun, that genetic information is encoded in DNA, that the Earth is 4.5 billion years old, and so on. It would certainly help if most people were aware of such things, but this attitude encourages the view that the task of science is to establish ‘the facts’, and that once these are known, that’s all there is to be said. Or, as a friend of mine’s Head of Department put it many years ago: ‘Our task as educators is to give the students the facts, and their job is to give them back to us in the exams.’ Whatever that process might be, it’s not education, and it’s not public understanding either. Though it does help to train a lot of ‘experts’ who think that their limited understanding of laboratory genetics qualifies them to pronounce on the effects of GMOs on the ecology. My view, for what it’s worth, goes something like this. Let me phrase it in the context of mathematics, for definiteness: much the same goes for other areas of science. All over the globe, every day of the week, mathematicians are carrying out research, proving new theorems, inventing new definitions, solving problems, posing new ones. The vast majority of the public have no idea that any of this is happening. They got excited by the TV programme on Andrew Wiles and Fermat’s Last Theorem, but that wasn’t because they thought it was the most interesting new idea in mathematics. They thought it was the only new idea in mathematics. What excited them was not a new breakthrough on an old problem, but the belief that for the first time in several centuries a new piece of mathematics had been brought into existence. So the primary objective, for the public understanding of mathematics, has to be to make people aware that new mathematics is constantly being created.


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This objective is more important than explaining what that new mathematics consists of, and it is more important than explaining what mathematics actually is. Only when people recognise that mathematicians are doing something do they start to get interested in what they are doing. Only when they’ve seen examples of what mathematicians are doing do they start to wonder what mathematics is. If by ‘the public’ we really do mean the typical, randomly chosen person on the street, then we have succeeded in improving their understanding of our subject as soon as they realise that there is more to it than they met at school. There is a more restricted subset of the public that requires, and should be given, more. These are the people who are actually interested in mathematics. They are the core audience for popularisation. For them, it is worth trying to convey more than the existence of new mathematics. It is possible to try to give a feeling for what it is. When you watch a football match on television, it is assumed that you enjoy football and have some general idea of the rules. The commentators do not explain that the round object is a ‘ball’ and that the aim is to get it into the net; nor do they point out that you have to choose the right net, and that the total number of ‘goals’ determines who wins. You are expected to know this. On the other hand, you are not expected to know the latest version of the offside rules. The commentators assume you are aware of the issue, but have temporarily forgotten the details. By reminding you of those, they can then engage your attention in a discussion of the issues. Too often, the media treat science very differently. You want to tell people about Fermat’s Last Theorem, but first you are obliged (so the producer or editor insists) to explain what a square and a cube is and who Pythagoras was. If you want to describe the latest work on polynomial-time algorithms for primality testing, you have to explain what a prime number is and what a polynomial is. In that case, the missing information can be sketched quite quickly, but it’s all too easy to find yourself in a situation where the main point you are trying to address is Galois theory, but all the programme manages to tackle is the concept of a square root. Better than nothing. . . but not what you intended, and not what is needed to break the mental link ‘mathematics = school’.

4. Be warned If you want to promote mathematical awareness among the public by making use of the mass media, you should be aware that it is not quite like standing in front of a blackboard or data projection screen and delivering a lecture to undergraduates. Rather different talents are needed, and in particular you have to be prepared to risk making a fool of yourself. I have dressed up in a white lab coat to talk about the probability theory of Friday 13th, presumably because the TV company concerned thought that was what mathematicians wear – or more likely thought that viewers thought that was what mathematicians wear. I have had my name up in lights on the scoreboard at Wembley football stadium, for a programme about crowd modelling

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that should have taken an hour to film and actually took five because the stadium – which was supposed to be empty – was full of schoolkids on their Easter break, and was being dismantled around our ears as well. I have spent a day lugging a stuffed duck-billed platypus round an ancient castle… a colleague, who often does TV biology, remarked ‘I’ve never done that.’ Pause. ‘Mine was a stuffed echidna.’ (We contemplated forming the Monotreme-Luggers’ Society.) I have sat in the hot sun, visibly becoming more and more sunburned as the filming progressed, because the topic was Maxwell’s equations and the backdrop of an array of radio telescopes was deemed essential to the atmosphere. I have stood in a huge supermarket at peak period to deliver five seconds of wisdom about the National Lottery to BBC news, live…terrified that the woman who was noisily changing thousands of coins at a nearby machine would still be doing so when we went on air. I have spent 16 hours in a muddy quarry filming the end of the world, and driving a battered VW beetle painted to resemble the world. Appropriately, its clutch-cable broke ten minutes into the filming, and I had to drive it all over the quarry, and a local farm, by crashing the gears. One attempt at a live broadcast for Irish local radio, about alien life forms, failed because they lost the connection. We did it again the next week. In another attempt at a live broadcast – I forget what about – I sat in a cramped studio for an hour, and my slot was then pre-empted by a news flash, so nothing went out at all. On the other hand, working with the media is occasionally wonderful. My most memorable moment ever was when we started a televised lecture by bringing a live tiger into the lecture room. (Warning: do not attempt this at home.) It’s a long story, but here are the bare bones. In 1826 Michael Faraday inaugurated a series of lectures on science for young people at the Royal Institution in London, where he was resident scientist. They have continued annually ever since, except for four years during World War II, and for almost 40 years they have been televised. Until recently they were recorded ‘as live’, meaning that most mistakes were left in, in front of an audience consisting mainly of schoolchildren. (Three things you should never do in show business: work with children, work with animals, work without a script. Christmas lecturers have to do all three simultaneously.) There are a few parents too, but they are placed out of sight of the cameras. Twice in the ensuing 180 years the topic has been mathematics. Christopher Zeeman delivered the first such series in 1978, and I gave the second in 1997. One of my lectures was on symmetry and pattern-formation. We decided to open the lecture with Blake’s ‘Tyger’ poem (‘dare frame they fearful symmetry’), which, although being a cliché, seemed unavoidable. Which meant, by the very direct logic of television, that we had to have a tiger. A month-long search yielded a baby puma, but no tiger. We had just about decided to go with the puma when my colleague Jack Cohen found us a tiger. More accurately, a six-month old tigress called Nikka. She was wonderful, a real pro – used to the lights and an audience. She had the requisite stripes (pattern-formation, remember?). For


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Health and Safety reasons she was separated from the audience by a row of upturned seats, while two burly young men held her on a chain. Apparently Health and Safety did not extend to presenters (me) so I delivered the relevant material squatting next to her. It was one of the most amazing experiences of my life, and I’ve never really been able to match it as a way of starting a lecture.

5. What the media want When we write research papers on mathematics, the main criteria for publication are that the paper should be competently written, new, true, and interesting. The criteria for acceptance of a newspaper article, a magazine article, a radio interview, or a TV broadcast are somewhat different. The most important difference is that you have to tell a story. A story has a beginning, an end, and a middle that joins them. Moreover, it should be clear at all times where the story is and where it is heading. This does not mean that you have to give away the punch line before you get to it: it means that the reader or listener must be made aware that a punch line is on its way. One way to describe the process is to say that the reader or listener needs to be given a ‘road map’, or at least a few signposts. My feeling is that in principle even a research paper ought to tell a story, but mathematicians are not trained in narrative thinking, and readers are generally able and willing to go over a research paper several times seeking understanding. This is not the case for a newspaper article or a radio broadcast. Readers or listeners are busy people, often on their way to or from work, and they expect to be able to follow the story as it unfolds. A few may read an article twice, or record a radio programme and listen to it again, but on the whole they will do this because they did understand it first time, not because they did not. As an example, suppose you want to write about Andrew Wiles’s proof of Fermat’s Last Theorem, for the Ghastliegh Grange Gazette. It does not work if you start with something like ‘Let E be an elliptic curve. . .’ or even something more civilised like ‘The key to proving Fermat’s Last Theorem is Galois cohomology. . .’ Instead, you need to structure the story around things the reader can readily identify with. The bare bones of the story might be something along the following lines: ‘Notorious puzzle that mathematicians have failed to solve for 350 years. . . Very simple to state but impossible to prove until now. . . After seven years of solitary research, major breakthrough by Wiles. . . Unexpectedly linked puzzle to a different area of mathematics, making breakthrough possible. . . Proof temporarily collapsed after being announced. . . After desperate last-ditch battle, proof repaired. . . Triumph!’ Notice that this summary of the narrative line does not include a statement of the theorem (though you would normally work this into the article somewhere) and in fact it does not even include Fermat (though again some historical background would be a good idea). It does not mention elliptic curves or Galois cohomology, and it absolutely does not define them. Your typical reader may well be a lawyer or a greengrocer, and

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these terms will be meaningless to them. If some technical idea is absolutely essential to your story, then you will have to find some way to make it comprehensible – but be aware that your readers have no idea what a function is, or a group, or even a rational number. This does not mean they are ignorant or unintelligent – after all, how much do you know about conveyancing or vegetables? It means that you are enticing them to venture into territory that is, for them, very new. They will need a lot of help. ‘Infinite intelligence but zero knowledge’ is a useful, though perhaps flattering, description. Another extraordinarily important aspect of a story, for the media, is timeliness. The editor or producer will not only ask ‘Why should I publish/broadcast this story?’ They will ask ‘Why should I publish/broadcast this story today?’ (Or ‘this week’ or ‘in the next available issue’ or whatever.) It is not enough for the material to be important or worthy. There has to be a ‘hook’ upon which the story can be hung. Typical hooks include: • Recent announcement of the relevant research. • Recent publication of the relevant research. • A significiant anniversary – 100 years since a major historical figure associated with the work was born, died, or made a key discovery. A genuine professional science writer will keep a diary of such occasions, and be ready for them as they come along. • A timely application (preferably related to stories currently considered newsworthy – such as cloning, nanotechnology, anything with a gene in it, mobile phones, computer games, the latest blockbuster movie. . .). • A current controversy – the media always go for a dust-up, and it seldom matters if the source of the dispute is totally obscure. Everyone understands a fight. There are other kinds of hook. With Christmas coming up, the TV programme Esther once decided to feature the science of Christmas, but much of it was deliberately spoof science – for example, my contribution was to point out that the aerodynamics of supersonic flight is very different from subsonic, so that at the hypersonic speeds employed by Santa Claus, reindeer antlers might be much more aerodynamic than they look. They aren’t, of course, but viewers knew it was a joke and subliminally took on board the message about supersonic flight changing the geometry. And they also got to see the back-of-the-envelope calculation that estimated Santa’s speed. One of the more bizarre hooks arose in 2003, when I received a phone call from the Daily Telegraph, one of the UK’s major newspapers. A reader had written a letter, recalling a puzzle with 12 balls that he had heard about as a boy. All balls have the same weight, except for one. He had been told that it was possible to work out which ball, and whether it was light or heavy, in 3 weighings with a balance but no weights. Could anyone tell him how?


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The response was remarkable. The newspaper reported [10] that ‘By teatime yesterday [7 February 2003], The Daily Telegraph had received its biggest mailbag in living memory and our telephones were still ringing off the hook’. But the editors had a serious problem: it was unclear to them whether any of the proposed solutions was correct. Could I supply a definitive answer? As it happened, Martin Golubitsky was visiting, and he remembered being inspired by this puzzle as a teenager. In fact, his success in solving it was one of the things that had made him decide to become a mathematician. We put our heads together and reconstructed one method for solving the puzzle. The newspaper duly published it, mainly as a way of ending the flow of letters and phone calls. There is, by the way, a more elegant solution than the one we devised. The puzzle has been discussed by O’Beirne [4], who gives a solution originating with ‘Blanche Descartes’ (a pseudonym for Cedric A. B. Smith), see Descartes [2]. Here the hero of the narrative, known as ‘F’, is inspired to write the letters F AM NOT LICKED on the 12 balls: ‘. . . And now his mother he’ll enjoin: MA DO LIKE ME TO FIND FAKE COIN’ The poetic solution lists a set of weighings (four balls in each pan) whose outcome is different for all 24 possible choices of the odd ball out and its weight. The problem with this answer, clever though it may be, is to motivate it. This is why we settled for the more prosaic ‘decision tree’ of weighings that you will find in the published article. We felt that readers would be more likely to follow the logic, even if our method was less elegant.

6. The media Let’s take quick trip through the main types of media outlet. There are others – webpages, CD-ROMs, DVDs, blogs, podcasting, whatever. 6.1. Magazines. Popular science magazines have the advantage that their readership is self-selected for an interest in science. Surveys have shown that mathematics is very popular among such readers. Each magazine has its own level, and its own criteria for what will appeal. Scientific American is justly famous for the ‘mathematical games’ columns originated by the peerless Martin Gardner, which unfortunately no longer run. In the UK there are New Scientist and Focus, which regularly feature mathematical items ranging from primality testing to su doku. If you are thinking of writing an article for such a magazine, it is always better to consult the editors as soon as you have a reasonably well formulated plan. They will be able to advise you on the best approach, and will know whether your topic

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has already been covered by the magazine – a problem that can sink an otherwise marvellous idea. Expect the editors and subeditors to rewrite your material, sometimes heavily. They will generally consult you about the changes, and you can argue your case if you disagree, but you must be prepared to compromise. Despite this editorial input, the article will usually go out under your name alone. There is no way round this: that’s how things are in journalism. 6.2. Newspapers. Few newspapers run regular features on mathematics, bar the odd puzzle column, but most ‘quality’ newspapers will run articles on something topical if it appeals to them. Be prepared to write 400 words on the Fields medallists with a four-hour deadline, though, if you aspire to appearing in the national news. 6.3. Books. Books, of course, occupy the other end of the deadline spectrum, typically taking a year or so to write and another year to appear in print. They really deserve an article in their own right, and I won’t say a lot about them here, except in Section 7 below. Sometimes expediency demands a quicker production schedule. I once wrote a book in 10 weeks. It was short, mind you: 40,000 words. The quality presumably did not suffer because it was short-listed for the science book prize. If you want to write semi-professionally, you will need an agent to negotiate contracts. At that level, book writing is much like getting a research grant. Instead of ploughing ahead with the book, you write a proposal and go for a contract with a specified advance on royalties. 6.4. Radio. Radio is my favourite medium for popularising mathematics. This is paradoxical, because radio seems to have all of the disadvantages (such as no pictures) and none of the advantages (such as being able to write things down and leave them in full view while you discuss them) of other media. However, it has two huge advantages: attention-span and imagination. Radio listeners (to some types of programme) are used to following a discussion for 30 minutes or longer, and they are used to encountering unfamiliar terminology. And radio has the best pictures, because each viewer constructs a mental image that suits them. On radio you can say ‘imagine a seven-dimensional analogue of a sphere’, and they will. It may not be a good image, but they’ll be happy anyway. Say the same on TV and the producer will insist that you build one in the studio for the viewers to see. TV removes choice: what you get is what they choose to show you. On radio, what you see is what you choose to imagine. 6.5. Television. Television is far from ideal as a medium for disseminating science, and seems to be becoming worse. As evidence: every year the Association of British Science Writers presents awards for science journalism in seven categories. In 2005 no award was made in the television category, and (Acker [1]) the judges stated:


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To say the quality of entrants was disappointing is an understatement. We were presented with ‘science’ programmes with virtually no science in them. Some were appalling in their failure to get across any facts or understanding. Whenever there was the possibility of unpicking a little, highly relevant, science, or research methodology, the programmes ran away to non-science territory as fast as possible, missing the whole point of the story as far as we were concerned. I still vividly recall a TV science programme which informed viewers that Doppler radar uses sound waves to observe the speed of air in a tornado. No, it uses electromagnetic waves – the word ‘radar’ provides a subtle clue here. Sound waves come into the tale because that’s where Doppler noticed them. The reasons why television is far from ideal as a medium for disseminating science are equally disappointing. It is not the medium as such that is responsible – although it does discourage attention-spans longer than microseconds. The responsibility largely rests with the officials who commission television programmes, and the companies who make them. Television changed dramatically in the 1980s, especially in the UK. Previously, most programmes were made ‘in-house’by producers and technicians with established track records and experience. Within a very short period, nearly all programming was subcontracted out to small companies (many of them set up by those same producers and technicians) on a contract-by-contract basis. This saved television companies the expense of pensions schemes for their employees (since they now had none) and protected them against their legal responsibilities as employers (ditto). But as time passed, contracts were increasingly awarded solely on the basis of cost. A new company would get the commission to make a programme, even if they had no experience in the area, merely because they were cheaper. Very quickly, most of the companies that knew how to make good science programmes were ousted by new kids on the block whose main qualifications were degrees in media studies and, the decisive factor, cheapness. Any lessons previously learned about how to present science on television were lost, and had to be re-learned, over and over again, by a system dedicated to the perpetual reinvention of the wheel. There is still some good TV science, but nowhere near as much as there ought to be given the proliferation of satellite and cable channels. The good news here is that TV is once again wide open as a medium for popular science, especially now that there are hundreds of channels desperate for content. But we will have to fight all the old battles again.

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7. Narrative imperative Sometimes an unexpected opportunity presents itself. The Science of Discworld and its sequels The Science of Discworld 2: The Globe and The Science of Discworld 3: Darwin’s Watch (see [5, 6, 7]) were written jointly with Jack Cohen, a biologist, and Terry Pratchett, one of the UK’s bestselling fantasy authors. They are superficially in the tradition established in The Physics of Star Trek, The Science of Jurassic Park, and The Science of Harry Potter, but on closer analysis they are distinctly different, and the difference is important. The latter three books all start from a popular television, film, or book series, and use that as a vehicle for explaining the alleged science that could actually make such things as space warps, resurrected dinosaurs, or flying broomsticks work. This approach may be an excellent way to interest non-scientists in Relativity, DNA, or anti-gravity, but it rests on a fundamental untruth: that today’s science tells us that such fiction could one day become fact. But typically the true link is rather more tenuous than that between a hang-glider and an interstellar spaceship. The Science of Discworld series takes the opposite stance. Instead of exploiting an existing body of fantasy as a basis for dubious science, it uses genuine science as a basis for new works of fantasy. In the three Science of Discworld books, we interwove entirely new fantasy stories with voyages through significant areas of modern science. We designed both aspects of the books to complement each other. The three authors worked together to plan the combined structure, choosing scientific topics that would lend themselves to a fantasy setting, tailoring the fiction to fit the facts, and selecting the facts for suitability as components of a work of fiction. If you’ve not encountered Discworld before, here’s a quick introduction. Pratchett’s Discworld series of humorous fantasy novels now comprises 31 novels, three graphic novels, four maps, 12 plays, two television animations, a cookery book, and countless spin-offs ranging from ceramics to computer-games. Its fans are numbered in the millions. Discworld is, as its name suggests, circular in form, and flat (though decorated with forests and oceans and deserts, hills and mountains dominated by the vast heights of the central Ramtops, where the gods live in an analogue of Valhalla). The disc is about 10,000 miles across, supported by four elephants standing on the back of the great turtle A’Tuin, who swims through space. Discworld is inhabited by people just like us, and by an assortment of wizards, witches, elves, trolls, zombies, ghosts, golems and vampires. Much of the action takes place in the city of Ankh-Morpork, where the wizards reside within the hallowed walls of Unseen University. It is a city of medieval proportions and Elizabethan filth. Discworld was originally conceived as a vehicle for poking fun at sword-andsorcery books, such as Robert Howard’s tales of Conan the Barbarian and Fritz Leiber’s ‘Fafhrd and Gray Mouser’ series set in the environs of Lankhmar, the model for Ankh-Morpork. But Discworld rapidly transmogrified into a vehicle for poking fun at everything, from Hollywood to the Phantom of the Opera, from religion to engineering, from the press to the police – even mathematics.


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Discworld has its own sideways logic, very appealing to mathematicians. It accepts the premises of fantasy (the Tooth Fairy really does come and take away teeth, leaving real cash) but asks hard questions (what’s in it for her?). It has no qualms about world-girdling turtles swimming through hard vacuum, but wonders what happens when they mate. It acknowledges the Butterfly Effect of Chaos Theory, but wonders which butterfly has this awesome power, and how to get the blighter. Discworld is our own planet, reified. Its driving forces are magic and Narrative Imperative. In magic, things happen because people want them to. In Narrative Imperative, things happen because the power of story makes them happen. The eighth son of an eighth son cannot avoid becoming a wizard – even if the midwife made a mistake and she was actually a girl. And so, in Equal Rites, the misogynist wizards of Unseen University have to come to terms with a female presence in the hallowed chambers. Some time in 1998 Cohen and I became convinced that there ought to be a book called The Science of Discworld. We broached the possibility to Terry, who pointed out the fatal flaw in the plan. There is no science in Discworld. In vain we argued that there is. When Greebo, the hyper-macho cat belonging to the witch Nanny Ogg is shut in a box, it rapidly emerges that there are three possible states for a cat in a box: alive, dead, and absolutely bloody furious. This is a profound comment on quantum superposition: what is the association between an object’s quantum state (wavefunction) and its macroscopic state (what we observe)? If you knew the cat’s wavefunction, could you tell whether it was alive or dead? My own view is that you couldn’t even tell that it was a cat. Terry gently explained why this approach would be misleading. On Discworld dragons do not breathe fire because of chemistry and genetics: they breathe fire because that’s what dragons do. What saved the idea was a concept breathtaking in its simplicity. ‘Terry: if there’s no science in Discworld, then you must put some there.’ Thus was born the Roundworld Project, in which the wizards of Unseen University set out to split the thaum (the fundamental unit of magic) and end up coming within a whisker of destroying the whole of the universe. As the magical reactor is about to go critical and explode, taking the universe with it, the computer Hex bleeds off the excess magic to create a small sphere, a magical containment field within which magic does not work. This is Roundworld, and it is our own universe. It runs not on magic, but on rules. It has helium and magnesium, but no narrativium. Things happen there because the rules say they must, not because someone wants them to. Oddly, this makes everything in Roundworld harder to understand, not simpler. If a person wants something built – a house, say – then they get some builders and up it goes. But if the rules want something built, such as a human being, then the construction process is much more obscure, involving big molecules and bacterial blobs and billions of years of nothing much happening; then blink your eye and the humans have come and gone, leaving only the ruins of the Space Elevator, and you can’t even be sure they were human.

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The best way to envisage the structure of the Science of Discworld books is to think of a novelette by Pratchett, set on Discworld, with its usual cast of characters and its usual narrative constraints, but with Very Big Footnotes by Cohen and Stewart. The novelette, which comprises the odd-numbered chapters of the book, is fantasy; the footnotes, comprising the even-numbered chapters, are the scientific commentary, and are typically between two and three times as long. This is a beautiful framework for writing about science, because the differences between magic and science are highly illuminating. Discworld is the perfect framework for a ‘What if?’ discussion of science – a well-established, self-consistent universe that can be used to ‘compare and contrast’. We managed to work quite a lot of mathematics into the books, too: chaos, complexity, Langton’s Ant, probability, phase spaces, combinatorics, information theory, infinity, and transfinite numbers. Not to mention scores of applications from astronomy to zoology. It was fun, too.

References [1] Acker, F., Dorks’ night out. The Science Reporter (Nov/Dec 2005) 1–2. [2] B. Descartes, B., The twelve coin problem. Eureka 13 (1950) 7, 20. [3] Hardy, G. H., A Mathematician’s Apology. Cambridge University Press, Cambridge 1940. [4] O’Beirne, T. H., Puzzles and Paradoxes. Oxford University Press, London 1965. [5] Pratchett, T., Stewart, I., and Cohen, J., The Science of Discworld. Ebury Press, London 1999. [6] Pratchett, T., Stewart, I., and Cohen, J., The Science of Discworld II: The Globe. Ebury Press, London 2002. [7] Pratchett, T., Stewart, I., and Cohen, J., The Science of Discworld III: Darwin’s Watch. Ebury Press, London 2005. [8] Stewart, I., Ah, Les Beaux Groupes! Belin, Paris 1983. [9] Stewart, I., Letters to a Young Mathematician. Basic Books, Cambridge 2006. [10] Uhlig, R., Odd ball letter starts maths puzzle mania. Daily Telegraph, 8 February 2003, 7. Mathematics Institute, University of Warwick, Coventry CV4 7AL, UK E-mail: [email protected]

Panel A Controversial issues in K-12 mathematical education Michèle Artigue (moderator) Ehud de Shalit and Anthony Ralston (panelists)

Abstract. This article sets the background for the panel session at the ICM on controversial issues in K-12 mathematics education. Three specific issues have been selected: Technology, skill building and the role of test and assessment. For each of these, a list of questions has been prepared. After introducing the three themes and the associated questions, this article presents the positions on these of the two panelists: Professor Anthony Ralston, from the State University of New York at Buffalo in the US, and Professor Ehud de Shalit from the Hebrew University of Jerusalem in Israel. The article ends with some personal comments from the coordinator of the panel: Professor Michèle Artigue from the University Paris 7 in France. Mathematics Subject Classification (2000). 97A80, 97D30, 97D40, 97D60. Keywords. Mathematics education, K-12 curriculum, technology, skill building, concept building, testing, assessment.

Introduction by Michèle Artigue K-12 mathematics education is obviously a controversial area, so much so that, in countries like the US, the term Math Wars has been used for describing the kind of conflicts between communities that has been generated in recent years. We all regularly hear colleagues complaining that the students they receive have not been adequately trained and that, every year, the situation becomes worse, or that they are not pleased with the kind of mathematics education their children receive etc. We all know that such feelings are not something new, but we cannot deny that in the last decade they have dramatically increased in intensity in many countries. Why does such a situation exist? What are the real challenges that K-12 mathematics education has to face at the beginning of the XXIst century? What can mathematicians do in order to enhance or support efficiently the necessary efforts, evolutions and changes of the whole educational community? These are the crucial issues that motivate the existence of a panel session on Controversial Issues in K-12 mathematical education at the ICM2006 in Madrid. It is certainly interesting to keep them in mind even if the panel does not address them all directly. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Panel A Controversial issues in K-12 mathematical education

For structuring this panel session, we have selected some particularly controversial issues, and will try to elaborate on these, with the support of the audience. These issues approach the current problems met by K-12 mathematics education through three different, but not independent, topics: technology, the place given to the learning of skills and techniques, and assessment and tests. Everyone will certainly agree that each of these is today a controversial topic, and that frequently in what we read or hear, it is advocated that the ways they have been dealt with in recent years or currently has resulted in some of the difficulties in K-12 mathematics education today. In what follows, we briefly introduce these three topics and articulate some questions that we would like to discuss for each of them. We then present the positions on these questions of the two panelists, Professor Anthony Ralston from the State University of NewYork at Buffalo, and Professor Ehud de Shalit from the Hebrew University in Jerusalem. The article ends with some general comments by the moderator of this panel session, Professor Michèle Artigue from the University Paris 7.

Topics and questions Technology. In 1985, the first study launched by ICMI entitled “The influence of computers and informatics on mathematics and its teaching” was devoted to computers and the ways the learning and the teaching of mathematics as well as this discipline itself was affected by technology. A second edition of the book issued from this study was prepared by B. Cornu and A. Ralston and published in 1992 in the Science and Education Series of UNESCO. As described in its introduction, the UNESCO book addresses the importance of the changes introduced by technology in professional mathematical practices and makes suggestions for new curriculum elements based on these new methods of doing mathematics. It is pointed out that even if these suggestions are judged by the reader to be stimulating and even persuasive as well as reasonably grounded, it is nevertheless the case that “such suggestions are fundamentally speculative at the level of large scale implementation – by which we mean that converting them into a well-developed and tested curriculum for the typical teacher and the typical student is still a major challenge.” Since that time, more and more sophisticated technological tools have continued to be developed for supporting the learning and teaching of mathematics, and their use is today encouraged by the K-12 mathematics curriculum in most countries. Nevertheless, in spite of the existence of an increasing amount of positive small-scale experiments, the real nature of the effect of technology on mathematics education in the large remains under discussion. The problems raised in the first ICMI study have not been solved, and the discourse of those who think that the impact of technology is globally negative and ask for a strict limitation of the use of calculators and software, and even for their banishment from mathematics education in the early grades, is opposed by those who consider that it does not make sense today to think about mathematics learning and teaching without taking into account the existence

Introduction

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of technology and without trying to benefit from the real and increasing potential it offers for mathematics education. Thus the first set of questions we propose to raise is: Up to what point should the changes introduced in social and professional mathematical practices by technology be reflected in mathematics education? What does technology have to offer today to K-12 mathematics education and why does it seem so difficult to have it benefit mathematics education in the large outside experimental settings? What could be done in order to improve the current situation? Is a strict limitation on the use of calculators and software a reasonable solution? Skill building. Every one of us certainly agrees that mathematical learning, as with any kind of human apprenticeship, requires skill building and also that it requires much more than that. In recent decades all over the world, K-12 mathematics curriculum developers, influenced by constructivist and socio-constructivist epistemologies of learning, by the results of cognitive research on learning processes, and also by the observed limitations of students’ achievements in mathematics, have stressed the necessity of moving some distance from teaching practices seen as too focused on drill and practice, and of getting a better balance between the technical and conceptual facets of mathematical learning. K-12 mathematics curricula have given increasing importance to exploration and work on rich and open problems in order to help students understand better the reasons for mathematical conceptualizations, and these conceptualizations themselves. They have also promoted teaching strategies that try to give more importance to the personal and collective elaborations of students in the development of classroom mathematical knowledge. Once more, the global effects of these curricular changes on K-12 mathematics education are a matter of controversy. Voices have arisen asking for a radical change in the role to be given to the learning and mastery of algorithms, with the long division algorithm often appearing as emblematic of the desired changes. In a similar vein, other voices denounce the dangers of what they see as a new “back to basics” program and the inability to understand that mathematics teaching has to take into account social and technological evolution, and the changes in scientific and mathematics culture needed in our societies today. Thus a second set of questions: What is the pertinence of the opposition between skill learning and the exploration of rich problems? Between techniques and concepts? What is the right balance to be achieved in K-12 mathematics education between the different facets of mathematical activity? How can this balance be achieved and what are the respective mathematics responsibilities to be given to the teachers and the students? Test and assessment. We are all aware of the influence that the form and the content of assessment have on any form of education and, thus, on K-12 mathematics education. We are also aware of the increasing importance given to national and

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international testing, as reflected for instance by the coverage in the media of the PISA enterprise of the OECD and TIMSS, and the influence that these results are taking in educational policies. The importance to be given to external assessment versus internal assessment, to international comparisons and standardized testing, to the effect of assessment on the mathematics learning of students, and to the effect of systematic testing on educational systems are all controversial issues, as are the discussions generated by the “No Child Left Behind” legislation in the US. Thus our third set of questions: How can we correctly reflect in assessment what we wish to achieve through mathematics education? Is standardized testing ever useful? For what purpose? Under what conditions? What exactly is tested by international assessments such as PISA or TIMSS? Do they represent the mathematical culture that we want K-12 mathematics education to develop? What can we learn from them?

A reform perspective by Anthony Ralston Preamble. I believe passionately that the K-12 mathematical curriculum, as it exists in most countries, needs substantial reform. But, because the notion of “reform curriculum” means different things to different people, I think I should begin by delineating the perspective from which I view the reform of mathematics curricula. First, neither constructivism nor its antithesis plays any role in my beliefs about reform. Thus, arguments about such things as discovery learning or about whether rote memorization is a good or a bad thing will play no role in what follows here. Next, I believe strongly that mathematics should be a demanding subject in all grades, probably the most demanding that students study in each grade. Thus, any suggestion that mathematics should be “dumbed down” at any level is anathema to me. Finally, I believe, as surely all attendees of ICM2006 do, that mathematics is a dynamic, growing subject with ever-changing opinions on what is more important or less important mathematical subject matter. But, perhaps in contradistinction to many ICM attendees, I think this perspective must include not just areas of research but also the entire K-12 curriculum. Thus, what is important subject matter in K-12 mathematics today may be – I think, is – different from what it was yesterday and no doubt is different from what it will be tomorrow. Technology. Mathematicians were slower than almost all scientists and engineers to make computing technology a part of their everyday working lives1 . Nowadays, how1 Mathematicians’ attitudes about technology as well as about other matters considered in this paper are discussed in A. Ralston, Research Mathematicians and Mathematics Education: A Critique, Notices Amer. Math. Soc. 51 (2004), 403–411.

A reform perspective

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ever, many research mathematicians use computers routinely for number crunching, for accessing computer algebra systems, and for using a variety of other computer software for both professional and non-professional purposes. Still, it appears that, even as most mathematicians now recognize computer technology as an indispensable tool for doing mathematics research, they resist the notion that computers should be widely used in mathematics education on the grounds that what is important in K-12 mathematics education has hardly changed in – dare one say it? – the past century. The crucial aspect of whether – and, if so, when – computers or calculators should be used in K-12 mathematics education has resulted in more controversy than any other aspect of mathematics education. I have written elsewhere about my belief that pencil-and-paper arithmetic (p-and-p, hereafter) should be abolished from the primary school curriculum in the sense that no level of proficiency in it should be expected of students although teachers should be free to use p-and-p examples as they wish. Since I published a paper to this effect in 19992 , I have seen no reason, cogent to me, to back off from this position3 . Of course, you must understand that, keeping in mind the position stated in the Preamble, I would replace a p-and-p-based curriculum with a rigorous curriculum emphasizing mental arithmetic while allowing free use of calculators in all grades. The goal of such a curriculum, as with any arithmetic curriculum in primary school, would be to achieve the number sense in students that would enable them to proceed successfully with secondary school mathematics. I cannot provide any evidence why a mental arithmetic, calculator based curriculum would work because it has not been tried but neither has anyone adduced a compelling reason why it should not work. Moreover, no one can give good reasons to continue the classical p-and-p curriculum which has never worked very well and must now be working more poorly than ever, given that almost all students will recognize that the classical curriculum tries to teach them a skill without practical value any longer. In addition, since students will almost universally use calculators outside the classroom, forbidding them inside the classroom is self-defeating. Only if it can be argued that a p-and-p-based curriculum is clearly the best way to prepare students for subsequent study of mathematics, can such a curriculum be justified in the 21st century. But I don’t believe any compelling argument of this nature can be made; all such attempts I’ve seen can only be described as feeble. Learning arithmetic – what the operations are, when to use them, place value etc. – is crucial for the study of all subsequent mathematics. But not only is p-and-p calculation not necessary to the goal of learning about arithmetic, it is positively destructive of that goal.

2A. Ralston, Let’s Abolish Pencil-and-Paper Arithmetic, Journal of Computers in Mathematics and Science Teaching 18 (1999), 173–194. 3An area of particular controversy is whether the traditional long division algorithm should be taught at all. My opinion on this can be found in A. Ralston, The Case Against Long Division, http://www.doc. ic.ac.uk/∼ar9/LDApaper2.html.

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Skill building. Skill building is of value in K-12 math education only insofar as the skills learned facilitate the doing of mathematics and the subsequent study of mathematics. It must be recognized that (almost?) none of the skills traditionally taught in K-12 mathematics have value any longer as skills per se. But, following the foregoing argument, if p-and-p skills are not to be taught, it is imperative that learning substantial mental arithmetic skills should be a major goal of primary school mathematics. These skills should include not just the obvious ones of immediate recall of the addition and multiplication tables and the ability to do all one-digit arithmetic mentally but also the ability to do substantial amounts of two-digit arithmetic mentally. It needs to be emphasized that the development of good mental arithmetic skills requires good coaching from a teacher about the various algorithms that can be used to do mental arithmetic and then hard work by the student. Mental arithmetic, say two-digit by two-digit multiplication, is hard4 . Learning to do it well involves much practice during which the student will decide which algorithm is most congenial to her/him. Teaching and learning mental arithmetic must be a joint responsibility of teacher and student. One advantage of learning to do two-digit arithmetic mentally is that such a skill requires a good grasp of place value, an important aspect of primary school mathematics in any case. Another advantage is that automaticity or near automaticity in oneand two-digit mental arithmetic allows students to be given demanding word problems. More generally, sound technique in mathematics must always be the forerunner of good conceptual understanding. A word about fractions. Primary school is certainly the place where students should learn about fractions, reciprocals and the conversion of fractions to decimals and vice versa. But I doubt it is the right place for them to learn fraction arithmetic except perhaps in some simple cases. When students get to secondary school, they will need to do arithmetic on algebraic fractions. This would be the best time to teach the arithmetic of both numeric and algebraic fractions since, in any case, few students will remember the arithmetic of numeric fractions from when it may have been taught in primary school. Test and assessment. The standardized testing culture that has swept over the United States and is rapidly advancing in the United Kingdom and other countries is perhaps the most serious threat of all to quality mathematics education throughout the world. The standardized testing requirements in the U. S. No Child Left Behind (NCLB) legislation will have the almost certain result that NCLB will be that act most destructive of quality education ever passed by the United States Congress. The pressure on schools and teachers for students to achieve high grades on stan4 Is there any reason why learning to perform two-digit by two-digit multiplication mentally should not be a realizable goal of school mathematics? I don’t think so. Some positive evidence is contained in D. Zhang, Some Characteristics of Mathematics Education in East Asia – An Overview from China, in Proceedings of the Seventh Southeast Asian Conference on Mathematics Education (N. D. Tri et al., eds.), Vietnamese Mathematical Society, Hanoi, 1997.

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dardized tests always leads to a number of evils that have been widely catalogued. Three of the worst are teaching to the test, emphasis on routine mathematics at the expense of advanced topics and problem solving, and the inordinate amount of time taken to prepare for these tests which not only drives important mathematics from the classroom but also often means decreased attention to science, history and the arts generally. Moreover, the inevitable result of emphasis on standardized tests is that scores increase without any concomitant increase in learning5 . I am not opposed to testing students. Quite the contrary. It is by far the best way for a teacher to assess the learning of her/his students. But in the not quite antediluvian past, the assessment task was left to individual teachers in their classrooms. Why have things changed so much? The answer in the United States and other countries appears to be that educational administrators, politicians and even parents no longer trust classroom teachers to do the assessment job themselves. This is not altogether wrongheaded. As I and others have argued elsewhere6 , the quality of K-12 mathematics teachers in, at least, American schools has been declining for half a century and, while there are still many excellent mathematics teachers in American schools, too many are not competent to teach the mathematics they are supposed to teach7 . But, if this is so, standardized testing will only exacerbate this problem by convincing too many who might become teachers that there is no scope for imagination or initiative in school mathematics teaching. The crucial point is that there is no sign whatever that standardized testing has ever been effective in increasing student learning. If all standardized testing in all subjects were abandoned at all levels short of university entrance, this would be an immediate boon to all education. I should say a word about TIMSS and PISA. Since both of these are essentially diagnostic tools given to a sampling of students, they do not suffer from most of the strictures above. For example, teachers cannot teach to the test because at most a very few students in each class will take these tests.

A traditional perspective by Ehud de Shalit The author of this essay is a mathematician who found himself involved in questions of mathematical education despite lack of formal background in the discipline. I make 5 See A.Ralston, The Next Disaster in American Education. The Sacramento Bee, 1 December 2002 (http:// www/doc.ic.ac.uk/∼ar9/NextDisaster.html). 6 See A. Ralston, The Real Scandal in American School Mathematics, Education Week, 27 April 2005 (http:// www/doc.ic.ac.uk/∼ar9/TeacherQual.html) and V.Troen and K.C.Boles, Who’s Teaching Your Children? Why The Teacher Crisis is Worse Than You Think and What Can Be Done about It, 2003, Yale University Press. 7 Indeed, while mathematicians generally choose to argue about something we may be knowledgeable about – curriculum – a far more serious problem with mathematics education in most countries is the inability to attract enough high quality people to teach school mathematics.

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no claim to know the literature of science education, and I am surely ignorant of important studies in the area. I nevertheless dare to participate in the discussion because I believe that educators and scientists alike should bear the burden of shaping our children’s education, listening to and learning from each other’s point of view. It is deplorable that recently, the two communities of math educators and mathematicians have been poised against each other, mostly, but not always, the first being portrayed as “reformers”, the latter as “traditionalists”8 . Emotions have run high, and the two communities found themselves in conflict, instead of joining forces towards a common cause. This being said, I also want to apologize for not having equally strong opinions on all issues. In fact, I will address two of the points raised by Prof. Artigue (the impact of technology and skill building), and make only minor remarks on the third (tests and assessment), which I consider to be a political issue more than a mathematical or educational one. I hope to make myself clear in due time. Moreover, depending on the circumstances, these three sample topics, important as they be, need not have a decisive affect on the success or failure of a given system. External factors such as class size, discipline, teacher training and resources, which vary considerably from state to state, are often of greater importance than questions of curriculum and methodology. However, unable to influence the first in a direct way, we, mathematicians, focus on the latter. My starting point is that mathematics teaching need not necessarily follow the rapid changes in the usage of the subject in society or technology. Its prime role is to imbed in our children a basic sense for, and understanding of numbers, symbols9 , shapes and other “mathematical objects”, together with skills in manipulating these objects, that are needed to develop what is commonly called “mathematical reasoning”. The objects to be chosen, the time devoted, and what is taught about them, should be dictated by their prominence in mathematics, and their epistemic and pedagogical value, and less so by their frequency in daily life. This does not mean, of course, that examples and applications of the material should not be updated and modernized, but I do preach respect for the traditional way of teaching, because more than it was based on old needs, it was based on inherent values that have not changed with time. A well-trained mathematical mind is a highly flexible system. If brought up correctly, it will find its way to adjust and analyze mathematical scenarios very different from the ones that surrounded it initially, while it was being shaped. As an example, consider the well trodden issue of long-division. I believe that the standard algorithm should be taught in elementary school, thoroughly explained and practiced not because of its practical value. Rather, it is important because it enhances the understanding of the decimal system, of zero as a place-holder, of the Euclidean algorithm, and is a necessary precursor for polynomial arithmetic. It allows 8 Those unaware of the ongoing controversies, can read David Ross’ article Math Wars (www.ios.org/articles/ dross_math-wars.asp) and the references therein. 9A. Arcavi, Symbol sense: informal sense-making in formal mathematics. For the Learning of Mathematics 14 (3) (1994), 24–35.


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the child to review the multiplication table and develop number sense while doing something else, more advanced, so it makes learning more interesting. Moreover, it is natural. It therefore agrees with mental arithmetic, and helps us visualize the process involved in division. For these and for many other reasons, well explained in10 and not mentioned here for lack of time, long division is a pedagogical gold mine. The abandoned algorithm for extracting the square root, often cited to justify abolishing long division as well, is in comparison a pedagogical swamp, was abandoned for this reason, and not because it became obsolete. Respect for traditional values in education has another advantage, that new theories are tested gradually, and radical potentially damaging changes are avoided. An ailing educational system need not be ailing because its underlying principles or methods are old-fashioned, and reform in itself is not an automatic cure, even where needed. More than often, the reason for failure is that good old principles stopped being implemented correctly, for various sociological reasons on which I do not want to elaborate here. The second general remark is that I do not believe in teaching in vacuum, or in a content-empty environment. Learning must focus on concrete concepts, methods, algorithms if necessary. Insight and creativity come with variations on a theme, not where there is no theme. Teaching “how to solve it” is not synonymous with dry cookbook mathematics. It can be fun and enlightening. Constructivism11 has led some educators to minimize teacher’s intervention in the learning process. Such an approach may be tried on a single-time basis, through enrichment activities. But it is time consuming, with the average teacher may lead to fixation of mistakes, and for anyone but the brightest students can be very frustrating. We simply cannot expect the children to come up with the great discoveries of arithmetic and geometry, let alone calculus, by pure exploration. A fundamental feature that distinguishes human beings from animals is that we can learn not only from our own experience, but also from that of our ancestors. To be illustrative, I think of the art of teaching as give-and-take. The teacher delivers a package of knowledge, bit by bit, each time taking back from the students their responses, their reflections, their mistakes. On these she or he builds up, shaping and manipulating the dialogue, until a deep understanding and the desired proficiency are achieved. To believe that these can spring up spontaneously, just by setting the stage and giving a slight stimulus, is to assume too much. Finally, a word about the term conceptual thinking. It is often brought up by advocators of certain approaches in education to distinguish their goals from those of others, who – so it is to be understood – lead to lower level thinking. I don’t know of any kind of thinking that is not conceptual. Abstraction, in language or in mathematics, making generalizations, or conversely, looking for examples, testing predictions and searching for the right vocabulary to communicate our mental processes, are all instances of conceptual thinking, namely thinking in terms of concepts. The contro10 The role of long division in the K-12 curriculum, by D. Klein and J. Milgram, ftp://math.stanford.edu/pub/ papers/milgram/long-division/longdivsiondone.htm. 11 Constructivism is the cognitive theory based on the idea that knowledge is constructed by the learner.

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versy, in my view, is not about whether conceptual thinking is more or less important than basic skills, but whether acquiring those skills is part of conceptual thinking, as I want to argue, or not12 . Skill building. Drill and practice. Like a swimmer or a pianist the student of mathematics has to absorb great ideas, but also to practice hard to be able to use them efficiently. Contrary to the common belief, the primary reason for skill building is not the need to perform mathematical tasks with great precision and speed, because in our age these human qualities have been surpassed by machines, and we need not regret it. I see three important reasons to promote skill building. The first is that skill building is essential for forming a sense for numbers, and later on for symbols, functions, or geometry. Subtle instances of insight and analogy, are woven into a web of images and associations in one’s mind, and cannot be classified and taught sequentially. They are only the product of long-term practicing and skill building. The distance between knowing something in principle and mastering it is very big in mathematics. The second reason is that our mind functions on several levels simultaneously, and we are not always aware of the sub-conscious levels that are “running in the background”, if I may use a metaphor from computer science. To be able to free the thinking creative part of our mind, to let it form the web of links needed for exploration and discovery, we must defer to the background more routine tasks, that in the past occupied the front, but should now be performed semi-automatically. The last reason in favor of skill building is rarely mentioned, and might seem to you heretical. Experience has taught me that many children, especially those suffering from math phobia or learning disabilities, are highly rewarded psychologically by success in performing a routine algorithm, such as long division, and by acquiring proficiency in a given task. Such a reward for them is a higher boost than the ability to understand the theory behind it, or the fun in discovering a method by themselves. Once they know the “how” they are lead to ask “why”. I would not rule out an approach that harnesses skill building before understanding, if the teacher feels that it suits the child better. Needless to say, both aspects should eventually be covered, and bright children who have mastered the technique and eagerly ask good questions should not be hindered. Skill building is often confined – by those promoting “conceptual understanding” as a substitute – to algorithmic skills, and algorithmic skills are then downgraded to mere rote. While algorithmic skills are very important, and the algorithmic approach to arithmetic is something to be cherished, as I made clear in the example of long division, mathematical skills are by no means only algorithmic. The ability to translate a word problem into arithmetic, or later on into algebra, is a well-defined skill. Analyzed closely, it consists of many sub-skills, like distinguishing relevant information from irrelevant data, choosing the variables cleverly, translating prose into algebra, and 12 See Basic skills versus conceptual understanding, a bogus dichotomy in mathematics education, by H. Wu (http://www.aft.org/pubs-reports/american_educator/fall99/wu.pdf).


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finally the technique of solving, say, a system of linear equations. Geometric skills, drawing to scale, recognizing hidden parts, decomposing and assembling figures, as well as computational skills of area and volume, form another category. Given my earlier criticism of the constructivist approach, it will not come as a surprise that I believe in standard algorithms. It is true, students who come up with their own (correct!) algorithms should never be scolded, but eventually standard algorithms are more efficient, help in the process of automatization of algorithmic tasks discussed above, and also serve an important purpose of establishing a common language. As an example, after a certain amount of preparatory classes meant to clarify the distributive law, which may include both manipulations of brackets and geometric representation by rectangles, I would simply teach the standard algorithm for “vertical multiplication”. I do not see the benefit in letting the students make up their own algorithms, where inevitably many will multiply units with units, tens with tens etc. and then add them up. To expect from fourth graders to come up with what was one of the main achievements of the Hindus and the Arab scholars in the Middle Ages is unrealistic. However, once the algorithm has been explained, both the how and the why, and practiced, there are many subtle questions that can be left for discussion and discovery. Would it always be more economical to apply the algorithm as is, or perhaps switching the position of the two numbers to be multiplied saves some operations? How can we estimate in advance the order of magnitude to save us from potential pitfalls, what double-checks should we make etc. etc. Anthony Ralston, in his paper “Let’s abolish pencil and paper arithmetic”13 advocates to abolish basic algorithmic skills that were the bread-and-butter of elementary school arithmetic for centuries. He summarizes his discussion by saying “Since no one argues any longer that knowledge of PPA (pencil-and-paper arithmetic) is a useful skill in life (or, for that matter, in mathematics), the question is only whether such ’deprivation’ could leave students without the understanding or technique necessary to study further mathematics.” Even if we accept the premises, doubtful in my mind, I think he misses the point. First, any attempt to separate understanding from technique is artificial. Second, it is the miracle of the subject that the very same principles underlying higher mathematics, or fashionable topics such as geometry and statistics, often quoted as benefitting from the time freed by the abolishment of PPA, are manifested in their purest and simplest form in these basic skills. A person not knowing how to calculate what 35 of 4 27 kg of rice are, will not have the technique to analyze the changes in the school budget of England. Nor will he have developed enough intimacy with numbers to estimate those changes in advance, or tell instantly, if his calculator-based computations make sense or not. As a substitute to PPA, Ralston elevates mental arithmetic to a central position in his proposed program. To give examples, he expects elementary school students to 13 In Journal of Computers in Mathematics and Science Teaching 18 (2) (1999), 173–194.

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perform two-digit by two-digit multiplication mentally, and high-school students “to be able to factor a variety of three term quadratics mentally”. To succeed, he admits, mental arithmetic should be practiced in calculator-free environment. I wholeheartedly agree with the importance of mental mathematics, both for developing number (and symbol) sense, and for practical purposes, estimation and checks. I do not understand though the reluctance to allow one to put things on paper. PPA does not contradict mental arithmetic. It records it, something we shouldn’t be ashamed of, and without which we cannot communicate or analyze peacefully what we have done. It also helps in visualizing graphically the steps carried in our mind, and it allows us to organize little mental steps into a larger procedure, without putting too heavy a burden on our memory. The impact of technology. There are two somewhat separate questions here. The first is to what extent should the curriculum be dictated by the way mathematics is used in technology, and to what extent should we conform to requests coming from the changing society, rather than teach basic principles and skills14 . I have expressed my opinion about this question in the opening statement. Contrary to the quotation just mentioned, I believe that education in the large, ought to enrich the child and teach him or her basic skills, knowledge, values and understanding that are absolute. If carried out correctly, they will inevitably produce a knowledgeable, thinking, skilled and creative citizen. If tailored to the needs of a certain industry or society, rather to these absolute values, they will produce poor technocrats. The second question involved in the issue of technology is to what extent do technological innovations influence the way we teach in class. This concerns mostly calculators in elementary school, but also the use of graphic calculators in calculus, Excel sheets in statistics and computers in general. It would be wrong to ignore the changes in technology, the challenges that they bring about, and the opportunities which they provide for demonstration and practice. However, we should clearly define our mathematical goals, phrase them in mathematical terms and avoid as much as possible slogans, even if we agree with their general mood. We should distinguish mathematical goals from educational goals. Only then may we look at issues of technology, and decide whether they help steering math education the right way, or not. To understand the effect this process of analyzing the role of technology has, consider the following example. The child will be able to derive qualitative and quantitative information form graphs such as a graph displaying the change of temperature with altitude. I hope everybody agrees with the statement as a basic goal of K-12 mathematical education. The terms qualitative and quantitative information demand further elaboration, but I shall not go into it. Now suppose we have to choose between graphic 14 Judah Schwartz, in his essay Intellectually stimulating and socially responsible school curricula – can technology help us get there? writes: “By far the dominant expectation of education in most societies, at least as articulated by political leaders and by the print and electronic press, is to prepare people for the world of work.”


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calculators and pencil-and-paper, for a first encounter with graphs as a tool to communicate observations and measurements. Have we phrased our goal as The child will learn to appreciate the use of graphs in natural sciences such as climatology, we might be inclined to favor graphic calculators. They are attractive, have the fragrance of modernism, and provide vast opportunities that pencil and paper do not provide. But are they as good in conveying first principles? Can the child learn from them where to choose to draw the axes, what scale to use, and how to plot the data? Even the mere physical act, the hand-eye coordination in handling the ruler, is fundamental in my eyes to the learning process. Feeding the data into a calculator, then pressing a button, produces wonderful results, but has its pedagogical drawbacks. This does not mean I would discard graphic calculators. At a later stage they can be helpful in adding visual affects that are difficult to achieve without them – zooming in and out, changing scale, flipping the axes, to name a few. I would simply be careful in my choices, which tool to apply first in class. While I can see the benefits of graphic calculators in middle-school in studying functional dependence, I am much less excited by the use of ordinary calculators in elementary school arithmetic. At this early stage building number sense is the teacher’s number one task. I still have to hear one good argument in favor of calculators in this regard. I need no proof for how destructive they can be. Even those opposed to PPA value mental arithmetic, as means for estimation and double-check. Unfortunately we have witnessed all around us, at school and at the university, a significant decline in these skills over the last two decades, that I can only attribute to the introduction of calculators. Whoever agrees that skill building is an important component of mathematical understanding, and cannot be separated from conceptual thinking, must also confess that calculators at an early age are impeding normal mathematical development. Those advocating early use of calculators necessarily advocate early emphasis on decimals at the expense of simple fraction arithmetic. Is it right? From the point of view of technology, simple fractions are probably obsolete. From the point of view of their pedagogical value, in understanding basic principles of arithmetic, such as ratio and proportionality, or unique factorization, and in anticipating similar structures in algebra, they are indispensable. For all these reasons I would happily ban the use of calculators in class until a solid understanding of arithmetic has been achieved, and the associated skills have been built. I am not in a position to judge whether these happen at the end of fifth, sixth or seventh grade, but the general spirit is clear to me. Two arguments that are often heard in favor of technology at school are (a) that to oppose it is a lost battle and (b) that technological skills are so important in society, that not teaching them early would deprive certain children, especially those coming from poor families, of future opportunities. To the first argument I have nothing to say, except that if we adopt it we shouldn’t be here today. As for the second, I must admit I am very sensitive to the social obligations of educators. Fortunately or unfortunately, home computers are not anymore the sign of a privileged family, much as TV is not a sign of progress, and I honestly believe that mastering Excel carries no

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more mathematical value than mastering a microwave manual. Finally, a comment on a growing trend among educators to write computer-assisted material or use sophisticated software, such as Dynamic Geometry Software, in conjunction with the standard curriculum. Some of it is very well made, enriches the learning environment, and I have no objection to computers per se. But from the little I have seen in this medium, in terms of cost-benefit analysis, the added value is not big, so I will never substitute a computer for the informal contact with a talented teacher. When it comes to political decisions, where to invest the money, my preferences are clear, at least in the country I come from. To this one should add that computers are not just a tool to convey the same message more efficiently. Learning in a computerized environment affects our perception of the objects of study. Good or bad, this has to be analyzed before a new computerdependent program is adopted. Tests and assessment. Testing is a controversial issue among educators. There is a whole separate session at ICM2006 devoted to two competing international comparative tests – PISA and TIMSS. It is well known that certain educators detest testing altogether, while others build their whole curriculum around it. The more I think on it the more I become convinced that testing is a political issue, namely an issue that has to be decided by policy makers, based on an ideology, and taking into account factors that are only remotely related to math education. An excellent example is the controversy around US government act “No Child Left Behind” from 2002. Testing takes various shapes. It can be comprehensive or diagnostic. You may test accumulated knowledge, or you may test the potential of a student. You may test algorithmic skills, or you may test insight and creativity. (Even though, as I said above, the former are indispensable for developing the latter, when it comes to testing, they are quite different.) A math test can be phrased in formal language or in prose. A test can be confined to one school, to one state, or to a nation. Studies show that the framework within which a problem is set affects the rate of success, and this effect changes with gender and origin. I have not mentioned more radical views which claim that western societies test only “western intelligence”, and blame the relative failure of certain minorities on the dominant western frame of mind. Testing can also serve a variety of goals. It may be purely informative, or can serve to rank, for purpose of admissions or stipends. It can test the students, but it can also test teachers success, and inform them of potential problems. Testing can be used for comparing alternative programs, or it may be needed to impose discipline on students, and on educators. I regard all these goals as legitimate, and every kind of test welcome, provided one knows what kind of information to expect form it. A company recruiting civil engineers will probably test different mathematical skills than a software developer, and a matriculation exam summarizing the achievements of a student in high school need not be similar to entrance exams at a university, where a greater emphasis may be put on the student’s potential and creativity.

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Obviously teaching should center on the subject-matter and not only prepare for tests, but a change in curriculum often requires frequent testing to make sure the message gets across. Where there is a good tradition, and little intervention is needed, testing can be kept at a minimum. Under different circumstances tests may become a central integral part of the program. Mathematical education will benefit from an open discussion of the issues raised here and others. It is important that mathematicians will express their views, paying respect to educators, and share their convictions with them. It is important to get to the bottom of examples, and refrain from vague statements. It is important to let changes happen, with ample time if needed, but refrain from changes that are made for the sake of reform alone. Changes must be gradual, and objectively followed. Most new ideas succeed when pushed vigorously with a small group of dedicated teachers, and with a fat budget. The problem is what happens when a case-study involving a dozen schools is over, and those ideas are adopted across the board. Do they carry enough weight to keep the momentum? Are the teachers qualified to spread the gospel?

Concluding remarks by Michèle Artigue In this panel session, we focus on only a few of many possible controversial issues: technology, skill building, test and assessment. For each theme, as the coordinator of this panel, I articulated a short list of questions and asked the two panelists to express their positions. As could be expected, these positions are quite different, as they probably would be on the following fundamental issues: What do we want to achieve today through elementary and secondary mathematics education? What mathematics should be taught in order to achieve these goals? And how should we teach this mathematics? What are the relationships between mathematics education and the society at large? K-12 mathematics education does not serve a unique goal. It aims at the transmission from one generation to the next one of a cultural heritage, which is one of the great achievements of humankind, and at the development of the logical reasoning competence which is so strongly attached to it. It aims at providing students with efficient means for understanding the world in which they live, and play their proper role in it. It aims at preparing and making possible the training of future mathematicians and scientists who will be in charge of the development of mathematics and scientific knowledge, and of the teachers who will have the responsibility of the transmission of this knowledge. Such ambitions can be seen as general invariants, but what is certainly not an invariant is the way we understand each of these components, at a given moment, in a given context; the way we understand the adequate balance between these, and last but not least what we consider the most appropriate strategies for achieving these ambitions. Educational systems try to adapt to this variation mainly

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through curricular changes. The turbulence and controversies we regularly observe attest to the difficulty of this adaptation, and also the fact that the curricular lever chosen is not necessarily the best one. As a mathematician who has worked in the area of mathematics education for more than 20 years now, I am struck by the simplistic way in which the complex problems that K-12 mathematics education faces today are often approached; the existing tendency to give the same value to rough affirmations and anecdotes as to well founded analysis and discussions; the persistent belief in the existence of easy and immediate solutions; the brutality of the changes imposed on educational systems, without considering their real cost, and without developing the necessary means for understanding observed success and failure. Education in the large seems a world where opposition and slogans are in front of the stage, hiding shades of meaning and dialectic visions. Slogans used by those favoring or opposing the use of technology, opposing positions on the development of concepts and of techniques are typical examples of these. Even educational research, in its attempts to reach a larger audience, does not always avoid undue simplifications and oppositions15 . For improving the current situation, we need to overcome such a state, and will try to do so in the ICM panel associated with this contribution. But in order to solve the complex and difficult problems that K-12 mathematics education faces today in many countries, we need to do more than express well-articulated positions on controversial issues and the rationale for these. We need coherent and long-term programs, taking into account the specificities of the different contexts and the existing material and human resources. We need exchanges on our respective situations and experiences for improving these, being aware that solutions in mathematics education are always local ones in terms both of space and time, that it is nearly impossible to determine what is the exact field of validity of a given observed result, the field of extension of a given regularity. We need the collaboration of all those who are involved in mathematics education: mathematicians, mathematics educators, teachers and teachers educators, each of whom can contribute different kinds of expertise. One of the ambitions of ICMI, through its series of ICMI Studies, is to foster such exchanges among all those interested in mathematics education and to make clear what is the state of the international reflection on some selected critical issues, what has been achieved and what is needed16 . I would like to add to these short comments that curricular choices are certainly important but that the dynamics of complex systems, such as educational systems, is not just a matter of curricular choices. The quality of teachers and of teacher education, both pre-service and also in-service, is certainly as important if not more 15 See for instance M. Artigue, Learning Mathematics in a CAS environment: The Genesis of a Reflection About Instrumentation and the Dialectics Between Technical and Conceptual Work, International Journal of Computers for Mathematics Learning 7 (2002), 245–274. 16 Themes for the most recent ICMI studies have been: The teaching and learning of mathematics at university level, the future of the teaching and learning of algebra, mathematics education in different cultural traditions – a comparative study of East Asia and the West, applications and modelling in mathematics education, the professional education and development of future teachers of mathematics.

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important than curricular choices. From this point of view, the fact that mathematics educational research, which has for a long time focused on students, has in the last decade paid increasing attention to the teacher and to teacher education, is a promising evolution. Research tries today to understand the coherence underlying observed teachers’ practices17 the kind of precise mathematical knowledge the profession requires, how it can be developed, how this mathematical knowledge interacts with other forms of professional knowledge, and how these complex interactions influence teachers’ practices and students’ learning. Interesting results begin to be obtained, which at the same time help us understand better what can be realistic dynamics for change. The final success of the enterprise requires the collaboration of those with diverse expertise18 . IUFM of Mathematics, Université Paris VII, France E-mail: [email protected] State University of New York, Buffalo, U.S.A. E-mail: [email protected] Institute of Mathematics, Hebrew University, Jerusalem, Israel E-mail: [email protected]

17A. Robert and J. Rogalski, Le système complexe et cohérent des pratiques des enseignants de mathématiques: une double approche, La revue canadienne des sciences, des mathématiques et des technologies 2.4 (2002), 505–528. 18An example of such a collaboration is given by the Mathematics and Sciences Research Institute in Berkeley which has created an education advisory board and organizes workshops involving mathematicians, mathematics educators, teachers, policy makers etc. on critical issues. The themes of the first two were the assessment of students’ mathematical knowledge and the mathematics knowledge for K-8 teachers.

Panel B What are PISA and TIMSS? What do they tell us? Lee Peng Yee (moderator) Jan de Lange and William Schmidt (panelists)

Abstract. This is a panel discussion on PISA and TIMSS, two international comparative studies in educational achievement. The panelists are Jan de Lange of the Freudenthal Institute, the Netherlands, for PISA, and William Schmidt of Michigan State University, the United States, for TIMSS, with Lee Peng Yee of National Institute of Education, Singapore, as a moderator. They are to explain the nature, the aims, and the conclusions of the two studies, and to argue over their relative merits. This document contains three initial statements from the above-mentioned participants respectively. Mathematics Subject Classification (2000). Primary 00A35; Secondary 00A05. Keywords. PISA, TIMSS, comparative studies.

Introduction by Lee Peng Yee One area of interest in education is comparative studies in educational achievement, in particular, in mathematics, science and reading. There are two such international studies involving mathematics, namely, PISA and TIMSS. PISA stands for the Programme for International Student Assessment. It is better known in Europe. TIMSS stands for the Trends in International Mathematics and Science Study. TIMSS was previously known as the Third International Mathematics and Science Study. Each study of PISA or TIMSS involves approximately 50 countries and thousands of students in each participating country. The studies generated volumes of publication and numerous related research projects. The fact that some Asian countries topped the achievement list in TIMSS amazed many people and drew the attention of the industrial countries. Consequently it induced the study on these high-performing Asian countries, namely, China, Korea, Japan, and Singapore. Further a country could do well in TIMSS but not in PISA. This phenomenon is now known as PISA shock. Hence in addition people are also interested in the comparison of these two international studies. The impact of PISA and TIMSS has gone way beyond the mathematics and science community. It even Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Panel B What are PISA and TIMSS? What do they tell us?

influences the policy makers of a country. It is timely that we have a panel discussion on the topic. TIMSS. The study was commissioned by IEA, the International Association for the Evaluation of Educational Achievement. The first round of TIMSS took place in 1995 and the second round in 1999. It was the third round that made TIMSS famous world wide It collects data on educational achievement from students at the fourth and eighth grades. It also collects extensive information from students, teachers and school principals about the teaching and learning of mathematics. The test items are matched against those in the standards or syllabus. Then the data are analyzed and the reports published. The next round will take place in 2007. For details, see [1]. PISA. The study was initiated by the OECD countries. OECD stands for Organisation for Economic Co-operation and Development whose member countries were originally countries from Western Europe but now they are all over the globe. PISA was conducted every three years in 2000, 2003 and the next one in 2006. The tests are administered to 15-years-old students. The tests are supposed to assess how well students are prepared for their full participation in society. Similarly, the data are analyzed and the reports published. As we can see, PISA differs from TIMSS in methodology and aims. For details, see [2]. Benchmarking. Both PISA and TIMSS have been used by many countries for benchmarking. Roughly speaking, TIMSS is grade-based, that is, testing students of Grade 4 and Grade 8, whereas PISA is age-based, that is, testing the 15-years-old students. The 15-years-old students are those who are near the end of their compulsory education. Test items in TIMSS are more content or standards orientated, whereas those in PISA are more literacy orientated. TIMSS assesses how much students have achieved in schools. PISA assesses how well students are prepared for the outside world. Of course, this is an over-simplified view of the differences between the two studies. It does give a general idea about the two studies. Panelists. They are Jan de Lange of the Freudenthal Institute, the Netherlands, speaking for PISA, and William Schmidt of Michigan State University, the United States, speaking for TIMSS. Jan de Lange is Director of the Freudenthal Institute and a full professor at University of Utrecht, the Netherlands. He was a member of the National Advisory Board of the Third International Mathematics and Science Study, and is currently Chair of the Mathematical Functional Expert Group of the OECD-PISA. William Schmidt is a professor at the College of Education, Michigan State University, and the national research coordinator and executive director of the United States National Research Center which oversees the United States’ participation in the Third International Mathematics and Science Study. At the panel discussion, they are to present what PISA and TIMSS are respectively, and what they are for. Then they will discuss and possibly answer questions from the audience.

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Issues for discussion. The issues for discussion include at least some or all of the following questions. The questions are divided into three categories. First, what are PISA and TIMSS? • Does PISA or TIMSS really serve the purpose intended? • Why do we need PISA when we already had TIMSS? • What are the good points or bad points of PISA and TIMSS? Secondly, what do they tell us? • Is it really meaningful to use PISA or TIMSS for benchmarking? • Some countries did well in TIMSS but not in PISA. Why? • Both PISA and TIMSS have collected a vast amount of data. Are they useful for other researchers? What can they do with the data? The last question above was previously raised at the International Round Table in Tokyo 2000 [3]. Thirdly, what is the future? • The learning process of a student is a long-term affair. Perhaps the threeyear cycle or four-year cycle is simply too short to measure the progressive achievement of a student. Do we need to measure so frequently? • Will there be PISA or TIMSS 20 years from now? This short statement serves as an introduction to the panel discussion to be held on 28 August 2006 in Madrid, Spain. Other statements from the panel speakers follow.

References [1] TIMSS 2003, Trends in International Mathematics and Science Study. International Mathematics reports released 14 December 2004. Website: www.timss.com. [2] PISA 2003 technical report, OECD Programme for International Student Assessment, Website: www.pisa.org. [3] Lee, Peng Yee, International Round Table, Proceedings of the International Congress on Mathematics Education, Tokyo 2000.

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TIMSS as a study of education: why should we care? by William H. Schmidt Comparative studies of education often seem to evoke a “so what?” or “who cares?” reaction. Studies of students’ achievement in different countries may leave one wondering what practical importance such differences hold in the real world or work and commerce. Descriptions that highlight differences in common educational practices may appear intriguing and stimulate curiosity but may leave one wondering what the relevance is to what happens (or should happen) at the school down the street. The value of such studies is almost assumed to be self-evident given, it seems, by the sort of attention the media frequently affords them. Reports of rankings along with comparisons of scores with countries x, y, and z reduce the entire endeavor to a sort of education Olympics or horserace. The value, obviously, lies in the comparisons! Who is on first? Who is doing it right? As intriguing and entertaining as some comparisons may be – “Wow! Teachers in country x never assign homework!” or “Students in country z have to go to school on Saturday!” – these are practices that must be understood within their particular social, cultural, and educational contexts. Attempting to copy or transplant the practices of one country into another will not likely have the desired effect: alien grafts rarely take without extensive preparation and effort. Value of international comparative studies. The real value in international studies lies not in the comparisons themselves, but in the insights we may gain into our own common practices. International comparisons hold up and frame what’s familiar against a background of a considerable range of alternatives. This can lead to a thoughtful reconsideration of our rationale for doing things the way we do – or even initiate a thoughtful evaluation of something never before considered. Many people, for example, are familiar with schools. They know what schools are and what happens in the classrooms inside the schools. Schools are schools; whether they are urban Paris or rural Montana. We began our involvement with international education research in the days leading up to TIMSS with a similar assumption about the nature of schools in various countries. We discovered that school has commonalities everywhere. What is common wherever schools are found are students, teachers, and textbooks. How these commonalities interact and work within a larger education system, however, can vary considerably. We discovered, for example, that in Norway primary teachers typically stay with the same group of students for the first five or six years of students’ formal school experience. We also learned that in Switzerland, ‘schools’only exist in large cities. The majority of students and teachers meet together in rooms located in buildings that are not necessarily dedicated to housing educational activities. Furthermore, school administrators and other support personnel are only found in such dedicated facilities which generally house the upper secondary grades or are located in the cities.

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Clearly there are a number of ways to conduct school. The examples mentioned here were not selected to suggest that all countries change either the nature of their school buildings nor the length of time primary teachers typically work with the same group of students. Some consideration of these issues may be fruitful, but the point to be made here is that these schooling practices represent options – choices that have been made about how school gets done. The more we can see the way we do things as choices, the better position we are in to consider and construct profitable change or reform. What we can learn from TIMSS. In the Third International Mathematics and Science Study (TIMSS) the focus was not on the structural aspects of school such as the previously mentioned examples, although these were a part of the study. The focus, rather, was on the substance of education, the school curriculum, the content that’s at the center of what teachers and students do in schools. Previous international studies led us to suspect that the achieved curriculum, what students demonstrate that they know, varies from one country to another. TIMSS assessed this aspect of curriculum in the context of an extensive investigation of the intended curriculum, what systems intend their students to learn, along with the implemented curriculum, what is taught in the classroom. Measurements of these curriculum aspects led to one unmistakable conclusion: the mathematics taught and studied in the schools of one country can differ substantially from what exists in the schools of another. In short, there are many ways to do mathematics education. More specifically, this curriculum measurement in TIMSS led to some thought provoking insights into the U.S. mathematics curriculum. For example, the U.S. intends teachers and students to study two to three times the number of topics in the first through eighth grade as is typical in other countries. Consistent with this breadth, U.S. textbooks are truly first in the world in their size, weight, and scope. Not too surprisingly, given these intentions and resources, the U.S. teachers tend to spend some time on every intended topic typically without emphasizing any small number of topics as is common in other countries. All of this contributes to the “mile wide, inch deep” nature of the U.S. curriculum. These insights were possible because TIMSS was designed from the start to examine the relationship among the various aspects of the curriculum: the intended, the implemented, and the attained. These insights have also led to several efforts to thoughtfully revise the U.S. mathematics curriculum. So, what is the value of international study? Certainly not to obtain bragging rights for the top spot on some list nor even to identify specific practices that we may want to copy. The real value stems from obtaining a fresh perspective on the array of choices embedded in our own approach to education. Thoughtful and principled insights stimulated by examples from other systems can lead to powerful revision in our quest to provide a challenging and equitable education for all students.

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PISA: promises, problems and possibilities by Jan de Lange

PISA versus TIMSS. According to the OECD: The OECD’s Programme for International Student Assessment (PISA) is a collaborative effort among the member countries of the OECD to measure how well young adults, at age 15 and therefore approaching the end of compulsory schooling, are prepared to meet the challenges of today’s knowledge societies. The assessment is forward looking, focusing on young people’s ability to use their knowledge and skills to meet real-life challenges, rather than on the extent to which they have mastered a specific school curriculum. This orientation reflects a change in the goals and objectives of curricula themselves, which are increasingly concerned with what students can do with what they learn at school, and not merely whether they have learned it. The term ‘literacy’ is used to encapsulate this broader conception of knowledge and skills. The first PISA survey was carried out in 2000 in 32 countries, including 28 OECD member countries. Another 13 countries completed PISA 2000 in 2002, and from PISA 2003 onwards more than 45 countries will participate ‘representing more than one third of the world population’. PISA 2000 surveyed reading literacy, mathematical literacy, and scientific literacy, with the primary focus on reading. In 2003 the main focus was on mathematical literacy (published in 2004), and in 2006 scientific literacy will be highlighted. It will be clear that TIMSS and PISA have a lot of similarities resulting in improper identification of the two series of studies in the media, which is undesirable and confusing. But the descriptions of the organizations that are responsible, show that they both claim similar relevance for the studies. Even for the expert it will be difficult to relate the following either to TIMSS or to PISA: ‘Countries participating in this study will have information at regular intervals about how well their students read and what they know and can do in mathematics and science.’ Both studies do this and do it, methodologically speaking, in a very similar way (based on Item Response Theory, IRT). Even the reporting tables in the respective reports look very similar. If there is a problem that both studies share, it is the design of the measuring instrument in relation to the validity of the outcomes. Traditionally, validity concerns associated with tests have centered about test content, meaning how the subject domain has been sampled. Typically evidence is collected through expert appraisal of alignment between the content of the assessment tasks and the curriculum standards (in case of TIMSS) and ‘subject matter’ assessment framework (PISA). Nowadays, empirical data are often used before an item is included in a test. Traditionally validation emphasized consistency with other measures, as well as the search for indirect indicators that can show this consistency statistically. More

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recently is the recognition that these data should be supplemented with evidence of the cognitive or substantive aspect of validity. Or as Knowing What Student Knows (2001) summarized: ‘The trustworthiness of the interpretation of test scores should rest in part on empirical evidence that the assessment tasks actually tap the intended cognitive process.’ One method to do this is a protocol analysis in which students are asked to think aloud as they solve problems; another is an analysis of reasons in which students are asked to provide rationales for their responses; and a third method is an analysis of errors in which one draws inferences about processes from incorrect procedures, concepts, or representations of problems. Although some of these methods are applied only after the test is administered, there is a trend that large-scale assessments like TIMSS and PISA use these methods as well. The use of cognitive laboratories to gauge whether students respond to the items in ways the developers intended has become a new instrument in the developmental process. The use of double-digit coding is another sign of interest in the process of problem solving instead of just judging whether an answer is incorrect or correct. A ‘correct’ or ‘partly correct’ score given not only to each work of the student, but also to which strategy was used or where in the process the students ‘lost track’. Validity. The validity of the test instrument remains a complex issue. It goes without saying that there is an inherent tension between the traditional choice of item formats, usually with very restricted time (1–2 minutes per item), and the rather ambitious definitions of what the instrument is intended to measure. But not only the concern about ‘errors’ plays an important role in relying so much on multiple-choice, it is also an economic issue: Many countries participating in these large cooperative studies are unwilling or unable to fund much more expensive multiple marker studies, even if such studies have demonstrated their efficacy. PISA 2003 also had a problem solving component. Many of the items would fit the mathematics Framework, and given the fact that the instrument for problem solving (PS) had much more open ‘constructive’ items, a study relating the math items and the PS items could be very helpful in advancing the discussion on item instruments and their restrictions in large-scale international studies. According to the PISA report on problem solving: ‘The items for problem solving give a first glimpse of what students can do when asked to use their total accumulated knowledge and skills to solve problems in authentic situations that are not associated with a single part of the school curriculum.’ One can easily argue that this is always the case in a curriculum: For mathematical literacy, mathematics as taught at school will not suffice. Students need to read, need to interpret tables and graphs (seen by many as belonging to reading literacy), and, indeed, need problem-solving strategies. But seen from the perspective of promising developments on item formats and item quality, the problem-solving component of PISA is interesting, at least. And if TIMSS implements their intent to ‘place more emphasis on questions and tasks that offer better insights into students’ analytical,

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problem-solving, and inquiry skills and capabilities,’ innovation in large-scale assessments could materialize. PISA versus TIMSS. The main differences between TIMSS and PISA seem to be the following: – curricular emphasis for TIMSS versus functional aspect (literacy) for PISA; – grade-specific structure of TIMSS versus age-specific structure of PISA. TIMSS uses the curriculum as the major organizational aspect. The TIMSS curriculum model has three aspects: the intended curriculum, the implemented curriculum, and the achieved curriculum. These represent, respectively, the mathematics and science intended for students to learn, and how the education system should be organized to facilitate this learning: what is actually taught in the classrooms, who teaches it, and how is it taught; and finally, what it is that students have learned, and what they think about those subjects. International curricular diversity was a serious point of concern to the TIMSS study. The goal was to develop an international test that would be equally fair to all participating countries. Therefore subject-matter specialists from all countries were consulted and asked to contribute to the process of test development. Most countries participating in TIMSS had an intended mathematics curriculum that matched with more than 90% of the items. The outliers were the United States and Hungary with 100% matching, and the Netherlands, with 71% matching. Insiders have discussed the procedure and its validity of this equally unfair analysis. The question not satisfactorily answered is how the mathematics education communities in the different countries were involved, and how representative they were. But if these numbers are accepted, in this context it is worth looking at the minimal matching result of the Netherlands. It was expected that students of other countries would outperform Dutch students. However, contrary to expectations, in 1995 Dutch grade 8 students performed well on the TIMSS test. Their score was significantly above the international average, just below the four Asian top-scoring countries. After some additional research it was concluded that somehow the Dutch students were knowledgeable about the 29% of test items that were remote from their intended curriculum. In the end it was concluded that the students had the abilities for transfer of their knowledge and skills to items that did not match with their intended curriculum. It can be very appropriate to test students on material they have not been taught, if the test is used to find out whether the schools are doing their job. PISA takes this point even further: It is based on a dynamic model of lifelong learning in which new knowledge and skills necessary for successful adaptation to a changing world are continuously acquired throughout life. It focuses on young people’s ability to use their knowledge and skills to meet real-life challenges, rather than on the extent to which they have mastered a specific school curriculum. The two different approaches can both be critiqued: What does it mean that the Netherlands scored so high with the minimal relation with its curriculum? What does

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it mean if PISA will not constrain itself to any national curricula? It is clearly not true that international studies of student achievement may be unintentionally measuring little more than the degree of alignment between the test instrument and the curriculum. What it does measure is still a question open to interpretation. Another indication that shows how difficult it is to make statements that go beyond well-intended opinions can be found in the observation of Westbury in 1992, in relation to SIMS, when he observed that the lower achievement of the United States is the result of curricula that are not as well matched to the SIMS test as are the curricula of Japan. But in TIMSS the match was 100% (see earlier), and still the United States did not perform very well. Impact. The Germans produced a national PISA 2000 report of 550 pages, the international OECD report was 330 pages, and the Dutch report a mere 65 pages. Most countries had something around 150 pages. It is not the statistics that are interesting here, but the message from the report and what has been selected to be included. Even a superficial analysis, which was carried out for this article with the reports mentioned and the one from the United States, makes significant differences visible. There is a common myth that numbers do not lie. It is now widely accepted that data can be gathered, processed, mathematized, and interpreted in a variety of ways. So a key issue is the question of who influences this process, for what reasons, and through what means. The studies just mentioned underscore this concern apart from the fact that even numbers can lie. Back to the very gründliches German report. Not only did the German PISA Konsortium do an excellent and thoughtful job, it also made recommendations for immediate improvement, including ones that directly affect the content. The changes should include: – more integration of inner- and outer-mathematical ‘networks’; – fewer calculations; – more thinking activities and student mental ‘constructions’; – more reflection; – more flexible use of schoolbooks. These goals can be reached when the recommendations that were formulated after TIMSS are implemented: – development of a different math-problems culture: more open-ended, more ‘real-world’; – a new teaching-and-learning culture, with a more exiting cognitive school environment; – more and different professionalization of teachers, emphasizing teamwork. PISA adds to these recommendations a ‘very different conceptualization’ of mathematical concepts and emphasis of modeling and mathematization, situated in contexts. And, argued the report, the Germans have definitely not reached the optimum in using different representations as a tool to build better conceptual understanding.

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Mathematics education is in a state of transition, in part because of the fact that both TIMSS and PISA were taken seriously. Surprisingly the shock and catastrophe that struck Germany as some kind of natural disaster, if one had only the popular media as a resource, has resulted in a government-supported nationwide action-plan with a very strong content part that will result in a different mathematics education culture at schools. Of course, the success of these changes will be measured by PISA 2003, 2006, 2009, and so on. At least in part. The future of PISA. It is very hard to predict the future of PISA. Of course it is a very successful project if one looks at the number of countries participating: 58 in 2006 and growing. And there are many opportunities to make PISA more successful from the content point of view. If PISA is able to include longer and more complex items, as it did with its Problem Solving study in 2003, if technology gets a proper place (as is intended), if group-work can be included in some way PISA would make itself much more rewarding for policy makers and practitioners alike. PISA will also start a study for the 9-year olds, in the near future. In short the OECD definitely has the intention to continue PISA for the next decade at least. And if the instrument keeps improving, it seems worth the effort – although OECD has to be more clear about the fact that PISA measures mathematical literacy, and not curricular mathematics – and how to deal with this principle in the future. PISA will have to address the problem of the Horse-race – a very undesirable aspect that draws a lot of criticism – and rightfully so. Another format of the international report with portraying country by country would not only be more informative, but also would give a more valid picture: one number cannot represent the quality of an educational system. Validity issues have to be addressed, even if PISA is using state-of-the-art methodology. Not only the methodology should be of the highest quality, but also the content – and improvement should be on the agenda continuously. And of course: communication between all parties should improve: Math educators and research mathematicians feel as being watchers of a game they hardly feel any ownership for. This is undesirable: PISA should not address just policy makers if it really wants to make a difference: the data of PISA are in the public domain and any country can analyse these data for its own purpose. This opportunity should not be lost. The meaning of PISA can be co-defined by its users. National Institute of Education of Singapore, Singapore 637616, Singapore E-mail: [email protected] Freudenthal Institute, University of Utrecht, 3561 GE Utrecht, The Netherlands E-mail: [email protected] Center for the Study of Curriculum, Michigan State University, East Lansing, MI 48824-1034, U.S.A. E-mail: [email protected]

Panel C The role of mathematicians in K-12 mathematics education Fr. Ben Nebres (moderator) Shiu-Yuen Cheng, Konrad Osterwalder, and Hung-Hsi Wu (panelists)

Abstract. The need for mathematics educators, schoolteachers and mathematicians to work together to improve K-12 mathematics education continues to be a great concern throughout the world. The main paper for this panel discussion proposes a paradigm or perspective within which to organize this working together of the different groups. This is to see mathematics education as mathematical engineering. From this perspective, the challenge of the mathematics educator and the schoolteacher is to customize mathematics to students’ needs. The role, in turn, of the university mathematician is to customize mathematics courses for teachers so that they in turn may be able to customize the mathematics for the different needs of their students. This paradigm is then discussed in different contexts, in the United States, Hong Kong, Switzerland, and the Philippines. The paradigm is seen to be fruitful in these different contexts. Mathematics Subject Classification (2000). Primary 97D20; Secondary 97B99. Keywords. Mathematical Engineering, curriculum, assessment, professional development, student creativity, gymnasium mathematics.

Introduction and overview by Ben Nebres, S. J. The theme of this panel discussion is “How mathematicians contribute to K-12 mathematics education.” Three distinguished mathematicians, coming from different contexts and different mathematics education traditions, provide challenging and helpful insights into this theme. Because the contexts in which they write are quite different (United States, China, Switzerland), it was decided to present the papers separately. This introductory note is meant to highlight the main proposal and perspective coming from Prof. Wu’s paper and relate the contributions from the other two panelists to it. I also add a few comments from the context of a developing country, the Philippines. First, a brief note on the diversity of contexts. In terms of mathematics achievement based on international comparative studies such as TIMSS, the United States ranks in the middle, while Hong Kong and Switzerland rank towards the top. The Philippines ranks towards the bottom. In terms of educational systems, the U.S. is quite decentralized with great diversity in terms of curriculum, textbooks, teacher Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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Panel C The role of mathematicians in K-12 mathematics education

training, while Hong Kong schools would have greater commonality in terms of curriculum, textbooks, assessment. Some would say that it may be better to compare not the performance of all U.S. schools, but to take account of the diversity of systems and compare blocks of schools (by states or groups of school districts). In terms of resources, the U.S., Switzerland and Hong Kong have first world resources, while Philippine schools operate in the context of great scarcity: classes of 80 students in rooms built for 40, one textbook shared by 5 or 6 pupils and so forth. One can even look at the differences in mathematics education between Hong Kong and the U.S. and Switzerland and the Philippines from the point of view of mathematics education cultures. This is discussed in the recently published “Mathematics Education in Different Cultural Traditions: A Comparative Study of East-Asia and the West”, edited by Frederick Leung, Klaus Dieter Graf and Frances Lopez-Real, Volume 9 in the New ICMI Study Series. My own role as a mathematician in helping improve mathematics education in the Philippine context of poverty of resources is described in a chapter entitled “Philippine Perspective on the ICMI Comparative Study” in this volume of the ICMI Study Series. Despite this diversity of contexts, there is agreement in all the three papers (and in the Philippine experience as well) on the importance of the role of mathematicians in K-12 mathematics education and on a particular paradigm or perspective (mathematics education as mathematical engineering) on how mathematicians can effectively fulfill this role. In the main paper for this panel presentation and discussion, Prof. Hung-Hsi Wu of the University of California Berkeley, proposes a re-conceptualization of mathematics education as mathematical engineering: “Thus chemical engineering is the science of customizing chemistry to solve human problems… . I will put forth the contention that mathematics education is mathematical engineering, in the sense that it is the application of basic mathematical principles to meet the needs of teachers and students.” This is somewhat different from the suggestion of Hyman Bass to look at mathematics education as a branch of applied mathematics. In engineering, what is important is the customization of scientific principles to address human needs. Similarly, in mathematics education as mathematical engineering, what is crucial is the customization of mathematical principles to address the needs of teachers and pupils. From this viewpoint, the challenge is to work out the role of mathematicians in mathematics education analogous to that of physicists in engineering. Just as the roles of physicists and engineers in engineering are deeply intertwined, so should the roles of mathematicians and mathematics educators be in mathematics education. Right now the two worlds are separate and do not communicate well with each other. Prof. Wu writes: “… if mathematicians want to participate in serious educational work in K-12, … the most important thing is the awareness that K-12 mathematics education is not a subset of mathematics, and that there is quite a bit to learn about the process of customization that distinguishes K-12 mathematics education from mathematics.” In my communication with Prof. Wu, we agreed that it is important that the term “mathematics educator” include both the university mathematics educator as

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researcher and the school mathematics teacher as practitioner. While the university mathematics education researcher is an expert on teaching and learning theories, the mathematics master teacher is most knowledgeable about actual teacher, student, and classroom contexts. Success in improving mathematics education will require good communication and working together among mathematicians, university mathematics educators and school mathematics teachers. To properly customize mathematics in different student contexts, a mathematics teacher needs: solid mathematical knowledge, clear perception of the setting defined by students’ knowledge, and flexibility of mind to customize this mathematics knowledge for use in this particular setting. In this model of mathematics education as mathematical engineering, the role of the mathematician is to provide the solid mathematical knowledge. This should be done in such a way that the teacher is provided with different ways of understanding and approaching a mathematics concept so that he can have a repertoire to draw from in customizing the mathematics for different student contexts. Prof. Wu gives examples such as in the teaching of fractions or in providing intervention for students at-risk. The paper of Prof. Shiu-Yuen Cheng of the Hong Kong University of Science and Technology picks up from the “mathematics education as mathematical engineering” framework of Prof. Wu and sets it in the context of mathematics education in Hong Kong. He notes “the main factors for providing an effective mathematics education as curriculum design, teacher competence and assessment methods.” He says that most important is teacher competence and it is to this factor that mathematicians can contribute the most. They can contribute in the university curriculum for mathematics teacher programs and in in-service workshops for mathematics teachers. Together with Prof. Wu (and from my experience as well) he points out that the university curriculum for mathematics teachers, which is usually a combination of courses for mathematics majors and education courses, “do not serve the purpose of providing the necessary understanding to be a competent mathematics teacher.” (Reasons for this are very well argued in Liping Ma, “Knowing and Teaching Elementary Mathematics.”) Prof. Cheng points out that the Hong Kong mathematics education context is one which shows great success as shown by the excellent performance of Hong Kong students in international comparative studies. “In Hong Kong, Johnny can add! In fact, Johnny can do fractions and decimals quite well.” There is, however, a downside to this achievement. Prof. Cheng is concerned that this is at great cost, particularly in “suffocating students’ creativity and motivation for learning.” He stresses the important role of mathematicians in communicating effectively to the public and to decision-makers these important concerns for mathematics education. (This balance between effective mastery of fundamentals and the need to foster creativity has been an important recent concern in East Asia and was the theme of the ICMI-East Asian Regional Conference on Mathematics Education in Shanghai in August 2005.) Prof. K. Osterwalder of the ETH Zurich writes in the context of the upper years of the Swiss gymnasium (years 9–12) and the role of mathematicians in universities

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such as ETH Zurich in preparing mathematics teachers for these upper years. He points out that in Switzerland, students do quite well in international comparative studies in mathematics. Teachers are well trained. They are required to get a masters degree at a level where they could equally opt to go into industry as mathematicians. On the role of mathematicians in K-12 mathematics education, he agrees that the main contribution of research mathematicians is in the education of mathematics teachers. He focuses in a special way on the “Specialized Mathematics Courses with an Educational Focus” taken by mathematics teachers in the university. He provides various examples of course material, from linear equations and linear algebra to noting recent research breakthroughs accessible to gymnasium students, where these course materials “narrow the gap between Gymnasium mathematics and University mathematics” in the spirit of Felix Klein’s “Elementary Mathematics from a Higher Viewpoint.” The paradigm or perspective of “mathematics education as mathematical engineering” proposed by Prof. Wu is thus seen to be quite fruitful in these different contexts. They all point to the central role of the mathematics teacher, whose challenge is to customize the mathematics to the students’ needs. The important role of the university mathematician is then to customize mathematics courses for these teachers in such a way that they may in turn be able to customize the mathematics needed by students in different contexts and with different needs.

How mathematicians can contribute to K-12 mathematics education by Hung-Hsi Wu “To overcome the isolation of education research, more effective links must be created between educational faculties and the faculties of universities. This could allow scholars of education better acquaintance with new developments in and across the disciplines and other professional fields of the university, while also encouraging discipline-based scholars with interests in education to collaborate in the study of education.” Lagemann [14], p. 241. I would like to make a general disclaimer at the outset. I think I should only talk about things I know firsthand, so I will limit my comments to the K-12 mathematics education in the U.S. rather than take a more global view. Such a restriction is not necessarily fatal since a friend of mine observed that what takes place in the U.S. tends also to take place elsewhere a few years later. For example, in France there is now a Math War that resembles the American Math Wars of the nineties (Education Week [7]). We live in a global village after all. Let me begin with a fairy tale. Two villages are separated by a hill, and it was decided that for ease of contact, they would drill a tunnel. Each village was entrusted

How mathematicians can contribute to K-12 mathematics education

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with the drilling of its own half of the tunnel, but after both had done their work, it was discovered that the two halves didn’t meet in the middle of the hill. Even though a connecting tunnel between the two lengths already built could be done at relatively small expense, the two villages, each in defense of its honor, prefer to continue the quarrel to this day. This fairy tale is too close to reality for comfort when the two villages are replaced by the education and mathematics communities, with the former emphasizing the overriding importance of pedagogy and the latter, mathematical content.1 Mathematics education rests on the twin pillars of mathematics and pedagogy, but the ongoing saga in mathematics education is mostly a series of episodes pitting one against the other. There is probably no better proof of the disunity between these communities than the very title of this article. Indeed, if someone were to write about “How chemists can contribute to chemical engineering”, that person would be considered a crank for wasting ink on a non-issue. Chemical engineering is a well-defined discipline, and chemical engineers are perfectly capable of doing what they are entrusted to do. They know the chemistry they need for their work, and if there is any doubt, they would freely consult with their colleagues in chemistry in the spirit of cooperation and collegiality. Therefore, the fact that we are going to discuss “How mathematicians can contribute to K-12 mathematics education” in the setting of the International Congress speaks volumes about both mathematics education and mathematicians. In matters of education it is of course natural for the power structure to hold the reins, just as in matters of engineering they are held by engineers. But while the chemical engineers are glad to have chemists down the hall, and glad to learn what they can use in their work, the corresponding relationship has not been the case for mathematics educators. Since education research is thriving and research funding is ample, it is not surprising that educators want to protect their intellectual independence in the university environment. Rumblings about how mathematically unqualified teachers or deficient curricula are undercutting mathematics learning do surface from time to time, but we have not witnessed the expected aggressive action agitating for collaboration with mathematicians. Other troubling issues related to mathematics content, such as the presence of incorrect assessment items in standardized tests, likewise fail to arouse genuine concern in the mathematics education community. To an outsider, the protection of the “education” enclave seems to matter more to university educators than collaboration with the research mathematics community that could strengthen K-12 mathematics education. By contrast, if the department of chemical engineering consistently produces engineers with a defective knowledge of chemistry, or if accidents occur in its laboratories with regular frequency, would the chemical engineering faculty not immediately spring to action? This question prompts the thought that maybe we no longer know what mathematics education is 1 In writing about sociological phenomena, especially education, it is understood that all statements are statistical in nature unless stated to the contrary, and that exceptions are part and parcel to each statement. In fact, there are striking (though isolated) exceptions in the present context. The reader is asked to be aware of this caveat for the rest of this article.

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about and it is time for us to take a second look. One meaning of the word “engineering” is the art or science of customizing scientific theory to meet human needs. Thus chemical engineering is the science of customizing chemistry to solve human problems, or electrical engineering is the science of customizing electromagnetic theory to design all the nice gadgets that we have come to consider indispensable. I will put forth the contention that mathematics education is mathematical engineering, in the sense that it is the customization of basic mathematical principles to meet the needs of teachers and students.2 I will try to convince you that this is a good model for the understanding of mathematics education before proceeding to a discussion of how mathematicians can contribute to K-12 mathematics education. The far-from-surprising conclusion is that, unless mathematicians and educators can work as equal partners, K-12 mathematics education cannot improve. Regarding the nature of mathematics education, Bass made a similar suggestion in [5] that it should be considered a branch of applied mathematics.3 What I would like to emphasize is the aspect of engineering that customizes scientific principles to the needs of humanity in contrast with the scientific-application aspect of applied mathematics. Thus, when H. Hertz demonstrated the possibility of broadcasting and receiving electromagnetic waves, he made a breakthrough in science by making a scientific application of Maxwell’s theory. But when G. Marconi makes use of Hertz’s discovery to create a radio, Marconi was making a fundamental contribution in electrical engineering, because he had taken the extra step of harnessing an abstract phenomenon to fill a human need.4 In this sense what separates mathematics education as mathematical engineering from mathematics education as applied mathematics is the crucial step of customizing the mathematics, rather than simply applying it in a straightforward manner to the specific needs of the classroom. There is no better illustration of this idea of customization than the teaching of fractions in upper elementary and middle schools, as I now explain. Students’ failure to learn fractions is well-known. School texts usually present a fraction as parts of a whole, i.e., pieces of a pizza, and this is the most basic conception of a fraction for most elementary students. However, when fractions are applied to 2After the completion of this article, Skip Fennell brought to my attention the article “Access and Opportunities to Learn Are Not Accidents: Engineering Mathematical Progress in Your School” by William F. Tate, which is available at: http://www.serve.org/_downloads/publications/AccessAndOpportunities.pdf. Tate is concerned with equity and uses “engineering” as a metaphor to emphasize the potential for designing different educational policies and pedagogical activities to promote learning, but without addressing the mathematics. On the other hand, the present article explains why mathematics education is the engineering of mathematics. 3 Hy Bass lectured on this idea in December of 1996 at MSRI, but [5] seems to be a convenient reference. After the completion of this article, Zalman Usiskin informed me that in the Proceedings of the U.S.-Japan workshop on the mathematics education of teachers in 2000 that followed ICME-9 in Japan, he had written that “ ‘Teachers’ mathematics’ is a field of applied mathematics that deserves its own place in the curriculum.” Along this line, let it be mentioned that the paper of Ferrini-Mundy and Findell [8] made the same assertion and, like Bass, it does not touch on the engineering aspect of mathematics education. The need for mathematicians and educators to work on equal footing in mathematics education is likewise not mentioned by these educators. 4 The invention was actually due to N. Tesla, but like many things in life, popular preception displaces the truth. I am indebted to S. Simic for pointing this out to me.


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everyday situations, then it is clear that there is more to fractions than parts-of-awhole, e.g., if there are 15 boys and 18 girls in a classroom, then the ratio of boys to girls is the fraction, which has nothing to do with cutting up a pizza into 18 equal parts and taking 15. In the primary grades, it is not a serious problem if students’knowledge of fractions is imprecise and informal, so that a fraction can be simultaneously partsof-a-whole, a ratio, a division, and an operator5 , and a number. Children at that age are probably not given to doubts about the improbability of an object having so many wondrous attributes. At some stage of their mathematical development, however, they will have to make sense of these different “personalities” of a fraction. It is this transition from intuitive knowledge to a more formal and abstract kind of mathematical knowledge that causes the most learning problems. This transition usually takes place in grades 5–7. There is by now copious mathematics education research6 on how to facilitate children’s learning of the fraction concept at this critical juncture in order to optimize their ability to use fractions efficiently. At present, what most children get from their classroom instruction on fractions is a fragmented picture of a fraction with all these different “personalities” lurking around and coming forward seemingly randomly. What a large part of this research does is to address this fragmentation by emphasizing the cognitive connections between these “personalities”. It does so by helping children construct their intuitive knowledge of the different “personalities” of a fraction through the use of problems, hands-on activities, and contextual presentations. This is a good first step, and yet, if we think through students’ mathematical needs beyond grade 7, then we may come to the conclusion that establishing cognitive connections does not go far enough. What students need is an unambiguous definition of a fraction which tells them what a fraction really is. They also need to be exposed to direct, mathematical, connections between this definition and the other “personalities” of a fraction. They have to learn that mathematics is simple and understandable, in the sense that if they can hold onto one clear meaning of a fraction and can reason for themselves, then they can learn all about fractions without ever being surprised by any of these other “personalities”. From a mathematician’s perspective, this scenario of having to develop a concept with multiple interpretations is all too familiar. In college courses, one approaches rational numbers (both positive and negative fractions) either abstractly as the prime field of characteristic zero, or as the field of quotients of the integers. The problem is that neither is suitable for use with fifth graders. This fact is recognized by mathematics education researchers, as is the fact that from such a precise and abstract definition of rational numbers, one can prove all the assorted “personalities” of rational numbers. If I have read the research literature correctly, these researchers despair of ever being able to offer proofs once they are forced to operate without an abstract definition, and 5 For example, the fraction can be regarded as a function (operator) which associates to each quantity three-

quarters of the same quantity. 6 Here as elsewhere, I will not supply explicit references because I do not wish to appear to be targeting specific persons or works in my criticism. I will be making generic comments about several general areas.

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that is why they opt for establishing cognitive, rather than mathematical connections among the “personalities” of rational numbers. The needs of the classroom would seem to be in conflict with the mathematics. At this point, engineering enters. It turns out that, by changing the mathematical landscape entirely and leaving quotient fields and ordered pairs behind, it is possible to teach fractions as mathematics in elementary school, by finding an alternate mathematical route around these abstractions that would be suitable for consumption by children in grades 5-7. Without going into details, suffice it to say that at least the mathematical difficulties can be overcome, for example, by identifying fractions with certain points on the number line (for this systematic development, see, e.g., Jensen [11], or Wu [25]). What is of interest in this context is that this approach to fractions is specific to the needs of elementary school and is not likely to be taught, ever, in any other situation. In addition, the working out of the basic properties of fractions from this viewpoint is not quite straightforward, and it definitely requires the expertise of a research mathematician. As to the further pedagogical implementation to render such an approach usable in grades 5–7, the input of teachers and educators would be absolutely indispensable.7 We therefore get to witness how mathematicians and educators are both needed to turn a piece of abstract mathematics into usable lessons in the school classroom. This is customization of abstract theory for a specific human need, and this is engineering at work. Through this one example of fractions, we get a glimpse of how the principles of mathematical engineering govern the design of a curriculum. Less obvious but of equal importance is the fact that even mathematics education research cannot be disconnected from the same principles. If, for example, a strong mathematical presence had been integral to the research on fractions and rational numbers, it would be very surprising that the research direction would have developed in the direction it did. Compare the quote by Lagemann at the beginning of this article as well as Lagemann [14]. An entirely analogous discussion of customization can be given to any aspect of mathematics education, but we single out the following for further illustrations: (a) The design of an “Intervention Program” for at-risk students. Up to this point, the methods devised to help these students are largely a matter of teaching a watered-down version of each topic at reduced pace; this is poor engineering from both the theoretical and the practical point of view. In Milgram-Wu [18], a radically different mathematical engineering design is proposed to deal with this problem. (b) The teaching of beginning algebra in middle school. The way symbols are usually handled in such courses, which necessitates prolix discussions in the research literature of the subtlety of the equal sign, and the way variable is introduced as the central concept in school algebra are clear indications that the algebra we teach students at present has not yet been properly customized 7 Some teachers who have worked with me are trying out this approach with their students in San Francisco.


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for the needs of school students. See the Preface and Sections 1 and 2 of Wu [30], and also Wu [31], for a more detailed account of both the problems and their proposed solutions. (c) The writing of mathematics standards at the national or state level. This is an example of what might be called “practical optimization problems”, which customize the mathematics to meet diverse, and at times conflicting, needs of different clientele. Cf. Klein [13]. The concept of mathematics education as mathematical engineering also sheds some light on Lee Shulman’s concept of pedagogical content knowledge ([20]). There has been a good deal of interest in precisely describing the kind of knowledge a teacher should possess in order to be effective in teaching. In the field of mathematics, at least, this goal has proven to be elusive thus far (but cf. Hill-Rowan-Ball [9]), but Shulman’s intuitive and appealing formulation of this concept crystallizes the diverse ideas concerning an essential component of good teaching. From the point of view of mathematical engineering, one of the primary responsibilities of a teacher is to customize her mathematical knowledge in accordance with the needs of each situation for students’ consumption. This particular engineering knowledge is the essence of pedagogical content knowledge. Although this approach to pedagogical content knowledge does not add anything new to its conception, it does provide a framework to understand this knowledge within mathematics, one that is different from what one normally encounters in educational discussions. It makes explicit at least three components to effective teaching: a solid mathematical knowledge, a clear perception of the setting defined by students’ knowledge, and the flexibility of mind to customize this mathematical knowledge for use in this particular setting without sacrificing mathematical integrity. The idea of customizing mathematics “without sacrificing mathematical integrity” is central to mathematical engineering. In engineering, it is obvious that, in trying to customize scientific principles to meet the needs of humanity, we cannot contradict nature regardless of how great the human needs may be. In other words, one respects the integrity of science and does not attempt anything so foolish as the construction of anti-gravity or perpetual-motion machines. Likewise, as mathematical engineering, mathematics education accepts the centrality of mathematics as a given. Again using the example of teaching fractions, a mathematics educator would know that no matter how one tries to teach fractions, it must be done in a way that respects the abstract meaning of a fraction even if the latter is never used explicitly. If, for instance, an educator catches himself saying that children must adopt new rules for fractions that often conflict with well-established ideas about whole numbers, then he knows he is teaching fractions the wrong way because, no matter what efforts one puts into making fractions intuitive to children, one cannot do violence to the immutable fact that the rational numbers contain the integers as a sub-ring. The need to teach the arithmetic of fractions as a natural extension of the arithmetic of whole numbers has gone unnoticed for far too long, with the result that too many of our students begin to

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harbor the notion that, after the whole numbers, the arithmetic of fractions is a new beginning. Such bad mathematical engineering in curricular designs is unfortunately a common occurrence. The only way to minimize such engineering errors is to have both mathematicians and educators closely oversee each curricular design. In fact, if we believe in the concept of mathematics education as mathematical engineering, then the two communities must work together in all phases of mathematics education: Any education project in mathematics must begin with a sound conception of the mathematics involved, and there has to be a clear understanding of what the educational goal is before one can talk about customization. In this process, there is little that is purely mathematical or purely educational; almost every step is a mixture of both. Mathematics and education are completely intertwined in mathematical engineering. Mathematicians cannot contribute to K-12 mathematics education if they are treated as outsiders.8 They have to work alongside the educators on equal footing in the planning, implementation, and evaluation of each project. But this is far from the reality at present. For at least three decades now, the mathematics and K-12 education communities in the U.S. have not been on speaking terms in the figurative sense. (Cf. Washington Post [21].) The harm this communication gap has brought to K-12 mathematics education can be partially itemized, but before doing that, let me point out three general consequences of a philosophical nature. The first one is that the isolation of the education community from mathematicians causes educational discussions to over-focus on the purely education aspect of mathematics education while seemingly always leaving the mathematics untouched. The result is the emergence of a subtle mathematics avoidance syndrome in the education community, and this syndrome will be seen to weave in and out of the following discussion of the specific harmful effects of this communication gap. Given the central position of mathematics in mathematical engineering, it would be noncontroversial to say that this syndrome should vanish from all discussions in mathematics education as soon as possible. The fact that many mathematicians teach mathematics and design mathematics courses throughout their careers seems to escape the attention of many educators. Here is a huge reservoir of knowledge and experience in mathematical engineering on tap. The chasm between the two communities in effect denies educators access to this human resource at a time when educators need all the engineering help they can get. The final consequence can best be understood in terms of the Darwinian dictum that when a system is isolated and allowed to evolve of its own accord, it will inevitably mutate and deviate from the norm. Thus when school mathematics education is isolated from mathematicians, so is school mathematics itself, and, sure enough, the latter evolves into something that in large part no longer bears any resemblance to mathematics. Correct definitions are not given, or if given, they are not put to use (Milgram-Wu [18], Wu [23], [27] and [29]). The organic coherence of mathematics is 8 This only tells half the story about mathematicians. See the comments near the end of this article.


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no longer to be found (Wu [23]), or when “mathematical connections” are intentionally emphasized, such “connections” tend to be the trivial and obvious kind. Logical deduction becomes an afterthought; proofs, once relegated to the secondary school geometry course, were increasingly diluted until by now almost no proofs at all are found there, or anywhere else in the schools (Wu [26]). And so on. This development naturally brings down the quality of many aspects of mathematics education. The absence of dialog between the two communities has led to many engineering errors in mathematics education, one of them being the unwelcome presence of mathematically incorrect test items in state and other standardized tests (Milgram [16] and [17]). The same kind of defective items also mar many teachers’ credentialing tests (Askey [1] and [2]). A more subtle effect of the absence of mathematical input on assessment is the way test scores are routinely misinterpreted. The low test scores have been used to highlight students’ dismal mathematical performance, but little or no thought is given to the possibility that they highlight not necessarily students’ achievement (or lack thereof) but the pervasive damage done by defective curricular materials, or even the chronic lack of effective teaching. Such a possibility may not be obvious to anyone outside of mathematics, but to a mathematician, it does not take any research to confirm the fact that when students are taught incorrect mathematics, they learn incorrect mathematics. Garbage in, garbage out. If the incorrect mathematics subsequently shows up in students’ test scores, how can we separate the errors due to the incorrect information students were given, from the errors due to students’ own misconceptions? A more detailed examination of this idea in the narrow area of school algebra is given in Wu [31]. The need for mathematicians’ participation in all phases of assessment is all too apparent. The lack of collaboration between mathematicians and mathematics educators affects professional development as well. The issue of teacher quality is now openly acknowledged and serious discussions of the problem are beginning to be accepted in mathematics education (cf. Ma [15], and Conference Board of the Mathematical Sciences [6]9 ). As a result of the inadequate mathematics instruction teachers receive in K-12, their knowledge of mathematics is, by and large, the product of the mathematics courses they take in college.10 In very crude terms, the number of such required mathematics courses is too low, and in addition, these courses are taught either by mathematicians who are not in close consultation with teachers, and are unaware as to what is needed in the school classroom, or by mathematics educators who are not professional mathematicians. The former kind of course tends to be irrelevant to the classroom, and the latter kind tends to be mathematically shallow or incorrect. It is only natural that teachers coming out of such an environment turn out to be mathematically ill-prepared. 9 Whatever reservations one may have concerning the details of its content, it is the fact that such a volume could be published under the auspices of a major scientific organization that is important. 10 It may be useful to also take note of what may be called “the second order effect” of university instruction: teachers’ knowledge of mathematics is also conditioned by their own K-12 experiences, but these teachers’ teachers were themselves products of the mathematics courses they took in the university.

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Similar woes persist in in-service professional development, thereby ensuring that teachers have little access to the mathematical knowledge they need for their profession. For example, the last decade has witnessed the appearance of case books consisting of actual records of lessons given by teachers.11 The idea is to invite teachers to analyze these lessons, thereby sharpening their pedagogical sensibilities. In too many instances, however, blatant mathematical flaws in the cited cases are overlooked in the editors’ commentaries. This raises the specter of bringing up a generation of teachers who are proficient in teaching school students incorrect mathematics. In this instance, it would appear that the need to respect mathematical integrity in mathematical engineering has been all but forgotten. The most divisive outcome of the noncommunication between the two communities in the U.S. is undoubtedly the conflict engendered by the new (reform) curricula written in the past fifteen years. I take up this discussion last, because it brings us face to face with some subtle issues about mathematicians’ participation in K-12 mathematics education. The prelude to the writing of these curricula is the unchecked degeneration in the mathematical integrity of the existing textbooks from major publishers over the period 1970–1990, a fact already alluded to above. This degeneration triggered the reform spearheaded by NCTM (National Council of Teachers of Mathematics [19]). Rightly or wrongly, the new curricula were written under the banner of the NCTM reform, and the manner in which some of the reform texts were imposed on public schools led eventually to the well-known Math Wars (Jackson [10]). The root of the discontent over these texts is the abundance of outright mathematical errors12 , as well as what research mathematicians perceived to be evidence of a lack of understanding of the mathematics. An example of the latter was the promotion of children’s invented algorithms at the expense of the standard computation algorithms in the elementary mathematics curriculum. Although the promotion was partly an overreaction to the way the standard algorithms were often inflicted on school children with nary a word of explanation, it also reflected a lack of awareness of the central importance of the mathematical lessons conveyed by the reasoned teaching of these algorithms. The “subtle issues” mentioned above stem from the fact that the writing of some of the new reform curricula actually had the participation of a few mathematicians. The first thing to note is that the latter are the rare exceptions to the general noncommunication between the mathematics and education communities. The noncommunication is real. At the same time, these exceptions seem to point to an apparent contradiction: How would I reconcile my critical stance toward these reform curricula with the principal recommendation of this article, namely, that mathematicians be equal partners with educators in the mathematics education enterprise? The answer is that there is no contradiction at all. The participation by mathematicians is, in general terms, a prerequisite to any hope of success in K-12 mathematics education, but in no way 11 Let it be noted explcitly that I am discussing the case books in K-12 mathematics education only. 12 These errors tend to be different from the earlier ones to be sure, but errors they are.


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does it guarantee success. It is helpful in this context to recall similar discussions that routinely took place some eight years ago when some mathematicians first went public with the idea that mathematics teachers must have a solid content knowledge. The usual rejoinder at the time was that “knowing mathematics is not enough (to be a good teacher)”. This is a common confusion that mistakes a necessary condition for a sufficient condition.13 There is no quick fix for something as complex as mathematics education. Getting mathematicians to fully participate is only the beginning; the choice of the mathematicians and the hard work to follow will have a lot to say about the subsequent success or failure. It is appropriate at this point to recall what was said at the beginning of the article about the power structure of mathematics education: thus far, educators get to make the decisions. Granting this fact, I should amplify a bit on the difficulties of choosing the right mathematicians for education work. Mathematicians have a range of background and experiences and, consequently, often have a range of opinions on matters of education as well. It is important that the range of these opinions be considered in all aspects of education. Many of the less happy incidents of the recent past in K-12 mathematics education were the result of choosing mathematicians of a particular persuasion. In addition, educators must make their own judgement on which among the mathematicians interested in K-12 are knowledgeable about K-12. Among the latter, some possess good judgment and leadership qualities while others don’t. Educators must choose at each step. If there are algorithms for making the right choices, I don’t happen to know them. Every mathematician potentially has something to offer in K-12 mathematics education: even an occasional glance at textbooks to check for mathematical correctness can be very valuable. However, if mathematicians want to participate in serious educational work in K-12, what must they bring to the table? I believe the most important thing is the awareness that K-12 mathematics education is not a subset of mathematics, and that there is quite a bit to learn about the process of customization that distinguishes K-12 mathematics education from mathematics. In particular, much (if not most) of the mathematics they teach in the university cannot be brought straight to the school classroom (Wu [22]; Kilpatrick et al. [12], Chapter 10 and especially pp. 375–6), but that it must first go through the engineering process to make it suitable for use in schools. If I may use the example of fractions once again, mathematicians interested in making a contribution to K-12 may find it instructive to get to know the reason that something like “equivalence classes of ordered pairs of integers” is totally opaque to students around the age of twelve. They would also want to know the reason that students of that age nonetheless need a definition of a fraction which is as close to parts-of-a-whole as possible. They should also get to know the appropriate kind of mathematical reasoning for students in this age group, because they will ultimately be called upon to safeguard such reasoning in the curriculum and 13And need I point out, there are some who intentionally use this confusion to reject that mathematical content knowledge is important for teachers, or that getting mathematicians to participate in mathematics education is critical for its success.

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assessment for these students. Mathematicians may regard school mathematics as technically primitive (in the sense of skills), but they must take note of its conceptual sophistication (Jensen [11]; Wu [24], [25] and [30]; cf. also Aharoni [1]). Above all, they must know that school mathematics is anything but pedagogically trivial: There is absolutely nothing trivial about putting any material, no matter how simple, into a correct mathematical framework so that it may be profitably consumed by school students. Mathematicians who want to contribute to K-12 mathematics education have to be constantly on the alert to ensure that the minimum requirements of their profession – the orderly and logical progression of ideas, the internal cohesion of the subject, and the clarity and precision in the presentation of concepts, – are still met in mathematics education writings. This is no easy task. If mathematicians want to enter K-12 mathematics education as equal partners with educators, then it is incumbent upon them to uphold their end of the bargain by acquiring this kind of knowledge about mathematical engineering. The concept of mathematics-education-as-mathematical-engineering does not suggest the creation of any new tools for the solution of the ongoing educational problems. What it does is to provide a usable intellectual framework for mathematics education as a discipline, one that clarifies the relationship between the mathematics and the education components, as well as the role of mathematicians in mathematics education. For example, it would likely lead to a better understanding of why the New Math became the disaster that it did. Most importantly, this concept lays bare the urgent need of the mathematical presence in every aspect of K-12 mathematics education, thereby providing a strong argument against the self-destructive policy of keeping mathematicians as outsiders in mathematics education. The chasm between mathematicians and educators must be bridged if our children are to be better served. I am cautiously optimistic14 that there are enough people who want to rebuild this bridge (cf. Ball et al. [4]), all the more so because the indications are that the NCTM leadership is also moving in the same direction. I look forward to a future where mathematics education is the joint effort of mathematicians and educators. Acknowledgement. I am first of all indebted to my colleague Norman E. Phillips for providing a critical piece of information about chemistry that got this article off the ground. The suggestion by Tony Gardiner to re-organize an earlier draft, and the penetrating comments on that draft by Helen Siedel, have left an indelible imprint on this article. Tom Parker, Ralph Raimi, and Patsy Wang-Iverson gave me very detailed corrections. David Klein also made corrections and alerted me to one of the references. In addition, the following members of the e-list mathed offered suggestions for improvement: R. A. Askey, R. Bisk, E. Dubinsky, U. Dudley, T. Foregger, T. Fortmann, K. Hoechsmann, R. Howe, W. McCallum, J. Roitman, M. Saul, D. Singer, A. Toom. Cathy Seeley and Skip Fennell also made similar suggestions. It gives me pleasure to thank them all. 14 In January of 2006.


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References [1] Aharoni, R., What I Learned in Elementary School. American Educator, Fall, 2005; http://www.aft.org/pubs-reports/american_educator/issues/fall2005/aharoni.htm. [2] Askey, R. A., Learning from assessment. In volume of the 2004 presentations in the Mathematical Sciences Research Institute, to appear. [3] Askey, R. A., Mathematical content in the context of this panel. In Proceedings of the Tenth International Congress on Mathematical Education (ed. by Mogens Niss et al.), 2006. [4] Ball, D. L., Ferrini-Mundy, J., Kilpatrick, J., Milgram, J. R., Schmid, W., Schaar, R., Reaching for common ground in K-12 mathematics education. Notices Amer. Math. Soc. 52 (2005), 1055–1058. [5] Bass, H., Mathematics, mathematicians, and mathematics education. Bull. Amer. Math. Soc. 42 (2005), 417–430. [6] Conference Board of the Mathematical Sciences. The Mathematical Education of Teachers, CBMS Issues in Mathematics Education 11, Amer. Math. Soc., Providence, RI, 2001. [7] Education Week. A Purge at the French High Committee for Education (HCE). Education Week, November 27, 2005. http://www.educationnews.org/A-Purge-at-the-French-HighCommittee-for-Education-HCE.htm. [8] Ferrini-Mundy, J. Findell, B., The mathematics education of prospective teachers of secondary school mathematics: old assumptions, new challenges. In CUPM Discussion Papers about Mathematics and the Mathematical Sciences in 2010: What Should Students Know? Washington DC: Mathematical Association of America, Washington DC, 2001. [9] Hill, H., Rowan, B., Ball, D. L., Effects of teachers’ mathematical knowledge for teaching on student achievement, 2004. http://www-personal.umich.edu/dball/ BallSelectPapers TechnicalR.html [10] Jackson, A., The Math Wars: California battles it out over mathematics education reform. Notices Amer. Math. Soc. 44 (1997), Part I, 695–702; Part II, 817–823. [11] Jensen, G., Arithmetic for Teachers. Amer. Math. Soc., Providence, RI, 2003. [12] Kilpatrick, J. Swafford, J. Findell, B., eds., Adding It Up. National Academy Press, Washington DC, 2001. [13] Klein, D. et al., The state of State MATH Standards. Thomas B. Fordham Foundation, Washington DC, 2005. http://www.edexcellence.net/foundation/publication/publication.cfm? id=338. [14] Lagemann, E. C., An Elusive Science: The Troubling History of Education Research. The University of Chicago Press, Chicago, London 2000. [15] Ma, L., Knowing and Teaching Elementary Mathematics. Lawrence Erlbaum Associates, Mahwah, NJ, 1999. [16] Milgram, R. J., Problem solving and problem solving Models for K-12: Preliminary Considerations. 2002; http://math.stanford.edu/ftp/milgram/discussion-of-well-posedproblems.pdf. [17] Milgram, R. J., Pattern recognition problems in K-12. 2003; http://math.stanford.edu/ftp/ milgram/pattern-problems.pdf. [18] Milgram, R. J. and Wu, H., Intervention program. 2005; http://math.berkeley.edu/~wu/.

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[19] National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. National Council of Teachers of Mathematics, Reston, VA, 1989. [20] Shulman, Lee, Those who understand: Knowledge growth in teaching. Educational Researcher 15 (1986), 4–14. [21] Washington Post. An Open Letter to United States Secretary of Education, Richard Riley. November 18, 1999; http://mathematicallycorrect.com/nation.htm. [22] Wu, H., On the education of mathematics teachers (formerly entitled: On the training of mathematics teachers). 1997; http://math.berkeley.edu/~wu/. [23] Wu, H., What is so difficult about the preparation of mathematics teachers? 2001; http:// math.berkeley.edu/~wu/. [24] Wu, H., Chapter 1: Whole Numbers (Draft). 2001; http://math.berkeley.edu/~wu/. [25] Wu, H., Chapter 2: Fractions (Draft). 2001; http://math.berkeley.edu/~wu/. [26] Wu, H., Geometry: Our Cultural Heritage – A book review. Notices Amer. Math. Soc. 51 (2004), 529–537. [27] Wu, H., Key mathematical ideas in grades 5-8. 2005; http://math.berkeley.edu/~wu/. [28] Wu, H., Must content dictate pedagogy in mathematics education? 2005; http://math. berkeley.edu/~wu/. [29] Wu, H., Professional development: The hard work of learning Mathematics. 2005; http:// math.berkeley.edu/~wu/. [30] Wu, H., Introduction to School Algebra (Draft). 2005; http://math.berkeley.edu/~wu/. [31] Wu, H., Assessment in school algebra. 2006, to appear.

The role of mathematicians in K-12 education: a personal perspective by Shiu-Yuen Cheng This draft was written after I read Prof. H. Wu’s draft on “How mathematicians can contribute to K-12 mathematics education”. I therefore have the advantage of adopting the same terms and scope of discussions in writing this draft. For example, I will be using Prof. Wu’s definition of the word “mathematician” to mean “research mathematicians”. Also, I am impressed and I agree with Prof. H. Wu’s philosophical idea of regarding mathematics education as mathematical engineering. I will in the following outline the roles that mathematicians can play for the enhancement of K-12 education. Frequently, I will come back to Prof. Wu’s idea of mathematical engineering so that we can do a good job. The main factors for providing an effective mathematics education are curriculum design, teacher competence and assessment methods. Among these three factors, I think the most important one is teacher competence. I think this is the factor that mathematicians can contribute the most. The processes of designing the curriculum and assessment mechanism vary from place to place and are greatly influenced by the local bureaucratic and political system. In most places, mathematicians do not

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get to play much of a role in the design of curriculum and the assessment mechanism. However, this does not mean that we should fold our hand and watch on the sideline. We should always engage in these two factors and make our contributions whenever possible. On the other hand, any curriculum design or assessment method in mathematics would need or welcome mathematicians’ stamp of approval. Mathematicians will definitely be involved but we have to vigilantly and patiently engage in the process. In Hong Kong, the relation between educators and mathematicians is much better than in the US. However, the educators do not call the shot. Instead, the government officials set the agenda and play the most influential role. On paper, it does not seem so because things are supposedly done through committees consisting of teachers, principals, educators and mathematicians. The government officials serve as secretariats in the committees. As the committee members are all busy people taking time off from their work to attend the committee meetings, the secretariat then get to draft all the papers and the agenda. They naturally become most influential. Moreover, as a consequence of the composition of the committees mathematicians are minority. To make things worse, they have few allies. The teachers, school principals in the committee usually assume the mathematicians in the committee have a secret agenda to tailor the curriculum for attracting students to be mathematics majors. Additionally, the educators talk the language of education officials. Their inputs are more helpful to the education officials in filling the reports with popular education jargons. It is then natural that mathematicians’ views usually do not prevail. Instead, some compromise can usually be reached if mathematicians engage in the process. I believe that teacher competence is most important factor as teachers are at the frontline implementing the curriculum and delivering the mathematics education. No matter how hard we work, usually the curriculum and assessment mechanism are far from perfect. A competent teacher can exercise discretions to compensate the inconsistencies and incompleteness of the curriculum and make them work. On the other hand, a teacher who has little confidence and competence in subject knowledge can easily turn a well-designed curriculum or assessment mechanism into disasters. In the area of teacher competence, mathematicians can contribute in two main areas: the university curriculum for mathematics teacher program, and courses and workshops for in-service mathematics teachers. Mathematicians can play major roles in these two areas and can get more colleagues to participate. However, we usually do not pay much attention or do not do the right thing. It was pointed out by Prof. H. Wu and many others that the university curriculums for mathematics majors do not serve the purpose of providing the necessary understanding to be a competent mathematics teacher. The main reason is that the curriculum for mathematics majors is designed with an aim to train research mathematicians. As for courses and workshops for inservice teachers, we need more colleagues to participate and contribute. The ball is in our court but so far we have not made the right play. In Hong Kong, about fifteen to twenty percent of mathematics graduate become mathematics teachers. This is not a small percentage and is in fact higher than the

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percentage of students going for postgraduate study in mathematics. However, the curriculum for mathematics major does not offer much help for those who will pursue the career of a mathematics teacher. Mainly, most mathematicians do not see the necessity of designing and offering some new courses to provide a profound understanding of school mathematics. It is assumed that our courses in abstract algebra, analysis and geometry will do the job and hence nothing needs to be done. As for courses and workshops for in-service teachers, the sad thing is that Hong Kong government does not provide much of this kind of opportunity. Mathematicians have to shoulder this task on a volunteer basis. There are mathematicians willing to contribute but in order to make it sustainable we need to convince the government and the mathematics community the importance of providing courses and workshops for deepening teachers’ understanding of the subject knowledge. To do this effectively, we need to communicate well to the community about the concerns of mathematicians about mathematics education. In many places, people are alarmed because “Johnny can’t add”. In Hong Kong, Johnny can add! In fact, Johnny can do fractions and decimals quite well. In many international studies about the mathematics attainment of school students, Hong Kong routinely occupies one of the top positions. On paper, we should congratulate ourselves and should not even attempt to touch the system as things are not broken. However, anyone in the university or familiar with the situation of the Hong Kong school system knows that the Hong Kong mathematics education is far from achieving the goals. We are able to train our students to do arithmetic and some simple algebra at a tremendous cost. In the process, we suffocate students’ creativity and motivation for learning. So far we have not been successful in documenting and communicating our concerns to the Hong Kong public and the government. Mathematics education is then getting little resource from the government as it is doing quite well comparing to our language education. I believe communicating effectively to the public about our views for enhancing mathematics education is crucial and should fall into one of the sub-areas of Prof. Wu’s framework of Mathematics Engineering. The banner of Mathematics Engineering is useful for setting a clear goal and rallying support of our fellow mathematicians to contribute to mathematics education.

The role of mathematicians in K-12 mathematics education by U. Kirchgraber and K. Osterwalder We begin with a few remarks on the Swiss educational system and on teaching and learning of mathematics in Switzerland. Then we focus on teacher training in Mathematics at the Swiss Federal Institute of Technology (ETH). Finally we sketch an answer to the question posed to the panel. In international comparative studies like TIMS and PISA Swiss students have demonstrated reasonable achievements in Mathematics. Without overestimating such

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results15 one may wonder whether some specific features of the Swiss educational system might be responsible for this relative success and which measures could serve a further improvement of the results. As we will see some of the possible explanations are related to the topic of the panel. It is well known Switzerland is not rich in natural resources. This is usually claimed to be one of the major reasons why education is quite highly valued in this country, with a number of important implications: the vast majority of schools are public, teachers on all levels are well trained, the profession of teacher is quite respected, teachers (on all levels) are well paid, schools are well equipped, school buildings are kept in good shape. The Swiss educational system leaves considerable freedom to teachers on all levels and in particular in upper secondary school, to which we will refer to as Gymnasium16 level17 . Gymnasium teachers in general and Gymnasium Mathematics teachers in particular have to follow a certain core curriculum. Yet beyond this guide line they are fairly free to include additional topics, there are hardly any restrictions concerning the type of pedagogy adopted, and teachers are even quite free as to the number and type of tests and examinations they will administer. As to the final examination, in many schools every single mathematics teacher is free to assign a selection of, say, 4-8 problems of his or her choice and depending on the topics he or she has covered in class to his or her students on which they will work during 4 hours. Eventually he or she will correct and grade these works. ETH offers Gymnasium Teacher Training Programs in the following fields: Biology, Chemistry, Earth Sciences, Mathematics, Physics and Sports. In the following we discuss some features of the Gymnasium Mathematics Teacher Training Program (GMTTP). A prerequisite for completing the GMTTP is a Master’s Degree in Mathematics, though the students are permitted to start with the GMTTP in the third year of the Bachelor’s program. Average duration time for completing the GMTTP is six months18 , if studied full time. The fact that Swiss Gymnasium Mathematics teachers must hold a Master’s Degree in Mathematics has – we suppose – far reaching professional and psychological consequences. Having completed a Master’s program has at least two implications which, we think, are important for a future Mathematics teacher: a) During the first two years of studies Mathematics students encounter many topics they have seen before yet dealt with in way that is qualitatively very different from what they had 15 Compared to the host of highly sophisticated tools to measure many quantities in the Sciences and in particular in Physics that have evolved since the time of Gallilei, measuring effectiveness of teaching and learning and similar variables is probably still in its infancy, yet is a fascinating and challenging enterprise. 16 There are some 150 Gymnasia in Switzerland, every year some 15000 students graduate from the Gymnasia, between 1800 and 2300 enroll at ETH. 17 This corresponds to grades 9–12. 18 Starting in fall 2006 federal requirements request studies twice as long. ETH’s GMTTP will be extended accordingly and will be renamed as “Master of Advanced Studies in Secondary and Higher Education in Mathematics”.

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experienced previously. b) In the third year of the Bachelor’s and during the Master’s program they are exposed to advanced fields, an indispensable experience for gaining a faithful picture of what Mathematics is about. Based on four and a half years of studies these teacher students can at least potentially be expected to dispose of a degree of mathematical expertise and mathematical maturity which is covered by the terms “content knowledge” and/or “deep understanding” in the Mathematics Education research literature. As to some practical implications: Being trained as full fledged mathematician a Gymnasium Mathematics teacher may leave school after a few years and start a career in Industry or elsewhere, or vice versa. Therefore requiring a Master’s Degree as a prerequisite for teacher training has the benefit of not excluding Gymnasium Mathematics teachers in an early stage of their professional development from the full scale of professional opportunities offered to mathematicians nowadays. The GMTTP includes courses in the Educational sciences, in Mathematics Education (Didactics of Mathematics), (a small amount of) guided teaching practice, and a fourth component, called Specialized Mathematics courses with an Educational Focus. It is of utmost importance that these components are excellently tuned and multiply intertwined. Moreover they should be accompanied by plenty of student activities19 . Since the late eighties the Educational course at ETH was designed and continuously updated. The basic concept was to make available both to the Science and Mathematics Educators as well as to their teacher students research grounded results from areas such as psychology, the cognitive sciences, etc. Over the years quite a number of teaching techniques and teaching methods were implemented and probed since they are known – on the basis of meta analyses20 – to enhance learning21 . Guided Learning programs22 for instance are self-contained study materials for pupils covering a learning unit of some 3–30 lessons with the following features: Precisely defined prerequisites, well structured and comprehensibly written explanations, explicitly stated learning goals, adjunct questions and their answers, learning aids, chapter tests to fulfill the so-called Mastery Learning Principle. According to Kulik, Kulik and Bangert-Drowns23 the effect size of this type of teach ware is of the order of 0.5 in Mathematics and of the order of 0.6 in the Sciences. A few examples of Guided Learning Programs in Mathematics and the Sciences (in German) can be found on www.educeth.ethz.ch24 . 19As research has shown, just attending lectures has little impact on future teaching. 20 See for instance Fraser B. J., Walberg H. J., Welch W. W., Hattie J. A., Syntheses of Educational Productivity

Research. International J. of Educational Research 11 (1987), 145–252; Walberg, H. J., Productive Teaching and Instruction: Assessing the Knowledge Base. University of Illinois at Chicago, School of Education, 1988, 18 p, mimeographed. 21 I.e. they have noteworthy effect sizes. 22 In German: Leitprogramme. 23 Kulik F. S., Kulik J. A., Bangert-Drowns R. L., Effectiveness of Mastery Learning Programs: A MetaAnalysis. Review of Educational Research 60 (1990), 265–299. 24 EducETH, a service of ETH to the Public, is ETH’s educational server providing teaching materials primarily for upper secondary schools.


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In the Mathematics Education courses the thrust is on domain specific aspects of the teaching and learning enterprise. Of course, goals, standards, competencies to be achieved are discussed, subject matter analysis with diverse approaches to selected topics is of central concern and textbooks25 are analyzed. The teacher students are exposed to Mathematics Education research concepts that prove useful to explain certain phenomena, for instance Talls’s and Vinner’s distinction between concept definition and concept image which helps to understand students’ misconceptions of the notion of function. Topics like the “Expert Blind Spot”-Hypothesis26 , or the influence of teacher’s pedagogical content beliefs on learning outcomes27 , and many others are treated. Videos and their transcriptions are analyzed to provide insight into the unpredictability and fragility of learning processes, among other things. We now turn to the Specialized Mathematics courses with an Educational Focus already mentioned before. It is by now generally accepted that transfer achievements quite often do not emerge automatically. F. Weinert, summarizing years of research at the Max Planck Institut für psychologische Forschung in Munich, explains it roughly speaking as follows. Knowledge a learner acquires systematically – for instance in Mathematics courses as they are usually organized – is likely to be structured and organized in the learners brain in a way not easily retrievable, amenable if the learner is put in a problem situation in which he/she should apply this body of knowledge. Thus, knowledge, which is available in principle, remains inert and unused, though it would be useful and even necessary to handle a certain situation. Weinert’s conclusion: To build up an intelligent, flexibly applicable knowledge base the learner needs both, systematic as well as situated learning. The Specialized Mathematics courses with an Educational Focus take place in the third year (in Switzerland: the last year) of the Bachelor’s and during the Master’s program. They are open to all students in the Bsc/Msc Mathematics program, but are compulsory for candidates in the GMTTP. These courses with an Educational focus were installed many years ago at ETH (long before there were courses on Mathematics Education!) and have their origin in lectures given by Felix Klein early in the 20th century in Göttingen under the title “Elementarmathematik vom höheren Standpunkt” (elementary mathematics from an advanced point of view) and directed to future Gymnasium Mathematics teacher. The Specialized Mathematics courses with an Educational Focus serve several goals. Very much in the spirit of Klein’s concept they attempt to narrow the gap between Gymnasium Mathematics and University Mathematics. Take a core topic in Mathematics, present at all levels: equations. Linear equations are a topic in grade 9, 25After what has been said earlier in this paper it will not come as a big surprise for the reader that Swiss Gymnasium Mathematics teacher are not obliged to use any particular text books. Many in fact do not use a textbook at all but use a variety of sources to assemble handouts, etc., for their students. 26 See: M. J. Nathan, A. Petrosino: Expert Blind Spot among Pre-service Teachers. Amer. Educ. Res. J. 40 (2003), 905–928. 27 See: Staub, F. C. and Stern, E., The Nature of teachers’ Pedagogical Content Beliefs Matters for Students’ Achievement Gains. J. Educational Psychology 93 (2002), 344–355.

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in a course on Linear Algebra in the first year of the Bachelor’s program, as well as in specialized courses on Numerical Linear Algebra: Relate the various aspects and draw conclusions for the future teaching of linear equations in grade 928 ! Pupils usually encounter nonlinear equations first in connection with the quadratic equation. Most emphasis is usually put on reducing a general quadratic equation29 to a “purely quadratic” equation30 . It is of course a marvelous discovery that arbitrary quadratic equations can be reduced to purely quadratic ones. It is a challenging design task for teacher students to compose a series of assignments that guides pupils to discover this phenomenon by themselves. Yet from a more general point of view the question of solving purely quadratic equations is even more intriguing. One encounters a pattern that is prevalent in (University) Mathematics: Equations are not always solvable. More often than not mathematicians have to invent a setting such that the equation becomes solvable: Loosely speaking – mathematicians make equations solvable! A second such instance comes up when complex numbers are invented to make all quadratic equations solvable with the totally unexpected benefit that in this setting all polynomial equations have solutions. The Fundamental Theorem of Algebra brings up another very interesting phenomenon: we may be able to prove that an equation has a solution and even that this solution is unique without being able to compute the solution. If complex numbers are treated at the Gymnasium31 level an intuitive proof of the Fundamental Theorem of Algebra can be offered to Gymnasium students. In a Specialized Mathematics course with an Educational Focus dedicated to equations, teacher students would not only design a learning unit for pupils around such a heuristic proof, but learn in addition how such a proof is made rigorous (not an easy task!), topics like Rouché’s Theorem would have to be discussed and an introduction to the Brouwer and Leray–Schauder Degree theory with applications to periodic solutions of differential equations would allow for a glance of the breadth of the field. Another aspect to which Specialized Mathematics courses with an Educational Focus can contribute concerns curricular development. School Mathematics curricula are often blamed for covering material only that was invented centuries ago. Of course, most subjects that are hot research topics in Mathematics are far too remote and far too specialized for being accessible at the Gymnasium level. Yet there are marvelous exceptions: The Diffie-Hellman Key Exchange and RSA Cryptography, invented in 28 Maybe you conclude that the 9th grade program on linear equation should contain a modest introduction to Computerized Tomography, as we did, see the Leitprogramm entitled “Gleichungen” at www.educeth.ethz.ch. Maybe you conclude that the program, in addition to Gaussian elimination, should include a homeopathic introduction to solving linear equations by iteration (a topic you might touch on again, when you treat Banach fixed point iteration in one dimension in connection with Kepler’s equation). Maybe you conclude that the question of what it means that two linear systems of equations are equivalent, and how one obtains equivalent systems from a given one, deserves to become (a small?) Mathematics Education research project. 29 I.e. one including a linear term with respect to the unknown. 30 I.e. one in which the linear term is absent. 31 Gymnasia have various different profiles in Switzerland. Some concentrate on Mathematics and Physics. There complex numbers are treated.


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the late seventies, are well suited to give 10th graders an idea of the mathematical enterprise32 . Euler buckling is probably the earliest example of a bifurcation problem. It was only during the last two or three decades, however, that bifurcation theory became a systematically developed branch of analysis. In a Specialized Mathematics course with an Educational Focus dedicated to an introduction to bifurcation theory as background it is well possible that teacher students adapt some of the material for Gymnasium students, hereby heavily drawing on a classic school subject: the study of the geometrical properties and graphing of functions defined by simple expressions. Finally we mention ill-posed inverse problems. This again is a relatively new field of Applied Mathematics. It is of great theoretical and practical interest, the way ill-posed inverse problems are treated mathematically is surprising and they lend themselves outstandingly for treatments on various different levels33 . We expect that Specialized Mathematics courses with an Educational Focus deepen the teacher students’ mathematical expertise, that they strengthen the link between University and Gymnasium level Mathematics, that they contribute to develop the secondary school Mathematics curricula, that they support the prospective teachers to teach Mathematics at the same time more mathematically and in such a way that their students can learn to value Mathematics as a human activity and for its significance in our world. What is eventually the role of mathematicians in K-12 in our system? Research mathematician can and do contribute in a number of ways. Via the Bachelor’s and Master’s program they shape lastingly the knowledge base, the picture and the skills our teacher students develop. They can substantially contribute to the Specialized Mathematics courses with an Educational Focus. They can contribute to the design of substantial teaching units. In Mathematics Education research highly interesting developments have just begun. We mentioned the paper by Staub and Stern entitled: “The Nature of teachers’ Pedagogical Content Beliefs Matters for Students’Achievement Gains.” Another paper in the same realm is by Hill, Rowan and Ball34 . It is entitled: “Effects of teachers’ mathematical knowledge for teaching on student achievement.” These papers provide results on primary school teachers and primary school pupils. We certainly need many more results on “what has which impact on student learning” and very much so in higher grades. In fact, very little seems to be known as to the upper Gymnasium level. 32 Here are some aspects: Fermat’s (little) Theorem on which RSA cryptography is based, though elementary,

is far from being obvious. The way it is proven illustrates the power of mathematical ideas. 350 years after its discovery it became the key ingredient to affirmatively answer a questions one hardly dares to ask: Is it possible that two persons, who cannot communicate but publicly, can agree on keys which permit them to exchange messages which cannot be decoded except by the person who is entitled to read the message? 33 See Kirchgraber, U., Kirsch, A., Stoffer, D. : Schlecht gestellte Probleme – oder wenn das Ungenaue genauer ist. Math. Semesterber. 51 (2004), 175–2005. 34 Hill, H. C., Rowan, B., Ball, D. L.: Effects of teachers’ mathematical knowledge for teaching on student achievement. Amer. Educ. Res. J. 42 (2005), 371–406.

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Earlier we noted that educational measurement techniques are probably still in their infancy. How can we suitably measure mathematical achievements, teachers’ pedagogical content knowledge, the nature of teachers’ pedagogical content beliefs, and many more variables of this type? This is certainly a field, where research mathematician can and should contribute. Research mathematicians in Switzerland are welcome as members in school boards, and/or as experts in the final examinations at Gymnasium Schools. Research mathematicians are welcome to offer lectures to in-service teachers or to participate in study weeks for Gymnasium students, or to visit schools and give talks. To summarize: The main contribution of research mathematicians to the second half of K-12 is to train Mathematics teachers as knowledgeable mathematicians and to develop with them methods to narrow the gap between “Gymnasium Mathematics” and University Mathematics. Other possible contributions are manifold, crucial and indispensable.

Ateneo de Manila University, Quezon City 1108, Philippines E-mail: [email protected] Department of Mathematics, Hong Kong University of Science and Technology, Hong Kong, China E-mail: [email protected] ETH-Zentrum, Eidgenössische Technische Hochschule Zürich, 8092 Zürich, Switzerland E-mail: [email protected], [email protected] Department of Mathematics, University of California, Berkeley, CA 94720-3840, U.S.A. E-mail: [email protected]

On the origins of Hilbert’s sixth problem: physics and the empiricist approach to axiomatization Leo Corry

Abstract. The sixth of Hilbert’s famous 1900 list of twenty-three problems is a programmatic call for the axiomatization of physical sciences. Contrary to a prevalent view this problem was naturally rooted at the core of Hilbert’s conception of what axiomatization is all about. The axiomatic method embodied in his work on geometry at the turn of the twentieth-century originated in a preoccupation with foundational questions related with empirical science, including geometry and other physical disciplines at a similar level. From all the problems in the list, the sixth is the only one that continually engaged his efforts over a very long period, at least between 1894 and 1932. Mathematics Subject Classification (2000). Primary 01A60; Secondary 03-03, 70-03, 83-03. Keywords. David Hilbert, axiomatization, physics.

1. Introduction Of the many important and brilliant plenary talks delivered in ICMs ever since the inception of this institution in 1897 in Zurich, none has so frequently been quoted and, possibly, none has had the kind of pervasive influence, as the one delivered by David Hilbert in 1900 at the second ICM in Paris, under the title of “Mathematical Problems”. Rather than summarizing the state of the art in a central branch of mathematics, Hilbert attempted to “lift the veil” and peer into the development of mathematics in the century that was about to begin. He chose to present a list of twenty-three problems that in his opinion would and should occupy the efforts of mathematicians in the years to come. This famous list has been an object of mathematical and historical interest ever since. The sixth problem of the list deals with the axiomatization of physics. It was suggested to Hilbert by his own recent research on the foundations of geometry. He proposed “to treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part.” This problem differs from most others on Hilbert’s list in essential ways, and its inclusion has been the object of noticeable reaction from mathematicians and historians who have discussed it throughout the years. Thus, in reports occasionally written about the current state of research on the twenty-three problems, the special status of the sixth problem is readily visible: not only has it been difficult to decide to what extent the problem was actually solved (or not), but one gets the impression that, of all the problems on the list, this one received Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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the least attention from mathematicians throughout the century and that relatively little effort was directed at solving it ([11], [25]). Many a historical account simply dismissed the sixth problem as a slip on Hilbert’s side, as a curiosity, and as an artificial addition to what would otherwise appear as an organically conceived list, naturally connected to his broad range of mathematical interests (e.g., [26], p. 159). In fact, this is how Hilbert’s interest in physical topics in general as well as his few, well-known incursions into physical problems have been traditionally seen. According to this view, these are seen as sporadic incursions into foreign territory, mainly for the purposes of finding some new applications to what would otherwise be purely mathematically motivated ideas. This is the case, for instance, with Hilbert’s solution of the Boltzmann equation in kinetic theory of gases in 1912. Starting in 1902, most of Hilbert’s mathematical energies had been focused on research related with the theory of linear integral equations, and his solution of the Boltzmann equation could thus be seen as no more than an application of the techniques developed as part of that theory to a particular situation, the physical background of which would be of no direct interest to Hilbert. An account in this spirit appears in Stephen G. Brush’s authoritative book on the development of kinetic theory, according to which: When Hilbert decided to include a chapter on kinetic theory in his treatise on integral equations, it does not appear that he had any particular interest in the physical problems associated with gases. He did not try to make any detailed calculations of gas properties, and did not discuss the basic issues such as the nature of irreversibility and the validity of mechanical interpretations which had exercised the mathematician Ernst Zermelo in his debate with Boltzmann in 1896–97. A few years later, when Hilbert presented his views on the contemporary problems of physics, he did not even mention kinetic theory. We must therefore conclude that he was simply looking for another possible application of his mathematical theories, and when he had succeeded in finding and characterizing a special class of solutions (later called “normal”) …his interest in the Boltzmann equation and in kinetic theory was exhausted. ([4], p. 448) A further important physical context where Hilbert’s appeared prominently concerns the formulation of the gravitational field-equations of the general theory of relativity (GTR). On November 20, 1915, Hilbert presented to the Royal Scientific Society in Göttingen his version of the equations, in the framework of what he saw as an axiomatically formulated foundation for the whole of physics. During that same month of November, Einstein had been struggling with the final stages of his own effort to formulate the generally covariant equations that lie at the heart of GTR. He presented three different versions at the weekly meetings of the Prussian Academy of Sciences in Berlin, before attaining his final version, on November 25, that is, five days after Hilbert had presented his own version.

On the origins of Hilbert’s sixth problem

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Einstein had visited Göttingen in the summer of 1915 to lecture on his theory and on the difficulties currently encountered in his work. Hilbert was then in the audience and Einstein was greatly impressed by him. Earlier accounts of Hilbert’s involvement with problems associated with GTR had in general traced it back to this visit of Einstein or, at the earliest, to the years immediately preceding it. As in the case of kinetic theory, this contribution of Hilbert was often seen as a more or less furtive incursion into physics, aimed at illustrating the power and the scope of validity of the “axiomatic method” and as a test of Hilbert’s mathematical abilities while trying to “jump onto the bandwagon of success” of Einstein’s theory. In biographical accounts of Hilbert, his lively interest in physics has never been overlooked, to be sure, but it mostly has been presented as strictly circumscribed in time and scope. Thus for instance, in his obituary of Hilbert, Hermann Weyl ([24], p. 619) asserted that Hilbert’s work comprised five separate, and clearly discernible main periods: (1) Theory of invariants (1885–1893); (2) Theory of algebraic number fields (1893–1898); (3) Foundations, (a) of geometry (1898–1902), (b) of mathematics in general (1922–1930); (4) Integral equations (1902–1912); (5) Physics (1910–1922). Weyl’s account implies that the passage from any of these fields to the next was always clear-cut and irreversible, and a cursory examination of Hilbert’s published works may confirm this impression. But as Weyl himself probably knew better than many, the list of Hilbert’s publications provides only a partial, rather one-sided perspective of his intellectual horizons, and this is particularly the case when it comes to his activities related to physics. Recent historical research has brought to light a very different picture of Hilbert’s involvement with physics, and in particular of the real, truly central place of the ideas embodied in the sixth problem within the general edifice of Hilbert’s scientific outlook. Hilbert’s involvement with physical issues spanned most of his active scientific life, and the essence of his mathematical conceptions cannot be understood without reference to that involvement. More importantly, the famous “axiomatic approach” that came to be identified with Hilbert’s mathematical achievements and with his pervasive influence on twentieth-century mathematics is totally misunderstood if it is not seen, in the first place, as connected with his physical interests. Under this perspective, the involvement with kinetic theory and GTR are seen as a natural outgrowth of the development of Hilbert’s world of ideas, and by no means as sporadic, isolated incursions into unknown territories. Moreover, contrary to a commonly held view, the sixth problem is the only one in the entire list of 1900 that refers to an idea that continually engaged the active attention of Hilbert for a very long period of time, at least between 1894 and 1932 ([5]). The key to a balanced understanding of the role of physics within Hilbert’s intellectual horizon is found not so much in his publications, as it is in the complex academic network of personal interactions and diverse activities that he was continually part of. Especially worthy of attention is his teaching, first at Königsberg and – more importantly – after 1895 at Göttingen. At the mathematical institute established by Felix Klein, Hilbert became the leader of a unique scientific center that brought

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together a gallery of world-class researchers in mathematics and physics. One cannot exaggerate the significance of the influence exerted by Hilbert’s thought and personality on all who came out of this institution. More often than not, these lectures were far from systematic and organized presentations of well-known results and established theories. Rather, Hilbert often used his lectures as a public stage where he could explore new ideas and think aloud about the issues that occupied his mind at any point in time. In a lecture held in commemorating his seventieth birthday, Hilbert vividly recalled how these lectures provided important occasions for the free exploration of yet untried ideas. He thus said: The closest conceivable connection between research and teaching became a decisive feature of my mathematical activity. The interchange of scientific ideas, the communication of what one found by himself and the elaboration of what one had heard, was from my early years at Königsberg a pivotal aspect of my scientific work. …In my lectures, and above all in the seminars, my guiding principle was not to present material in a standard and as smooth as possible way, just to help the student keep clean and ordered notebooks. Above all, I always tried to illuminate the problems and difficulties and to offer a bridge leading to currently open questions. It often happened that in the course of a semester the program of an advanced lecture was completely changed, because I wanted to discuss issues in which I was currently involved as a researcher and which had not yet by any means attained their definite formulation. ([16], p. 79) The collection of Hilbert’s lecture notes offers an invaluable source of information for anyone interested in understanding his scientific horizon and contributions.

2. Axiomatics and formalism A main obstacle in historically understanding the significance of the sixth problem has been the widespread image of Hilbert as the champion of formalism in modern mathematics. The traditional association of Hilbert’s name with the term “formalism” has often proved to be misleading, since the term can be understood in two completely different senses that are sometimes conflated. One sense refers to the so-called “Hilbert program” that occupied much of Hilbert’s efforts from abut 1920. Although involving significant philosophical motivations, at the focus of this program stood a very specific, technical mathematical problem, namely, the attempt to prove the consistency of arithmetic with strictly finitist arguments. The point of view embodied in the program was eventually called the “formalist” approach to the foundations of mathematics, and it gained much resonance when it became a main contender in the so-called “foundational crisis” in mathematics early in the twentieth century. Even though Hilbert himself did not use the term “formalism” in this context,


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associating his name with term conceived in this narrow sense seems to be essentially justified. It is misleading, however, to extend the term “Hilbert program” – and the concomitant idea of formalism – to refer to Hilbert’s overall conception of the essence of mathematics. Indeed, a second meaning of the term formalism refers to a general attitude towards the practice of mathematics and the understanding of the essence of mathematical knowledge that gained widespread acceptance in the twentieth century, especially under the aegis of the Bourbaki group. Jean Dieudonné, for instance, explained what he saw as the essence of Hilbert’s mathematical conceptions in a well-known text where he referred to the analogy with a game of chess. In the latter, he said, one does not speak about truths but rather about following correctly a set of stipulated rules. If we translate this into mathematics we obtain the putative, “formalist” conception often attributed to Hilbert ([6], p. 551): “mathematics becomes a game, whose pieces are graphical signs that are distinguished from one another by their form.” Understanding the historical roots and development of the sixth problem goes hand in hand with an understanding of Hilbert’s overall conception of mathematics as being far removed from Dieudonné’s chess-game metaphor. It also comprises a clear separation between the “Hilbert program” for the foundations of arithmetic, on the one hand, and Hilbert’s lifetime research program for mathematics and physics and its variations throughout the years, on the other hand. In this regard, and even before one starts to look carefully at Hilbert’s mathematical ideas and practice throughout his career, it is illustrative to look at a quotation from around 1919 – the time when Hilbert began to work out the finitist program for the foundations of arithmetic in collaboration with Paul Bernays – that expounds a view diametrically opposed to that attributed to him many years later by Dieudonné, and that is rather widespread even today. Thus Hilbert said: We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. ([16], p. 14) The misleading conflation of the formalist aspect of the “Hilbert program” with Hilbert’s overall views about mathematics and its relationship with physics is also closely related with a widespread, retrospective misreading of his early work on the foundations of geometry in purely formalist terms. However, the centrality attributed by Hilbert to the axiomatic method in mathematics and in science is strongly connected with thoroughgoing empiricist conceptions, that continually increased in strength as he went on to delve into ever new physical disciplines, and that reached a peek in 1915–17, the time of his most intense participation in research associated with GTR. The axiomatic approach was for Hilbert, above all, a tool for retrospectively investigating the logical structure of well-established and elaborated scientific theories, and the possible difficulties encountered in their study, and never the starting point for

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the creation of new fields of enquiry. The role that Hilbert envisaged for the axiomatic analysis of theories is succinctly summarized in the following quotation taken from a course on the axiomatic method taught in 1905. Hilbert thus said: The edifice of science is not raised like a dwelling, in which the foundations are first firmly laid and only then one proceeds to construct and to enlarge the rooms. Science prefers to secure as soon as possible comfortable spaces to wander around and only subsequently, when signs appear here and there that the loose foundations are not able to sustain the expansion of the rooms, it sets about supporting and fortifying them. This is not a weakness, but rather the right and healthy path of development. ([5], p. 127)

3. Roots and early stages Physics and mathematics were inextricably interconnected in Hilbert’s scientific horizon ever since his early years as a young student in his native city of Königsberg, where he completed his doctorate in 1885 and continued to teach until 1895. Hilbert’s dissertation and all of his early published work dealt with the theory of algebraic invariants. Subsequently he moved to the theory of algebraic number fields. But his student notebooks bear witness to a lively interest in, and a systematic study of, an astounding breadth of topics in both mathematics and physics. Particularly illuminating is a notebook that records his involvement as a student with the Lehrbuch der Experimentalphysik by Adolph Wüllner (1870). This was one of many textbooks at the time that systematically pursued the explicit reduction of all physical phenomena (particularly the theories of heat and light, magnetism and electricity) to mechanics, an approach that underlies all of Hilbert’s early involvement with physics, and that he abandoned in favor of electrodynamical reductionism only after 1912. In the intimate atmosphere of this small university, the student Hilbert participated in a weekly seminar organized under the initiative of Ferdinand Lindemann – who was also Hilbert’s doctoral advisor – that was also attended by his good friends Adolf Hurwitz and Hermann Minkowski, by the two local physicist, Woldemar Voigt and Paul Volkmann, and by another fellow student Emil Wiechert, who would also become Hilbert’s colleague in Göttingen and the world’s leading geophysicist. The participants discussed recent research in all of branches of mathematics and physics, with special emphasis on hydrodynamics and electrodynamics, two topics of common interest for Hilbert and Minkowski throughout their careers. From very early on, fundamental methodological questions began to surface as part of Hilbert’s involvement with both mathematics and physics. On the mathematical side one may mention the intense research activity associated with the names of Cayley and Klein in projective geometry, concerning both the main body of results and the foundations of this discipline; the questions sparked by the discovery and publication of non-Euclidean geometries, which raised philosoph-


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ical concerns to a larger extent than they elicited actual mathematical research; the introduction by Riemann of the manifold approach to the analysis of space and its elaboration by Lie and Helmholtz; the question of the arithmetization of the continuum as analyzed by Dedekind, which had also important foundational consequences for analysis; the gradual re-elaboration of axiomatic techniques and perspectives as a main approach to foundational questions in mathematics, especially in the hands of Grassmann and of the Italian geometers. Hilbert’s intellectual debts to each of these traditions and to the mathematicians that partook in it – even though more complex and subtle than may appear on first sight – belong to the directly visible, received image of Hilbert the geometer. What is remarkable, and virtually absent from the traditional historiography until relatively recently, is the extent to which similar parallel developments in physics played a fundamental role in shaping Hilbert’s views on axiomatization. Very much like geometry, also physics underwent major changes throughout the nineteenth century. These changes affected the contents of the discipline, its methodology, its institutional setting, and its image in the eyes of its practitioners. They were accompanied by significant foundational debates that intensified considerably toward the end of the century, especially among German-speaking physicists. Part of these debates also translated into specific attempts to elucidate the role of basic laws or principles in physical theories, parallel in certain respects to that played by axioms in mathematical theories. As with geometry, foundational questions attracted relatively limited attention from practitioners of the discipline, but some leading figures were indeed involved in them. From about 1850 on, physics became focused on quantification and the search for universal mathematical laws as its fundamental methodological principles, on the conservation of energy as a fundamental unifying principle, and very often on mechanical explanation of all physical phenomena as a preferred research direction. If explanations based on imponderable “fluids” had dominated so far, mechanical explanations based on the interaction of particles of ordinary matter now became much more frequent. In particular, the mechanical theory of ether gave additional impulse to the concept of “field” that would eventually require a mechanical explanation. Likewise, the kinetic theory of gases gave additional support to the foundational role of mechanics as a unifying, explanatory scheme. On the other hand, these very developments gave rise to many new questions that would eventually challenge the preferential status of mechanics and lead to the formulation of significant alternatives to it, especially in the form of the so-called “electromagnetic worldview”, as well as in the “energicist” and the phenomenological approaches. Beginning in the middle of the century, several physicists elaborated on the possibility of systematically clarifying foundational issues of this kind in physical theories, based on the use of “axioms”, “postulates” or “principles”. This was not, to be sure, a really central trend that engaged the leading physicists in lively discussions. Still, given the vivid interest on Volkmann in the topic, Hilbert became keenly aware of many of these developments and discussed them with his colleagues at the seminar.

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Above all, the ideas of Heinrich Hertz and Ludwig Boltzmann on the foundations of physics strongly influenced him, not only at the methodological level, but also concerning his strong adherence to the mechanical reductionist point of view. The lecture notes of courses in geometry taught by Hilbert in Königsberg illuminatingly exemplify the confluence of the various points mentioned in the preceding paragraphs. Central to this is his conception of geometry as a natural science, close in all respects to mechanics and the other physical disciplines, and opposed to arithmetic and other mathematical fields of enquiry. This was a traditional separation, adopted with varying degrees of commitment, among the German mathematicians (especially in Göttingen) since the time of Gauss. Even geometers like Moritz Pasch, who had stressed a thoroughly axiomatic approach in their presentations of projective geometry [20], would support such an empiricist view of geometry. In the introduction to a course taught in 1891, for instance, Hilbert expressed his views as follows: Geometry is the science dealing with the properties of space. It differs essentially from pure mathematical domains such as the theory of numbers, algebra, or the theory of functions. The results of the latter are obtained through pure thinking …The situation is completely different in the case of geometry. I can never penetrate the properties of space by pure reflection, much the same as I can never recognize the basic laws of mechanics, the law of gravitation or any other physical law in this way. Space is not a product of my reflections. Rather, it is given to me through the senses. ([5], p. 84) The connection between this view and the axiomatic approach as a proper way to deal with this kind of sciences was strongly supported by the work of Hertz. Hilbert had announced another course in geometry for 1893, but for lack of students registered it was postponed until 1894. Precisely at this time, Hertz’s Principles of Mechanics [13] was posthumous published, and Hilbert got enthusiastic notice of the book from his friend Minkowski. Minkowski had been in Bonn since 1885 where he came under the strong influence of Hertz, to the point that the latter became his main source of scientific inspiration ([15], p. 355). In the now famous introduction to his book, Hertz described physical theories as “pictures” (Bilder) that we form for ourselves of natural phenomena, and suggested three criteria to evaluate among several possible images of one and the same object: permissibility, correctness, and appropriateness. Permissibility corresponds very roughly to consistency, whereas correctness and appropriateness are closer to the kind of criteria that will appear later on in Hilbert’s Grundlagen der Geometrie (GdG – see below). In the lecture notes of his 1893–94 course, Hilbert referred once again to the natural character of geometry and explained the possible role of axioms in elucidating its foundations. As he had time to correct the notes, he now made explicit reference to Hertz’s characterization of a “correct” scientific image (Bild) or theory. Thus Hilbert wrote ([5], p. 87):


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Nevertheless the origin [of geometrical knowledge] is in experience. The axioms are, as Hertz would say, images or symbols in our mind, such that consequents of the images are again images of the consequences, i.e., what we can logically deduce from the images is itself valid in nature. Hilbert also pointed out the need of establishing the independence of the axioms of geometry, while alluding, once again, to the kind of demand stipulated by Hertz. Stressing the objective and factual character of geometry, Hilbert wrote: The problem can be formulated as follows: What are the necessary, sufficient, and mutually independent conditions that must be postulated for a system of things, in order that any of their properties correspond to a geometrical fact and, conversely, in order that a complete description and arrangement of all the geometrical facts be possible by means of this system of things. The axioms of geometry and of physical disciplines, Hilbert said, “express observations of facts of experience, which are so simple that they need no additional confirmation by physicists in the laboratory.” The empirical character of geometry has its clear expression in the importance attributed to Gauss’s measurement of the sum of angles of a triangle formed by three mountain peaks in Hannover. Hilbert found these measurements convincing enough to indicate the correctness of Euclidean geometry as a true description of physical space. Nevertheless, he envisaged the possibility that some future measurement would yield a different result. This example would arise very frequently in Hilbert’s lectures on physics in years to come, as an example of how the axiomatic method should be applied in physics, where new empirical facts are often found by experiment. Faced with new such findings that seem to contradict an existing theory, the axiomatic analysis would allow making the necessary modifications on some of the basic assumptions of the theory, without however having to modify its essential logical structure. Hilbert stressed that the axiom of parallels is likely to be the one to be modified in geometry if new experimental discoveries would necessitate so. Geometry was especially amenable to a full axiomatic analysis only because of its very advanced stage of development and elaboration, and not because of any other specific, essential trait concerning its nature that would set it apart from other disciplines of physics. Thus, in a course on mechanics taught in 1899, the year of publication of GdG, he said: Geometry also [like mechanics] emerges from the observation of nature, from experience. To this extent, it is an experimental science.…But its experimental foundations are so irrefutably and so generally acknowledged, they have been confirmed to such a degree, that no further proof of them is deemed necessary. Moreover, all that is needed is to derive these foundations from a minimal set of independent axioms and thus to construct the whole edifice of geometry by purely logical means. In this way [i.e., by means of the axiomatic treatment] geometry is turned into a pure mathematical science. In mechanics it is also

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the case that all physicists recognize its most basic facts. But the arrangement of the basic concepts is still subject to changes in perception …and therefore mechanics cannot yet be described today as a pure mathematical discipline, at least to the same extent that geometry is. ([5], p. 90. Emphasis in the original) Thus, at the turn of the century, Hilbert consolidated his view of the axiomatic method as a correct methodology to be applied, in parallel and with equal importance, to geometry and to all other physical disciplines. The publication of GdG helped spread his ideas very quickly and in strong association with geometry alone. But the idea of applying the same point of view to physics, although made known to the public only in the 1900 list of problems, was for him natural and evident from the outset. In his course of 1899, Hilbert devoted considerable effort to discussing the technical details of, as well as the logical and conceptual interrelations among, the main principles of analytical mechanics: the energy conservation principle, the principle of virtual velocities and the D’Alembert principle, the principles of straightest path and of minimal constraint, and the principles of Hamilton and Jacobi. All of this will appear prominently in Hilbert’s later own elaboration of the program for the axiomatization of physics.

4. Grundlagen der Geometrie Hilbert’s Grundlagen der Geometrie embodied his first published, comprehensive presentation of an axiomatized mathematical discipline. Based on a course taught in the winter semester of 1898–99, it appeared in print in June of 1899. The declared aim of the book was to lay down a “simple” and “complete” system of “mutually independent” axioms, from which all known theorems of geometry might be deduced. The axioms were formulated for three systems of undefined objects named “points”, “lines”, and “planes”, and they establish mutual relations that these objects must satisfy. The axioms were grouped into five categories: axioms of incidence, of order, of congruence, of parallels, and of continuity. From a purely logical point of view, the groups have no real significance in themselves. However, from the geometrical point of view they are highly significant, for they reflect Hilbert’s actual conception of the axioms as an expression of spatial intuition: each group expresses a particular way that these intuitions manifest themselves in our understanding. Hilbert’s first requirement, that the axioms be independent, is the direct manifestation of the foundational concerns that guided his research. When analyzing independence, his interest focused mainly on the axioms of congruence, continuity and of parallels, since this independence would specifically explain how the various basic theorems of Euclidean and projective geometry are logically interrelated. This requirement had already appeared – albeit more vaguely formulated – in Hilbert’s early lectures on geometry, as a direct echo of Hertz’s demand for “appropriateness” of physical theories (i.e., the demand of “distinctness and simplicity” for the axioms


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of the theory). This time Hilbert also provided the tools to prove systematically the mutual independence among the individual axioms within the groups and among the various groups of axioms in the system. However, this was not for Hilbert an exercise in analyzing abstract relations among systems of axioms and their possible models. The motivation for enquiring about the mutual independence of the axioms remained, essentially, a geometrical one. For this reason, Hilbert’s original system of axioms was not the most economical one from the logical point of view. Indeed, several mathematicians noticed quite soon that Hilbert’s system of axioms, seen as a single collection rather than as a collection of five groups, contained a certain degree of redundancy ([19], [23]). Hilbert’s own aim was to establish the interrelations among the groups of axioms, embodying the various manifestations of space intuition, rather than among individual axioms belonging to different groups. The second one, simplicity is also related to Hertz’s appropriateness. Unlike the other requirements, it did not become standard as part of the important mathematical ideas to which GdG eventually led. Through this requirement Hilbert wanted to express the desideratum that an axiom should contain “no more than a single idea.” However, he did not provide any formal criterion to decide when an axiom is simple. Rather this requirement remained implicitly present in GdG, as well as in later works of Hilbert, as a merely aesthetic guideline that was never transformed into a mathematically controllable feature. The idea of a complete axiomatic system became pivotal to logic after 1930 following the works of Gödel, and in connection with the finitist program for the foundations of arithmetic launched by Hilbert and his collaborators around 1920. This is not, however, what Hilbert had in mind in 1899, when he included a requirement under this name in the analysis presented in GdG. Rather, he was thinking of a kind of ”pragmatic” completeness. In fact, what Hilbert was demanding here is that an adequate axiomatization of a mathematical discipline should allow for an actual derivation of all the theorems already known in that discipline. This was, Hilbert claimed, what the totality of his system of axioms did for Euclidean geometry or, if the axiom of parallels is ignored, for the so-called absolute geometry, namely that which is valid independently of the latter. Also the requirement of consistency was to become of paramount importance thereafter. Still, as part of GdG, Hilbert devoted much less attention to it. For one thing, he did not even mention this task explicitly in the introduction to the book. For another, he devoted just two pages to discussing the consistency of his system in the body of the book. In fact, it is clear that Hilbert did not intend to give a direct proof of consistency of geometry here, but even an indirect proof of this fact does not explicitly appear in GdG, since a systematic treatment of the question implied a full discussion of the structure of the system of real numbers, which was not included. Rather, Hilbert suggested that it would suffice to show that the specific kind of synthetic geometry derivable from his axioms could be translated into the standard Cartesian geometry, if the axes are taken as representing the entire field of real numbers. Only in the second edition of GdG, published in 1903, Hilbert added an additional axiom,

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the so-called “axiom of completeness” (Vollständigkeitsaxiom), meant to ensure that, although infinitely many incomplete models satisfy all the other axioms, there is only one complete model that satisfies this last axiom as well, namely, the usual Cartesian geometry. Hilbert’s axiomatic analysis of geometry was not meant to encourage the possibility of choosing arbitrary combinations of axioms within his system, and of exploring their consequences. Rather, his analysis was meant to enhance our understanding of those systems with a more intuitive, purely geometrical significance – Euclidean geometry, above all – and that made evident the connection of his work with longstanding concerns of the discipline throughout the nineteenth century [8]. As already stressed, the definition of systems of abstract axioms and the kind of axiomatic analysis described above was meant to be carried out always retrospectively, and only for “concrete”, well-established and elaborated mathematical entities. The publication of the Grundlagen was followed by many further investigations into Hilbert’s technical arguments, as well as by more general, methodological and philosophical discussions. One important such discussion appeared in the correspondence between Hilbert and Gottlob Frege. This interchange has drawn considerable attention of historians and philosophers, especially for the debate it contains between Hilbert and Frege concerning the nature of mathematical truth. But this frequentlyemphasized issue is only one side of a more complex picture advanced by Hilbert in his letters. In particular, it is interesting to notice Hilbert’s explanation to Frege, concerning the main motivations for undertaking his axiomatic analysis: the latter had arisen, in the first place, from difficulties Hilbert had encountered when dealing with physical, rather than mathematical theories. Echoing once again ideas found in the introduction to Hertz’s textbook, and clearly having in mind the problematic conceptual situation of the kinetic theory of gases at the turn of the century, Hilbert stressed the need to analyze carefully the process whereby physicists continually add new assumptions to existing physical theories, without properly checking whether or not the former contradict the latter, or consequences of the latter. In a letter of December 29, 1899, Hilbert wrote to Frege: After a concept has been fixed completely and unequivocally, it is on my view completely illicit and illogical to add an axiom – a mistake made very frequently, especially by physicists. By setting up one new axiom after another in the course of their investigations, without confronting them with the assumptions they made earlier, and without showing that they do not contradict a fact that follows from the axioms they set up earlier, physicists often allow sheer nonsense to appear in their investigations. One of the main sources of mistakes and misunderstandings in modern physical investigations is precisely the procedure of setting up an axiom, appealing to its truth, and inferring from this that it is compatible with the defined concepts. One of the main purposes of my Festschrift was to avoid this mistake. ([9], p. 40)


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In a different passage of the same letter, Hilbert commented on the possibility of substituting the basic objects of an axiomatically formulated theory by a different system of objects, provided the latter can be put in a one-to-one, invertible relation with the former. In this case, the known theorems of the theory are equally valid for the second system of objects. Concerning physical theories, Hilbert wrote: All the statements of the theory of electricity are of course valid for any other system of things which is substituted for the concepts magnetism, electricity, etc., provided only that the requisite axioms are satisfied. But the circumstance I mentioned can never be a defect in a theory [footnote: it is rather a tremendous advantage], and it is in any case unavoidable. However, to my mind, the application of a theory to the world of appearances always requires a certain measure of good will and tactfulness: e.g., that we substitute the smallest possible bodies for points and the longest possible ones, e.g., light-rays, for lines. At the same time, the further a theory has been developed and the more finely articulated its structure, the more obvious the kind of application it has to the world of appearances, and it takes a very large amount of ill will to want to apply the more subtle propositions of [the theory of surfaces] or of Maxwell’s theory of electricity to other appearances than the ones for which they were meant …([9], p. 41) Hilbert’s letters to Frege help understanding the importance of the link between physical and mathematical theories on the development of his axiomatic point of view. The latter clearly did not involve either an empty game with arbitrary systems of postulates nor a conceptual break with the classical, nineteenth-century entities and problems of mathematics and empirical science. Rather it sought after an improvement in the mathematician’s understanding of the latter. This motto was to guide much of Hilbert’s incursions into several domains of physics over the years to come.

5. Physics and the 1900 list of problems In the introductory section of his Paris talk, Hilbert stressed the important role he accorded to empirical motivations as a fundamental source of nourishment for what he described as a “living organism”, in which mathematics and the physical sciences appear tightly interrelated. The empirical motivations underlying mathematical ideas, Hilbert said, should by no means be taken as opposed to rigor. On the contrary, contrasting an “opinion occasionally advocated by eminent men”, Hilbert insisted that the contemporary quest for rigor in analysis and arithmetic should in fact be extended to both geometry and the physical sciences. He was alluding here, most probably, to Kronecker and Weierstrass, and the Berlin purist tendencies that kept geometry and applications out of their scope of interest. Rigorous methods are often simpler and easier to understand, Hilbert said, and therefore, a more rigorous treatment would

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only perfect our understanding of these topics, and at the same time would provide mathematics with ever new and fruitful ideas. In explaining why rigor should not be sought only within analysis, Hilbert actually implied that this rigor should actually be pursued in axiomatic terms. He thus wrote: Such a one-sided interpretation of the requirement of rigor would soon lead to the ignoring of all concepts arising from geometry, mechanics and physics, to a stoppage of the flow of new material from the outside world, and finally, indeed, as a last consequence, to the rejection of the ideas of the continuum and of irrational numbers. But what an important nerve, vital to mathematical science, would be cut by rooting out geometry and mathematical physics! On the contrary I think that wherever mathematical ideas come up, whether from the side of the theory of knowledge or in geometry, or from the theories of natural or physical science, the problem arises for mathematics to investigate the principles underlying these ideas and to establish them upon a simple and complete system of axioms, so that the exactness of the new ideas and their applicability to deduction shall be in no respect inferior to those of the old arithmetical concepts. (Quoted from [12], p. 245) Using a rhetoric reminiscent of Volkmann’s work, Hilbert described the development of mathematical ideas as an ongoing, dialectical interplay between the two poles of thought and experience. He also added an idea that was of central importance to Göttingen scientists for many decades, namely, the conception of the “pre-established harmony” between mathematics and nature ([21]). The importance of investigating the foundations of mathematics does not appear as an isolated concern, but rather as an organic part of the manifold growth of the discipline in several directions. Hilbert thus said: Indeed, the study of the foundations of a science is always particularly attractive, and the testing of these foundations will always be among the foremost problems of the investigator …[But] a thorough understanding of its special theories is necessary for the successful treatment of the foundations of the science. Only that architect is in the position to lay a sure foundation for a structure who knows its purpose thoroughly and in detail. (Quoted from [12], p. 258) The first two problems in Hilbert’s list are Cantor’s continuum hypothesis and the compatibility of the axioms of arithmetic. In formulating the second problem on his list, Hilbert stated more explicitly than ever before, that among the tasks related to investigating an axiomatic system, proving its consistency would be the most important one. Yet, Hilbert was still confident that this would be a rather straightforward task, easily achievable “by means of a careful study and suitable modification of the known methods of reasoning in the theory of irrational numbers.” Clearly Hilbert meant his remarks in this regard to serve as an argument against Kronecker’s negative


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reactions to unrestricted use of infinite collections in mathematics, and therefore he explicitly asserted that a consistent system of axioms could prove the existence of higher Cantorian cardinals and ordinals. Hilbert’s assertion is actually the first published mention of the paradoxes of Cantorian set theory, which here were put forward with no special fanfare ([7], p. 301). He thus established a clear connection between the two first problems on his list through the axiomatic approach. Still, Hilbert was evidently unaware of the difficulties involved in realizing this point of view, and, more generally, he most likely had no precise idea of what an elaborate theory of systems of axioms would involve. On reading the first draft of the Paris talk, several weeks earlier, Minkowski understood at once the challenging implications of Hilbert’s view, and he hastened to write to his friend: In any case, it is highly original to proclaim as a problem for the future, one that mathematicians would think they had already completely possessed for a long time, such as the axioms for arithmetic. What might the many laymen in the auditorium say? Will their respect for us grow? And you will also have a though fight on your hands with the philosophers. ([22], p. 129) Frege’s reaction to the GdG proved Minkowski’s concern to be justified, as his main criticism referred to the status of axioms as implicit definitions. The next three problems in the list are directly related with geometry and, although not explicitly formulated in axiomatic terms, they address the question of finding the correct relationship between specific assumptions and specific, significant geometrical facts. The fifth problem, for instance, relates to the question of the foundations of geometry as it had evolved over the last third of the nineteenth century along two parallel paths. On the one hand, there was the age-old tradition of elementary synthetic geometry, where the question of foundations more naturally arises in axiomatic terms. On the other hand, there was the tradition associated with the Helmholtz– Lie problem, that derived directly from the work of Riemann and that had a more physically-grounded orientation connected with the question of spaces that admit the free mobility of rigid bodies. Whereas Helmholtz had only assumed continuity as underlying the motion of rigid bodies, in applying his theory of groups of transformations to this problem, Lie was also assuming the differentiability of the functions involved. Hilbert’s work on the foundations of geometry, especially in the context that led to GdG, had so far been connected with the first of these two traditions, while devoting much less attention to the second one. Now in his fifth problem, he asked whether Lie’s conditions, rather than assumed, could actually be deduced from the group concept together with other geometrical axioms. As a mathematical problem, the fifth one led to interesting, subsequent developments. Not long after his talk, in November 18, 1901, Hilbert himself proved that, in the plane, the answer is positive, and he did so with the help of a then innovative, essentially topological, approach [14]. That the answer is positive in the general case was satisfactorily proved only in 1952 ([10], [18]). The inclusion of this problem in

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the 1900 list underscores the actual scope of Hilbert’s views over the question of the foundations of geometry and over the role of axiomatics. Hilbert suggested here the pursuit of an intricate kind of conceptual clarification involving assumptions about motion, differentiability and symmetry, such as they appear intimately interrelated in the framework of a well-elaborate mathematical theory, namely, that of Lie. This quest, that also became typical of the spirit of Hilbert’s axiomatic involvement with physical theories, suggests that his foundational views on geometry were very broad and open-ended, and did not focus on those aspects related with the synthetic approach to geometry. In particular, the fifth problem emphasizes the prominent role that Hilbert assigned to physical considerations in his approach to geometry. In the long run, this aspect of Hilbert’s view resurfaced at the time of his involvement with GTR ([5], Ch. 7–8). In its more immediate context, however, it makes the passage from geometry to the sixth problem appear as a natural one within the list. Indeed, if the first two problems in the list show how the ideas deployed in GdG led in one direction towards foundational questions in arithmetic, then the fifth problem suggests how they also naturally led, in a different direction, to Hilbert’s call for the axiomatization of physical science in the sixth problem. The problem was thus formulated as follows: The investigations on the foundations of geometry suggest the problem: To treat in the same manner, by means of axioms, those physical sciences in which mathematics plays an important part; in the first rank are the theory of probabilities and mechanics. (Quoted in [12], p. 258) As examples of what he had in mind Hilbert mentioned several existing and well-known works: the fourth edition of Mach’s Die Mechanik in ihrer Entwicklung, Hertz’s Prinzipien, Boltzmann’s 1897 Vorlesungen Über die Principien der Mechanik, and also Volkmann’s 1900 Einführung in das Studium der theoretischen Physik. Boltzmann’s work offered a good example of what axiomatization would offer, as he had indicated, though only schematically, that limiting processes could be applied, starting from an atomistic model, to obtain the laws of motion of continua. Hilbert thought it convenient to go in the opposite direction also, i.e., to derive the laws of motions of rigid bodies by limiting processes, starting from a system of axioms that describe space as filled with continuous matter in varying conditions. Thus one could investigate the equivalence of different systems of axioms, an investigation that Hilbert considered to be of the highest theoretical importance. This is one of the few places where Hilbert emphasized Boltzmann’s work over Hertz’s in this regard, and this may give us the clue to the most immediate trigger that was in the back of Hilbert’s mind when he decided to include this problem in the list. Indeed, Hilbert had met Boltzmann several months earlier in Munich, where the latter gave a talk on recent developments in physics. Boltzmann had not only discussed ideas connected with the task that Hilbert was now calling for, but he also adopted a rhetoric that seems to have appealed very much to Hilbert. In fact, Boltzmann


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had suggested that one could follow up the recent history of physics with a look at future developments. Nevertheless, he said, “I will not be so rash as to lift the veil that conceals the future” ([2], p. 79). Hilbert, on the contrary, opened the lecture by asking precisely, “who among us would not be glad to lift the veil behind which the future lies hidden” and the whole trust of his talk implied that he, the optimistic Hilbert, was helping the mathematical community to do so. Together with the well-known works on mechanics referred to above, Hilbert also mentioned a recent work by the Göttingen actuarial mathematician Georg Bohlmann on the foundations of the calculus of probabilities [1]. The latter was important for physics, Hilbert said, for its application to the method of mean values and to the kinetic theory of gases. Hilbert’s inclusion of the theory of probabilities among the main physical theories whose axiomatization should be pursued has often puzzled readers of this passage. The notes of a course taught in 1905 on the axiomatic method show that this was a main point in Hilbert’s views on physics because of the use of probabilities also in insurance mathematics and in problems of observational error calculation in astronomy. It is also remarkable that Hilbert did not mention electrodynamics among the physical disciplines to be axiomatized, even though the second half of the Gauss– Weber Festschrift, where Hilbert’s GdG was published, contained a parallel essay by Wiechert on the foundations of electrodynamics. At any rate, Wiechert’s presentation was by no means axiomatic, in any sense of the term. On the other hand, the topics addressed by Wiechert would start attracting Hilbert’s attention over the next years, at least since 1905. This sixth problem is not really a problem in the strict sense of the word, but rather a general task for whose complete fulfillment Hilbert set no clear criteria. Thus, Hilbert’s detailed account in the opening remarks of his talk as to what a meaningful problem in mathematics is, and his stress on the fact that a solution to a problem should be attained in a finite number of steps, does not apply in any sense to the sixth one. On the other hand, the sixth problem has important connections with three other problems on Hilbert’s list: the nineteenth (“Are all the solutions of the Lagrangian equations that arise in the context of certain typical variational problems necessarily analytic?”), the twentieth (dealing with the existence of solutions to partial differential equations with given boundary conditions), closely related to the nineteenth and at the same time to Hilbert’s long-standing interest on the Dirichlet Principle, and, finally, the twenty-third (an appeal to extend and refine the existing methods of variational calculus). Like the sixth problem, the latter two are general tasks rather than specific mathematical problems with a clearly identifiable, possible solution. All these three problems are also strongly connected to physics, though unlike the sixth, they are also part of mainstream, traditional research concerns in mathematics. In fact, their connections to Hilbert’s own interests are much more perspicuous and, in this respect, they do not raise the same kind of historical questions that Hilbert’s interest in the axiomatization of physics does. A balanced assessment of the influence of the problems on the development of mathematics throughout the century must take into account not only the intrinsic

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importance of the problems, but also the privileged institutional role of Göttingen in the mathematical world with the direct and indirect implications of its special status. However, if Hilbert wished to influence the course of mathematics over the coming century with his list, then it is remarkable that his own career was only very partially shaped by it. Part of the topics covered by the list belonged to his previous domains of research, while others belonged to domains where he never became active. On the contrary, domains that he devoted much effort to over the next years, such as the theory of integral equations, were not contemplated in the list. In spite of the enormous influence Hilbert had on his students, the list did not become a necessary point of reference of preferred topics for dissertations. To be sure, some young mathematicians, both in Göttingen and around the world, did address problems on the list and sometimes came up with important mathematical achievements that helped launch their own international careers. But this was far from the only way for talented young mathematicians to reach prominence in or around Göttingen. But, ironically, the sixth problem, although seldom counted among the most influential of the list, can actually be counted among those that received greater attention from Hilbert himself and from his collaborators and students over the following years.

6. Concluding remarks For all its differences and similarities with other problems on the list, the important point that emerges from the above account is that the sixth problem was in no sense disconnected from the evolution of Hilbert’s early axiomatic conception at its very core. Nor was it artificially added in 1900 as an afterthought about the possible extensions of an idea successfully applied in 1899 to the case of geometry. Rather, Hilbert’s ideas concerning the axiomatization of physical science arose simultaneously with his increasing enthusiasm for the axiomatic method and they fitted naturally into his overall view of pure mathematics, geometry and physical science – and the relationship among them – by that time. From 1900 on, the idea of axiomatizing physical theories was a main thread that linked much of Hilbert’s research and teaching. Hilbert taught every semester at least one course dealing with a physical discipline, and by the end of his career he had covered most of the important fields that were at the cutting edge of physics, currently attracting the best research efforts of young and promising minds (see the appendix to this article). The axiomatic point of view provided a unifying methodology from which to approach many of the topics in which Hilbert became interested. In 1905 he taught a course on the axiomatic method where he presented for the first time a panoramic view of various physical disciplines from an axiomatic perspective: mechanics, thermodynamics, probability calculus, kinetic theory, insurance mathematics, electrodynamics, psychophysics. The variety of physical topics pursued only grew over the years. The extent of the influence of Hilbert’s ideas on physics on contemporary research is a more complex question that cannot be discussed here for lack


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of space. Still, it is relevant to quote here an account of Hilbert’s ideas as described by the physicist on whom Hilbert’s influence became most evident, Max Born. On the occasion of Hilbert’s sixtieth birthday, at a time when he was deeply involved together with Bernays in the technical difficulties raised by the finitist program, Born wrote the following words: The physicist set outs to explore how things are in nature; experiment and theory are thus for him only a means to attain an aim. Conscious of the infinite complexities of the phenomena with which he is confronted in every experiment, he resists the idea of considering a theory as something definitive. He therefore abhors the word “Axiom”, which in its usual usage evokes the idea of definitive truth. The physicist is thus acting in accordance with his healthy instinct, that dogmatism is the worst enemy of natural science. The mathematician, on the contrary, has no business with factual phenomena, but rather with logic interrelations. In Hilbert’s language the axiomatic treatment of a discipline implies in no sense a definitive formulation of specific axioms as eternal truths, but rather the following methodological demand: specify the assumptions at the beginning of your deliberation, stop for a moment and investigate whether or not these assumptions are partly superfluous or contradict each other. ([3]) The development of physics from the beginning of the century, and especially after 1905, brought many surprises that Hilbert could not have envisaged in 1900 or even when he lectured at Göttingen on the axioms of physics in 1905; yet, Hilbert was indeed able to accommodate these new developments to the larger picture of physics afforded by his program for axiomatization. In fact, some of his later contributions to mathematical physics, particularly his contributions to GTR, came by way of realizing the vision embodied in this program.

7. Appendix: Hilbert’s Göttingen courses on physics (and related fields): 1895–1927 For an explanation on the sources used for compiling this list, see [5], p. 450 (WS = Winter Semester, SS = Summer Semester, HS = Special Autumn [Herbst] Semester). WS 1895/96 SS 1896 SS 1898 SS 1899 WS 1900/01 SS 1901 WS 1901/02

Partial Differential Equations Ordinary Differential Equations Mechanics Variational Calculus Partial Differential Equations Linear Partial Differential Equations Potential Theory

1716 SS 1902 WS 1902/03 SS 1903 WS 1903/04 WS 1904/05 SS 1905 SS 1905 WS 1905/06 WS 1905/06 SS 1906 WS 1906/07 SS 1907 WS 1909/10 SS 1910 WS 1910/11 SS 1911 WS 1911/12 SS 1912 SS 1912 SS 1912 WS 1912/13 WS 1912/13 WS 1912/13 SS 1913 SS 1913 WS 1913/14 WS 1913/14 WS 1913/14 SS 1914 SS 1914 WS 1914/15 SS 1915 WS 1915/16 SS 1916 SS 1916 WS 1916/17 SS 1917 SS 1918 WS 1918/19 WS 1918/19 HS 1919 WS 1920 SS 1920

Leo Corry

Selected Topics in Potential Theory Continuum Mechanics - Part I Continuum Mechanics - Part II Partial Differential Equations Variational Calculus Logical Principles of Mathematical Thinking (and of Physics) Integral Equations Partial Differential Equations Mechanics Integral Equations Continuum Mechanics Differential Equations Partial Differential Equations Selected Chapters in the Theory of Partial Differential Equations Mechanics Continuum Mechanics Statistical Mechanics Radiation Theory Ordinary Differential Equations Mathematical Foundations of Physics Molecular Theory of Matter Partial Differential Equations Mathematical Foundations of Physics Foundations of Mathematics (and the axiomatization of Physics) Electron Theory Electromagnetic Oscillations Analytical Mechanics Exercises in Mechanics (together with H. Weyl) Statistical Mechanics Differential Equations Lectures on the Structure of Matter Structure of Matter (Born’s Theory of Crystals) Differential Equations Partial Differential Equations Foundations of Physics I (General Relativity) Foundations of Physics II (General Relativity) Electron Theory Ordinary Differential Equations Space and Time Partial Differential and Integral Equations Nature and Mathematical Knowledge Mechanics Higher Mechanics and the New Theory of Gravitation


WS 1920/21 SS 1921 SS 1921 SS 1922 WS 1922/23 WS 1922/23 WS 1922/23 SS 1923 WS 1923/24 SS 1924 WS 1926/27 SS 1930 WS 1930/31 WS 1931/32

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Mechanics and the New Theory of Gravitation Einstein’s Gravitation Theory. Basic Principles of the Theory of Relativity On Geometry and Physics Statistical Mechanics Mathematical Foundations of Quantum Theory Knowledge and Mathematical Thought Knowledge and Mathematical Thought Our Conception of Gravitation and Electricity On the Unity of Science Mechanics and Relativity Theory Mathematical Methods of Quantum Theory Mathematical Methods of Modern Physics Nature and Thought Philosophical Foundations of Modern Natural Science

References [1] Bohlmann, G., Ueber Versicherungsmathematik. In Über angewandte Mathematik und Physik in ihrer Bedeutung für den Unterricht an den höheren Schulen (ed. by F. Klein & E. Riecke), Teubner, Leipniz, Berlin 1900, 114–145. [2] Boltzmann, L., Über die Entwicklung der Methoden der theoretischen Physik in neuerer Zeit (1899). In L. Boltzmann Populäre Schriften, J. A. Barth, Leipzig 1905, 198–277. [3] Born, M., Hilbert und die Physik. Die Naturwissenschaften 10 (1922), 88–93. (Reprint in Born. M., Ausgewählte Abhandlungen, Vol. 2, Vandenhoek & Ruprecht, Göttingen 1963, 584–598.) [4] Brush, S. G., The Kind of Motion we Call Heat - A History of the Kinetic Theory of Gases in the 19th Century. North Holland Publishing House, Amsterdam, New York, Oxford 1976. [5] Corry, L., David Hilbert and the Axiomatization of Physics (1898–1918): From Grundlagen der Geometrie to Grundlagen der Physik. Archimedes: New Studies in the History and Philosophy of Science and Technology 10, Kluwer Academic Publishers, Dordrecht 2004. [6] Dieudonné, J., Les méthodes axiomatiques modernes et les fondements des mathématiques. In Les grands Courants de la Pensée Mathématique (ed. by F. Le Lionnais), Blanchard, Paris 1962, 443–555. [7] Ferreirós, J., Labyrinths of Thought. A History of Set Theory and its Role in Modern Mathematics. Sci. Networks Hist. Stud. 23, Birkhäuser, Boston 1999. [8] Freudenthal, H., Zur Geschichte der Grundlagen der Geometrie. Zugleich eine Besprechung der 8. Auflage von Hilberts ‘Grundlagen der Geometrie’. Nieuw Archief voor Wiskunde 4 (1957), 105–142. [9] Gabriel, G. et al. (eds.), Gottlob Frege - Philosophical and Mathematical Correspondence. The University of Chicago Press, Chicago 1980. [10] Gleason, A., Groups without Small Subgroups. Ann. Math. 56 (1952), 193–212.

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[11] Gnedenko, J., Zum sechsten Hilbertschen Problem. In Die Hilbertsche Probleme (ed. by P. Alexandrov), Ostwalds Klassiker der exakten Wissenschaften 252, Leipzig 1979, 144–147. [12] Gray, J. J., The Hilbert Challenge. Oxford University Press, New York 2000. [13] Hertz, H., Die Prinzipien der Mechanik in neuem Zusammenhange dargestellt. Leipzig 1984. [14] Hilbert, D., Über die Grundlagen der Geometrie. Math. Ann. 56 (1902), 233–241. [15] Hilbert, D., Hermann Minkowski. Math. Ann. 68 (1910), 445–471. [16] Hilbert, D., Natur und Mathematisches Erkennen: Vorlesungen, gehalten 1919-1920 in Göttingen. Nach der Ausarbeitung von Paul Bernays. Edited and with an English introduction by David E. Rowe, Birkhäuser, Basel 1992. [17] Hilbert, D., Über meine Tätigkeit in Göttingen. In Hilbert: Gedenkband (ed. by K. Reidemeister). Springer-Verlag, Berlin, Heidelberg, New York 1971, 79–82. [18] Montgomery, D., Zippin, L., Small Subgroups of Finite-dimensional Groups. Ann. Math. 56 (1952), 213–241. [19] Moore. E. H., Projective Axioms of Geometry. Trans. Amer. Math. Soc. 3 (1902), 142–158. [20] Pasch, M., Vorlesungen über neuere Geometrie. Teubner, Leipzig 1882. [21] Pyenson, L., Relativity in Late Wilhelmian Germany: the Appeal to a Pre-established Harmony Between Mathematics and Physics. In L. Pyenson The Young Einstein: The Advent of Relativity, Adam Hilger Ltd., Bristol, Boston 1985, 137–157. [22] Rüdenberg L., Zassenhaus, H., Hermann Minkowski - Briefe an David Hilbert. SpringerVerlag, Berlin, New York 1973. [23] Schur, F., Über die Grundlagen der Geometrie. Math. Ann. 55 (1901), 265–292. [24] Weyl, H., David Hilbert and his Mathematical Work, Bull. Amer. Math. Soc. 50 (1944), 612–654. [25] Wightman, A. S., Hilbert’s Sixth Problem: Mathematical Treatment of the Axioms of Physics. In Mathematical Developments Arising from Hilbert Problems (ed. by F. E. Browder), Symposia in Pure Mathematics 28, Amer. Math. Soc., Providence, RI, 1976, 147–240. [26] Yandell, B. H., The Honors Class: Hilbert’s Problems and Their Solvers. AK Peters, Natick, MA, 2002. Cohn Institute for History and Philosophy of Science, Tel-Aviv University, Tel-Aviv 69978, Israel E-mail: [email protected]

Method versus calculus in Newton’s criticisms of Descartes and Leibniz Niccolò Guicciardini

Abstract. In my talk I will consider Newton’s views on mathematical method. Newton never wrote extensively about this issue. However, in his polemic writings addressed against Descartes and Leibniz he expressed the idea that his method was superior to the ones proposed by the French and the German. Considering these writings can help us in understanding the role attributed to algebra and calculus in Newton’s mathematical thought. Mathematics Subject Classification (2000). Primary 01A45; Secondary 00A30. Keywords. 17th century, philosophy of mathematics, Newton, Descartes, Leibniz.

1. Newton’s memorandum on his early discoveries Newton blossomed as a creative mathematician in 1665–1666, the so-called anni mirabiles, about four years after matriculating at Cambridge.1 A Newtonian memorandum, written about fifty years later, gives an account that has been basically confirmed by manuscript evidence: In the beginning of the year 1665 I found the Method of approximating series & the Rule for reducing any dignity of any Binomial into such a series. The same year in May I found the method of Tangents of Gregory & Slusius, & in November had the direct method of fluxions & the next year in January had the theory of Colours & in May following I had entrance into ye inverse method of fluxions. And the same year I began to think of gravity extending to ye orb of the Moon […] All this was in the two plague years of 1665–1666. For in those days I was in the prime of my age for invention & minded Mathematicks & Philosophy more than any time since. ([1]) There would be much to say in order to decipher and place into context Newton’s discourse. For instance, the task of commenting on the meaning of the term ‘philosophy’ would require space and scholarship not at my disposal [2]. 1 Readers interested in Newton’s mathematics should read Tom Whiteside’s introductions and commentaries in [9].


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Niccolò Guicciardini

Figure 1. Newton’s home at Woolsthorpe where – he claimed – he made his early discoveries in mathematics and natural philosophy when Cambridge University was evacuated because of the plague during the biennium 1665–1666. As a matter of fact, he did important work in mathematics during periods in which he returned to the University. Further, his juvenile insights – particularly those concerning gravitation – had to be elaborated during the next decades. Source: [1], 54.

Let me note three things about the above memorandum. The ‘Method of approximating series’ is the method of series expansion via long division and root extraction (as well as other methods which were later subsumed under more general techniques usually attributed to Puiseux) that allowed Newton to go beyond the limitation of what he termed ‘common analysis’ – where ‘finite equations’ were deployed – and express certain curves locally in terms of infinite fractional power series, which Newton called ‘infinite equations’. The ‘Rule for reducing any dignity of any Binomial’ is what we call the ‘binomial theorem’. Such methods of series expansion were crucial for attaining two goals: the calculation of areas of curvilinear surfaces and the rectification of curves (see Figure 2). Notice that Newton does not talk about a theorem, but rather about ‘methods’ and a ‘rule’. This last fact is of utmost importance and deserves our commentary in Sections 2, 3, and 4, before turning in Section 5 to the direct and inverse methods of fluxions which are the Newtonian equivalent of the Leibnizian differential and integral calculus.2 2 For a recent evaluation of Newton’s early mathematical researches see [3].

Method versus calculus in Newton’s criticisms of Descartes and Leibniz

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Figure 2. Calculation √ √ of areas of hyperbolic and circular surfaces via extraction of root of aa + xx = y and aa − xx = y. This technique of series expansion and termwise integration was basic in Newton’s early mathematical work and was displayed in a tract entitled On the analysis by means of infinite equations (written in 1669, but printed only in 1711), an extension of ‘common analysis’ which proceeds via ‘finite equations’ only. Source: [8], vol. 1, 8.

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2. Pappus on the method of analysis and synthesis Newton belonged to a mathematical community in which the distinction between theorems and problems was articulated according to criteria sanctioned by the venerated Greek tradition. Most notably in the work of the late Hellenistic compiler Pappus entitled Mathematical Collection which appeared in 1588 in Latin translation Newton – who avidly read this dusty work – could find a distinction between ‘theorematic and problematic analysis’. In the 7th book of the Collection there was a description of works (mostly lost and no longer available to early modern mathematicians) which – according to Pappus – had to do with a heuristic method followed by the ancient geometers. The opening of the seventh book is often quoted. It is an obscure passage whose decoding was top in the agenda of early modern European mathematicians, convinced as they were that here lay hidden the key to the method of discovery of the ancients. Given the importance this passage had for Newton, it is worth quoting at length: That which is called the Domain of Analysis, my son Hermodorus, is, taken as a whole, a special resource that was prepared, after the composition of the Common Elements, for those who want to acquire a power in geometry that is capable of solving problems set to them; and it is useful for this alone. It was written by three men: Euclid the Elementarist, Apollonius of Perge, and Aristaeus the elder, and its approach is by analysis and synthesis. Now analysis is the path from what one is seeking, as if it were established, by way of its consequences, to something that is established by synthesis. That is to say, in analysis we assume what is sought as if it has been achieved, and look for the thing from which it follows, and again what come before that, until by regressing in this way we come upon some one of the things that are already known, or that occupy the rank of a first principle. We call this kind of method ‘analysis’, as if to say anapalin lysis (reduction backward). In synthesis, by reversal, we assume what was obtained last in the analysis to have been achieved already, and, setting now in natural order, as precedents, what before were following, and fitting them to each other, we attain the end of the construction of what was sought. There are two kinds of analysis: one of them seeks after the truth, and is called ‘theorematic’: while the other tries to find what was demanded, and is called ‘problematic’. In the case of the theorematic kind, we assume what is sought as a fact and true, then advancing through its consequences, as if they are true facts according to the hypothesis, to something established, if this thing that has been established is a truth, then that which was sought will also be true, and its proof the reverse of the analysis; but if we should meet with something established to be false, then the thing that was sought too will be false. In the case of the problematic kind, we assume the proposition as something we know, then, proceeding through its consequences, as if true, to something


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established, if the established thing is possible and obtainable, which is what mathematicians call ‘given’, the required thing will also be possible, and again the proof will be the reverse of the analysis; but should we meet with something established to be impossible, then the problem too will be impossible. ([4]) Pappus here made a distinction between analysis and synthesis. Analysis (‘resolutio’ in Latin) was often conceived of as a method of discovery, or a method of problem solving, which, working step by step backwards from what is sought as if it had already been achieved, eventually arrives at what is known. Synthesis (‘compositio’ or ‘constructio’) goes the other way round: it starts from what is known and, working through the consequences, arrives at what is sought. On the basis of Pappus’ authority it was often stated that synthesis ‘reverses’ the steps of analysis. It was synthesis which provided the rigorous proof. Thus the belief – widespread in early modern Europe – that the ancients had kept the method of analysis hidden and had published only the rigorous synthesis, either because they considered the former not wholly demonstrative, or because they wanted to hide the method of discovery. Another distinction which was of momentous importance for early modern mathematicians is that between problems and theorems. A problem asks a construction for its solution. It starts from certain elements considered as already constructed either by postulate or by previously established constructions. Such elements are the ‘givens’ (in Latin the ‘data’) of the problem. A problem ends with a ‘Q.E.I.’ or with a ‘Q.E.F.’ (‘quod erat inveniendum’ – ‘what was to be discovered’–, and ‘quod erat faciendum’ – ‘what was to be done’–, respectively). A theorem asks for a deductive proof, a sequence of propositions each following from the previous one by allowed inference rules. The starting point of the deductive chain can be either axioms or previously proved theorems. A theorem ends with ‘Q.E.D.’ (‘quod erat demonstrandum’– ‘what was to be demonstrated’). According to Pappus, therefore, there are two kinds of analysis: the former referred to problems, the latter to theorems. But it is clear from classical sources that the most important, or at least the most practiced kind, was problematic analysis: and indeed early modern European mathematicians were mainly concerned with the analysis of geometrical problems. Another powerful idea that began to circulate in Europe at the end of the seventeenth century was that the analysis of the Greeks was not geometrical but rather symbolical: i.e. the Greeks were supposed to have had algebra and to have applied it to geometrical problem solving. The evidence that symbolic algebra was within the reach of the ancients was provided by a far from philological reading of the work of Diophantus and of parts of Euclid’s Elements. The approach of Renaissance culture towards the classics, in sculpture, architecture, music, philosophy, and so on, was characterized by admiration united to a desire to restore the forgotten conquests of the ancients. This approach often confined with worship, a conviction of the occurrence of a decay from a glorious, golden past. The works of Euclid, Apollonius, Archimedes were considered unsurpassable models by many Renaissance mathematicians. The question that often emerged was: how could the Greeks have achieved such a wealth

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of results? In the decades following the publication of the Collection the belief in the existence of a lost, or hidden, ‘Treasure of analysis’ promoted many efforts aimed at ‘restoring’ the ancients’ method of discovery. Not everybody trod in the steps of the classicists. Typically, many promoters of the new symbolic algebra were proud to define themselves as innovators, rather than as restorers. It was common, however, even among creative algebraists such as François Viète, John Wallis and Isaac Newton, to relate symbolic algebra to the ancient analysis, to the hidden problem solving techniques of the ancients.

3. Descartes’ method of problem solving and problem construction Newton was deeply embedded in the conceptual space defined by Pappus and by his readers, interpreters and critics. Mainly he referred his views on mathematical method to Descartes’ Géométrie (1637), an early source of inspiration for him and soon a target of his fierce criticisms ([5]). From this tradition Newton derived the idea that a problem, once analyzed (resolved), must be synthesized (composed or constructed). How did Descartes define his canon of problem solving and the role of algebra in the analysis and synthesis of geometrical problems? The historian who has done most to clarify this issue is Henk Bos. It is to his work that we now turn for advice ([6]). In book 1 of the Géométrie Descartes explained how one could translate a geometric problem into an equation. Descartes was able to do so by a revolutionary departure from tradition. In fact he interpreted algebraic operations as closed operations on segments. For instance, if a and b represent lengths of segments the product ab is not conceived by Descartes as representing an area but rather another length. As he wrote: ‘it must be observed that by a 2 , b3 , and similar expressions, I ordinarily mean any simple lines’, while before the Géométrie such expressions represented an area and a volume respectively (see Figure 3). Descartes’ interpretation of algebraic operations was indeed a gigantic innovation, but he proceeded wholly in line with Pappus’ method of analysis and synthesis, to which he explicitly referred. In fact, according to Descartes, one has – following Pappus’ prescriptions– to ‘start by assuming that the problem was solved and consider a figure incorporating the solution’.3 The segments in the figure are then denoted by letters, a, b, c, . . . , for segments which are given, x, y, z, . . . , for segments which are unknown. Geometrical relationships holding between the segments are then translated into corresponding equations. It is thus that one obtains a system of equations which symbolically express the assumption that the problem is solved. In fact, here we are at the very beginning of the analytic process: the unknown segments are treated as if they were known and manipulated in the equations on a par with the givens of the problem. The resolution of the equation allows the expression of the unknown x in terms of given segments. We have thus moved from the assumption that the problem is solved 3 [6] on p. 303.


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Figure 3. Descartes’ geometric interpretation of algebraic operations. He writes: ‘For example, let AB be taken as unity, and let it be required to multiply BD by BC. I have only to join the points A and C, and draw DE parallel to CA; and then BE is the product of BD and BC’. So, given a unit segment, the product of two segments is represented by another segment, not by a surface. The second diagram is the construction of the square root of GH . Given GH and a unit segment F G, one draws the circle of diameter F G + GH and erects GI , the required root. Source: [5], 4.

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(the first step of the analysis) to a reduction of the unknown, sought magnitude to the givens. This is why Descartes, and the other early-modern promoters of algebra, associated algebra with the method of analysis. The resolution of the equation is not, however, the solution of the problem. In fact, the solution of the problem must be a geometrical construction of the sought magnitude in terms of legitimate geometrical operations performed on the givens (‘Q.E.F.’!). We now have to move from algebra back to geometry again. Descartes understood this process from algebra to geometry as follows: the real roots of the equation (for him if there are no real roots, then the problem admits no solution) must be geometrically constructed. After Descartes, this process was known as the ‘construction of the equation’. This is where the synthetic, compositive part of the whole process begins. Descartes accepted from tradition the idea that such constructions must be performed by intersection of curves. That is to say, the real roots are geometrically represented by segments, and such segments are to be constructed by intersection of curves. As a matter of fact, the construction of the equation presented the geometer with a new problem: not always an easy one. One had to choose two curves, position and scale them, such that their intersections determine points from which segments – whose lengths geometrically represent the roots of the equation – can be drawn (see Figure 4).

Figure 4. Construction of a third-degree equation in Descartes’ Géométrie. The problem of trisecting angle NOP is resolved (‘resolutio’ is the Latin translation of the Greek ‘analysis’) by a third-degree equation. Descartes constructs the roots (‘constructio’ or ‘compositio’ translate ‘synthesis’) via intersection of circle and parabola. The segments kg, KG and LF represent two positive and one negative root. The smaller of the two positive roots kg must be ‘taken as the length of the required line NQ’. KG is equal to NV , ‘the chord subtended by one-third the arc NV P ’. Source: [5], 208.

The synthetic part of Descartes’ process of problem-solving gave rise to two questions: which curves are admissible in the construction of equations? which curves,


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among the admissible, are to be preferred in terms of simplicity? In asking himself these questions Descartes was continuing a long debate concerning the role and classification of curves in the solution of problems. A tradition that, once again, stems from Pappus, and the interpretations of Pappus given by mathematicians such as Viète, Ghetaldi, Kepler, and Fermat. His answer was that only ‘geometrical curves’ (we would say ‘algebraic curves’) are admissible in the construction of the roots of equations and that one has to choose the curves of the lowest possible degree as these are the simplest. Descartes instead excluded ‘mechanical curves’ (we would say transcendental curves) as legitimate tools of construction. Notice that Descartes presented his canon of problem resolution and construction in aggressively anti-classicist terms. His algebraic method, he claimed, was superior to the ones followed by the ancients. He gave pride of place to a problem discussed in Pappus’ Mathematical Collection that – according to Descartes – neither Euclid nor Apollonius could solve. He proudly showed to the readers of the slim Géométrie that, by applying algebra to geometry, he could easily achieve a solution not included in the ponderous Pappusian tomes.4

4. Newton versus Descartes Newton sharply criticized Descartes’canon of problematic analysis and construction.5 Newton’s point was that geometrical constructions have to be carried on in terms independent from algebra. Newton elaborated his criticism to Descartes in his Lucasian Lectures on Algebra which were held before 1684 and which, in somewhat modified form, appeared in 1707 as Arithmetica Universalis ([8], vol. 2, 3–135). The Arithmetica Universalis ends with an Appendix devoted to the ‘construction of equations’ which abounds with oft-quoted statements in favour of pure geometry and against the ‘Moderns’ (read Descartes) who have lost the ‘Elegance’ of geometry: Geometry was invented that we might expeditiously avoid, by drawing Lines, the Tediousness of Computation. Therefore these two sciences [Geometry and Arithmetical Computation] ought not be confounded. The Ancients did so industriously distinguish them from one another, that they never introduced Arithmetical Terms into Geometry. And the Moderns, by confounding both, have lost the Simplicity in which all the Elegance of Geometry consists.6 4 Briefly said, Pappus problem requires the determination of the locus of points P such that their distances di (i = 1, 2, 3, 4) from four lines given in position are such that d1 d2 = k(d3 d4 ). In the Géométrie Descartes introduces a system of oblique coordinates, and notices that the distance of a point from a line is given by an expression of the form ax + by + c. Therefore Pappus 4-lines locus has a second-degree defining equation: namely it will be a conic section. The algebraic approach immediately allowed Descartes to generalize Pappus problem for any number of lines. 5 Further information on Newton’s criticisms to Descartes can be gained from [7]. 6 [8], vol. 2, 228.

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Such statements have often puzzled commentators since they occur in a work devoted to algebra and in which the advantage of algebraic analysis is displayed in a long section on the resolution of geometrical problems. Why was Newton turning his back to ‘arithmetic’7 now saying that algebra and geometry should be kept apart? In order to understand this seemingly paradoxical position we have to briefly recall that according to Descartes the demarcation between admissible and inadmissible curves as means of construction was that between geometrical and mechanical curves. Ultimately, Descartes was forced to make recourse to algebraic criteria of demarcation and simplicity: in fact, algebraic curves coincided for him with the loci of polynomial equations, and the degree of the equation allowed him to rank curves in terms of their simplicity. As far as demarcation is concerned, in the Arithmetica Universalis Newton maintained that it would be wrong to think that a curve can be accepted or rejected in terms of its defining equation. He wrote: It is not the Equation, but the Description that makes the Curve to be a Geometrical one. The Circle is a Geometrical Line, not because it may be expressed by an Equation, but because its Description is a Postulate.8 Further, Descartes’ classification of curves in function of the degree of the equation – Newton claimed – is not relevant for the geometrician, who will choose curves in function of the simplicity of their description. Newton, for instance, observed that the equation of a parabola is simpler than the equation of the circle. However, it is the circle which is simpler and to be preferred in the construction of problems: It is not the simplicity of its equation, but the ease of its description, which primarily indicates that a line is to be admitted into the construction of problems. […] On the simplicity, indeed, of a construction the algebraic representation has no bearing. Here the descriptions of curves alone come into the reckoning.9 Newton observed that from this point of view, the conchoid, a fourth degree curve, is quite simple. Independently of considerations about its equation, its mechanical description – he claimed – is one of the simplest and most elegant; only the circle is simpler. Descartes’ algebraic criterion of simplicity is thus deemed alien to the constructive, synthetical, stage of problem solving. The weakness of Newton’s position is that the concepts of simplicity of tracing, or of elegance, to which he continuously refers are qualitative and subjective. One should be aware that no compelling reason is given in support of Newton’s evaluations on the simplicity of his preferred constructions: his are largely aesthetic criteria. Considering them is however crucial for our understanding of Newton’s views concerning mathematical method. 7 Notice that Newton employed the term ‘universal arithmetic’ for algebra, since it is concerned with the doctrine of operations, not applied to numbers, but to general symbols. 8 [8], vol. 2, 226. 9 [9], vol. 5, 425–7.


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As a matter of fact, Newton – this master of algebraic manipulations – in the mid 1670s developed a deep distaste for symbolism and distanced himself from the mathematics of the ‘moderns’. He wrote: The Modern Geometers are too fond of the Speculation of Equations. The Simplicity of these is of an Analytick Consideration. [in the Appendix to the Arithmetica Universalis] [w]e treat of Composition, and Laws are not given to Composition from Analysis. Analysis does lead to Composition: but it is not true Composition before it is freed from Analysis. If there be never so little Analysis in Composition, that Composition is not yet real. Composition in it self is perfect, and far from a Mixture of Analytick Speculations.10 This position, let me restate it, does not exclude the use of algebra in the analysis; it does, however, rule out algebraic criteria of demarcation and simplicity from the synthesis. As Newton was to affirm in a manuscript dating from the early 1690s: if a question be answered […] that question is resolved by the discovery of the equation and composed by its construction, but it is not solved before the construction’s enunciation and its complete demonstration is, with the equation now neglected, composed.11 But, around 1680, Newton moved a step forward in his opposition to the method proposed in the Géométrie: not only Cartesian synthesis, but also Cartesian analysis fell under his fierce attack. He developed a deep admiration for the ancient Greek mathematicians, while he criticized in bitter terms the symbolical analysis pursued by the moderns. He began to doubt that the analysis of the Greeks was algebraical, he rather suspected that Euclid and Apollonius possessed a more powerful geometrical analysis displayed in the three lost books on Porisms attributed to Euclid and described in Book 7 of the Mathematical Collection. So not only the composition (the synthesis) had to be freed from algebra, the algebraic calculus had to be avoided also in the process of resolution (the analysis). His target was often Descartes. For instance in the late 1670s, commenting on Descartes’ solution of Pappus problem, he stated with vehemence: To be sure, their [the Ancients’] method is more elegant by far than the Cartesian one. For he [Descartes] achieved the result by an algebraic calculus which, when transposed into words (following the practice of the Ancients in their writings), would prove to be so tedious and entangled as to provoke nausea, nor might it be understood. But they accomplished it by certain simple propositions, judging that nothing written in a different style was worthy to be read, and in consequence concealing the analysis by which they found their constructions.12 10 [8], vol. 2, 250. 11 [9], vol. 7, 307. 12 [9], vol. 4, 277.

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Newton was not alone in his battle against the algebraists. Similar statements can be found in the polemic works of Thomas Hobbes. But probably the deepest influence on Newton in this matter was played by his mentor Isaac Barrow. Newton’s quest for the ancient, non-algebraical, porismatic analysis led him to develop an interest in projective geometry (see Figure 5).

Figure 5. Newton was interested in using projective transformations as a heuristic analytic tool. Here we reproduce the diagram for Lemma 22, Book 1, of the Principia. In this Lemma we are taught how ‘To change figures into other figures of the same class’ (namely, algebraic curves of the same degree). The figure to be transmuted is the curve H GI . Draw the straight parallel lines AO and BL cutting any given third line AB in A and B. Then from some point O in the line AO draw the straight line OD. From the point d erect the ordinate dg (you can choose any angle between the ‘new ordinate’ dg and the ‘new abscissa’ ad). The new ordinate and abscissa have to satisfy the following conditions: AD = (AO × AB)/ad and DG = (AO × dg)/ad. These transformations are exactly those occurring between figures projected from one plane into another. Now suppose that point G ‘be running through all the points in the first figure [HGI] with a continual motion; then point g – also with a continual motion – will run through all the points in the new figure [hgi]’. Source: [11], 162.]

He convinced himself that the ancients had used projective properties of conic sections in order to achieve their results. Moving along these lines he classified cubics into five projective classes.13

13 From his work on cubics ([8], vol. 2, 137–161) Newton derived two lessons. First, Descartes’ classification of curves by degree is an algebraic criterion which has little to do with simplicity. Indeed, cubics have rather complex shapes compared to mechanical (transcendental) curves such as the Archimedean spiral. Second, it is by making recourse to projective classification that one achieves order and generality.


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5. Newton’s new analysis Now that we know more about Newton’s views concerning the role of algebraic symbolism in the method of problem solving, we are in the position to step back to Newton’s memorandum on his early mathematical discoveries that I quoted in Section 1. There he mentions the direct and the inverse methods of fluxions. The direct method allowed the determination of tangents (and curvature) to plane curves. Newton approached this problem by conceiving curves as generated by the continuous ‘flow’ of a point. He called the geometric magnitudes generated by motion ‘fluents’, while ‘fluxions’ are the instantaneous rates of flow. In the 1690s he denoted fluxions with overdots, so that the fluxion of x is x. ˙ He deployed a variety of strategies in order to determine tangents. Some of them are algorithmic, but in many cases Newton made recourse to kinematic methods. In Newton’s mathematical writings the algorithm is indeed deeply intertwined with geometrical speculations. By resolving motion into rectilinear components Newton could determine the tangent by composition of motions, even in the case of mechanical lines (see Figure 6). Indeed, the possibility to deal with transcendental curves (as the spiral and the cycloid) was top in Newton’s agenda. Or one could focus attention on the ‘moment of the arc’ generated in a very short interval of time (Newton termed the infinitesimal increment acquired in an infinitesimal interval of time a ‘moment’) and establish a proportion between the moment of the abscissa and the moment of the ordinate and other finite lines embedded in the figure. When the curve was expressed symbolically via an equation Newton had ‘rules’which allowed him to calculate the tangent (see Figure 7). One recognizes here rules which are ‘equivalent’ to those of the differential calculus; but the reader should be reminded that this equivalence was, and still is, object of debate. The inverse method of fluxions was Newton’s masterpiece. It is this method that allowed him to approach the problem of ‘squaring curves’. By conceiving a surface t as generated by the flow of the ordinate y which slides at a right angle over the abscissa z, he understood that the rate of flow of the surface’s area is equal to the ordinate (he stated t˙/˙z = y/1). This is how the idea of integration as anti-differentiation was born in Newton’s mind. His approach consisted in applying the direct method to ‘equations at will [which] define the relationship of t to z’. One thus obtains an equation for t˙ and z˙ , and so ‘two equations will be had, the latter of which will define the curve, the former its area’.14 Following this strategy Newton constructed a ‘Catalogue of curves’ which can be squared by means of ‘finite equations’ (see Figure 8). In Leibnizian terms, he built the first integral tables in the history of mathematics. Newton attached much importance to the inverse method. With almost visionary mathematical understanding of what is truly revolutionary, while still in his early years, he wrote: If two Bodys A & B, by their velocitys p & q describe ye lines x & y. 14 [9], vol. 3, 197.

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& an Equation bee given expressing ye relation twixt one of ye lines x, & ye ratio q/p of their motions q & p; To find ye other line y. Could this ever bee done all problems whatever might bee resolved.15

Figure 6. Newton’s early work (November 1666) on tangents to ‘mechanicall lines’ (i.e. transcendental plane curves). His technique consisted in conceiving curves as generated by motion and resolving motion into components. Source: [9], vol. 1, 378. 15 [9], vol. 1, 403.


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Figure 7. Newton’s algorithm for the direct method of fluxions. In this example he calculates the relation between fluxions (instantaneous speeds) x˙ and y˙ of fluent quantities (magnitudes changing continuously in time) x and y related by the equation x 3 − ax 2 + axy − y 3 = 0. Source: [8], vol. 1, 50.

In this context Newton developed techniques equivalent to integration by parts and substitution. Newton labelled the techniques of series expansion, tangent determination and squaring of curves as the ‘method of series and fluxions’. This was, he proudly stated, a ‘new analysis’ which extended itself to objects that Descartes had banished from his ‘common analysis’ – such as mechanical curves – thanks to the use of infinite series: And whatever common analysis performs by equations made up of a finite number of terms (whenever it may be possible), this method may always perform by infinite equations: in consequence, I have never hesitated to bestow on it also the name of analysis.16 According to Newton, the ‘limits of analysis are enlarged by […] infinite equations: […] by their help analysis reaches to all problems’.17

16 [9], vol. 2, 241. 17 [10]

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Figure 8. The beginning of Newton’s table of curves (an integral table, in Leibnizian terms), obtained thanks to understanding of what we call the ‘fundamental theorem of calculus’. Here Newton lists the first four ‘orders’. z is the abscissa, y the ordinate, t the area. In Newton’s notation t˙/˙z = y/1. Notice that d, e, f, g, h are constants (d is a constant!), η is integer or √ fractional, and R stands for e + f zη or e + f zη + gz2η . Source: [8], vol. 1, 105.

6. Newton’s synthetical method One should recall that the ‘new analysis’ occupied in Newton’s agenda a place which, according to the Pappusian canon, was subsidiary to the synthesis or construction, and that the construction had to be carried on in terms independent of algebraic criteria. For instance, as to the squaring of curves (in Leibnizian terms, integration) he wrote: After the area of some curve has thus been found, careful considerations should be given to fabricating a demonstration of the construction which


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as far as permissible has no algebraic calculation, so that the theorem embellished with it may turn out worthy of public utterance.18 Newton therefore devoted great efforts to providing geometrical demonstrations, somewhat reminiscent of Archimedean exhaustion techniques, of his ‘analytical’ quadratures. Only such demonstrations were deemed by him ‘worthy of public utterance’. It is in this context that Newton in the 1670s began reworking his early discoveries in ‘new analysis’ in terms that he conceived concordant with the constructive geometrical methods of the ancients. He termed this more rigorous approach the ‘synthetical method of fluxions’ and codified it around 1680 in a treatise entitled Geometria curvilinea ([9], vol. 4, 420–521). In this method no infinitesimals, or ‘moments’, occurred and no algebraic symbols were deployed. Everything was based upon geometric limit procedures that Newton termed the ‘method of first ratios of nascent quantities and last ratios of vanishing quantities’. It is this method that was widely deployed in the Principia (1687) (see Figure 9). It is somewhat astonishing to see one of the most

Figure 9. In Section 1, Book 1 of the Principia Newton lays down his ‘method of first and last ratios’, a geometric limit procedure that allows him to avoid infinitesimals. In Lemma 2 Newton shows that a curvilinear area AabcdE can be approached as the limit of inscribed AKbLcMdD or circumscribed AalbmcndoE rectilinear areas. Each rectilinear surface is composed of a finite number of rectangles with equal bases AB, BC, CD, etc. The proof is magisterial in its simplicity. Its structure is still retained in present day calculus textbooks in the definition of the Riemann integral. It consists in showing that the difference between the areas of the circumscribed and the inscribed figures tends to zero, as the number of rectangles is ‘increased in infinitum’. In fact this difference is equal to the area of rectangle ABla which, ‘because its width AB is diminished in infinitum, becomes less than any given rectangle’. In Newton’s terms AB is a ‘vanishing quantity’. Source: [11], 74. 18 [9], vol. 3, 279.

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creative algebraists of the history of mathematics spend so much time and effort in reformulating his analytical results in geometric terms, but Newton had compelling reasons to do so. First, Newton in his programme of reformation of natural philosophy attributed an important role to mathematics as a source of certainty. From the early 1670s he expressed his distaste for the probabilism and hypotheticism that was characteristic of the natural philosophy19 practiced at the Royal Society by people like Robert Hooke and Robert Boyle. His recipe was to inject mathematics into natural philosophy. As he stated: by the help of philosophical geometers and geometrical philosophers, instead of the conjectures and probabilities that are being blazoned about everywhere, we shall finally achieve a science of nature supported by the highest evidence. ([12]) But if mathematics has to provide certainty to natural philosophy her methods must be above dispute, and Newton was keenly aware of the fact that the new analysis was far from being rigorous. Second, Newton soon developed a deep anti-Cartesianism associated with a conviction of the superiority of the ancients over the moderns. From his point of view Descartes was the champion of an impious mechanistic philosophy which, conceiving nature as an autonomous mechanism, denied any role to God’s providence. Newton conceived himself as a restorer of an ancient, forgotten philosophy according to which nature is always open to the providential intervention of God. Indeed, he thought that, according to the theory of gravitation – which he was convinced the ancient Hebrews possessed–, the quantity of motion in the universe was bound to decline if divine intervention had not prevented the ‘corruption of the heavens’. The modern philosophers were dangerous from a theological point of view and had to be opposed on all grounds. Therefore, also in mathematics Newton looked with admiration to ancient exemplars and conceived himself as a restorer of their glory. It goes without saying that the above reasons led Newton into a condition of strain, since his philosophical values were at odds with his mathematical practice, which was innovative, symbolical, and – pace Newton – deeply Cartesian. Several hitherto unexplained aspects of Newton’s mathematical work are related to this condition of stress and strain that characterizes his thoughts on mathematical method. Why did Newton fail to print his method of series and fluxions before the inception of the priority dispute with Leibniz? Why did he hide his competence in quadratures when writing the Principia, which are written mostly in geometrical style? Even though there is no single answer to these vexed questions, I believe that Newton’s conviction that the analytical symbolical method is only a heuristic tool, 19 For Newton the aim of ‘natural philosophy’is to deduce the forces from phenomena established by experiment, and – once established the forces – to deduce new phenomena from them. Nowadays we would call this enterprise ‘physics’.


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not ‘worthy of public utterance’, can in part explain a policy of publication which was to have momentous consequences in the polemic with Leibniz.

7. Leibniz’s views When the war with Leibniz exploded in 1710 Newton had to confront an opponent who not only advanced mathematical results equivalent to his, but was promoting a different view concerning mathematics.20 The rhetoric on the novelty of the calculus pervades Leibniz’s writings. Reference to the ancient mathematicians generally took the rather abused form of a tribute to Archimedes’ ‘method of exhaustion’. Leibniz in most of his declarations concerning the calculus wished to highlight the novelty and the revolutionary character of his algorithm, rather than continuity with ancient exemplars. This approach is quite at odds with Newton’s ‘classicism’. Furthermore, Leibniz often referred to the heuristic character of the calculus understood as an algorithm independent from geometrical interpretation. It is exactly this independence that would render the calculus so efficacious in the process of discovery. The calculus, according to Leibniz, should also be seen as an ars inveniendi (an art of discovery): as such it should be valued by its fruitfulness, rather than by its referential content. We can calculate, according to Leibniz, √ with symbols devoid of referential content (for instance, with −1), provided the calculus is structured in such a way as to lead to correct results.21 Writing to Christiaan Huygens in September 1691, Leibniz affirmed with pride: It is true, Sir, as you correctly believe, that what is better and more useful in my new calculus is that it yields truths by means of a kind of analysis, and without any effort of the imagination, which often works as by chance. ([13]) 20 The circumstances surrounding the controversy between Newton and Leibniz have been analysed in detail by Rupert Hall [15] and Tom Whiteside [9], vol.8. In broad outlines let me recall a few bare facts. Newton formulated his method of series and fluxions between 1665 and 1669. Leibniz had worked out the differential and integral calculus around 1675 and printed it in a series of papers from 1684. It is clear from manuscript evidence that he arrived at his results independently from Newton. It is only in part in Wallis’ Algebra in 1685 and Works in 1693 and 1699, and in full in an appendix to the Opticks in 1704, however, that Newton printed his method. In 1710 a British mathematician, John Keill, stated in the Philosophical Transactions of the Royal Society that Leibniz had plagiarized Newton. After Leibniz’s protest a committee of the Royal Society secretly guided by its President, Isaac Newton, produced a publication – the so-called Commercium epistolicum (1713) – in which it was maintained that Newton was the ‘first inventor’ and that ‘[Leibniz’s] Differential Method is one and the same with [Newton’s] Method of Fluxions’. It was also suggested that Leibniz, after his visits to London in 1673 and 1676, and after receiving letters from Newton’s friends, and from Newton himself (in fact Newton addressed two letters to Leibniz in 1676) had gained sufficient information about Newton’s method to allow him to publish the calculus as his own discovery, after changing the symbols. It is only after the work of historians such as Fleckenstein, Hofmann, Hall and Whiteside that we have the proof that this accusation was unjust. Newton and Leibniz arrived at equivalent results independently and following different paths of discovery. 21 Complex numbers received a geometric interpretation only around 1800 thanks to Jean Robert Argand, Carl Friedrich Gauss, and Caspar Wessel.

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Leibniz was thus praising the calculus as a cogitatio caeca and promoted the ‘blind use of reasoning’ among his disciples. Nobody, according to Leibniz, could follow a long reasoning without freeing the mind from the ‘effort of imagination’.22 Leibniz conceived of himself as the promoter of new methods of reasoning, rather than ‘just’a mathematician. The calculus was just one successful example of the power of algorithmic thinking. The German diplomat was interested in promoting in Europe the formation of a group of intellectuals who could extend a universal knowledge achieved thanks to a new algorithm that he termed universal characteristic. He thus helped to form a school of mathematicians who distinguished themselves by their ability in handling the differentials and the integrals and by their innovative publication strategy. Thanks to Leibniz’s recommendation, they colonized chairs of mathematics all over Europe. The efficacy of this new algorithm was affirmed to be independent from metaphysical or cosmological questions. The persons who practised it had to be professional mathematicians, rather than ‘geometrical philosophers’, able to teach and propagate knowledge of calculus. A typical Leibnizian attitude emerges in the context of the vexed question of the existence of infinitesimals. The new calculus was often attacked, since – it was maintained – it employed symbols devoid of meaning, such as differentials ordered into a bewildering hierarchy of orders. Newton, as we know, was particularly sensitive to such criticisms, and tried in his synthetical method to dispense with infinitely small quantities. Leibniz, on the other hand, repeated many times that for him the question of the existence of infinitesimals had to be distinguished from that of their usefulness as algorithmic devices. While he was leaning, for philosophical reasons, towards a denial of the existence of infinitesimals, he also wanted to stress that this ontological question was somewhat extraneous to mathematics. A typical statement, written in the early years of the eighteenth century, is the following: We have to make an effort in order to keep pure mathematics chaste from metaphysical controversies. This we will achieve if, without worrying whether the infinites and infinitely smalls in quantities, numbers and lines are real, we use infinites and infinitely smalls as an appropriate expression for abbreviating reasonings. ([14]) Leibniz was thus leaving to his disciples the choice of maintaining, philosophically speaking, different approaches to the ontological question on the existence of infinitesimals. What he wished to defend was their utility as symbols in mathematical calculation.

8. The war against Leibniz: methodological aspects When Newton had to confront Leibniz in the squabble over priority he was concerned in building up a forensic and historical document whose purpose was to prove 22 [14], 205.


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Figure 10. A portrait of Newton in old age (Source: [1], 831). He proudly opens the Principia at a page devoted to the attraction of extended bodies. In dealing with this problem Newton made recourse to his ‘inverse method of fluxions’ (the equivalent of Leibniz’s integral calculus) which allowed him to ‘square curves’. As a matter of fact, only by making recourse to his tables of curves (‘integral tables’), see Figure 8, could Newton solve several problems in the Principia. Such analytic methods were not, however, made explicit to the reader. In the polemic with the Leibnizians – who claimed that absence of calculus from the Principia was proof positive of Newton’s ignorance of quadrature techniques prior to 1687 – Newton was forced to maintain, with some exaggeration, that ‘By the help of this new Analysis Mr Newton found out most of the Propositions in his Principia Philosophiae. But because the Ancients for making things certain admitted nothing into Geometry before it was demonstrated synthetically, he demonstrated the Propositions synthetically that the systeme of the heavens might be founded upon good Geometry. And this makes it now difficult for unskillful men to see the Analysis by wch those Propositions were found out.’ ([9], vol. 8, 599). On the issue of Newton’s use of analytic methods in the Principia see [16].

Leibniz’s plagiarism. But he did not do only this, he also wished to highlight the superiority of his method over Leibniz’s calculus. The mathematical programme that Leibniz was promoting with so much success was at odds with Newton’s deeply felt values. There is not only mathematics in this story, of course. Leibniz had to be opposed for a series of reasons that have to do with the Hannoverian succession. The German,

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in fact, who was employed by the Hannover family, wished to move to London as Royal Historian. The idea of having in England such a towering intellectual who was defending a philosophical view which contradicted Newton’s voluntarist theology and who was promoting the unification of the Christian Churches was anathema for Newton and his supporters. For our purposes, it is interesting to turn to some passages that Newton penned in 1715 contained in an anonymous ‘Account’ to a collection of letters, the Commercium epistolicum, that the Royal Society produced in order to demonstrate Leibniz’s plagiarism. In the ‘Account’, speaking of himself in the third person, Newton made it clear that Leibniz had only approached the analytical, heuristic part of the problem-solving method. He wrote: Mr. Newton’s Method is also of greater Use and Certainty, being adapted either to the ready finding out of a Proposition by such Approximations as will create no Error in the Conclusion, or to the demonstrating it exactly; Mr. Leibniz’s is only for finding it out.23 So according to Newton, Leibniz had achieved only the first stage of the Pappusian method and had not attained the rigorous, constructive demonstration. This, as we know, had to be carried on in purely geometric terms. Further, Newton insisted on the fact that the emphasis with which Leibniz praised the power of his symbolism was excessive. Algorithm is certainly important for Newton, but it has to be viewed only as a component of the method: Mr Newton — he wrote – doth not place his Method in Forms of Symbols, nor confine himself to any particular Sort of Symbols.24 Finally, Newton noticed that in his method of first and last ratios no infinitesimals occur, everything being performed according to limiting procedures. From Newton’s point of view the avoidance of infinitesimals and the possibility of interpreting algebraic symbols as geometric magnitudes had the double advantage of rendering his method endowed with referential content and consonant with ancient mathematics: We have no ideas of infinitely little quantities & therefore Mr Newton introduced fluxions into his method that it might proceed by finite quantities as much as possible. It is more natural & geometrical because founded on primae quantitatum nascentium rationes [first ratios of nascent quantities] wch have a being in Geometry, whilst indivisibles upon which the Differential method is founded have no being either in Geometry or in nature. […] Nature generates quantities by continual flux or increase, & the ancient Geometers admitted such a generation of areas & solids […]. But the summing up of indivisibles to compose an area or solid was never yet admitted into Geometry.25 23 Cited in [15], 296. 24 Cited in [15], 294.


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Nature and geometry are the two key concepts: they allow Newton to defend his method because of its continuity with ancient tradition as well as its ontological content. In his polemic writings against Leibniz Newton engineered an attack which was aimed at proving the German’s plagiarism. One of Newton’s priorities was to assemble evidence which proved Leibniz guilty, and he did so with means that show his ability to employ archival sources as well as his prejudice and egotism. However, Newton also defended positions concerning mathematical method that have deep roots in his protracted opposition against Descartes and the ‘modern mathematicians’ who, by confounding geometry and algebra, ‘have lost the Simplicity in which all the Elegance of Geometry consists’.

References [1] Add MS 3968.41 f. 85. Cited in Westfall, R. S., Never at Rest: a Biography of Isaac Newton. Cambridge University Press, Cambridge 1980, 143. This is the best biography of Newton. [2] On Newton’s philosophical programme one can read: Cohen, I. B., The Newtonian Revolution. Cambridge University Press, Cambridge 1980. Smith, G. E., The methodology of the Principia. In The Cambridge Companion to Newton (ed. by I. Bernard Cohen and George E. Smith). Cambridge University Press, Cambridge 2002, 137–173. Stein, H., Newton’s metaphysics. In The Cambridge Companion to Newton (ed. by I. Bernard Cohen and George E. Smith). Cambridge University Press, Cambridge 2002, 256–307. [3] Panza M., Newton et les Origines de l’ Analyse: 1664–1666. Blanchard, Paris 2005. [4] Pappus of Alexandria, Book 7 of the Collection (ed. by Alexander Jones). Sources in the History of Mathematics and the Physical Sciences 8, Springer, New York, Berlin, Heidelberg, Tokyo 1986, 82–4. [5] Descartes, R., The Geometry of René Descartes with a Facsimile of the First Edition (ed. by D. E. Smith and M. L. Latham). Dover, New York 1954. [6] Bos, H. J. M., Redefining Geometrical Exactness: Descartes’ Transformation of the Early Modern Concept of Construction. Springer-Verlag, New York, Berlin, Heidelberg, Tokyo 2001. [7] Galuzzi, M., I marginalia di Newton alla seconda edizione latina della Geometria di Descartes e i problemi ad essi collegati. In Descartes: il Metodo e i Saggi (ed. by G. Belgioioso, G. Cimino, P. Costabel, G. Papuli), Istituto dell’Enciclopedia Italiana, Roma 1990, 387–417. [8] Newton, I., The Mathematical Works of Isaac Newton (ed. by Derek T. Whiteside). 2 vols., Johnson Reprint Corp., New York, London 1964–1967. [9] Newton, I., The Mathematical Papers of Isaac Newton (ed. by Derek T. Whiteside). 8 vols., Cambridge University Press, Cambridge, London, New York 1967–1981. In quoting from this work we follow Whiteside’s translation from the Latin. [10] Newton, I., The Correspondence of Isaac Newton (ed. by H. W. Turnbull et al.). 7 vols., Cambridge University Press, Cambridge, 1959–1977, Vol. 2, 39. 25 Cited in [15], 295–6.

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[11] Newton, I., Philosophiae Naturalis Principia Mathematica. The Third Edition (1726) with Variant Readings (ed. byAlexandre Koyré and I. Bernard Cohen, with the assistance ofAnne Whitman). Cambridge University Press, Cambridge 1972. We follow the translation from the Latin provided in Isaac Newton, The Principia: Mathematical Principles of Natural Philosophy. A new translation by I. Bernard Cohen and Anne Whitman assisted by Julia Budenz, preceded by A guide to Newton’s Principia by I. Bernard Cohen, University of California Press, Berkeley, Los Angeles, London 1999. [12] Translated and cited in Shapiro, A., Fits, Passions, and Paroxysms: Physics, Method, and Chemistry and Newton’s Theories of Colored Bodies and Fits of Easy Reflection. Cambridge University Press, Cambridge 1993, 25. [13] Leibniz, G. W., Leibnizens mathematische Schriften (ed. by C. I. Gerhardt). 7 vols., Olms, Hildesheim 1971, Vol. 1 (2), 104. My translation. [14] Niedersächsische Landesbibliothek (Hannover) Lh 35 VIII 21, f. 1r. Quoted in E. Pasini Il Reale e l’Immaginario: la Fondazione del Calcolo Infinitesimale nel Pensiero di Leibniz. Sonda, Torino 1993, 149n. My translation. [15] Hall, A. R. Philosophers at War: the Quarrel between Newton and Leibniz. Cambridge University Press, Cambridge 1980. [16] Guicciardini, N. Reading the Principia: the Debate on Newton’s Mathematical Mathods for Natural Philosophy from 1687 to 1736. Cambridge University Press, Cambridge 1999. Dipartimento di Filosofia e Scienze Sociali, Università di Siena, via Roma, 47, 53100 Siena, Italy E-mail: [email protected]

e-learning mathematics∗ Sebastià Xambó Descamps† (moderator) Hyman Bass, Gilda Bolaños Evia, Ruedi Seiler‡, and Mika Seppälä§ (panelists)

Abstract. In addition to the current state of knowledge about the learning of mathematics and its aims in today’s society, the main purpose of this paper is discussing ways of improving the process of learning, and especially, in that regard, the role of e-learning technologies. We chart the situation of e-learning mathematics as of December, 2005, including distance-learning or open university courses, and then we consider a number of areas where e-learning is likely to develop. Finally, we assess the impact of e-learning on the role of the new educators in mathematics. Mathematics Subject Classification (2000). Primary: 97-xx, 97Uxx; Secondary: 00-xx. Keywords. Online material, distance learning, e-learning, metadata.

Presentation by Sebastià Xambó Descamps Following a suggestion of the Executive Committee (EC) of ICM2006 that came forth in the Fall of 2004, this panel has been promoted by the Conference of Spanish Mathematics’ Deans [1]. After having formally accepted the invitation on December 16, 2004, the CDM Executive Committee discussed possible topics, until “e-Learning Mathematics” (eLM) was chosen and approved by both the CDM and the EC of ICM2006. Names to be invited as panelists were also decided, and it is a great satisfaction, and an honour, to be able to say that all accepted. On behalf of the CDM, my sincerest thanks to all. If e-learning is learning by means of systems built on current computer and communications technologies, then the main interest of eLM is on what advantages e-learning can offer in the case of mathematics. The main reason for choosing eLM is that the accelerated evolution of the e-Learning field is having, and will most likely continue to have, a major worldwide impact ∗ Panel promoted by the Spanish Conference of Mathematics’ Deans. † Partially supported by the European e-Content project “Web Advanced Learning Technologies” (WebALT), Contract Number EDC-22253. ‡ Thanks for the support by the Bundesminister für Bildung und Forschung. § Partially supported by the European e-Content project “Web Advanced Learning Technologies” (WebALT), Contract Number EDC-22253.


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e-learning mathematics

on many aspects of the teaching-learning systems, at all levels, while offering, at the same time, new opportunities to professional mathematicians and to existing or new institutions, as for example in life-long learning. It is thus a topic that should greatly interest not only mathematicians in all walks of life, but also academic and political authorities everywhere. This is why we imagined that the panel could aim at describing the situation of eLM as of 2006, outlining the most likely trends of its evolution in the next few years, indicating what the strongest impacts (positive or negative) in the mathematics teaching-learning systems will be, and charting the sorts of opportunities that will arise. We are of course aware that such aims can only be attained by the panel in very broad terms, although this should be enough to bring forward a generally useful picture. For those wanting to have more detailed views, the references provided by the panelists should be a valuable resource to continue a journey that by all evidence has no return. For example, the articles in the recent book [2] will quite likely be serviceable to a wide range of readers seeking to know more about e-learning in general. Let me continue with a few general remarks on learning, teaching and e-learning. Mathematics, or mathematics knowledge, is a vast universe (let me call it M). It has many smaller interelated universes, of which we have a dim glimpse in the standard classifications. Because of the increasing number of research mathematicians, and the availability of ever more sophisticated computational and communication tools, M has undergone an extraordinary growth, and all indications are that this trend will continue in the coming years. To a large extend this blooming is explained because M is both a source of deep beauty and the only precision method we have for modelling the physical universe. In any case, the number of university students required to take mathematics courses is globally increasing, but at the same time the number of professional mathematicians that seek a teaching position is most likely decreasing, as there are, on one side, ever newer job profiles, and, on the other side, the number of students in mathematics degrees is decreasing in most countries. Moreover, in the last decade a steady decline in the mathematical skills of the students beginning higher education has been reported (see, for example, [3]). Can eLM help to face this situation in a more positive mood? The expectations created by e-learning are certainly high, at all levels, and we may wonder how much of it is going to be true, and up to what point can it help in the case of mathematics. The reasons behind the high expectations on e-learning stem from well-known characteristics of the e-learning systems: • In principle, access is possible from anywhere and at any time, thus making possible flexible (even just-for-me) and just-in-time courses of learning.

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• The teacher can also be anywhere and do most of his teaching job at any time (preparing materials or following-up and coaching his students). • It allows for synchronous activities of a teacher and a group (at an agreed time), but again without restriction on the location of the people involved, and, what is more, with the possibility of addressing a much larger audience than a conventional class. • Assessment can be automated to a large extend and final grading can be integrated seamlessly into the institution’s information system. • The learning materials and experiences can be richer in many ways, and they can be easily maintained and updated (as compared to preparing, say, a new edition of a paper book). • There are also indications that it may induce deeper understanding and stronger retention. So the main question is how can we harness all that potential for improving the quantity and quality of the learning of mathematics. Since there are many levels that we ought to consider, and many variations in each level, we cannot expect a universal recipe. And even if we restrict ourselves to a very particular situation, say remedial mathematics for freshman in engineering schools or mathematical modules for prospective secondary school teachers, we cannot expect a formula that would satisfy everybody. A sensible starting point is just looking at people, groups and institutions that are leading the way in one direction or another. This is the idea behind the purpose and composition of this panel. Since it is not feasible, and perhaps not even desirable, under the circumstances, to have a comprehensive survey of eLM, the best alternative is having experts in a few areas that have a major bearing on what eLM is and can be, and on how it is evolving. Before going into their reports, let me briefly introduce them. Hyman Bass Hyman Bass is Roger Lyndon Collegiate Professor of Mathematics and Professor of Mathematics Education at the University of Michigan. A graduate of Princeton, Dr. Bass earned his Ph.D. from the University of Chicago under Irving Kaplansky. He has had visiting appointments at sixteen different universities in ten countries. The many honors and prizes that Dr. Bass has received include the Cole Prize in algebra. He is an elected member of the American Academy of Arts and Sciences and the National Academy of Arts and Sciences, and the Third World Academy of Sciences, and was elected Fellow of the American Association for the Advancement of Science. He is former president of the American Mathematical Society and current president of ICMI. He has been both a Sloan and Guggenheim Fellow. Dr. Bass has published eighty-six papers in mathematics and seventeen in mathematics education.

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Gilda Bolaños Dr. Bolaños is a certified teacher and trainer in the didactical techniques of Problem Based Learning (PBL) and Project Oriented Learning (POL). She is the author of several certified Blackboard courses. With classroom technologies based on Maple and Minitab, she has worked extensively on problems and materials for her online courses. Ruedi Seiler Full Professor for Mathematics at the Technische Universität Berlin, Ruedi Seiler’s main fields of interest are Mathematical Physics, Quantum-Hall Systems, Information Theory, Data Compression, and E-Math: Teaching, Learning, Research. Member of the Research Center “Mathematics for Key Technologies” (DFG), and of the Executive Committee of the International Association of Mathematical Physics (IAMP), his most recent undertakings, culminating an extensive experience in organizing events and participating in projects, are MUMIE and MOSES. More specifically, he is leading, since 2001, the project “Multimedial Mathematical Education for Engineers”, a project developed in Cooperation between the Berlin University of Technology, the Munich University of Technology, the Aachen University of Technology and the University Potsdam (funded by the German Federal Ministry of Education and Research within the programme “New Media in Education”), and, within the program “Notebook-University” of the German Federal Ministry of Education and Research, he is co-manager, since 2002, of the TU Berlin project “MOSES – Mobile Service for Students”. Mika Seppälä Dr. Seppälä is Professor of Mathematics at Florida State University and Professor of Computer Aided Mathematics at the University of Helsinki. He was the Co-ordinator of the HCM network “Editing and Computing” (1995–1996) which initiated the development that lead to the MathML and OpenMath languages allowing the inclusion of mathematical formulae on the web pages in a meaningful way. He is currently the Secretary of the OpenMath Society, and the co-ordinator of the eContent Project “Web Advanced Learning Technologies” (WebALT). The main goal of the WebALT Project is to use MathML and OpenMath to create tools and content for multilingual on-line mathematics. Seppälä was the President of the Finnish Mathematical Society for the period 1992–1996. Sebastià Xambó Descamps Full Professor of Information and Coding Theory at the Universitat Politècnica de Catalunya (UPC, Barcelona, Spain), and former Full Professor of Algebra at the Departamento de Algebra of the Universidad Complutense of Madrid (1989–1993), is serving as Dean of the “Facultat de Matemàtiques i Estadística” of the UPC. Member

The instructional potential of digital technologies

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of the EU eContent Project “Web Advanced Learning Technologies”. In the period 1994-2000 led the team that developed the mathematical engine of Wiris ([4], [5]) and authored the e-book [6]. Cofounder of Maths for More ([7]). Has served as President of the Societat Catalana de Matemàtiques (1995–2002) and of the Executive Committee of the 3rd European Congress of Mathematics (Barcelona, 2000), and as Vicerector of Information and Documentation Systems of the UPC (1998–2002). Since the Fall of 2004 he serves as President of the Spanish Conference of Mathematics’ Deans.

References [1] http://www.usc.es/mate/cdm/. [2] Integrated E-learning. Implications for Pedagogy, Technology and Organization. Edited by Wim Jochems, Jeroen van Merriënboer and Rob Koper. RoutledgeFalmer, 2004. [3] http://www.engsc.ac.uk/index.asp. [4] Eixarch, R., Marquès, D., Xambó, S., WIRIS: An Internet platform for the teaching and learning of mathematics in large educational communities. Contributions to Sience 2 (2) (2002), 269–276. [5] Eixarch, R., Marquès, D., Xambó, S., Report on the positive effects for an educational community of having universal Internet access to a mathematical computational system. In preparation. [6] Xambó, S., Block Error-Correcting Codes: A Computational Primer. Springer-Verlag, 2003. Digital version at http://www.wiris.com/cc/. [7] http://www.mathsformore.com/.

The instructional potential of digital technologies by Hyman Bass Educational uses of technology. Digital technology continues to rapidly transform all aspects of life and work, even (and perhaps all the more so) in the developing world. It is designed, and presumed, to bring great benefit and empowerment to its users, as well as profit to its developers. Yet, as it opens new and even unanticipated possibilities, it poses as many problems as it solves, some new, and some technoversions of classical problems, all of them important and interesting. And technology, for its novelty and glamorous aspirations, is greedy for our attention, liking to take center stage in every arena it enters. Education, and mathematics education in particular, is the context in which this panel is examining these transformations. I find it helpful here to distinguish three broad kinds of roles that technology can play in mathematics education. They are of course not disjoint.

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I. Transmission: Use of technology (web, video conferencing, etc.) to transmit, perhaps interactively, instruction and/or instructional materials that are conceptually of a traditional genre – lectures, demonstrations, problem sets, assessments, etc. These are the kinds of uses that fundamentally support distance learning, for example. II. Power, speed, and visualization: Use of technology to carry out quickly and more accurately and completely, mathematical processes of a traditional nature – perform large or complex calculations, solve equations, approximate integrals, exhibit function graphs, study effects of variation of parameters, produce vivid and accurate images of geometric figures, etc. III. New ways to explore the (mathematical and experiential ) universes: Use of technology to do things we have never previously been able to do. Such capability affects mathematics itself, not just mathematics education. Examples include the study of long-term evolution of dynamical systems, and the images of fractal geometry that emerge there from. (This had an effect on dynamics comparable with that of the telescope in astronomy and the microscope in biology.) Software development gave life to the field of computational complexity, with its applications to coding and cryptography. Mathematical modeling and computer simulation supports a virtually empirical study of physical systems and designs. Dynamic geometry offers unprecedented opportunities to visually explore and analyze geometric structures, and to produce evocative imagery of dimensions three and four (using time). Computer algebra systems furnish unprecedented resources for solving equations. Much of this new technological power is now within reach of many students, and this raises possibilities of thereby expanding the horizons of the mathematics curriculum. At a pragmatic level, technology thus offers resources to address two fundamental challenges of contemporary education – distance and demographics. Distance because many learners in need are physically remote from the sources of quality instruction and materials. Gilda Bolaños offers us an excellent survey of diverse modes of distance learning formats. Demographics because class sizes, particularly in introductory level mathematics courses, are too large to afford adequate instructor attention to individual student learning. (Bounding class sizes is often done at the cost of using instructors of highly variable quality.) In this case, technology affords various interactive formats for student work and assessment. These include the “virtual laboratories” described by Ruedi Seiler, and the interactive online materials (lectures, automatically graded homework, etc.) discussed by Mika Seppälä. But independently of these practical needs, technology also offers possibilities for improving mathematics instruction itself. And the fundamental questions about the quality of teaching and learning do not recede when the instruction is mediated by technology; they only change their form.

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Instruction. By “instruction” I mean the dynamic interaction among teacher, content, and students. I rely here on the “instructional triangle” that Cohen and Ball use to depict the set of interactions that they call “instruction” (Cohen and Ball, 1999). teacher

students

content

students

Viewed in this way, instruction can go wrong in some simple but profound ways, for its quality depends on the relations among all of these three elements. When they misconnect, students’ opportunities for learning are impaired. For example, if a teacher is not able to make the content accessible to students, framing it in ways that are incomprehensible to them, the chances that they may misunderstand are great. If students’ interpretations of a task are different from the teacher’s or the textbook author’s intentions, then their work may be misrouted or take the work in unhelpful directions. It may seem slightly strange, in the context of this panel, to propose the above representation of instruction. For, if you think about it, most descriptions of instructional uses of technology appear to reside exclusively on the bottom edge of the instructional triangle, absent the teacher. A tacit premise of some of this thinking is that somehow, the technology, with its interactive features, actually substitutes for the teacher, or renders the teacher obsolete, except perhaps as a manager of the environment. The viability of this view is a deep and important question, one that I shall not enter here except to make a couple of observations. One is that, in the most successful models of distance learning, it was found to be essential to have a tutor or facilitator available at the remote sites of reception of the materials, to respond to the many questions and requests that students would have, and that were not adequately responded by the technology environment. In addition, it was found to be important to have real time online questioning of the primary source available at certain times. In other words, prepared and transmitted material alone no more teaches a learner than does a textbook, unmediated by a teacher. The other comment is that interactive technology formats can at best provide well-prepared instructional materials and tasks, and respond to the student productions and questions that the software developers have anticipated and for which they have programmed responses. There are many domains of procedural learning and performance where this can be somewhat successful, though

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the software, no more than a skilled teacher, cannot completely predict and prepare for all of what students may come up with. Moreover this uncertainty is all the greater once one enters into territory that is less procedural and involves more conceptual reasoning and problem solving. In what follows, I identify five persistent problems of mathematics instruction and discuss ways in which technology can be deployed to address these. How these are actually used, however, would affect the degree to which they were helpful, so for each case, I point out its possible pitfalls. 1. Making mathematically accurate and pedagogically skillful diagrams. One problem faced by mathematics teachers at all levels is how to make clear and accurate diagrams that make the essential mathematical ideas plain to learners, and how to do so in ways that are manipulable for mathematical reasoning. Doing this by hand is often no easy task, whether the sketch is of slices of an ellipsoid in calculus, or sixteenths of a rectangle in fifth grade. Mathematical accuracy is one dimension of the challenge; featuring is a second - that is, making the instructionally key features visible to learners. In addition, instructors must manage these challenges fluently, using class time effectively. An instructor who can make diagrams accurately and helpfully, but who must use 10 minutes of class time to do so, loses effectiveness. Diagrams are also used for a variety of purposes: explorationally, to investigate what happens if certain elements are allowed to vary, or presentationally, to demonstrate an idea, an explanation, or a solution. This means, sometimes, the need for dynamics - translations, rotations, rescaling, variation of parameters. Often diagrams must be made in ways that map clearly to algebraic or numerical representations. Drawing software, or other design tools, can help. Important is the capacity to produce carefully-scaled diagrams, with the capacity for color or shading, and to be able to move elements of a diagram. Its use must be fast and flexible, helpful both for carefully designed lectures and for improvisation on the fly, in response to a student’s question. Such software or tools can provide significant support for the use of diagrams in class, by both students and instructor. Making such software accessible to students increases their capacity for individual explorations and preparation for contributions in class. Students can quickly put their diagrams up for others’inspection, or support a point in class, in ways that are difficult to do when students go to the board to generate representations by hand. Using software tools to support the visual dimensions of mathematical work in instruction can significantly alter a major dimension of instruction and do so in ways that are mathematically accurate, pedagogically useful, and sensitive to the real-time challenges of classroom instruction where class periods are finite and time is a critical resource. Software tools to support the making of diagrams can create problems, too. For example, if the tools are rigid or interfere with the purposes for making diagrams, or cannot be manipulated as desired, the representations may not be as useful as needed. Another problem may be that the use of such tools inhibits students from developing personal skills of appraisal and construction. If the tools quickly make


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correct diagrams, students may not develop a critical eye with which to inspect them. If they never have to make a diagram themselves, they may remain entirely dependent on the software and not develop independent capacities for drawing. 2. Making records of class work and using them cumulatively across time. A second pervasive problem of mathematics instruction can be seen in the overflowing blackboards full of work and the slippery sheets of transparencies filled with notation and sketches, generated in class, and that vanish into weak memory when class ends. The record of class work (not just text or prepared materials), whether lecture, discussion, or exploration, is an important product of instruction. Under ordinary circumstances, this product vanishes and is thus unavailable for study or future reference, use, or modification. So acute is this problem that, too often, even during a single class, such work is erased (in the case of chalkboards) or slid away (as in transparencies). The work of that single class period is weakened for not being able to secure its place in evolution of ideas in the course. Moreover it is not available for students who may have missed a class. When the work done in class is created or preserved in digital form, an archive of the mathematical progress of the class can become a resource for ongoing learning. It can then be easily accessed and transmitted remotely to others. Doing it “live” in class requires skill and dexterity on the part of the instructor. Making records of classwork afterwards (i.e., photographing the board with a digital camera) is easier but possibly less manipulable for subsequent class work. Important, too, is that everyone who needs to access these records can work on a common platform or that the format will work reliably across platforms. 3. Alignment between classes and textbook. Instructors, perhaps in response to student ideas or productions, may choose to depart from the text - in topic treatment or sequencing, or even topic coverage, and in the design of student activities and tasks. If the instructor creates these variations and alternative paths in electronic form, then a new text is created based on the instructor’s design. This affords students access to the substance and course of the lessons. This gives license to flexible and innovative instruction, by affording the means to do so without disadvantaging students through disconnection from a text to be perused and revisited over time. 4. Ease of access to the instructor between classes. In the developed world, it is hard to imagine university instructors who do not maintain email (and web) connection with their students. This has made much more fluent and elastic the traditional functions of “office hours.” Most student questions can be handled expeditiously, in timely fashion (though asynchronously), by email (perhaps with attachments), thus greatly reducing the need for face-to-face meetings, with their scheduling difficulties. And, as with the discussion above, these exchanges can contribute significantly to the record of the student’s work and progress. When appropriate, an exchange between one student and the instructor can easily be made available to other students, thus changing an individual “office hour” into a group discussion. Pitfalls can exist with electronic

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communications, of course. Misunderstanding is frequent when communication is restricted to text, without gesture, intonation, and the ability to demonstrate or show. 5. The repetitive nature of individual outside-of-class sessions. One feature of traditional office hours, or help sessions, is that they tend to be repetitive, processing over and over again the same questions and assistance with each new student or group of students. When such assistance is administered electronically, and it is seen to be germane to the interests of the whole class, it is an easy matter to copy the whole class, or perhaps selected individuals, on such exchanges. This puts to collective profit the considerable instructional investment made in one student, or group of students, and everyone gains, not least the instructor. An important consideration here is sensitivity to privacy issues and confidentiality. In particular, making individual student communications requires prior consent. Conclusion. Technology continues to transform all aspects of our lives and work. It is already difficult to imagine how we once functioned without email and the web. We are still at the early stages of trying to understand and design the best uses of technology for mathematics instruction. I have pointed to some promising uses of technology to address some endemic problems of even traditional instruction. I have also tried to signal that the fundamental problem of developing quality teaching does not disappear just because instruction is mediated in technological environments. References Cohen, D. K., and Ball, D. L., Instruction, capacity, and improvement. (CPRE Research Report No. RR-043). University of Pennsylvania, Consortium for Policy Research in Education, Philadelphia, PA, 1999.

Distance learning today by Gilda Bolaños Evia The definition of distance learning has been modified over time, and today we have a variety of definitions. We will adopt the definition of Greenberg, in [Greenberg98], where contemporary distance learning is defined as “a planned teaching/learning experience that uses a wide spectrum of technologies to reach learners at a distance and is designed to encourage learner interaction and certification of learning”. In this section we will discuss the effects of some of the technologies used in distance learning education on mathematics and its effects on student’s knowledge. Video taped lectures. Since the introduction of videos to instruct students on different areas, many studies have been conducted to determine the effectiveness of these

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methods. Some examples are [Beare 89], [Moore 96], [Russell 97], and [Pflieger 61]. On all of these studies the conclusion is that there is no significant difference on the achievement of students on video classes and regular classes. A three-year study involving 200,000 students and 800 public schools states: “... whereas most comparisons showed no significant differences, 119 were significant in favor of TV-taught students, and 44 in favor of conventionally taught students.” [Pflieger 61]. We have to observe that on these studies the quality of the taught material was the same for video students and traditional students. Due to the lack of availability of similar studies for Latin America, we asked some professors and authorities that have been part of the VIBAS (video high school system) about the effectiveness of the system. In general they think that there is a significant difference in favor of traditional education, but this difference is not because of the video system, but mainly because of quality of materials and lack of availability of tutors. Moore and Kearsky converge to the same opinion in [Moore 96]. They also estimate that the difference is bigger in mathematics and physics. Coordinators of mathematics departments in public universities in Guanajuato State, Mexico, have noticed that students coming from video systems have a higher probability to fail its first math courses. They argue that their math knowledge is lower compared with regular students. In the opinion of these authorities video taped lectures will tend to disappear, but not in the near future, at least for underdeveloped countries, because it is one of the cheapest forms to deliver distance education. They will be replaced by technologies as videoconferences. At some universities video taped lectures are used inside the classroom for very specific concepts within the syllabus to present an expert opinion. Teachers at the Instituto Tecnológico y de Estudios Superiores de Monterrey (ITESM) highly recommend this instrument for advanced courses and also to present interesting and attractive applications on elementary courses. Video conference. Video conference has been used within higher education for more than a decade. Video conferencing is highly used for teaching sessions, teachers training, seminars and research. At many universities video conference is used as a tool to bring into the class an international experimented and recognized teacher to a large number of students. From the experience at ITESM, has been determined that the success of a video conference class depends on such factors as: a) Quality of sound, images and degree of interaction. b) Compatibility of the equipment with ingoing and outgoing signal places. c) Availability and quality of material presented in the video conference. d) Quick response to students questions.

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e) A tutor on the conference classroom. At ITESM and at the University of Salle Bajío, coordinators of the video conference programs have found that for subjects such as mathematics, statistics and classes with “heavy contents”, the presence of a tutor capable of answering students questions regarding the content of the conference makes a significant difference to the students learning and grades. f) Tutor-student oral communication is very important, because when listening to the student, the teacher might understand some questions better orally than using other types of methods like the internet. Especially in the case of mathematics and statistics, it is very hard for students to write down some of their doubts, and this may cause problems, like using the mathematical language improperly, or overcoming technological barriers that make an extremely difficult task to write down a mathematical sign in a computer. According to the Faculty of Education at The University of Plymouth [Plymouth], the future of videoconferencing is to incorporate video conference into web based systems, so teachers and presenters can sit in their own office or in a nearby studio and present a ‘live’lecture in front of a camera attached to a web server. Using a simple switching device and several cameras, the presenter can provide remote participants with graphics, whiteboard, flipchart and other visual aids as well as alternative views of the local classroom, lecture room, etc. Online courses. By experience at ITESM, the first step to success for online mathematics and statistics courses it is to convince students about the feasibility of the project. At this institution, full online courses are offered just for graduate students. It is also very important to have a quick response to student’s questions, so they feel that “there is someone supporting them on the other side of the line”. A second step is to make sure that students can manage technology properly and have all necessary means to remain on line and to send and download information, documents, graphics, etc. On a study conducted by Karr, Weck, Sunal and Cook [Karr 2003] at the University of Alabama to analyze the effectiveness of online learning in a graduate engineering mathematics course, they divided the class into three groups: Group A (Online course only), Group B (traditional for the first two thirds of the course and traditional and online for the final third of the course, Group C (traditional on the first third, online for the second third and traditional and online for the final third of the course). On this study they found that: a) Students perform better on the analytical portion of the course when they had used the online mode of delivery. According to the teachers and students feedback this is due to the consistency of online materials and the fact that they have to “face the problem on their own” b) Students taking the class by traditional mode perform better on the in-class portions of examinations. This might have been because of the instructor

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dropping inadvertently little hints about which aspects of the class might be on the test. c) The two groups with a traditional mode segment perform better when they have access to both modes of deliver, traditional and online. d) There was no significant difference on the overall performance of the groups. From my personal experience and from some non formal studies conducted on high school and undergraduate courses it is reasonable to believe that similar results will be obtained for high school and undergraduate mathematics courses. Many universities as ITESM consider, even for traditional courses, that online sections and online materials should be included to make courses more attractive to students and to enhance the student’s performance, especially in traditionally difficult courses as mathematics. Online problems and materials. Online problems are widely used to improve students learning on mathematics courses. Within the experience of Professor Maritza Sirvent and me, some advantages of using online problems on web based programs for mathematics problems are: a) The bank of problems is large and includes a big variety of questions. b) The students know immediately if their answer is correct, so they get engaged and they try the problem as many times as necessary to get the right answer. c) Some students feel that using the computer helps them in their homework. d) It is clear that the correctness of the problem is independent of the procedure used on the resolution. So they try their own ideas to solve the problem and use techniques as approximations using calculators. After that they study a method that will work at different situations. e) Problems solved for students at the same class are similar but not the same so they can’t copy the homework from a classmate. f) Student’s attitude toward mathematics problems seems to improve. A disadvantage of online problems might be that, when entering the answer to a problem, sometimes the student makes a typing mistake or forgets some parenthesis and then gets an incorrect answer even if he has solved the problem correctly. Also students are not forced to write down the complete procedure, so when they are tested on a traditional writing test, they have no training on that. WeBWorK is an internet based program to deliver homework to students on internet. It was designed by the University of Rochester. On a study conducted at Rutgers University to measure how effective WeBWorK was in improving learning measured by student’s performance in Calculus [Weibel 2002], they divided students in two sections: Sections where WeBWorK homework was required weekly and it counts as part of the final grade, and sections where traditional written homework was required. Two thirds of calculus students were on WeBWorK sections, and they found the following:

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a) Students in WeBWorK section did slightly better than students on traditional section. However, within WeBWorK sections, students who did over 80% of the WeBWorK problems performed dramatically better (by a full letter grade) than those who did less than half of the WeBWorK problems. b) First year calculus students were very responsive to WeBWorK and most of them attempted every problem. They found that there is a 2-letter grade difference (on the average, from B to D) between students who do well on WeBWorK and those who do not attempt it. For upper class students taking calculus there is a 3-letter grade difference (on the average, from B to F) between students who do well on WeBWorK and those who do not attempt it. These upper class students are not very responsive to WeBWorK. c) Students repeating calculus are not responsive to WeBWorK, and there is no significant difference on grades even for those that perform well on WeBWorK. Online didactical material helps students to understand some concepts that might be difficult to them. Some students express that it is easier for them to read online materials than books, because they are usually more attractive and often interactive. For them it is the perfect complement for text books. There is a bright future for online mathematics problems and didactical material. Each year the number of teachers convinced of the effectiveness of online mathematics problems and didactical material is increasing. Internet-based methods to deliver homework to students are improving and making it easier for teachers and students, saving a considerately amount of time on grading. For instance, projects such as WebALT [WebALT] aim at using existing technology standards for representing mathematics on the web and existing linguistic technologies to produce not just online mathematics problems, but language-independent mathematical didactical material. Problem based learning (PBL) and project oriented learning (POL). These learning methodologies have been applied from elementary school to graduate programs. It is based on the principle that learning occurs not by absorbing information but by interpreting it. These methodologies are ideal for distance learning, but require that students work in teams, an arrangement that may be very difficult for some students that prefer to work individually. With these didactical techniques, learning is generated by solving a realistic situation that requires learning new concepts and applying them to solve a problem. At some universities the full curricula is build around PBL or POL techniques, while at some other universities (as ITESM) these methodologies are mixed with traditional methods [Bolaños 2003], [Watson 2002]. PBL and POL are excellent tools to introduce students on the more difficult tasks of the syllabus. The results are excellent, as statistics show that students perform better with the concepts when introduced by PBL or POL than when introduced on traditional lectures. On these methodologies the role of the tutor is very important. The tutor is responsible for the direction of students and to help in team conflicts. The tutor has to

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Virtual labs in mathematics education: concepts and deployment

address the student’s efforts in the right direction and make suggestions about working lines. Students communicate online with their teammates and the tutor, also the final report of all teams is placed online, and so all teams might look at the similarities and differences with the solutions of the others teams. These are just some aspects of the big world of distance learning and were choosen because we consider that they might be applied on very different teaching environments. Distance learning will continue modifying our teaching practices.

References [Beare 89]

Beare, P. L., The Comparative Effectiveness of Videotape, Audiotape, and Telelectures in Delivering Continuing Teacher Education. The American Journal of Distance Education 3 (2) (1989), 57–66.

[Bolaños 2003]

Bolaños, G., Problem Based Learning for great statistics learning. In Proceedings of the Hawaii International Conference on Statistics and Related Fields, 2003.

[Greenberg 98]

Greenberg, G., Distance education technologies: Best practices for K-12 settings. IEEE Technology and Society Magazine (Winter) 36–40.

[Karr 2003]

Karr, C., Weck, B., Sunal, D. W., and Cook, T. B., Analysis of the Effectiveness of Online Learning in a Graduate Engineering Math Course. The Journal of Interactive Online Learning 1 (3), 2003.

[Moore 96]

Moore, M., and Kearsley, G., Distance Education: Wadsworth Publishing Company, Belmont 1996.

[Moore 97]

Moore, M., and Thompson, M., The Effects of Distance Learning, Revised Edition. Technical Report ACSDE Research Monograph (Number 15), American Center for the Study of Distance Education, The Pennsylvania State University, 110 Rackley Building, University Park, PA 16802-3202, 1997.

[Pflieger 61]

Pflieger, E. F., and Kelly, F. G., The National Program in the Use of Television in the Public Schools. Technical Report, Ford Foundation/FAE, 1961.

[Plymouth]

University of Plymouth web page, http://www2.plymouth.ac.uk/distancelearning/vidconf.html

[Russell 97]

Russell, T., The “No Significant Difference” Phenomenon. Retrieved from http://tenb.mta.ca/phenom/phenom.html, 1997.

[Watson 2002]

Watson, G., “Using technology to promote Success in PBL Courses”. The Technology Source, 2002.

[WebALT]

WebALT web page, http://webalt.math.helsinki.fi/content/index_eng.html.

[Weibel 2002]

Weibel, C., and Hirsch, L., WebWork Effectiveness in Rutgers Calculus. Retrieved October 27, 2005, from http://math.rutgers.edu/∼weibel/ww.html, 2002.

A Systems View.

[Whittington 89] Whittington, N., Is Instructional Television Educationally Effective? A Research Review. Readings in Principles of Distance Education. Pennsylvania State University, 1989.

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Virtual labs in mathematics education: concepts and deployment by Ruedi Seiler1 Background. The work field of engineers, as well as that of scientists and mathematicians, is undergoing drastical changes: as numeric software and computeralgebra-systems are capable of performing intensive and complex arithmetical calculations, other abilities, such as the fast acquisition of new knowledge and new methodologies, are growing in significance. Thus, learning and teaching methods that promote life-long, efficient and independent learning have to be conveyed. The traditional teaching methods employed at universities are of only limited success in this respect: Teacher-centered lessons provide the essential basic knowledge, but it hardly allows for a more active approach to the subject-matter. Classical experiments, in contrast, while targeted at independent knowledge acquisition, soon stumble across limits imposed by the reality of a university: high and constantly increasing numbers of participants in a course, limited access to and inadequate equipment. In addition, the experimental approach to knowledge acquisition in “real laboratories” is by its very nature limited to certain fields of studies, while more theoretical fields, such as mathematics and theoretical physics are either completely precluded or only peripherally touched upon by the existing experimental concepts. The deployment of new media and technology in class thus represents a turning point: Virtual Labs are environments based on physical labs in which computer aided experiments can be designed, created, implemented and evaluated. Experiments are implemented in the form of computer-based algorithms, representing either real tools and objects or even theoretical concepts and objects. Such explorative learning environments can be placed at the disposal of every student and teacher, independent of time and place. In the framework of the classical experimental sciences, virtual labs are capable of complementing real laboratories by allowing the concise elaboration of the actual “phenomenon” and diminishing the influence of metrological problems. As, however, the handling of the equipment and the mentioned problems represent a vital part of the acquired competence, real laboratory experiments should not be set aside completely in the experimental disciplines. In theoretical subjects, on the other hand, these technologies make abstract phenomena visually comprehensible. In this article, we will offer detailed requirements on Virtual Labs and describe the consequences of the implementation along the lines of a prototypical Virtual Lab for Statistical Mechanics. Pedagogical requirements. In the following, we present a list of pedagogical requirements we demand from modern e-learning technology, especially from virtual 1 In collaboration with Thomas Richter (TU, Berlin, [email protected]) and Sabina Jeschke (TU, Berlin, [email protected]).


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laboratories. In comparison with most other e-learning environments, though, virtual labs do not define learning goals by themselves. Rather, they put “learning spaces” at the disposal of teachers and students. A laboratory should provide the necessary equipment – or, in the case of virtual labs, the necessary algorithms – that facilitate the independent development and testing of problem solving strategies, incorporating typical problems of mathematics, physics and engineering science in order to prepare the student for his or her professional life. Laboratories offer students the unique opportunity to control their learning, without outside interference and consequently being able to make an independent decision about their learning process. We divide the support of self-directed learning into the following categories: First of all, (Virtual) Laboratories support explorative learning by allowing their users to work independently and efficiently with the technical equipment in order to investigate interconnections independently and to build an intuitive understanding of the subject. Therefore, it is vital that Virtual Laboratories should allow and encourage unconventional approaches, options, work flows, etc. Second, the support of different learning styles is one of the utmost features of the deployment of multimedia technologies in education, even though the first generation of e-learning technologies [1] did not yet allow individual approaches to the subject. Similarly, pre-fabricated experiments might not fit into the previous knowledge of the user, strictly limited specific environments and learning goals might not fit the individual interests, failing to motivate the user. Thus, virtual laboratories must enable the user to setup and control the experiment freely. Laboratories should ideally be adaptable to different application scenarios. This includes the deployment of the same basic lab in different courses, stressing different field-specific foci on the one hand, and the use in different scenarios ranging from demonstration through practice to examination on the other hand. For that reason, a virtual laboratory should not be limited to a fixed set of experiments or aimed at the requirements of one single lecture or one specific audience; for each different target audience arise different requirements. Typical application scenarios might reach from simple demonstrational support within lectures, over experiments in the classroom teaching for training and tutorials up to self-study and deployment in research applications. Both research and engineering achievements are increasingly the result of cooperations between distributed, separated teams. Thus, team work and team-oriented projects have to be an integral part of any modern scientific education, and thus must be actively supported by virtual laboratories as well. Laboratories must offer appropriate interfaces that will allow the integration of or linking with standard elements as Maple or Mathematica; experimental set-ups should include these elements correspondingly, for their use and handling should be a part of the scope of learning. Laboratory elements should be detachable from the actual lab through the application of open interfaces and thus should be reusable. Such requirements not only

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allow the efficient construction of new laboratories from existing elements, they also ease the integration of laboratories in more complex experiments requiring additional support from outside software components. Consequences for the implementation. The pedagogical requirements on virtual laboratories pose various demands on the software design which we demonstrate for the laboratory VideoEasel developed at the DFG Research Center. The technical focus of this laboratory is in a first, prototype phase with application to the field of statistical mechanics and related areas. Statistical problems are here modeled through the use of cellular automata, which are well-suited to design statistical models, covering many interesting areas ranging from the Ising model, statistical image denoising, lattice-gas models to Turing completeness. In order to be able to support different and varying deployment scenarios while imposing as few restrictions on the labs themselves, it must be possible to combine the elements of laboratory equipment flexibly and creatively. This leads to a “strictly anti-monolithic”, fine-granular software design, its basic structure characterized by the tripartition into simulation and arithmetic modules implementing the mathematical modules, an interface layer that serves as link between the equimentments, that allows the free combination of the software modules into an experiment, and last, graphical user interfaces allowing to control the experimental setup conveniently. The experiments in the lab VideoEasel are implemented as small, modular units, independent of the lab’s actual core, that can be created and loaded on demand. The elementary units can be separated into two distinct classes, “automata” for the algorithmic definition of physical phenomena – e.g. the Ising model – and “measuring tools” to measure certain quantities arising within the experiment – e.g. the Free Energy. VideoEasel offers basic methods for evaluation of measurements, but does not provide any numerical tools for more complex analysis or a build-in process control for more elaborate experiments. Such functions are taken over to specialized tools by utilizing the software interfaces of the laboratory, which are here realized in the middle-ware CORBA [2]. Mappings are available to many languages, such as Java, C and Python, thus facilitating the connection to various other external tools. Presently, in addition to the native Java-interfaces, there is a Python-connection for script-control, as well as a C-implementation of a Maple-connection available. Cooperative learning strategies in virtual laboratories imply in particular that several users from different working locations can work simultaneously on a single experiment while being well aware of the actions of their partners. Therefore, the need of designing the laboratory as a multi-part network application becomes self-evident: experiments are, for example, run on a server accessible by students. VideoEasel follows a classical client-server approach where the students control the simulations run on the server by Java front-ends. In the most simple case – as for support of a lecture in a auditorium – server and client are run on the same computer; in cooperative learning settings, the server synchronizes more clients.


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Reading the above arguments concerning the requirements in implementing a virtual laboratories drafted in the previous paragraph might create the impression of a “canonical” approach. However, most existing virtual laboratories posses a narrow technical focus on specific areas and follow a monolithic design. The second remark concerns tutorials, user guidance and the “usability” of such laboratories: The afore-mentioned flexible granular structure of the software inevitably leads to a more complex user interface and consequently to a higher adaptation time for the teaching staff as well as the students. Problems arising from the initial contact with technical problems present a prominent “motivation killer” in e-learning. In some cases, it is not easy to find the ideal compromises; to overcome this problem, one should then provide several, separate user interfaces, as for example found in VideoEasel: For simple demonstrational applications in lectures, a Java Applet is available that allows only minimal control of an experiment. For deployment in student groups and classroom teaching, a simple but efficient Java interface has been developed; it provides more options to influence the experiment, while keeping the complexity rather low. Additional menus allow the adjustment of all kinds of parameters within the experiment. The drawing surface, though, is very similar to the applet and mimics that of standard software tools. A more refined and complete interface was created through the Oorange toolkit [3] – also developed at the TU Berlin – allowing the purely graphic set-up of an experiment, as well as the integration and connection to other elements through “Java Beans” [4]. The server provides templates available for existing experiments, similar to the ones for the Java interfaces; these templates are transformed client-side into a Oorange compatible XML-representation. Different from the more basic interfaces, the user has the option of changing, modifying or completing the experiment at will. This access to VideoEasel does not have the pretense of being particularly easy to navigate, as it was conceived primarily for the use in research and not in teaching or in practice. Therefore, it is acceptable to require the user to go through a reasonable adaptation phase. Last but not least, VideoEasel is also completely controlable from within the computer algebra program Maple for applications whenever the Oorange toolkit is not able to deliver the mathematical algorithms required for research purposes. This interface uses, similar to all others, the CORBA technology to exchange data between the components. Now, in retrospective, we analyze how the required didactic concepts are implemented within VideoEasel: the field of cellular automata is rich enough to simulate interesting physical effects, yet straightforward enough to avoid undue obstacles in easy access. The basic principle of such automata can be learned quickly and allows for the execution of interesting (and esthetically pleasing) experiments through quite basic tools. Through the integration of time-proven, well known concepts – drawing programs and measuring tools – and the choice of an appropriate interface, the user is encouraged to experiment. Comprehension of the behavior of the effect to be under-

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stood is achieved through practice in the laboratory. Explorative learning is promoted through the connection of esthetical and academical contents. The availability of various surfaces allows us to address several user groups with very different demands on the laboratories and diverse application purposes ranging from pure demonstration to research applications. Cooperative deployment scenarios become viable through the two-part set-up as a client/server network architecture. Thus, acquisition and research between teams geographically far separated is feasible. Finally, CORBA-interfaces allow the docking and linking of the core laboratory with other laboratories, algebra-systems and connectors to demonstrate even more complex facts and to avoid locking the user in one single laboratory technology. Future developments. In conclusion, we will discuss some aspects of important relevance to our original aims, which are improving university education through the use of virtual laboratories: Virtual labs, including the presented VideoEasel, are still mostly at a prototype stage. Thus, practical experience about their deployment in e-learning environments are still rare. It has to be expected that use and evaluation will result in extensive adaptations and expansions of the existing concepts, particularly in the field of usability. To realize the pedagogical goals as presented above, it is necessary to integrate virtual laboratories into the framework of larger virtual knowledge spaces. VideoEasel does provide a number of generic interfaces which will have to be specified in more detail. More experiences with laboratories from other fields of science and engineering are necessary to define a standardized data-exchange between different laboratories. Finally, the virtual laboratories are becoming more and more complex to use as a direct result of the diversity of addressed learning scenarios, the desired interconnectability of different applications and the broad variety of the learning contents. To counter this effect it might be desirable to extend laboratories by “digital assistants” [5]. New concepts developed in the field of artificial intelligence in recent years have to be expanded and applied to virtual knowledge spaces and their components.

References [1] Jeschke, S., and Kohlhase, M., and Seiler, R., eLearning-, eTeaching- & eResearchTechnologien - Chancen und Potentiale für die Mathematik. DMV-Nachrichten, July 2004. [2] Scallan, T., A Corba Primer. http://www.omg.org/. [3] Oorange: The Oorange development environment. http://www.oorange.de/. [4] JavaBeans. http://java.sun.com/products/ejb/. [5] Jeschke, S., and Richter, T., and Seiler, R., Mathematics in Virtual Knowledge Spaces: User Adaption by Intelligent Assistents. In Proceedings of the 3rd International Conference on Multimedia and ICTs in Education, June 7–10, 2005.

Roles for the new mathematics educators

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Roles for the new mathematics educators by Mika Seppälä The future is here. It is just not evenly distributed. We are living interesting times! The industrial revolution is on its way in education, publishing, and business. Ways to conserve knowledge and transfer it from generation to generation are changing. Libraries are becoming digital and classes virtual. This development opens extraordinary opportunities to those willing and capable to profit from them. It also opens possibilities to spectacular failures of which we saw many some years ago. “Emergent technology is, by its very nature, out of control, and leads to unpredictable outcomes.” This certainly applies to the current development in e-learning, including e-learning mathematics. “The Future is here. It is just not evenly distributed.” Both quotes are by William Gibson. So in order to understand what lies in the future we can simply look at what our colleagues are doing today. There is no doubt that the information network and the advanced technology are going to change the way we write, publish and teach all disciplines, including mathematics, in the future. This will happen because it is possible, and because proper usage of technology will enhance our current ways to work. To understand how educators work in 2016, we simply need to understand which, of the currently existing ways to use information technology in education, have most potential. These are likely to emerge as general paradigms and set examples that many will follow. Changing the educational system. Not only instruction, but the whole educational system is changing. New interdisciplinary fields are emerging at a fast pace. Largely this is due to mathematics becoming more applicable thanks to the various advanced mathematics systems like Maple, Mathematica or Matlab. It is now possible to use mathematical modeling in a fundamentally deeper way than before. This is true in practically all fields, perhaps most notably in biology and medicine. In the past, applications of mathematics in biology or medicine have been, from the mathematical point of view, rather simple. Now more complex methods can be used. This requires expertise in mathematics, computer science, and in the subject matter to which mathematics is being applied. Hence interdisciplinary study programs have been created to educate experts capable of developing these new applications. The new roles of mathematics educators. In the past, and in many cases even today, the teaching of mathematics has been the responsibility of instructors, and the learning that of students. At most European universities, basic mathematics courses are being taught in very large sections. A typical undergraduate calculus class may have well over 100 students. In some cases these classes have hundreds of students.

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The instruction is lecturing with little or no personal interactive contact between the students and the professor. Instructors simply cannot follow the day-to-day progress of their students. Technology can be very useful here. Using systems like Maple TA or STACK, it is possible to offer automated private instruction to students and to monitor the progress of individual students even in large classes. This will empower professors and enhance traditional contact instruction in a dramatic way. Instruction, even in the case of large classes, becomes student centered instead of instructor centered. Professors will take responsibility of their students in a way that has not been usual in the past. The emerging new role of instructors is very similar to that of coaches. Athletes have their personal coaches, so will students as well. The future instructors work like sports coaches today assisting students to achieve goals they could not achieve on their own. Empowered with advanced learning technologies, instructors can provide individual assistance to their students in a way that was not possible earlier. Interactivity can now be provided, using the web, in a way that is likely to permanently change the way we work. Educating new educators. The inertia of the academia resists changes and delays the necessary development. Instructors in general are not ready to change the way they work. There is also a good reason for the resistance. Moving from traditional contact instruction to computer aided learning is not easy. The data in the table below are generally accepted estimates of the efforts needed for various types of teaching. All these forms of teaching, except lecturing and small group teaching, will require additional technical support. The large spreads in the first four items reflect the fact that experienced educators can work much faster than beginning professors. There is no spread in the table for computer aided learning and interactive video. Here also experience will eventually help, but for now there are not many instructors having extensive experience in computer aided learning. Academic work to produce one hour of student learning ([2]) Lecturing Small group teaching Videotaped lectures Authoring a text Computer aided learning Interactive video

2–10 1–10 3–10 50–100 200 300

Using the figures of the above table, the development of a typical one semester course will amount to over five years of full time work of the author in addition to the required technical support. Regardless of the above, some professors are developing content for computer aided learning. They are driven by the vision of greatly improved education once


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the necessary content is in place and available in the same ways as books are now available to students and professors. Metadata. Developing content for computer aided learning is very costly. Furthermore, today the materials developed by professors are mostly being used only by the authors themselves and their students. Sharing does not happen, not to speak of shared development of content. To address this problem, the European Commission is currently investing heavily into projects which enhance existing content with metadata. This metadata will make content cross border usable, and shared creation of content a real possibility. The development of metadata is likely to dramatically change the way we work. It will make the hard work to develop premium on-line content cost effective and worthwhile. Best practices. All of the above applies really to all disciplines. Problems related to the teaching of mathematics or of sciences more generally, focus on the presentation of mathematical formulae. Almost all mathematics is being written using LATEX today. Also these proceedings have been prepared by LATEX. LATEX and TEX generate extremely high quality typesetting of scientific text. These systems produce content ready for printing and publishing in the traditional way. New LATEX classes for producing high quality presentations have been created. Practically all mathematicians are using LATEX. Intelligent interactivity ([3] and [4]) requires that mathematical formulae are presented in the on-line materials so that the meaning of the formulae can be automatically understood. MathML and OpenMath make this possible. To embed mathematical formulae in a proper way to web content requires the usage of these languages. LATEX or TEX do not support MathML or OpenMath. In spite of the fact that TEX enthusiasts are working hard to develop solutions to this problem, the use of MS Word and PowerPoint together with products like MathType often makes the content development much easier. Searching the web one can find, for example, a variety of electronic presentations of calculus or linear algebra courses. Most of these are pdf presentations of printed materials, and are not designed to be studied from the computer monitor. The new media, the computer screen, requires a different presentation of the content than what is used on printed materials. The resolution of a printed page is much higher than the resolution of the best monitors. Hence printed pages are easier to read than computer monitors. To overcome this problem, content, for the computer screen, needs to be presented in a very condensed way. For instruction based on the computer screen, the presentation of the materials needs to follow the general design principles implemented, for example, in PowerPoint. On-line content has many important advantages which greatly overcome the handicap that computer monitors have with respect to printed pages. These advantages include hyperlinking, live interactive and adaptive content, student performance track-

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ing, and, most recently, multilinguality. The WebALT encoding of mathematical content uses an extension of OpenMath and is such that the content can be generated in many languages automatically. Hence the content is truly multilingual, or rather, language independent. This is a serious advantage in view of the high cost of the development of on-line content. A case study: on-line calculus at the University of Helsinki. The lesson learned from previous experiences at Florida State University was that on-line materials should use standard tools as much as possible, not require students to install new programs, and that the illustrations of mathematics should be done so that the required technicalities are completely hidden. With these points in mind, the development of new on-line materials for calculus was started at the University of Helsinki in the Fall of 2001. These materials consist of a collection of lectures presented by PowerPoint, a collection of PowerPoint presentations of solved problems, a collection of calculus calculators empowered by MapleNET, and a repository of problems delivered to students using Maple TA, a system for the delivery and automatic grading of homework, quizzes, and examinations. Students reactions to these new on-line materials have been overwhelmingly positive. During the Fall of 2004, a basic course in calculus was offered, at the same time, as a fully on-line course, and as a traditional lecture/problem session course. Both courses were based on the on-line materials, and had the same exercises and examinations. For the on-line students, the examinations were the only events that took place on campus and were proctored. The results were surprising: the on-line students fared better than the traditional students in both examinations, and the retention rate was higher among the on-line students than among the traditional students. Automatic assessment. Systems providing automatic assessment of homework problems, quizzes and examinations have been used in lower level mathematics instruction at Florida State University with spectacular results for several years. The failure rates of precalculus courses have gone down by about 50%. This is due to students being able to practice for examinations at home so that they get immediate feed-back from the system. Currently the most advanced automatic assessment systems are Maple TA, STACK, the forthcoming LeActive Math System and the WebALT System. Common to all of these is that they offer the possibility to create algorithmic problems which are programs that generate a different version of a problem every time the program is invoked. In addition to the others, the WebALT System will also be able to generate the problem in many languages. The algorithmic problems really make a difference. Consider, for example, the method of partial fraction decompositions. Students of calculus will have to learn that. It is relatively simple to write a program which generates over a million different but


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equally hard problems of partial fraction decompositions. Hence the examination about partial fraction decompositions can be published to the students before the test! Students can take the partial fraction decomposition test as many times as they want at home, get individual feed-back including full solutions. Learning by heart is not helpful, because regardless how many times they take the same test at home, they are going to get different questions in the examinations. Such algorithmic problems were used in instruction at Florida State University in Spring 2005. Most students reacted very positively, and used the system a lot to their benefit. Some students solved even hundreds of problems on computing limits, for example. Starting in Fall 2006, students at Florida State University are required to have a laptop computer. Then the automatic assessment systems can be used in class, and examinations can be based on the use of these systems. Conclusions. The development on-line education in mathematics at the university level has been very slow. Administrators at national agencies and ministries in various countries see the great potential that on-line content can bring to education, but largely this potential has not been realized in mathematics and, more generally, in sciences. This is partly due to problems that one has in the presentation of scientific content on the web. The majority of on-line materials present mathematical formulae as pictures only. This is not a satisfactory solution. One cannot use a picture as a key word in a database search. MathML and OpenMath provide solutions to this. Commercial editors, such as MS Word and PowerPoint together with MathType, provide a convenient way to produce content in which mathematics is embedded using MathML. Authoring tools are available, robust and easy to use. Missing synchronous interactivity has been another problem in on-line instruction. Together with the introduction of tools like Skype and the various easy-to-use conferencing systems, this problem has suddenly disappeared. Virtual on-line courses can provide more personal interaction between instructors and students than a regular class with hundreds of students attending the same lectures. This development is new, and we have not yet seen how that will change instruction. The effect is likely to be impressive, however. To use the available technology to the maximum places large demands on instructors. They have to rethink their roles and convert themselves from lecturers to coaches. And they have to be able to use technology in a fluent way. Most instructors resist doing this mainly because the transition requires a lot of work. The main remaining obstacle in this development is the fact that premium on-line content is expensive to produce and hard to find. Extensive funding programs, like the European Commission supported Content Enhancement Projects of eContent Plus, are likely to make a dramatic difference with respect to these remaining obstacles. The most important lessons learned were that it is necessary to keep the use of technology as simple as possible while still providing advanced functionalities. Pretty

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good is good enough. For the student, everything has to work right out of the box. Technicalities have to be hidden. On-line content satisfying this criteria is going to have large and permanent value. In 2016 we cannot understand how education without the information network and its services was possible.

References [1] Bass, H., Mathematics, mathematicians, and mathematics education, Bull. Amer. Math. Soc. 42 (2005), 417–430. [2] Boettcher, JudithV., Designing for Learning. http://www.designingforlearning.info/services/ writing/dlmay.htm. [3] Caprotti, O., Seppälä, M., Xambó, S., Using Web Technologies to Teach Mathematics. In Proceedings of SITE 2006 Conference, Orlando, FL, March 20–24, 2006. [4] Caprotti, O., Seppälä, M., Xambó, S., Mathematical Interactive Content: What, Why and How. To appear in Proceedings of the 1st WebALT Conference (held in the Technical University Eindhoven, January 5–6, 2006). [5] Grottke, S., Jeschke, S., Natho, N., Rittau, S., Seiler, R., mArachna: Automated Creation of Knowledge Representations for Mathematics. To appear in Proceedings of the 1st WebALT Conference (held in the Technical University Eindhoven, January 5-6, 2006). Facultat de Matemàtiques i Estadística, Universitat Politècnica de Catalunya, Barcelona, Spain E-mail: [email protected] School of Education, University of Michigan, Ann Arbor, Michigan, U.S.A. E-mail: [email protected] Instituto Tecnológico y de Estudios Superiores de Monterrey, Monterrey, México E-mail: [email protected] Institut für Mathematik, Technische Universität Berlin, Berlin, Germany E-mail: [email protected] Department of Mathematics and Statistics, University of Helsinki and Department of Mathematics, Florida State University Helsinki, Finland & Talahassee, USA E-mail: [email protected]

Author index

Agrawal, Manindra, 985 Artigue, Michèle, 1645 Barvinok, Alexander, 763 Bass, Hyman, 1743 Bianchini, Stefano, 147 Bochev, Pavel, 1137 Bolaños Evia, Gilda, 1743 Borkar, Vivek S., 1299 Bousquet-Mélou, Mireille, 789 Bovier, Anton, 499 Boyd, Stephen, 1311 Caflisch, Russel E., 1419 Candès, Emamnuel J., 1433 Caselles, Vicent, 1453 Cattaneo, Alberto S., 339 Cerf, Raphaël, 519 Chen, Zhiming, 1163 Cheng, Shiu-Yuen, 1673 Corry, Leo, 1697 de Lange, Jan, 1663 Dembo, Amir, 535 Derrida, Bernard, 367 de Shalit, Ehud, 1645 Donnelly, Peter, 559 Durán, Ricardo G., 1181 Dyn, Nira, 1201 Elworthy, K. David, 575 Emanouilov, Oleg Yu., 1321 Fan, Jianqing, 595 Fuchs, Jürgen, 443 Geelen, Jim, 827 Gérard, Patrick, 157 Gerards, Bert, 827

Ghrist, Robert, 1 Golse, François, 183 Gorodetski, Anton, 27 Griebel, Michael, 1473 Guicciardini, Niccolò, 1719 Guionnet, Alice, 623 Gunzburger, Max, 1137 Gursky, Matthew J., 203 Haiman, Mark, 843 Hamaekers, Jan, 1473 Holevo, Alexander S., 999 Hunt, Brian, 27 Ishii, Hitoshi, 213 Kaloshin, Vadim, 27 Kenderov, Petar S., 1583 Kerkyacharian, Gérard, 713 Kim, Jeong Han, 873 Kleinberg, Jon, 1019 Kra, Bryna, 57 Lalley, Steven P., 637 Le Bris, Claude, 1507 Le Calvez, Patrice, 77 Le Jan, Yves, 649 LeVeque, Randall J., 1227 Li, Runze, 595 Li, Xue-Mei, 575 Łuczak, Tomasz, 899 Maday, Yvon, 1255 Maillet, Jean Michel, 383 Mariño, Marcos, 409 McCullagh, Peter, 669 Nebres, Ben, 1673 Nowak, Martin A., 1523

1770 Nualart, David, 1541 Okounkov, Andrei, 687 Osterwalder, Konrad, 1673 Picard, Dominique, 713 Pulvirenti, Mario, 229 Ralston, Anthony, 1645 Reingold, Omer, 1045 Rodnianski, Igor, 421 Roughgarden, Tim, 1071 Rubinfeld, Ronitt, 1095 Runkel, Ingo, 443 Ruzsa, Imre Z., 911 Santos, Francisco, 931 Savin, Ovidiu, 257 Schaft, Arjan van der, 1339 Schmidt, William, 1663 Schweigert, Christoph, 443 Seiler, Ruedi, 1743 Seppälä, Mika, 1743 Serfaty, Sylvia, 267 Shub, Michael, 99

Author index

Siegel, Alan, 1599 Soffer, Avy, 459 Staffans, Olof J., 1367 Stewart, Ian, 1631 Süli, Endre, 1271 Szepessy, Anders, 1563 Thomas, Robin, 963 Trevisan, Luca, 1111 Trudinger, Neil S., 291 Vega, Luis, 303 Velázquez, Juan J. L., 321 Villani, Cédric, 473 Werner, Wendelin, 741 Whittle, Geoff, 827 Wu, Hung-Hsi, 1673 Xambó Descamps, Sebastià, 1743 Yang, Jie, 669 Yee, Lee Peng, 1663 Zorich, Anton, 121 Zuazua, Enrique, 1389

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