Frontiers And Prospects of Contemporary Applied Mathematics (Series in Contemporary Applied Mathematics Cam)

Series in Contemporary Applied Mathematics CAM 6

liiiiiiiiiiimiiMlIlIH Tatsien Li • P ingwen Zhang

frontiers and Prospects of Contemporary Applied Mathematics

Series in Contemporary Applied mathematics

CAM

Honorary Editor: Chao-Hao Gu (Fudan University) Editors: P. G. Ciarlet (City University of Hong Kong), Tatsien Li (Fudan University)

1. Mathematical Finance

Theory and Practice

(Eds. Yong Jiongmin, Rama Cont)

2. New Advances in Computational Fluid Dynamics Theory, Methods and Applications (Eds. F. Dubois, Wu Huamo)

3. Actuarial Science

Theory and Practice

(Eds. Hanji Shang, Alain Tosseti)

4. Mathematical Problems in Environmental Science and Engineering (Eds. Alexandre Ern, Liu Weiping)

5. Ginzburg-Landau Vortices (Eds. Haim Brezis, Tatsien Li)

6. Frontiers and Prospects of Contemporary Applied Mathematics (Eds. Tatsien Li, Pingwen Zhang)

Series in Contemporary Applied Mathematics CAM 6

Frontiers and Prospects of

editors

Tatsien Li Fudan University, China

Pingwen Zhang Peking University, China

Higher Education Press

\[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANCHAI • H O N G K O N G • TAIPEI • CHENNAi

Tatsien Li

Pingwen Zhang

Department of Mathematics

School of Mathematical Sciences

Fudan University

Peking University

Shanghai, 200433

Beijing, 100871

China

China

^ t t M t f c ^ Wftff&^MW. ^Frontiers and Prospects of Contemporary Applied Mathematics / ^^kM

(Li,

Tatsien), ft^fX (Zhang, Pingwen) if. —JfcJiC:

mmm^&mt, 2005.12 ISBN 7-04-018575-X i.Mf... n.©$. ..©&... —XM^X IV.29-53 ^ l a w s ^ i t CIP mm?f

m.mffi&-ffi% (2005) m 13795s ^

Copyright © 2005 by Higher Education Press 4 Dewai Dajie, Beijing 100011, P. R. China, and World Scientific Publishing Co Pte Ltd 5 Toh Tuch Link, Singapore 596224 All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from the Publisher. ISBN 7-04-018575-X Printed in P. R. China

Contents Preface Invited Talks Jin Cheng, Mourad Choulli, Xin Yang: An Iterative BEM for the Inverse Problem of Detecting Corrosion in a Pipe

1

Weinan E, Pinghing Ming: Analysis of the Local Quasicontinuum Method Houde Han: The Artificial Boundary Method

18 Numerical Solutions

of Partial Differential Equations on Unbounded Domains

33

Kerstin Hesse, Ian H. Sloan: Optimal order integration on the sphere

59

Jialin Hong: A Survey of Multi-symplectic Runge-Kutta Type Methods for Hamiltonian Partial Differential Equations

71

Ming Jiang, Yi Li, Ge Wang: Inverse Problems in Bioluminescence Tomography

114

Fangting Li, Ying Lu, Chao Tang, Qi Ouyang: Global Dynamic Properties of Protein Networks

149

Wei Li, Yunqing Huang: A Modified Adaptive Algebraic Multigrid Algorithm for Elliptic Obstacle Problems

160

Zeyao Mo: Parallel Algorithms and Implementation Techniques for Terascale Numerical Simulations of Typical Applications Yaguang Wang: Long Time Behaviour of Solutions to Linear

179

Thermoelastic Systems with Second Sound

191

Contributed Talks Hongxuan Huang, Changjun Wang: Distance Geometry Problem and Algorithm Based on Barycentric Coordinates

208

Jijun Liu: On IU-Posedness and Inversion Scheme for 2-D Backward Heat Conduction

227

Yirang Yuan, Ning Du, Yuji Han: Careful Numerical Simulation and Analysis of Migration-Accumulation

242

Rongxian Yue: Error Analysis on Scrambled Quasi-Monte Carlo Quadrature Rules Using Sobol Points

254

Preface During the period of the 8th Annual Conference of the China Society for Industrial and Applied Mathematics (CSIAM) held on August 24 30, 2004 in Xiangtan, Hunan Province, China, the Symposium on Frontiers and Prospects of Contemporary Applied Mathematics was held. About 300 representatives from over 100 domestic universities, scientific research institutions and enterprises and from abroad attended the conference. At the symposium some Chinese and foreign scholars and experts were invited to give plenary lectures. They introduced current progress and expressed their prospects on some important topics of the industrial and applied mathematics. Besides, at the section meetings many participants gave academic reports. Considering that these plenary lectures have high academic values due to, their representative and perspective, we collected them in a volume for publication. Meanwhile a small part of the academic reports provided in the sections was also selected for this volume. We hope that the publication of this book would effectively help readers understand the current situation of the industrial and applied mathematics and the hot issues in this area. Also we hope the publication of this book would be helpful in pushing the industrial and applied mathematics forward. We would like to take this opportunity to express our heartfelt thanks to all of the speakers at the symposium for their great support, especially to those speakers who wrote papers for this book. We would also like to show our sincere thanks and respect to the National Natural Science Foundation of China, the Mathematical Center of Ministry of Education of China and Xiangtan University for their financial help and support; and to Higher Education Press and World Scientific Publishing Company for their hard work and efforts in publishing this book.

Li Tatsien October 2005

1

An Iterative BEM for the Inverse Problem of Detecting Corrosion in a Pipe* Jin Cheng School of Mathematical Shanghai E-mail:

Sciences, 200433,

Fudan

University,

China.

[email protected]

Mourad Choulli Departement

de Mathematiques

du Saulcy,

Universite

57045 Metz cedex,

E-mail:

de Metz He

France.

[email protected]

Xin Yang Department

of Computer

State University, E-mail:

Science

University

and Engineering,

Park, PA 16802,

Penn USA.

[email protected] Abstract

In this paper, we consider an inverse problem of determining the corrosion occurring in an inaccessible interior part of a pipe from the measurements on the outer boundary. The problem is modelled by the Laplace equation with an unknown term 7 in the boundary condition on the inner boundary. Based on the Maz'ya iterative algorithm, a regularized BEM method is proposed for obtaining approximate solutions for this inverse problem. The numerical results show that our method can be easily realized and is quite effective.

1

Introduction

Detecting the corrosion inside a pipe is one of the most important topics in engineering, especially in the safety administration of the nuclear power station. There are several ways t o do this. In this paper, we will discuss the m a t h e m a t i c a l theory a n d numerical algorithm for a m e t h o d of detecting the corrosion by electrical fields. More exactly, we consider an *The authors are partly supported by NNSF of China (No. 10271032 and No. 10431030) and Shuguang Project of Shanghai Municipal Education Commission (N.E03004).

2

Jin Cheng, Mourad Choulli, Xin Yang

inverse problem of determining the corrosion occurring in an inaccessible interior part of a pipe from the measurements on the outer boundary. Our goal is to determine information about the corrosion that possibly occurs on an interior surface of the pipe, which is an 'inaccessible' part, and we collect electrostatic data on the part of the exterior surface of the pipe, which is an 'accessible' part. In the case that the thickness of the pipe is sufficiently small when compared with the radius of the pipe and the Cauchy data are given on the whole outer boundary, this inverse problem can be treated by the Thin Plate Approximation,method (TPA). The algorithm and numerical analysis can be found in [7]. But this algorithm works only under the assumption that the thickness is small enough when compared with the radius of the pipe. The case, in which the Cauchy data are given on part of the outer boundary and the smallness assumption is abandoned, has not been studied and it is obvious that it is of great importance for practice problems. The main difficulty for this inverse problem is the ill-posedness of the inverse problem. The measured data are given only on part of the outer boundary and we want to determine an unknown function in the inner boundary. Because of the ill-posedness, the errors in measured data will be enlarged in the numerical treatment if we do not treat it suitably. In this paper, based on the Maz'ya iterative method, we propose a new BEM algorithm for this inverse problem. It can be easily realized. The numerical results show the efficiency of this method. This paper is organized as follows: 1. Formulation of the inverse problem, 2. The iterative boundary element method, 3. Numerical examples, 4. Conclusions.

2

Formulation of the inverse problem

Suppose a domain fi = {x | n < \x\ < r2} C R 2 (see Figure 2.1) and the boundaries Ti = {x\ \x\ = ri}and T2 = {x\ \x\ = r2}-

3

An Iterative BEM for the Inverse Problem • • •

Assume that Q is a metallic body with constant conductivity. In the domain CI, we consider an electrostatic field. The electric potential u satisfies the Laplace's equation in Cl, i.e., Au = 0,

in

Cl.

(2.1)

Let To be an open set of the outer boundary T2 of Cl which is an 'accessible' part. On To, the Dirichlet data and the Neumann data of the electric potential u are given, i.e.,

u(x) = 4>(x),

x e r0,

(2.2)

x e T0,

(2.3)

uv{x) = ip(x),

where uv is the outer normal derivative of u on the boundary.^ We denote the rest part of the exterior boundary of Cl by I^,

f2 = r 2 \r 0 . We assume that the corrosion only happened on the interior boundary of the domain Cl and the corrosion can be described by a non-negative function 7 in the boundary condition on the interior boundary. That is, uv+ju

= 0,

on

TU

(2.4)

where 7 > 0 represents the corrosion damage. The inverse problem we discuss in this paper is to find the unknown coefficient 7 from the Cauchy data and ip on I V We will treat this inverse problem by the following steps: Step 1: Get the Cauchy data on the interior circle by solving the Cauchy problem for Laplace's equations. We use the iterative boundary element method to solve the Cauchy problem: 'Au(x) — 0, xeCl, < u(x) = (j)(x), x € To, (2.5) un(x) =ip(x), xeT0. Our goal is to get the Cauchy data on I?i: u(x) = 0 until a prescribed stopping criterion is satisfied. The stopping criterion we will use in this paper is \\uk+i — Mk||L2(riur2) < e, where s is a small positive number.

An Iterative BEM for the Inverse Problem • • •

5

Remark 3.1. The mixed boundary value problems (3.2) and (3.3) are well-posed problems. We solve the mixed boundary value problems (3.2) and (3.3) by the boundary element method, which can be found in a lot of guide books on the boundary element method, for example, [1]. In the following, we give only the outline of the iterative BEM form. Consider the following mixed boundary value problem in twodimensional case: Aw = 0, in Cl, u = f, on rD, (3.4) un = g, on I V As we have known, the foundational integral formula of the harmonic function u{Mi) = J

(u*^-

u^pj

dT,

Mi e Q,

(3.5)

where u* = •£- In —-— represents the foundational solution of the Laplace's equation. And the boundary integral formula is : *u{Mi) = / ( u* — - u — J dT,

Mi e dQ.

(3.6)

Equation (3.6) can be discretized as follows: CiUi + V

/

uq*dY -J2

u*qdT = 0.

(3.7)

The values of u and q in the integrands of (3.7) are constant within each element, and u and q consequently can be taken out of the integrals. This gives cm + J2\

I

Q*dT )uj-^2[

I u*dT j q3 = 0.

(3.8)

With the given boundary condition, we can rearrange equation (3.8) with all the unknowns on the left-hand side and a vector on the righthand side obtained by multiplying matrix elements with the known values. This gives

iUi + Y^l f q*dr)Uj-

^2 {[ u*dr\qj (3.9) u*dT )

j=m+l Vri

/

j=l V r .

qj.

6

Jin Cheng, Mourad Choulli, Xin Yang

The whole set of equations can be expressed in a matrix form as A(*D)=-B(UD) \uNJ \

A>0,

(312)

8

Jin Cheng, Mourad Choulli, Xin Yang

Theorem 3.2. Set ft be an annular domain, ft C R2. Let (f,g) be consistent Cauchy data and assume that the solution of the Cauchy problem (3.10) satisfies <j>-<j>o£ Hper, where cfio € H is some initial guess. Let fi > 2, (fe,ge) be some given noisy data with \\ze — Zfi9\\ < e, e > 0 and k(e,ze) be the stopping rule determined by the discrepancy principle k(e, z€) = min{fc € N\\\ze - (/ - Ti)4>% \\ < fie}.

(3.13)

Then there exists a constant C, depending on c/>o only such that i)

U-4>U of the Cauchy problem (3.10) in this domain satisfies f-foeH^,

(3.14)

where 4>o G H is some initial guess and H^er is the Sobolev spaces of periodic functions defined as in (3.11). This regularity assumption is equivalent to choosing some ip € Hper satisfying

where f is the logarithmic-type source conditions (3.12). Proof. For simplicity, we consider Cauchy problem (3.10) in the annular domain r1 = {(^^);^e(-7r,7r)},JR>l, r 2 = {(l,0);0€(-7r,7r)}, where f(9) = £f=i a,- sm(j0), g{6) = £ f = 1 bj sin(j0).

An Iterative BEM for the Inverse Problem • • • Given the Neumann data N

»o(0) = X>,:>8inO'0), 3=1

we can get

where

_ (R? - R-i)2 ~ (Ri + R-i)2'

j

For (fr — foe Hper, there exists a,j, (j = 1 • • • N) satisfying S , = i

a

| < °°>

N 1

- 0 o = ^2a,jj

sin(jy).

3=1

So we get N

J2(l+J2)a2r2 t n e n &.=

^

3

3f(l-\iY

From the estimate In (f^-)

>l-ln

(exp(l)

W - Br' RP + R-J

2R~i ^Ri + R-i > 2jlnR - 1, -In

In

(!?m € H°er.

D

Lemma 3.5. Let (f,g) be consistent Cauchy data and assume that the solution 4> of the fixed point equation satisfies the source condition 4>- 4>o = f(I - Ti)ip,

for

some

ip e H,

where o G H is some initial guess and f is the function defined in (3.12) with p > 1. Let /J, > 2, (fe,g€) be some given noisy data with \\ze — Zftg\\ < e, e > 0 and k(e,ze) the stopping rule determined by the discrepancy principle. Then there exists a constant C, depending on p and \{ijj\\only such that

i)

U-4>l\\%\\
R m such that \\y\\k,P < 00 (see [1] for definition). We write Wl>P{Q) for W ^ t f j R 1 ) and '^{Q) for Wl'2{n). In particular, W#' p (£?;R m ) denotes the sub1p n space of W ' {n;W ) with the same trace on the opposite faces of dfi. Summation convention will be used. We will use | • | to denote the absolute value of a scalar quantity, the Euclidean norm of a vector and the volume of a set. In several places, we denote by | • \g2 the £2 norm of a vector to avoid confusion. For a vector v, Vv is the tensor with components (V«)y = djVi; for a tensor field S, div S is the vector with components djSij. Given any function W: Mdxd —> R, we define

D w

* w-(is;)

-"

Diw(A)

~(j&)-

Analysis of the Local Quasicontinuum Method

23

where Mmxn denotes the set of real mxn matrices. For any p > d and m > 0, define X: = W m + 2 ' p (/2;R d ) n

W^p(n-M.d),

imdY: = Wm>p(n;] Given the total energy functional /(»):= f(WcB(Vv(x))-f(x)-v(x)^dx,

(2.1)

n where WCB(VW) is given by (1.3) or (1.4) with A = Vv, we seek a solution u — B • x G X such that I(u) =

min I(v). v-BxeX

The Euler-Lagrange equation of the above minimization problem is: (C(v): = -div(DAWCB(Vv))=f, 1

in Q,

v — B • x is periodic,

on dfi.

As to the atomistic model, we consider the following minimization problem: E m o • ^i1 , ^ ,VN}(2-3) n iVi> -y—x—Bxis

periodic and 2_,4 Vi=0

The existence result is based upon the following two assumptions: Assumption A: W(A,p) satisfies the generalized Legendre-Hadamard condition at the undeformed configuration: There exist two constants Ai and A2, independent of e, such that for all £,77, £ S R d , there holds {DlW(0,Po) W(0,p VKDpA W ( 0 ,0P) O ) PA

DApW(0,Poy

^D ^2(pW(0,p 0 , p 00)) y V C

where p0 is the shift at the undeformed configuration. The second assumption is: Assumption B: There exist two constants Ai and A2 such that the acoustic branch and the optical branch of the phonon spectrum satisfy w 0 (fc)>Ai|fc|

and

u0(k)>A2/e,

(2.4)

respectively, where k belongs to the first Brillouin zone, and o;a(fc), a;0(fc) are respectively the acoustic and the optical branches of the phonon spectrum.

24

Weinan E, Pingbing Ming

T h e o r e m 2 . 1 . [6, Theorem 2.1, Theorem 2.2] If Assumption A holds and p > d, m > 0, then there exist three constants >c\, X2 and S such that for any B e M+Xd with ||B|| < tt\ and for any f eY with ||/||u"».p(.r2) < >ci, problem (2.2) has a unique solution UQB that satisfies \\UCB - B • x\\w™+z.p(n) < 8, and UQB is o, Wl'°°-local minimizer of the total energy functional (2.1). Moreover, if Assumption B holds and p > d, m > 6, then there exist two constants M\ and M 2 such that for any B £ M^ xd satisfying ||B|| < Mi and for any f EY with \\f\\wm'P(n) 5= M2, problem (2.3) has a local minimizer y that satisfies \\V-VCB\U d, there exist two constants H > 0 and A > 0 such that for any \\f\\Lp(n) < *> A(UCB;

3

v, v) > A\\v\\i

for all v e X.

Local quasicontinuum method

The original local Q C ' ^ i s based on the Cauchy-Born rule, which can be formulated as

Analysis of the Local Quasicontinuum Method

25

Problem 3.1. Find UH G XH such that IH(UH)

= jam

IH(V),

where IH(V): = % ( V V )

- J f(x) • V(x) dx, n

with

WQC(VV) = J2 " * W C B ( V V ) . KGTH

The Euler-Lagrange equations associated with the above minimization problem is of the form: Find UH € XH such that AH(UH,V)

= (f,V)

for all V 6 XH,

(3.1)

where AH is defined for all V, W G X # as i4ji(V,W):=

£ (njf/|lT|) Ker„

[DAWCB(W)VWdx. J K

For any t>, « # , u; G X , define R(v, VH, W): = A(VH, W) — A(v, w) — A(v; VH — v, w).

(3.2)

Here R satisfies for ejj: = v — VH and i + ^ = 1, (p,q> 1), |-R(w,w»,«»)[ < C(Af)||VeH||§>2P||V«»||o,,

(3.3)

with any v and vH satisfying ||w||i l0o + ||VH||I,OO < M . The existence and the local uniqueness of the solutions of (3.1) are established in the following lemma, which is similar to [7, Theorem 5.1]. We only give proof for the case d = 2. The other cases are the same except that the estimate for the discrete Green's function changes into: | | G h | | M < C,

d = l,3.

Lemma 3.2. Assume that UCB £ W2,p{fi) with p > d is the solution of (2.2). There exists a constant Ho such that for all 0 < H < Ho, problem (3.1) has a solution UH satisfying \\UH - J W C B | | I , O C < e(QC) 1 / 2 +

H1-^,

\\UCB ~ UH\\i,oc < C(e(QC) 1 /2 + j j i - a / p ) , where PHUCB

is defined as

A(UCB;PHUCB,V)

= A{UCB;UCB,V)

for all

VeXH.

(3.5)

26

Weinan E, Pingbing Ming

Moreover, if there exists a constant rj(M) with 0 < 7/(M) < 1 such that e(QC) £

[(DAWCB(VV)-DAWcB(VW))VZdx 0 such that for 0 < H < H0, A{U

™;X,W)

sup

>CWVh for all 7 6 l f f .

W

W€XH

(3.7)

\\ \\l

Hence there is a unique solution PHU CB satisfying (3.5) and \\UCB

< CH1-^.

- PHUCB\\I,OO

Define a nonlinear mapping T: XH —> ^

(3.8)

by

i ( t / C B ; T(V), W ) = i ( t / C B ; C^CB, W ) - R(UCB,

+

V, W)

A(V,W)-AH(V,W)

for any W € XH- Obviously T is continuous. Define the set B: = {VGXH

\ ||V-PHC/cB||i)oc<e(QC)1/2 + F 1 - d / p } .

We claim that there exists a constant HQ > 0 such that for all 0 < H < Ho, T{B) C B. Notice that A(UCB;T(V)

- PHUCB, W) = -R(UCB,V, +

W) A(V,W)-AH(V,W).

Taking W = GH, where GH is the discrete regularized Green's function'12] , and using the classical estimate for the Green's function' 12 ', we obtain \\T(V) - PHUCB\\I,OO

< C\\nH\ \\UCB - V|| 2 ] 0 0 + Ce(QC)|lnff|

< C ( | | J 7 C B - i^U"cB||?,oo + WPHUCB ~ V|| 2 ; 0 0 + Ce(QC)|lnff |) < C(e(QC) + H2-2d/p <e(QG)

1/2

+

1 d p

H-/.

+ H)\\nH\

Analysis of the Local Quasicontinuum Method

27

An application of Brouwer's fixed point theorem gives the existence of UH e B such that T(UH) = UH- By the definition, UJJ satisfies (3.4)i. An application of the triangle inequality and (3.8) yields (3.4)2Suppose that both UH and UH are solutions of (3.1). Then we have r\m

C\\UH

IT II {xi)ijj(xi).

x,eLnD

For each node x of TH, define a cluster Br(x) = :{xi G L \ \xi — x\ < r}. For any domain D\, \DX denotes its characteristic function. We let all the nodes as {xi}^x, and the corresponding basis function is {i}i^iThe weight associated with the node Xi is defined as rii, and let n = (ni, • • • , UM)T- The cluster summation rule can be formulated as Bn = g,

(4.1)

where B is an M x M matrix with B^ — {i,XBr(x))L and the M x 1 vector g is defined as gi — (fa, 1)LTo get the weights we have to solve a system of M x M linear algebraic equations, which is very expensive in particular for big N. Therefore, the mass lumping is commonly employed in practice, which amounts to assembling all entries on each line of B into the diagonal entry, namely, we need to solve the following simple linear equations: Bn = g,

(4.2)

Analysis of the Local Quasicontinuum Method

29

with Bu = & / ( E j l i Ba) a n d Bij = 0 for i ^ j . With the above consideration, the energy IH is defined as M

lH(yV)

= ^2niWi(yV),

(4.3)

t=i

where Wi(VV) = ^

J^

IBrix^DKjlWcB^Vi)

is the energy associated with the i—th node, where 3A/3/(27T) is a scaling factor. Here Mi is the set of elements sharing the common node ajj. For any element K e T g , assembling the energy contribution of each vertices in K, we rewrite (4.3) into IH(VV)

Yl

= ^ n

^nKABrixJnKlWcBiWi),

KeTH i=i

where {«K,i}f=i denotes three weights associated with three vertices of the element K. If we define *K = ^Y,nK,i\Br{xi)r\K\l\K\,

(4.4)

i=l

then the energy can be rewritten as /fl(VV) = Y, ^ ^ ( V V ) . KerH This is similar to the original QC formulation. In what follows, we estimate e(QC) for the case when all elements K are equal and the lattice summation rule in [9] is employed (see Fig.4.1). We define Lo to be the number of atoms over each edge and roe the cluster radius. Lemma 4.1. If all elements K £ Tg are equal and the first order lattice summation rule of Knap and Ortiz [9] is employed, then

Proof. Let be the linear base function associated with the center of the hexagonal. A direct calculation gives

Y xecnM

0(x) = l + 6 ^ ( * - l ) ( L o + l - « ) / i o = i§. »=i

30

Weinan E, Pingbing Ming

Figure 4.1 A special cluster-based summation rule in 2-D.Here L0 = 4 and ro — 1
2)...}6-

for

(3-17)

The exact boundary condition (3.16) was obtained by Feng (1980) [13]. The exact boundary conditions (3.14) and (3.15) were given by Han and Ying [54] in 1980, using Hilbert transformations. Formulations (3.8)-(3.10) were discovered by Han and Wu (1985) [50]. The above discusion on the Steklov-Poincare mapping is mainly from the work ( [50],1985) by Han and Wu. At the same time formulation (3.8) was proposed by Yu (1985) [72]. In 1989, formulation (3.8) appeared in the paper [19] by Givoli and Keller in which formulation (3.8) is called DtN mapping. The Steklov-Poincare mappings (3.8)-(3.10) and (3.14)-(3.16) are exact boundary conditions on the given artificial boundary TR, which are global.

3.2

T h e reduced problem on t h e b o u n d e d computational domain f2j

Any one of the six equivalent Steklov-Poincare mappings can be used to reduce the original problem (P) to a problem on the bounded computational domain Qi: (Pi)

Au == f(x), u

=9(x),

(3.18) (3.19)

=SMrR)

(3.20)

\r-

du dn r

infii,

forj=l,2,---,6. For j=l,2,- • • ,6 problem(Pj) is defined on the bounded computational domain J2» and with the exact global boundary condition (3.20). It is straightforward to check that problem (Pj) is equivalent to the original

40

Houde Han

problem (P) in the following sense. If u is the solution of the original problem (P),then the restriction of u is the solution of problem (Pj); otherwise, if u is the solution of problem (Pj), then u is the restriction of the solution of problem (P). Therefore we only need to solve problem (Pj) on the bounded computational domain fij, and then we can obtain the solution of original problem (P). But we need to pay the added expense to compute the singular integrals or infinite series, which are from the exact global boundary conditions (3.20). In practical applications, we take first few terms of the series in the exact boundary conditions (3.8)-(3.10), and then we obtain a sequence of the approximate artificial boundary conditions on FR, which are also global.

N

du

Y. j

rR du dn

^«(«.*)\xi=L = i>L=

\ U00(s)ds, Jo

—— dx

-OO < Si < + 0 0 ,

= 0 , — oo < xi < +oo,

(5.5) (5.6) (5.7)

2\x2=0,L

ip = const,

7— = 0, on

on

8Q,Q,

(5.8)

I-X2

4> —> ^00(^2) = /

Uoo(s)ds, when \x±\ —• +00,

w —*• 0^00(^2) = — ^ ( x s ) , when \xi\ —> +00.

(5.9) (5.10)

When \b\,c are large, in the domains Qb and fic the flow is close to Poiseuille flow, and the N-S equations can be linearized on domain ftc

47

The Artificial Boundary Method • • • (or Qb) as . . du dp . «oo(a;2)^ k — = I'Ait, oxi axi . . dv dp "00(^2)^ 1-^—=^Aw, aa;i 0x2 9M

. in

_, ilc,

(5.11)

. in

S2C,

(5.12)

fic.

(5.13)



-5 h ^ — = 0, in OTi OT2

Therefore linear N-S equations (5.11)-(5.13) with the boundary conditions in stream-function vorticity are given as u

'oo02)^-^

u00(x2)-^-+

OX1OX2

vAw = 0,

infic,

(5.14)

in

(5.15)

OXi

A ^ + w = 0, ^1x2=0 = 0,

fic,

tp\x2=L = i>L, c < a ; i < + o o , -~\X2=Q,L

= 0, c < a ; i < + o o ,

(5.16) (5.17)

i> —»• i>oa{x2), u —> u>oo{x2), when x\ —• +00. (5.18) Since the boundary conditions on the artificial boundary Tc are unknown, problem (5.14)-(5.18) is an incompletely posed problem and it can not be solved independently. Let tp\Xl=c = 4>c{x2),

0<x2* = {4,x%),j

= 0,l,2,---,k

=

0,l,2,-.-,N

with x[ =c + j61,

j = 0,1,2, ••• ,

48

Houde Han X2=k52, fc = 0,1,2, ••• ,N.

Using the following difference approximations: U

d2^

/ i k\

U

L(X2),,

i

°°^X2' dx dx '(xJ'k) ~

45 5

Wj+1>k+1 -vj+i,k-i

- Vj-i,fc+i +ipj-i,k-i],

ksdu.

U00O2) r

Uoo(^) — \(xi,k) «

i

[u;j + i, fc -u;j-i,fej,

2

Au,U» «

^

+

52

.

A^,,*

^

+

^

,

then we obtain the infinite difference scheme: 45i5; 1!

-[>Pj+l,k+l - 1pj+l,k-l

(T

—°l\

~ i>j-l,k+l

+

1pj-l,k-l]

)

2

x

( 5 - 21 )

K'+i.fc ~ Uj-i,k]

r^j+i,fe ~ 2^j,fc + ^j-i,fc , ^j.fc+i — 2^j,fc + Uj,k-i i _ n + J U " 5? 5| - ' ^j+i, fc - 2ipjtk + V>,-_i|fc V'j.fc+i - 2^ j|fc + ^ | f c _ ! _ +

L

for j = 1,2, • • • , k = 1, • • • , N — 1 with boundary conditions V>i,o = 0, ^o = - ^ - , ^

= - ^ _

ipj,N = tpL,

3 1

3

% ^

(

^ ^ -

1 +

+

j = 0,1,2, • • • ,

l )

, l

i^O.1,2,..., )

,

j =0,1,2,...,

limj.^+oo Vj,fc = ^ o o ^ l ) . limj^+oo Wj,fe = Woo (2:2 )> ipo,k = ipc(x2), w 0 ,fc=w c (x§), fc = 1,2, - • • ,iV - 1.

(5.23) (5.24) (5.25) ( 5 - 26 ) (5.27)

Let X

J = k j . i i ' • - ' VJ,N-I; ipj,i, • • • , V'j.Ar-iF G R 2 J V _ 2 ,

X ^ = [ ^ ( 4 ) , • • • . W o o ^ - 1 ) ; V'oo(^), • • • , V o o ^ - 1 ) ] 7 " € R 2Ar ~ 2 . The infinite difference equations (5.21)-(5.27) are equivalent to the following system of linear algebraic equations including infinitely many

The Artificial Boundary Method • • •

49

unknowns X i , X2, • • •: f For given X 0 , X ^ € R2N~2, find (PD) I AoXj-i + B0Xj + C0Xj+1 = D0,

{Xi, X 2 , • • •} such that j = 1,2, • • • ,

^ hnij—xx) X j = XQO ,

where A0,BQ, CQ are three (2iV — 2) x (2iV — 2) matrices with constant elements and Do £ R 2 -^ - 2 . Ao,-Bo,Co and Do are obtained from the difference equations (5.21)-(5.22) and we know that [47] (A0 + B0 + C 0 )X o o = D0.

(5.28)

Let Yj = Xj - Xoo for j = 0,1,2, • • • . Then {Yj,j = 0, • • •} satisfies:

{

For given Y 0 6 R 2JV ~ 2 , find {Yj, j = 1,2 • • •} such that A0Yj^ + B0Yj + C0Yj+1 = 0, j = 1,2, • • • , Hindoo Yj = 0.

Problem (-Pp) can be solved numerically by a fast iteration method [47], and we obtain Y i ~ -TY0 where T e

R^2N'2^2N'2\

Return to the vectors Xo, X i , and we have X i « - T X o + (/ + T)Xoo. Let W

rduj(c,x\) dx G R2N~2.

dw{c,X2~l) ' dx

dw(c,x\) dx '

dw(c,x^~x) ' dx

T

Then approximately we obtain

W:

•1

—

Xo

Si

Finally on the artificial boundary r c we have a discrete artificial boundary condition W = ~(T

+ I)(X0-Xoo).

(5.29)

On the artificial boundary Ti, we can also obtain a similar discrete artificial boundary condition, and then the original problem can be reduced to the bounded computational domain in a finite difference formulation. Furthermore the discrete artificial boundary conditions are given on a polygonal artificial boundary for the exterior problem of Poisson equation [34], the problem of infinite elastic foundation [6] and the problem with interface [39,42].

50

6

Houde Han

Implicit boundary conditions

We now consider the typical problem (P) again: '-Au (P)
oo.

Introduce an artificial boundary T (arbitrary shape)C fl (see Fig. 6.1) such that r divides the domain fi into two parts: the bounded part fi; and the unbounded part £le and f(x) = 0,

Vxe

ne.

On the artificial boundary T, the exact boundary condition is the

Figure 6.1 Steklov-Poincare mapping, namely, for given u\r we solve the following problem: Au = 0, in £le, u = u\r,

on r ,

u is bounded when la;I —> oo.

(6.1) (6.2) (6.3)

By the solution u of problem (6.1)-(6.3), Steklov-Poincare mapping is given by du

dn r

=S(ur).

(6.4)

51

The Artificial Boundary Method • • •

Unfortunately, since the shape of the artificial boundary T is arbitrary, we can not find the explicit formulation of the Steklov-Poincare mapping in general. In this section we will discuss the implicit boundary conditions on any shape of artificial boundary T. Let A = f^|r, n denotes the outward unit normal to T = •• («> %y

The equalities (6.7) and (6.8) are two implicit boundary conditions on the given artificial boundary T and are satisfied by u(x), the solution of problem (P). By a combination of the implicit boundary (6.7) and (6.8), the typical problem (P) can be reduced to the bounded computational domain fi; with unknown functions u, A and unknown constant a [10,30]. This approach is considered as the symmetric coupling method of finite element and boundary element, which is one of the two principal classes of FEM-BEM formulation [64] introduced independently by Costabel [10] and Han [30].

7

Conclusions

The artificial boundary method has been established for computing the numerical solutions of partial differential equations on unbounded domains. The key points of this method are to find the exact boundary conditions or the approximate artificial boundary conditions on the

52

Houde Han

given artificial boundary for various problems arising in many fields of science and engineering. In general, the artificial boundary conditions can be classified into implicit boundary conditions and explicit boundary conditions including global artificial boundary conditions, local artificial boundary conditions and discrete artificial boundary condition. The explicit artificial boundary conditions are more convenient in applications, but the implicit artificial conditions can handle the artificial boundary with any shape.This method has attained successful applications in many fields in science and engineering and has shown wider and wider application prospect. In this field there are still many open problems, for example, how to solve the nonlinear partial difference equations on unbounded domains numerically? It is an interesting and important problem waiting for solving.

References [1] B. Alpert, L. Greengard and T. Hagstrom. Nonreflecting boundary conditions for the time-dependent wave equations. J. Comput. Phys., 180:270-296, 2002. [2] W. Z. Bao. The approximations of the exact boundary condition at an artificial boundary for linearized incompressible viscous flows. J. Comput. Math., 16:239-256, 1998. [3] W. Z. Bao. Artificial boundary conditions for incompressible NavierStokes equations: A well-posed result. Comput. Methods. Appl. Meth. Engrg., 188:595-611, 2000. [4] W. Z. Bao and H. D. Han. Nonlocal artificial boundary conditions for the incompressible viscous flow in a channel using spectral techniques. Journal of Computational Physics, 126:52-63, 1996. [5] W. Z. Bao and H. D. Han. Local artificial boundary conditions for the incompressible viscous flow in a slip channel. Journal of Computational Mathematics, 15:335-343, 1997. [6] W. Z. Bao and H. D. Han. The direct method of lines for the problem of infinite elastic foundation. Comput. Methods Appl. Mech. Engrg., 175:157-173, 1999. [7] W. Z. Bao and H. D. Han. High-order local artificial boundary conditions for problems in unbounded domains. Comput. Methods Appl. Mech. Engrg., 188:455-471, 2000. [8] W. Z. Bao and H. D. Han. Error bounds for the finite element approximation of an incompressible material in an unbounded domain. Numerische Mathematik, 93:415-444, 2003.

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[9] W. Z. Bao, H. D. Han and Z. Y. Huang. Numerical simulations of fracture problems by coupling the FEM and the direct method of lines. Comput. Methods Appl. Mech. Engrg., 190:4831-4846, 2001. [10] M. Costabel. Symmetric Methods for the coupling of finite elements and boundary elements, in Boundary Elements IX, volume 1. Springer-Verlag, Berlin, 1987. [11] Q. Du and X. N. Wu. Numerical solution for the three-dimensional Ginzberg-Landau models using artificial boundary. SIAM J. Numer. Anal., 36:1482-1506, 1999. [12] B. Engquist and A. Majda. Absorbing boundary conditions for the numerical simulation of wave. Math. Comput., 31:629-651,1977. [13] K. Feng. Differential vs. integral equations and finite vs. infinite elements. Math. Numer. Sinica, 2:1:100-105, 1980. [14] K. Feng. Canonical boundary reduction and finite element method. In Proceedings of International Invitational Symposium on the Finite Element Method, Hefei, 1981. Science Press, Beijing, 1982. [15] K. Feng. Finite element method and natural boundary reduction. In Proceedings of the International Congress of Mathematicians, pages 1439-1453, Warszawa, 1983. [16] K. Feng. Asymptotic radiation conditions for reduced wave equation. J. Comput. Math., 2:2:130-138, 1984. [17] K. Feng and D. H. Yu. Canonical integral equations of elliptic boundary value problems on the finite element method. In Proceedings of International Invitational Symposium on the Finite Element Method, pages 211-252, Beijing, 1982. Science Press, Beijing, 1983. [18] G. N. Gatica, L. F. Gatica and E. P. Stephan. A FEM-DtM formulation for a non-linear exterior problem in incompressible elasticity. Math. Meth. Appl. Sci., 26:151-170, 2003. [19] D. Givoli and J. B. Keller. A finite element method for large domains. Comput. Methods. Appl. Meth. Engrg., 76:4-66, 1989. [20] D. Givoli and J. B. Keller. Nonreflecting boundary conditions for elastic waves. Wave Motion, 12:261-279, 1990. [21] D. Givoli and J. B. Keller. Special finite elements for use with highorder boundary conditions. Comput. Methods. Appl. Meth. Engrg., 119:199-213, 1994. [22] D. Givoli, I. Patlashenko and J. B. Keller. High-order boundary conditions and finite elements for infinite domains. Comput. Methods. Appl. Meth. Engrg., 143:13-39, 1997.

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[23] C. I. Goldstein. A finite element method for solving Helmholtz type equations in waveguides and other unbounded domains. Math. Comput, 39:309-324, 1982. [24] I. S. Gradshteyn and I. M. Kyzhik. Tables of Integrals, Series and Products, Sixth Edition. Academic Press, 2000. [25] M. J. Grote and J. B. Keller. On nonreflecting boundary conditions. J. Comput. Phys., 122:231-243, 1995. [26] T. Hagstrom, S. I. Haraharan and D. Thompson. High-order radiation boundary conditions for the convective wave equation in exterior domains. SI AM J. Sci. Comput, 25:1088-1101,2003. [27] T. Hagstrom and H. B. Keller. Exact boundary conditions at an artificial boundary for partial differential equations in cylinders. SIAM J. Math. Anal., 17:322-341, 1986. [28] T. Hagstrom and H. B. Keller. Asymptotic boundary conditions and numerical methods for nonlinear elliptic problems on unbounded domains. Math. Comput, 48:449-470, 1987. [29] L. Halpern and M. Schatzman. Artificial boundary conditions for incompressible viscous flows. SIAM J. Math. Anal, 20:308-353, 1989. [30] H. D. Han. A new class of variational formulations for the coupling of finite and boundary element methods. Journal of Computational Mathematics, 8:223-232, 1990. [31] H. D. Han. Boundary integro-differential equations of elliptic boundary value problems and their numerical solutions. Scientia Sinica, 31:1153-1165, 1998. [32] H. D. Han and W. Z. Bao. An artificial boundary condition for the incompressible viscous flows in a no-slip channel. Journal of Computational Mathematics, 13:51-63, 1995. [33] H. D. Han and W. Z. Bao. The artificial boundary conditions for incompressible materials on an unbounded domain. Numerishe Mathematik, 77:347-363, 1997. [34] H. D. Han and W. Z. Bao. The discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of Poisson equation by using the direct method of lines. Comput. Methods Appl. Mech. Engrg., 179:345-360, 1999. [35] H. D. Han and W. Z. Bao. Error estimates for the finite element approximation of problems in unbounded domains. SIAM J. Numer. Anal, 37:1101-1119, 2000.

The Artificial Boundary Method • • •

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[36] H. D. Han and W. Z. Bao. Error estimates for the finite element approximation of linear elastic equations in an unbounded domain. Mathematics of Computation, 70:1437-1459, 2001. [37] H. D. Han, W. Z. Bao and T. Wang. Numerical simulation for the problem of infinite elastic foundation. Computer Methods in Applied Mechanics and Engineering, 147:369-385, 1997. [38] H. D. Han, C. H. He and X. N. Wu. Analysis of artificial boundary conditions for exterior boundary value problems in three dimensions. Numer. Math., 85:367-386, 2000. [39] H. D. Han and Z. Y. Huang. The direct method of lines for the numerical solutions of interface problem. Comput. Methods Appl. Mech. Engrg., 171:61-75, 1999. [40] H. D. Han and Z. Y. Huang. A semi-discrete numerical procedure for composite material problems. Mathematical Sciences and Applications, 12:35-44, 1999. [41] H. D. Han and Z. Y. Huang. The direct method of lines for incompressible material problems on polygon domains. In T. Chan, T. Kako, H. Kawarada and O. Pironneau, editors, 12th International Conference on Domain Decomposition Methods, pages 125132, 2001. [42] H. D. Han and Z. Y. Huang. The discrete method of separation of variables for composite material problems. International Journal of Fracture, 112:379-402, 2001. [43] H. D. Han and Z. Y. Huang. A class of artificial boundary conditions for heat equation in unbounded domains. Computers & Mathematics with Applications, 43:889-900, 2002. [44] H. D. Han and Z. Y. Huang. Exact and approximating boundary conditions for the parabolic problems on unbounded domains. Computers & Mathematics with Applications, 44:655-666, 2002. [45] H. D. Han and Z. Y. Huang. Exact artificial boundary conditons for Schrodinger equation in R2. Comm. Math. Sci., 2:79-94, 2004. [46] H. D. Han, Z. Y. Huang and W. Z. Bao. The discrete method of separation of variables for computation of stress intensity factors. Chinese J. Comput. Phy., 17:483-496, 2000. [47] H. D. Han, J. F. Lu and W. Z. Bao. A discrete artificial boundary condition for steady incompressible viscous flows in a no-slip channel using a fast iterative method. Journal of Computational Physics, 114:201-208, 1994. [48] H. D. Han and X. Wen. The local artificial boundary conditions for numerical simulations of the flow around a submerged body. Journal of Scientific Computation, 16:263-286, 2001.

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[49] H. D. Han and X. Wen. The global artificial boundary conditions for numerical simulations of the 3d flow around a submerged body. Journal of computational Mathematics, 21:435-450, 2003. [50] H. D. Han and X. N. Wu. Approximation of infinite boundary condition and its applications to finite element methods. Journal of Computational Mathematics, 3:179-192, 1985. [51] H. D. Han and X. N. Wu. The mixed finite element method for stokes equations on unbounded domains. Journal of Systems Sci.and Mathematical Sci, 5:121-132, 1985. [52] H. D. Han and X. N. Wu. The approximation of the exact boundary conditions at an artificial boundary for linear elastic equations and its application. Mathematics of Computation, 59:21-37, 1992. [53] H. D. Han and X. N. Wu. A fast numerical method for the blackscholes equation of american options. SIAM J. NUMER. ANAL., 41:2081-2095, 2003. [54] H. D. Han and L. A. Ying. Large elements and the local finite element method. Acta Mathematicae Applicatae Sinica, 3:237-249, 1980. [55] H. D. Han and C. X. Zheng. High-order local artificial boundary conditions of the exterior problem of poisson equations in 3-d space. Numer. Math. (A Journal of Chinese Universities), 23:290304, 1999. [56] H. D. Han and C. X. Zheng. Mixed finite element and high-order local artificial boundary conditions of elliptic equation. Comput. Methods Appl. Mech. Engrg., 191:2011-2027, 2002. [57] H. D. Han and C. X. Zheng. Mixed finite element method and higher-order local artificial boundary conditions for exterior 3-d Poisson equation. Tsinghua Science and Technology, 7:228-234, 2002. [58] H. D. Han and C. X. Zheng. Exact nonreflecting boundary conditions for acoustic problem in three dimensions. Journal of computational Mathematics, 21:15-24, 2003. [59] S. D. Jiang and L. Greengard. Fast evaluation of nonreflecting boundary conditions for the Schrodinger equations in one dimension, journal. [60] C. Johnson and J. C. Nedelec. On the coupling of boundary integral and finite element methods. Math. Comp.,, 35:1063-1079, 1980. [61] J. B. Keller and D. Givoli. Exact nonreflecting boundary conditions. J. Comput. Phys., 82:172-192, 1989.

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[62] A. Kirsch and P. Monk. A finite element method for approximating electromagnetic scattering from a conducting object. Numer. Math., 92:501-534, 2002. [63] Z. P. Li and X. N. Wu. Multi-atomic Young measure and artificial boundary in approximation of micromagnetics. Appl. Numer. Math., 51:69-88, 2004. [64] S. Meddahi, M. Gonzalez, and P. Perez. On a FEM-BEM formulation for an exterior quasilinear problem in the plane. SIAM J. Numer. Anal, 37:1820-1837, 2000. [65] F. Nataf. An open boundary condition for the computation of the steady incompressible Navier-Stokes equations. J. Comput. Phys., 85:104-129, 1989. [66] T. Ushijima. And FEM-CSM combined method for planar exterior Laplace problems. Japan J. Indust. Appl. Math, 18:359-382, 2001. [67] X. N. Wu and H. D. Han. A finite-element method for Laplace and Helmholtz type boundary value problems with singularities. SIAM J. Numer. Anal, 34:1037-1050, 1997. [68] X. N. Wu and H. D. Han. Discrete boundary conditions for problems with interface. Comput. Methods Appl. Mech. Engrg., 190:49874998, 2001. [69] X. N. Wu and Z. Z. Sun. Convergence of difference scheme for heat equation in unbounded domains using artificial boundary conditions. Appl. Numer. Math., 2004:261-277, 50. [70] D. H. Yu. Canonical integral equations of biharmonic elliptic boundary value problems. Math. Numer. Sinica, 4:330-336, 1982. [71] D. H. Yu. Numerical solutions of harmonic and biharmonic canonical integral equations in interior or exterior circular domains. J. Comput. Math., 1:52-62, 1983. [72] D. H. Yu. Approximation of boundary conditions at infinity for a harmonic equation. J. Comput. Math., 3:219-227, 1985. [73] D. H. Yu. Canonical integral equations of Stokes problem. J. Comput. Math., 4:62-73, 1986. [74] D. H. Yu. The approximate computation of hypersingular integrals on interval. Numer. Math. J. Chinese Univ., 1:114-127, 1992. [75] D. H. Yu. The coupling of natural BEM and FEM for Stokes problem on unbounded domain. Math. Numer. Sinica, 14:371-378,1992. [76] D. H. Yu. The mathematical theory of the natural boundary element method. In Monograph on pure mathematics and applied mathematics, No. 26. Science Press, Beijing, 1993.

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[77] D. H. Yu. The computation of hypersingular integrals on a circle and its error estimates. Numer. Math. J. Chinese Univ., 16:332-337, 1994. [78] D. H. Yu. Natural Boundary Method and Its Applications. Kluwer Academic Publishers and Science Press, 2002. [79] C. X. Zheng and H. D. Han. High-order artificial boundary conditions for ideal axisymmetric irrotational flow around 3d obstacles. International Journal of Numerical Methods in Engineering, 54:1195-1208, 2002. [80] C. X. Zheng and H. D. Han. Artificial boundary method for exterior Stokes flow in three dimensions. International Journal for Numerical Method in Fluids, 41:537-549, 2003.

59

Optimal order integration on the sphere Kerstin Hesse, Ian H. Sloan School of Mathematics, Sydney

The University NSW

2052,

of New South

Wales,

Australia.

Abstract This paper reviews some recent developments in cubature over the sphere S2 for functions in Sobolev spaces. More precisely, for an m-point cubature rule Qm we consider the worst-case (cubature) error, denoted by E(Qm; Hs), of functions in the unit ball of the Sobolev space H" = HS(S2) with s > 1. The following recent results are reviewed in this paper: For any sequence (Q m ( n ))neN of positive weight m(n)-point cubature rules Q m ( n ), where Qm(n) integrates all spherical polynomials of degree < n exactly, the worstcase error in H" satisfies the estimate E(Qm(ny, Hs) < cs n~s with a universal constant c„ > 0. Whenever m{n) = 0(n2) we deduce E{Qm(n)\ H") < c3 m(n)~3'2, where the constant cs now depends on the constant in m(n) = 0(n2). This rate of convergence is optimal since it has also been shown that there exists a universal constant c 3 > 0 such that for any m-point cubature rule Qm, the worst-case error in H" with s > 1 satisfies E(Qm; H3) > c 3 m~s^2. For example, sequences (Qm(n)) of positive weight product rules with m(n) = 0(n2) achieve the optimal order of convergence 0(m{n)~3!2). So too, if the weights are all positive, do sequences ()dtd) '•= / ( V l — t2 cos(p, V l - t2 sin ,£), the most natural integration rules that come to mind are so-called product rules, the 'product' of two one-dimensional integration rules. (The substitution t = cos 6, 6 G [0,7r], gives the usual polar coordinates.) For our purposes product rules of the following type are of interest:

where Qm(n)

1S

the product of

which integrates all trigonometric polynomials of degree < n exactly and a rule n'

„i

5^MiM*0» / • ' -

»=i

h(t)dt,

(2.2)

1

which is assumed to have positive weights \ii, i = 1 , . . . , n', and to integrate all algebraic polynomials of degree < n exactly. We also demand that n' = 0(n) in order that m(n) = (n + l)n' = 0(n2). There are many possible choices of rule (2.2) with the mentioned properties (see for example [2,15]). The product rule Qm(n) u s e s Tn(n) = (n + l)n' = 0(n2) points, and the properties of the two one-dimensional rules imply that the product rule (2.1) has positive weights, and that it integrates any spherical polynomial of degree < n exactly. More precisely, Qm(n)P = IP

Vp € P„,

where P„ is the space of all spherical polynomials of degree < n, that is, the restriction to S2 of all polynomials on R 3 of degree < n. The dimension of P n is dn := dim(P„) = (n + 1) 2 . An unattractive feature of product rules is that they have an uneven point distribution: the points cluster at the poles. In contrast to the point sets of product rules, so-called extremal fundamental systems possess a much nicer geometric point distribution. Let {3>j | j = 1 , . . . , dn} be any basis of P„. A fundamental system of degree n G No is a point set {x, | i = 1 , . . . , dn} of dn points for which d e t l ^ x O l ^ ^ ^ O .

(2.3)

The interpolation problem based on a fundamental system is: given a continuous function / , find L n / G P„ such that Lnf{xi)

= /(xj),

i = l,...,dn.

(2.4)

62

Kerstin Hesse, Ian H. Sloan

It follows from (2.3) that this problem always has a unique solution. (Note that Ln : C(S2) -» P„ is a projector, that is, L„ = Ln, and in particular Lnp = p for all p e P n .) Of course, the interpolation problem (2.4) can be arbitrarily badly conditioned if the condition number of the matrix [$j(x»)]i,j=i,...,d„ is large. A fundamental system {xt \i = 1,... ,dn} (of degree n) is called an extremal fundamental system (of degree n) if it maximizes the determinant of the interpolation matrix, that is, if det [$j(xj)l. . ,

. =

max

det [,• (y,)]. . ,

. . (2.5)

It should be noted that both (2.3) and (2.5) do not depend on the choice of the basis {$j | j = 1 , . . . , dn} of P„. The interpolatory cubature rule with respect to an (extremal) fundamental system of degree n is defined by integrating the interpolating polynomial Lnf e P„ of a continuous function / exactly, that is, Qm(n)f-= ! Lnf{x)du>(x). JS2

(2.6)

The interpolant Lnf S P n can be represented with the help of the Lagrange polynomials lj e P n ) j = 1, • • • , 4 , defined by £j(xi) = 5ji for % ^ 1 , . . . , dni as

Lnf = JT/f(xj)ej.

(2.7)

Substituting (2.7) into (2.6) yields Qm(n)f = Y]wjf(xj)

with

Wj ••= £j(x)dw(x).

(2.8)

Prom the definition of the weights in (2.8) it is not at all clear whether the weights are non-negative. However for the computed extremal fundamental systems up to degree n = 50 the computed weights uij are all positive and lie in the interval [\ j 1 , | j1-] (see [14]). This is strong numerical evidence that interpolatory cubature rules with respect to extremal fundamental systems have positive weights, although we do not have a proof. The cubature rule Q m ( n ), given by (2.8), uses m(n) = (n+ l ) 2 = 0(n2) points, and from (2.6) it is clear that it integrates all p e P n exactly. Extremal fundamental systems lead to a very well-conditioned polynomial interpolation problem (2.4). They can also possess a very nice geometric point distribution, as shown by the following two results. In [11, Theorem 5.1] Reimer, generalizing an argument from [17], showed that

Optimal order integration on the sphere

63

Figure 2.1 GauB-Legendre product rule points (left) and extremal fundamental system (right) for degree n = 41.

for any cubature rule that is exact for polynomials of degree < n, and that has points {x, | i = 1 , . . . , m} and positive weights, max

min

xGS 2 j = l , . . . , m

arccos(x • x,) < arccos(z n ),

(2.9)

where zn is the largest zero of the Legendre polynomial P\n/i\ • The expression on the left of (2.9) is called the mesh norm. From the known asymptotics of the zeros of the Legendre polynomials we have , , 4.8096 arccos(z n ) ~ . n Essentially (2.9) tells us that there are no 'large holes' in extremal fundamental systems if the corresponding weights are positive. The other result (due to [10], using a result of [12]) asserts that the points of an extremal fundamental system are well separated, in that for an extremal fundamental system {x* | i = 1 , . . . , dn} of degree n, 7T

arccos(xi • x,) > — 2n

for j , i = 1 , . . . , dn, with i ^ ?'.

Figure 2.1 illustrates that extremal fundamental systems have a much nicer geometric point distribution than the point sets of product rules. In terms of the point number m(n) = 0(n2) and the degree of polynomial exactness n a product rule (2.1) and the interpolatory cubature rule (2.8) with respect to an extremal fundamental system (of degree n) have comparable properties.

64

3

Kerstin Hesse, Ian H. Sloan

Optimal order estimates for cubature in Sobolev spaces

To formulate and explain our cubature results we need some notations. For more details the reader is referred to [3,8,10]. The restriction to S2 of airy homogeneous harmonic polynomial of exact degree t is called a spherical harmonic of degree £. The space of all spherical harmonics of degree £ has the dimension 2£ + 1, and we choose an orthonormal set {Y^ \k = 1 , . . . , 2^+1} of spherical harmonics of degree £ with respect to the L2(S2) inner product ( / . 5 ) £ 2 = / 2 /(x)s(x)dw(x). Js Then U^ 0 {l^fc | k = 1 , . . . , 2£ + 1} is a complete orthonormal set in L2{S2), and n

P„ = span \J{Yek \k =

l,...,2£+l}.

Any / € L,2(S2) can be expanded into a Fourier series (or Laplace series) with respect to this complete orthonormal set of spherical harmonics: oo 2^+1 f=0 fc=l

with the Fourier coefficients given by ftk-=

/ Js2

f(x)Yik(x)(Lj(x).

The equality in (3.1) is to be understood in the -^(-S^-sense. The Sobolev space Hs = HS(S2), s > 0, is now defined as the completion of s p a n ( J ^ l 0 P n with respect to the norm

(

oo 2t+X

\

1

/2

e=o fe=i / The space Hs is a Hilbert space with the inner product oo 2t+l €=0 fc=l

which induces the norm (3.2). Clearly, H° = L2(S2). For s > 1 the space Hs is embedded into the space C(S2) of continuous functions on S2,

Optimal order integration on the sphere

65

endowed with the supremum norm, and in this situation the Sobolev space Hs is also a reproducing kernel Hilbert space, a fact which plays an important role in the proof (see [5,6]) of the upper bound of the worst-case cubature error (see Theorem 3.1 below). Numerical experiments (see [14]) showed that for typical n up to degree 191 E(Qm{n);H3/2)

« 3.217 (n + l)" 1 - 5 0 0 6 = 3.217m(nT 0 - 7 5 0 3 ,

where Qm{n) is the interpolatory cubature rule with respect to an extremal fundamental system of degree n. Note that 1.5006 ss 3/2. That experimental result motivated first the conjecture and then the proof of the following theorem from [5,6]: Theorem 3.1. For each s > 1, there exists a constant cs > 0 such that for any m-point cubature rule Qm, which has positive weights and satisfies Qmp = Ip for all p € P n , the worst-case cubature error in Hs satisfies the estimate E(Qm;Hs) 1, but using the knowledge of this representation we conjectured in [6] an analogous result for arbitrary s > 1, and then verified that result, enabling us in [6] to prove Theorem 3.1 for general s> 1. For a sequence (Qm(n)) of m(n)-point cubature rules with the property Qm(n)P = Ip f° r ail p € P„ we have always m{n) > en2 for some constant c > 0, that is, n2 is the lowest possible order of m(n). If m(n) = 0(n2) for a sequence {Qm{n)) of rules satisfying the conditions in Theorem 3.1 then E(Qm(ny,Hs) 0 depends on the constant in m{n) = 0(n2). The following result, from [7], shows that (3.4) is indeed the optimal order of convergence. Theorem 3.2. For each s > 1, there exists a constant cs > 0 such that for any m-point cubature rule Qm on S2, E{Qm-Hs)>csm-s'2.

(3.5)

Theorem 3.2 is a 'negative' result that shows the limitations of m-point cubature rules in Hs. The estimate (3.5) is sharp (or optimal) because Theorem 3.1 with the additional assumption m{n) = 0(n2) identifies sequences of cubature rules that achieve this optimal rate of convergence. We now give a brief sketch of the proof of Theorem 3.2 in [7]. The proof was inspired by the method of the proof for lower bounds for the worst-case cubature error in certain spaces of continuous functions on the unit cube [0, l ] 2 (see [1,9]). Sketch of the proof of Theorem 3.2 The idea is to construct for each m-point cubature rule Qm a function fm 6 Hs such that 7^-TT

\Qmfm ~ Ifm\

> Cs m " ^ 2

(3.6)

\\Jm\\s

with a constant cs > 0 that is independent of Qm and m but depends on s. As the cubature error of / m / | | / m | | s is a lower bound for E(Qm; Hs), (3.6) implies (3.5). First, we pack the sphere S2 with 2m spherical caps S(yj,am)

:= { z e S2 | arccos(z • yj) < am} ,

j = 1 , . . . , 2m, where the spherical angle am satisfies ci (2m)" 1 / 2 %%*) \bl'ddxQmkJ

( dJ^k \)

+ ~bn}a£l -

V^bV)d(^Kk)K^d^k)

+(brna$n + &n a&m - ifflbh)d{dxQnk) --—*

)

+ (6m Omn + &n Onm ~ bfflbn

Hffiffl

Kid{dxQmk) •

)d{dxPnk)

- bttdttl - b^~a{xi)d(d^Qnk)T

T

K0d(dxQmk)

K0d{dxPmk). (2.32)

A Survey of Multi-symplectic Runge-Kutta • • •

83

If for k = 1,2, • • •, r and m = 1,2, • • •, s, b£)=bV=bk,

~bW =~b% = ~hm.

(2.33)

Then the corresponding multi-symplectic conservation law of the method (PRK) is s

r

bk5xKk = 0.

Ax ^2 bmSttOm + At ^ m=l

(2.34)

fe=l

Consequently, in this case, it is sufficient for (2.34), which holds, that 7i = 0

and

h = 0,

(2.35)

where

h= (A*)2 E ^ M M ? + b?aiS -^b^)d(dtPmj)TM1(i(dtPmk) +(4 2) 4? + bfafk ~ bk2)bf)d{dtQmj)TM2d{dtQmk) +(4 2) 4? + bfafk - b^h^dijhP^j) M0d(dtQmk) +(bf)bk1) - bfVj* -

b^a^did^u/ModidtPrnk)) (2.36)

and

h= (Ax)2 E m „ = l ( ( ^ ) « ^ + ^ 2 ^ ' ——

+(fem)a£& + 6l2)ai2^ - b$b$)d(dxQnk) { )

&^)d(dxPnk)TK^d^rnk)

T

K2d(dxQmk)

——~——

T

+ ( ^ 0 ^ 7 1 + i f t S - V$b k )d(dxPnk)

K0d(dxQmk)

+ (&n2)6m) - ftrn'omn ~ b^Okm)d{dxQnk)

K0d(dxPmk)). (2.37)

We let (Mi)fcj (M2)fcj

-W>.

- 6 (1) a (1) + 6 (1) a (1) -- °fe % + °j ajk J,(2)-(2) , u(2) (2) _ - °fc % + °j ajk

-bfbf\ -b?b?\ { 4- ftWfiW -b X\

- (2) (l) (l) (2)_ (M3)fcj ~b °k akj ++b °j a ajk ("l)ran (^)mn (^Imn

= 6 (1) a( 1 ) =

"m amn

u%w - b{1)h{1) - fe(1)s(1) - o

(3-24) C3 2«rt

£(2)r(2)

(onp.\

?(2)~(2)

E(2)-(2) _ „

l(l) _ £(2) _ 1(1) _ 1(2) _ I "m — "m — "TO — "m — "tin

to o 7 \ V°"^V

Jialin Hong

90

bi1] = b^ = bk,

(3.28)

for all A;, j = 1 , . . . , r, m, n = 1 , . . . , s, then the method (PRK1) is multi-symplectic with a discrete conservation law

Az E m = i M * ? m A dpi - dq°m A dp°m)+ A* £ L i bk(dqk A dvk - dq% A du§ + dpj A dwf - dp§ A dwft) = 0. (£>CL1) The outline of the proof. The method (PRK1) implies d(dtQmk) A dPmk + dQmk A d(dtPmk) + dQmk A d(dxVmk) + dPmk A d(dxWmk)

= 0.

After we give the variational equations, a straightforward calculation leads to

dqm A dpm - dq%, A dp°m = AfLUiipPdidtQmk)

A dPmk + bk2)dQmk A d(dtPmk))

+(A*)2 £ U i ( 6fc1)6S2) - hi1)ak] - bf^MdtQmk)

A d(3tPmfe).

Similarly, we have dqk A dvk - dq£ A dv£ = Ax £ „ + (Ax)2 ^

(

W

-

fc^fi^,

= 1

V$dQmk

A d{dxVmk)

- VP&fnWxQn*)

A

d(dxVmk)

and dpk A dwk - dpg A dwft = Ax J2m=i bm dPmk A

d(dxWmk)

+ ( A x ) 2 Em „ = 1 $ M 2 ) - & M » - ^ ^ J d ^ Q m f c ) A

d(dxWmk).

Therefore, the conditions (3.24-3.28) imply the discrete conservation law (DCL1). This completes the proof.

A Survey of Multi-symplectic Runge-Kutta • • •

3.2

91

Multi-symplecticity of Runge-Kutta-Nystrom methods for the Schrodinger equations

In this subsection we restate the main results in [16]. For (3.2-3.3), in ^-direction by applying an s-stage Nystrom method over [-L, L] with coefficients {am,j},{bm}, {/?m} and cm — E j = i a m ? ' and in t-direction by applying an r-stage Runge-Kutta method over [0, At] with coefficients {o-k,i\, {bk} and dk = E J = i ^kj, it is concluded that

Qtm = Ql + CmAxvf

+^2 E;=I amii-dtPti - vaQfj)2+(Pi^Qt), P k

l

m

= Pf + CmAxwf

+Ax2ZUa m^dtQij-V'dQir v

l+l

= V? +

wf+^wf +

+ iPtj^Ptj),

^ZSm=I bmi-dtPtm - V{{Qlm? +

(Pt^QtJ,

Ax^ =1 bm(dtQlm - V'iiQtJ* + {Pt,mf)Ptm\ Axv

Qi+i = 1i +

i

+Ax 2 E; = 1 /? m {-dtPtm ~ v'WtJ2

pf+1 = pf +

+ {P£m)2)Qtn),

Axw

t

+A^2 EJUl PmidtQtm - V'HQU2 + {Pt,m?)P?,m), -Qlm + pfc

1l,m

A

tEWakidtQlm,

= Plm + ^EU~^idtPi,m, = $m + &tZl=1bkdtQtm>

P},m= plm +

^EUhdtPlkm.

The method in the above box is denoted by (RKN1). The notations above are in the following sense: Qfm « q((l+cm)Ax, J fe At), qf w q(lAx,dkAt), $ Q * m « $g((Z + c m )Az, d fc At), J ^ m « p((f + c m )Ax,d f e At), pf « P(lAx,dkAt), $P,* m « ftp((i + c m ) A z , 4 At), i f w dxv(lAx, 4 A t ) .

92

Jialin Hong The variational equations corresponding to (RKNl) are dQlm = dqk + cmAxdvk + Ax2 J2sj=1 amjdUfj,

(3.29)

dPkm = dpk + cmAxdwk + Ax2 £ * = 1 OmjdV^,

(3.30)

dvk+1 = dvk + Ax £ m = 1 bmdUkm,

(3.31)

dwk+l = dwk + Ax £ m = 1 bmdVkm,

(3.32)

dqk+1 = dqk + Axdvk + Ax2 ESm=i PmdUkm,

(3.33)

dpk+1 = dpk + Axdwk + Ax2 £ m = 1 pmdVkm

(3.34)

dQtm = dq?,m + At £ [ = 1 akiddtQlm,

(3.35)

dPtm = dplm + A* £!=i 5feiddtPz*m,

(3.36)

rfff/U = dQi,m + A* E U MdtQf,ro, *j,m = dP?,m + At E L i ^

^

(3.37) (3-38)

Here 'ere dUfj = -ddtPfc - V'dQl)2 -V'HQlf

+ (PfjWQfjVQfjdQ^ +

iP^dQl,

dVft = ddtCtfj - V"((Qkj)2 + (P^P^Q^dQ^ -V\{Q^)2

+

+ 2 i * dP&)

+ 2P£dP£)

{Ptj)2)dP^.

Theorem 3.2.[16] In the method (RKNl), if Pm = bm{l-Cm), bm(/3j-amj) = bj(Pm-ajm), for m,j = 1,2,.. .s, and hbi = bkhki + haik, for i, k — 1,2,.. .r, then the method (RKNl) is multi-symplectic with the discrete multisymplectic conservation law r

Ax J2 bk[dqi+l A dvk+l - dqk A dv? + dpk+1 A dwk+1 - dpk A dwk] fc=i s

+ At H bm[dqlm A d p ^ - d^°m A d p £ j = 0. m=l

A Survey of Multi-symplectic Runge-Kutta • • •

93

Proof. It follows from the variational equations that dft+i A dv,*+1

= dq1^ A dvf + Ax Y bmdqf A dU^m s

+ Axdvf A dvf + Ax2 Y

bmdv\ A dV^m

s

s

s

+ Ax2 J2 PmdUtm A dvf + Ax* J2 P™ Y W * m A dtf&. m=l

ro=l

j=l

We solve dgf from (3.29) and insert it into the second part of the above expression. This yields dqi+i A dv,fc+1 S

= dq\ A dv\ + Ax Y, bmdQlm A dtfm m=l s

+ Ax2 Y

(-bmcm + bm~ Pm)dvf A dU£m s

+ Ax3 Y

( M & - amj))dUtj A dU^m.

m,j=l

By using the conditions /3 m = bm(l - cm) and 6j(/?m - o,-m) = 6m(/3j ), we obtain s

dft+i A dv,fc+1 = dg,fe A du,fc + Aa; ^

bmdQlm

A d^ f e m .

m=l

Similarly, it can be concluded that * ! + i A dt«f+1 = dp? A dwf + Aa; Y ftmdiftn A dV£m. From the combination of the above two equations, it follows that m A dplm + At £ bk{ddtP^m A dQlm + P*m A ddtQ\m) k=\ r

+ At2 J2 (i>kbi ~ biaik - bkaki){ddtQlm A ddtPi,m)k,i=l

In terms of the condition bkbi — bidik — bkaki = 0, the corresponding conservation property is obtained: d

Qi,m A dplm - dqlm A dp?>m

= At E L i hiddtP^ A dQlm + P*m A ddtQlJ. Combining the above results yields the discretized multi-symplectic conservation law r

Ax ] T bk[dqk+1 A dvk+1 - dqk A dv? + dpk+1 A dw?+1 - dp? A dwk] fe=i s

+ At Y, bm[dqtm A dplm - dqlm A dp£m] = 0, which is what we want. The following result reveals an interesting intrinsic character of (RKN1): Theorem 3.3.[16]/n the method (RKN1), assume that (3m = bm(l-Cm),

bm((3j-amj)

= bj(f3m-ajm),

for m,j = 1,2,.. .s,

and bkh = bkaki + haik, for i, k = 1,2,.. .r. Then for equation (3.1) with periodic boundary condition or zero boundary conditions, i.e., V'jv = V'oi 9xip% = dxipk or ip^ = ipk = 0, the method satisfies the discrete charge conservation law, that is, N-l

s

Yl Yl hmWlm? 1=0 m=l

= COnst.

A Survey of Multi-symplectic Runge-Kutta • • • Proof.

95

The method (RKN1) implies

W, J 2 - KJ2 = i>l„M,m - < m < = AtJ2h(^mdM,m

+d^lm^J

fe=l

+ At2 J ^ (bkk - bkaki -

biaik)dt*i,mdtV\,m.

i,k=l

By using bkbi = bkaki + biCHk and taking the sum of the above equation over the spatial grid points, we can get

7"

~

= At E h Eilo1 EJU M*U$*L +

dt^m^j.

fc=l

On the other hand, it follows that

Wf+1)tf+i - Wf)tf

bxtftf + AxJ2 M*f, Jd***t m=l

+Az 2 ^

/M0***?, m )^ + Ax2 £

m=l

+Az3 £

&m(l -

cjtfd^

m=l

M&-amj)(&**f,jR**tn.

m,j=l

By using /3m = bm(l - cm), we have M* + i)tf + i - ( # ) t f = Ax|^|2 + A x ^ 6 m ( * f ) m ) a x x * t m=l

+2A* 2 J ] A„»((a i x ** m )0f) m—1

+Ax 3 Y^ bm(Pj ~

amj){dxx*?j)dxx*?im,

m,j=l

where 3i(u) denotes the real part of the complex u.

96

Jialin Hong Secondly, we can get

M+i)#+i - M*)# = W+i)ti+1 - Wfk\Ak M

Axltff +

Ax^b^lJd^l m=l

+2Az2 £ /^((Sx**?, J $ ) m=\

+Ax3 £

bjiPm-ajmXd^l^y?^.

m,j=l

Subtracting (3.6) from (3.5), summing the results over the spatial grid points and use the condition bj([3m — o-jm) = bm(0j — amj), we obtain N-l (=0 N-l

s

=E E

bmm,jdxx*im-(dxxytj*tj.

i=0 m = l

We know that *m satisfy the nonlinear Schrodinger equations lu

t^l,m

u

xx^ltm

^

v

)vl,m-

\\^l,m\

It follows from the above equalities that

Combining the above equalities leads to

E E Mltf,Ja " Kmf) r

= -iAt^hl^N

-i%4)

- (^N

- N = il>o, O

or

^

=

^o=0.

then

£ £ MlViJ 2 -1<J2) = o. /=0 m = l

This completes the proof. This result shows that the multi-symplectic Runge-Kutta-Nystrom methods preserve exactly the important normalization in quantum physics.

A Survey of Multi-symplectic Runge-Kutta • • •

4

97

Multi-symplectic Runge-Kutta methods for one dimensional Dirac equations

In this section, we consider the one-dimensional nonlinear Dirac equation (2.43). Under appropriate conditions, (2.43) has the conservation of the charge Q, the linear momentum V and the energy E, where

' eW0(*) = /fl(hM*,t)l2 + \Mx,t)\2)dx, < V(m) = / « 9 ( ^ i ^ i + *h&ih)dx,

(4.1)

, E^)(t) = /« W i | ^ 2 + th&fr) + /(iV'il2 - \ih\2))dx, where 9(u) and u denote the imaginary part and the conjugate of the complex u, respectively, and / is a primitive function of / , namely,

/(*)= f fir)dr.

Jo In the sequel, we will focus on an important particular case of (2.43) ^

+ d-t + im^i + 2 i A ( | ^ | 2 - |Vi|2)V>i = 0, (4.2) 2

2

^T + ^ t - *™/>2 + 2iA(|Vi| - l ^ l ) ^ = 0, that is, f(s) = m — 2As in (2.43), where m and A are real constants. The results obtained can be easily extended to the general case (2.43). Proposition 4.1 [20] If the solution ip of the Dirac equation (4.2) satisfies lim

\ip(x,i)\=0,

uniformly for

t G R,

(4.3)

\x\—++oo

then Q(V)(t) = Q(),

(4.4)

where Q W ( t ) = f(\n - 5 ( 4 ) - h t fc=l

L

(4.47) bm{dxZt)TVzS{Zkm)

m=l

Following the second half of the proof of Theorem 4.2 leads to

\S(z1) - S(z*) -hJ2

bm{dxZkm)TVzS{Zkm)\ < Ch\

(4.48)

A Survey of Multi-symplectic Runge-Kutta • • •

107

According to (4.47) and (4.48), we have |M l e | < Crh3,

(4.49)

where C is a constant as stated above. Remark 4.5. As same as stated in Remark 4.2, M\e is the discrete approximation of the integration (4.44) of the local momentum conservation law. Similarly, we make use of

M. =

^J'(4J-/a)) + ^ t > (Q«)-e(4» m=l

(450)

fc=l

to approximate the momentum conservation law (1.12), and under the assumptions of Theorem 4.5, we have |M,;| < Ch?,

(4.51)

where C is a constant as same as mentioned in (4.46) or (4.49). Remark 4.6. As same as stated in Remark 4.3, the local error estimates (4.46) and (4.51) can be extended to any multi-symplectic system with the regularity conditions as required in Remark 4.3. Now we turn to the discussion of the total momentum. First, integrating the local momentum conservation law (1.12) over the spatial interval [-L/2, L/2] gives

= I-L/2 %dx

+ G (V2, t) - G(-L/2, t)

(4.52)

where we make use of the periodic boundary conditions. It implies that the total momentum is conserved in the continuous case. In our RungeKutta methods, we define the total momentum at time U as N-l

s

(ldLy = h Y, £ bml(ztm),

(4.53)

1=0 m = l

where z\ m and i have the same meaning as before. Theorem 4.6. [20] Under the assumptions of Theorem 4.5 and with the periodic boundary condition, if dxz is a periodic function on the spatial

Jialin Hong

108

interval [—L/2,L/2], namely, dxz(-L/2,t) = dxz(L/2,t) for all t, then we have the following discrete total momentum conservation law: d\0 &

)

=

&

)

(4.54)

•

Proof. From (4.15)-(4.20), with the similar calculation of (4.22)-(4.27), we get

I(zlJ-I(z°J

= \[{ziJTMdxzlm

(zlJTMdxz°m

-

= I E hUdtZ^fMd^^

+

iZ^fMY^

fc=i

(4.55)

where Ytkm denotes (dtdxZ)lm or From (4.55), it follows that

(dxdtZ)lm.

= f E &*{ " E 1 E bm Udtzkm)TMdxzkm + (zkm)TMYkm]}. fc=l

L

I. Z=0 m = l

-> J

(4.56) On the other hand, it is deduced that (zk+1)TMdxzf+1

bm[(dtZkm)TMdxZkm+(Zkm)TMYkm

= h± L

m=l

(zk)TMdxzk

-

(4.57)

Combining (4.56) and (4.57), we have

{nf - pi? = § E h \ N£ k=i

L i=o

k

T

\(z?+1)TMdxz*+1

L

= E bk\{z N) Mdxz% fe=i

-

-

(zk)TMdxzk

(4.58)

k T

(z ) Mdxz$

L

= 0, where the last equality comes from the periodic boundary conditions of z and dxz. The proof is finished. Remark 4.7. For any multi-symplectic system if the phase variable z and the first order derivatives of z with respect to spatial variables are periodic in the spatial domain, then applying the multi-symplectic RK methods to this system, we can get the discrete total momentum

A Survey of Multi-symplectic Runge-Kutta • • •

109

conservation law with the same form of (4.54). For the Runge-Kutta discretization of the nonlinear Dirac equation, the symplecticity both in temporal and spatial directions implies the multi-symplecticity of the integrator. The preservation of charge, energy and momentum conservation laws is very important under the structurepreserving discretization. A known result that the multi-symplectic integrator preserves the local energy and momentum exactly if the multisymplectic Hamiltonian is of quadratic has been contained in our energy and momentum analysis. In particular, from the main results in this section, it follows that, under given conditions, there exists a constant C > 0 such that for sufficiently small r and h, we have \Eie + Mle\ 2 space dimensions, Appl. Math. Lett. 15 (2002) 1005-1011. [22] A. Iserles, A First Course in the Numerical Analysis of Differential Equations, Cambridge University Press, Cambridge 1996. [23] A.L.Islas, D. A.Karpeev and C.M.Schober, Geometric integrators for the nonlinear Schrodinger equation, J. Comput. Phys. 173 (2001) 116-148. [24] A. L. Islas and C. M. Schober, On the preservation of phase space structure under multisymplectic discretization, J. Comput. Phys. 197 (2004) 585-609. [25] F. M. Lasagni, Canonical Runge-Kutta methods, ZAMP 39 (1988) 952-953. [26] J. E. Marsden & M. West, Discrete mechanics and variational integrators, Acta Numerica 10 (2001) 1-158. [27] J. E. Marsden, S. Pekarsky, S. Shkoller and M. West, Variational methods, multi-symplectic geometry and continuum mechanics, J. Geom. and Phys. 38 (2001) 253-284. [28] R. I. McLachlan, Symplectic integration of Hamiltonian wave equations, Numer. Math. 66 (1994) 465-492. [29] B. Moore and S. Reich, Multi-symplectic integration methods for Hamiltonian PDEs, Future Gener. Comput. Syst. 19 (2003) 395402. [30] B. Moore and S. Reich, Backward error analysis for multi-symplectic integration methods, Numer. Math. 95 (2003) 625-652. [31] S. Reich, Multi-symplectic Runge-Kutta collocation methods for Hamiltonian wave equations, J. Comp. Phys. 157 (2000) 2, 473499. [32] J. M. Sanz-Serna, Runge-Kutta schemes for Hamiltonian systems, BIT 28 (1988) 877-883.

A Survey of Multi-symplectic Runge-Kutta • • •

113

[33] J. M. Sanz-Serna, The numerical integration of Hamiltonian systems, In: Computational Ordinary Differential Equations, ed. by J.R. Cash & I.Gladwell, Clarendon Press, Oxford, 1992, 437-449. [34] J. M. Sanz-Serna and M.P.Calvo, Numerical Hamiltonian Chapman & Hall, London, 1994.

Systems,

[35] G. Sun, A simple way of constructing symplectic Runge-Kutta methods, J. Comput. Math. 18 (2000) 61-68. [36] G. Sun, Symplectic partitioned Runge-Kutta methods, J. Comput. Math. 11 (1993) 365-372. [37] Y. Sun and M. Qin, Construction of multi-symplectic schemes of any finite order for modified wave equation, J. Math. Phys. 41 (2000) 7854-7868. [38] Y. B. Suris, On the conservation of the symplectic structure in the numerical solution of Hamiltonian systems (in Russian), In: Numerical Solution of Ordinary Differential Equations, ed. by S.S. Filippov, Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, Moscow, 1988, 148-160. [39] Y. B. Suris, Hamiltonian methods of Runge-Kutta type and their variational interpolation (in Russian), Math. Model. 2 (1990) 78-87. [40] Y. Wang, B. Wang and M. Qin, Numerical implementation of the multi-symplectic Preissmann scheme and its equivalent schemes, Appl. Math. Comput. 149 (2004) 299-326. [41] P. Zhao and M. Qin, Multi-symplectic geometry and multisymplectic Preissmann Scheme for the KdV equation, J. Phy. A: Math. Gen. 33 (2000) 3613-3626.

114

Inverse Problems in Bioluminescence Tomography Ming Jiang* LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China. E-mail: [email protected]

YiLit Department of Mathematics, Hunan Normal Changsha, China. Email: [email protected]

University,

Ge Wang E-mail: [email protected]

1

Introduction

Gene therapy is a breakthrough in the modern medicine, which promises to cure diseases by modifying gene expression. A key for the development of gene therapy is to monitor the in vivo gene transfer and its efficacy in the mouse model. Traditional biopsy methods are invasive, insensitive, inaccurate, inefficient and limited in the extent. To map the distribution of the administered gene, reporter genes such as those producing luciferase are being used to generate light signals within a living mouse, which can be externally measured. A highly sensitive CCD camera has been built to take a 2D external view of expression of the bioluminescent signal [1]. Such a 2D image of photon emission is then registered with a 2D visible light picture of the mouse for the localization of the reporter gene activity. In addition to gene therapy, this new imaging tool has great potentials in other various biomedical applications as well [2-6]. However, this 2D bioluminescence imaging technique, like the *M. Jiang and G. Wang are at CT/Micro-CT Laboratory, Department of Radiology, University of Iowa, Iowa City, IA 52242, USA. tY. Li and G. Wang are at Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA.

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traditional radiography, is incapable of 3D characterization of internal source features of interest. To address the needs for 3D localization and quantification of a bioluminescent source distribution in a small animal, a bioluminescence tomography (BLT) system is being developed [7-10]. Mathematically, BLT is an inverse problem to recover an internal bioluminescent source distribution subject to Cauchy data for the diffusion equation. Traditionally, optical diffuse tomography utilizes incoming visible or near infra-red light to probe a scattering object, and reconstructs the 3D distribution of internal optical properties, such as one or both of absorption and scattering coefficients [11,12]. In contrast to this active imaging mode, BLT reconstructs an internal bioluminescent source distribution, generated by luciferase induced by reporter genes, from external optical measurement. In BLT, the complete knowledge on the optical properties of anatomical structures of the mouse has been established based on an independent tomographic scan, such as a CT/micro-CT scan, by image segmentation and optical property mapping. That is, we can segment the CT/micro-CT image volume into a number of anatomical structures, and assign optical property values to each structure by using a database of the optical properties compiled for this purpose [7,8]. The organization of the paper is as follows. In §2 we introduce the forward process for BLT by the radiative transfer equation and its diffusion approximation. In §3 we discuss briefly the optical diffuse tomography problem. The BLT problem is formulated in §4. In §5, we reformulate the BLT problem in an abstract operator form by the Dirichlet-to-Neumann map. We review recent results on the uniqueness of the solution for BLT and demonstrate that the minimal norm solution is not physically favorable for BLT. And in §6 we give a detailed analysis of the structure of the solution for BLT in §7 for sources as a linear combination of points or ball-like sources or general radial basis functions. Then, we propose several iterative reconstruction algorithms for BLT in §8. Finally, we discuss remaining issues and future directions in §9.

2 2.1

Radiative transfer equation and diffusion approximation Radiative transfer equation (RTE)

Let ft be a domain in the three contains the object to be imaged. 0 € S2 at x 6 ft, where S2 is the migration in a random medium

dimensional Euclidean space R 3 that Let u(x, 9) be the light flux in direction unit sphere. A general model for light is the radiative transfer equation, or

116

Ming Jiang, Yi Li, Ge Wang

Boltzmann equation [11-17]: ldu - — (x,9,t)+0-

Wxu(x,9,t)

+ n(x)u(x,0,t)

,^ ^

= Hs{x) JS2 v{0 • 0')u(x, 0', t) tiff + q(x, 6, t) for t > 0 and i e ( l , where c denotes the particle speed, fj, = /xa + /xs with Ha and /i s being the absorption and scattering coefficients respectively, the scattering kernel r] is normalized such that J r)(0 • 6')d0' = 1, and q s2 is the internal light source. In (2.1), the radiance u(x,6,t) is in the unit of W c m - 2 sr _ 1 , the source term q(x, 0, t) is in the unit of W c m - 3 s r - 1 , the scattering coefficient /t s and the absorption coefficient \ia both are given in the unit of c m - 1 , and the scattering phase function r\ is in the unit o f s r - 1 [18]. One widely assumed kernel is the Henyey-Greenstein scattering [11, 12],

M-hii+yx,*)**'

(2 2)

-

The parameter g € (—1,1) is a measure for anisotropy, with g = 0 corresponding to isotropic scattering. Other scattering kernels can be found in [15,16]. The initial condition for u is u(x,e,o)

= o, xen,oes2.

(2.3)

The boundary condition for u represents the incoming flux g~

u(x,e,t) = g-(x,e,t),

x e an, e e s2, v{x) • e < 0, t > 0, (2.4)

where v is the exterior normal on dCl. The problem (2.1), (2.3) and (2.4) admits a unique solution under appropriate assumptions on fx, /xa and r? [17]. The homogeneous condition g~(x,9,t) = 0 specifies that no photons travel in an inward direction at the boundary, except for the source terms [11, p. R50].

2.2

Diffusion approximation

Typical values of \xa and fxs in optical tomography for biological tissues are fia = 0.1 ~ 1.0mm - 1 , /zs = 100 ~ 200mm _ 1 , respectively. This means that the mean free path of the particles is between 0.005 and 0.01 mm, which is very small compared to a typical object. Thus, the predominant phenomenon in optical tomography is scatter rather than transport. Therefore, one can replace the transport equation (2.1) by a much simpler diffusion equation [12]. The diffusion theory, therefore,

Inverse Problems in Bioluminescence Tomography

117

becomes an appropriate approximation for many biomedical applications [19]. In the following we present the derivation of the diffusion approximation from the RTE under the above conditions in [12]. Other approaches can be found in [11,13,16]. Due to the prevalence of scatter, the flux is essentially isotropic within a small distance away from the sources, i.e., it depends only linearly on 6. Thus we may describe the process adequately by the first few moments u0(x,t)

= — u(x,e,t)d9, 47T Js?

(2.5)

u1(x,t)

= —

6u(x,6,t)d6,

(2.6)

09*u(x,e,t)d9

(2.7)

u2(M) = 7- / 47T JS2

of u. Note that uo(x,t) is the photon density and u\(x, t) is the photon current, which define the measurability [11, p. R46]. Integrating (2.1) over S2 and using the normalization of 77, we obtain - — (x,t) + V -ui{x,t)

+ /j,a(x)u0{x,t)

=q0(x,t),

(2.8)

where q0(x,t) = ^-

f

q{x,9,t)d6.

(2.9)

47T JS2

qx and 92 are defined similarly as that for u. Similarly, multiplying (2.1) with 6 and integrating over S2 yield —^-(x,t)+\7-u 2 (x,t)+fi(x)u 1 (x,t)

= fjns(x)ui(x,t)+q1(x,t),

(2.10)

where fj is the mean scattering cosine fj=^-

J 6-e'r)(6-6')d6',

(2.11)

47T JS2

which does not depend on 6 and equals to g for the Henyey-Greenstein scattering kernel in (2.2) and qi{x,t) = — I 9q(x,Q,t)d6.

(2.12)

47T JS2

Introducing the reduced scattering coefficient /i'a = ( l - J j K ,

(2.13)

118

Ming Jiang, Yi Li, Ge Wang

we can write the equations for uo and u\ in the more concise forms 1 /-I

--zr(x,t) --^-(x,t)

+ V -ui(x,t)

+ V-u2(x,t)

+ fj,a(x)uo(x,t)

+ (fia(x) + fJ,'s(x))Ul(x,t)

=q0(x,t),

(2.14)

= qx{x,t).

(2.15)

Now we assume that u depends only linearly on 9, u(x,9,t)

=auo(x,i)+/36-ui(x,t).

(2.16)

For the constants a and (3, we easily obtain a = 1 and (3 = 3 by computing the moments. By expressing u2 by (2.16) in terms of UQ and u\, it follows that V • u2 = ~Vu 0 . Eliminating V • 1*2 from (2.15), we obtain a closed system for uo and u\\ --7r-(x,t) —J£(x,t)

+ V -ui(x,t)

+ -Vu0

+ ^a(x)u0{x,t)

+ (j*a(x) + n's(x))Ul(x,t)

=q0(x,t),

(2.17)

= «i (*,*)•

( 2 -!8)

To obtain the diffusion approximation, we go one step further, assuming that u is almost stationary in the sense that ^f-(x,t) is negligible in (2.18) and that q\ = 0. We obtain approximately Ul

= -DVu0,

(2.19)

where D(x) = n. . *—-r^T. (2.20) y 3(Ha(x) + n's(x)) ' This is called Fick's law, and D is the diffusion coefficient. Inserting (2.19) into (2.17), we finally arrive at the diffusion approximation - - ^ - V • (DVu0) + »au0 = go-

(2.21)

This is the most commonly used forward model for photon migration in tissue. It is used almost exclusively in optical tomography and is also called the Pi-approximation [11,12,20-22]. The initial and boundary conditions for (2.21) can be obtained similarly. From (2.3), we immediately obtain u0(x,0)=0,

xefi.

(2.22)

From the boundary condition (2.4), we obtain v(x) • f

9u(x, 6,t)d6 = v(x) • f

9g~ (x, 9, t) d9

Inverse Problems in Bioluminescence Tomography

119

for x e 80, and t > 0. From (2.16) and (2.19), we get u(x,6,t)

= u0(x,t)

- 30 •

D(x)Wu0{x,t).

Observing that p{x)-

I 9de = -n, Jv(x)e