Inverse Problems for Electrical Networks

INVERSE PROBLEMS FOR

ELECTRICAL NEWORKS

SERIES ON APPLIED MATHEMATICS Editor-in-Chief: Frank Hwang Associate Editors-in-Chief: Zhong-ci Shi and U Rothblum

Vol. 1

International Conference on Scientific Computation eds. T. Chan and Z.-C. Shi

Vol. 2

Network Optimization Problems — Algorithms, Applications and Complexity eds. D.-Z. Du and P. M. Pandalos

Vol. 3

Combinatorial Group Testing and Its Applications by D.-Z. Du and F. K. Hwang Computation of Differential Equations and Dynamical Systems eds. K. Feng and Z.-C. Shi

Vol. 4 Vol. 5

Numerical Mathematics eds. Z.-C. Shi and T. Ushijima

Vol. 6

Machine Proofs in Geometry by S.-C. Chou, X.-S. Gao and J.-Z. Zhang

Vol. 7

The Splitting Extrapolation Method by C. B. Liem, T. Lu and T. M. Shih

Vol. 8

Quaternary Codes by Z.-X. Wan Finite Element Methods for Integrodifferential Equations by C. M. Chen and T. M. Shih

Vol. 9 Vol. 10

Statistical Quality Control — A Loss Minimization Approach by D. Trietsch

Vol. 11

The Mathematical Theory of Nonblocking Switching Networks by F. K. Hwang

Vol. 12

Combinatorial Group Testing and Its Applications (2nd Edition) by D.-Z. Du and F. K. Hwang

Vol. 13

Inverse Problems for Electrical Networks by E. B. Curtis and J. A. Morrow

Series on Applied Mathematics Volume 13

INVERSE PROBLEMS FOR

ELECTRICAL NETWORKS Edward B. Curtis James A. Morrow Mathematics Department University of Washington, Seattle USA

VX World Scientific Singapore • NewJersey• London •Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Farrer Road, Singapore 912805 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-Publication Data Curtis, Edward B., 1933Inverse problems for electrical networks / Edward B. Curtis and James A. Morrow. p. cm. (Series on applied mathematics - Vol. 13) Includes bibliographical references and index. ISBN 9810241747 (alk. paper) 1. Electric networks. 2. Inverse problems (Differential equations) 3. Matrices. Morrow, James A., 1941II. Tide. III. Series. TK454.2.C87 2000 621.3'01'51535-dc21 99-053875

British Library Cataloguing-in-Publicatlon Data A catalogue record for this book is available from the British Library.

Copyright © 2000 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore by Uto-Print

Preface This book is the result of an accumulation of work done by the authors and their students over the past twelve years. In each of the years listed, 8-10 students were brought to the University of Washington for a summer REU (Research Experience for Undergraduates) program, supported by a REU Grant from the NSF. We want to thank the NSF for its support during this period. And we want to thank the students for their enthusiasm, their ded­ ication, and their individual contributions, without which this book would not have been possible. R E U Students 1988 Sara A. Beavers, Thaddeus J. Edens, Jeffrey E. Eldridge, Troy B. Holly, Christina H. Lamont, Olga M. Simek, Laura A. Smithies, Hsi-Jung Wu, Matthew J. Curland. 1989 Michael C. Carini, Robert A. Coury, Richard P. Dechance, Peter L. Engrav, Matthew G. Hudelson, Jeannie C. Mah, John Morgan Oslake, Michael J. Parks. 1990 Eric J. Auld, John T. Guthrie, Joshua K. Landrum, Adrian V. Mariano, Edith A. Mooers, Miriam A. Myjak, Brett A. Sovereign, Peter Staab, Stefan G. Treatman. v

VI

1991 Michael W. Buksas, Jim L Carr, Peter B. Gilbert, Benjamin Thaddeus Kosnik, Victor Lee, Nancy McNally, Michael J. Mills, Jami M Moksness. 1992 David Dorrough, Kristine Fromm, David Ingerman, Kurt Krenz, Keli Kringle, Justin Mauger, Julie Olsen, Kevin Rosema, Konrad Schroder, James Warren. 1993 Christopher Cook, Andrew Iglesias, Laura Judd, Matthew Munro, Aleksandr Murkes, David Muresan, Chris Higginson, Konrad Schroder, Neil York, Leonid Zheleznyak. 1994 Nathan E. Bramall, Sean P. DeMerchant, Darin Diachin, James A. Herzog, Todd Hollenbeck, Keith Johnson, Michael McLendon, Erika L. Schubert, Tung T. Tran, David Vanderweide. 1996 Margaret Chaffee, Amy Ehrlich, Mark Hoefer, Derek Jerina, Dmitriy Leykekhman, Phillip Lynch, Marc Pickett, Aubin Whitley. 1997 James Bisgard, Benjamin Blander, Ryan Daileda, Darren Lo, Sreekar M. Shastry, Spencer Shepard, Chris Staskewicz, Ryan Yamachika. 1998 Tarn Adams, Neil Burrell, Laura Kang, Kjell Konis, Jeffrey Mermin,

vii Amanda Mueller, Laura Negrin, Derek Newland, Julie Rowlett, Sturgell.

Ryan

1999 Ingrid Abendroth, Thomas Carlson, Rod Huston, Carla Pellicano, Chris Romero, Mike Usher, John Thacker, Christopher Twigg.

This page is intentionally left blank

Contents Preface

v

1

Introduction 1.1 Electrical Networks 1.2 Other Topics

1 1 9

2

Circular Planar Graphs 2.1 Connections 2.2 Y - A Transformations 2.3 Edge Removal 2.4 Trivial Modifications 2.5 Well-connected Graphs

11 11 14 16 18 20

3

Resistor Networks 3.1 Conductivities on Graphs 3.2 The Response Matrix 3.3 The Kirchhoff Matrix 3.4 The Dirichlet Norm 3.5 The Schur Complement 3.6 Sub-matrices of the Response Matrix 3.7 Connections and Determinants 3.8 Recovery of Conductances I

27 27 32 33 35 40 47 49 55

4

Harmonic Functions 4.1 Harmonic Continuation 4.2 Recovering Conductances from A 4.3 Special Functions on Networks

59 59 62 67

IX

x

CONTENTS CONTENTS 4.4 4.5 4.6

Special Functions on G4m+3 Recovery of Conductances II The Differential of L

71 74 77

5

Characterization I 5.1 Properties of Response Matrices 5.2 Some Matrix Algebra 5.3 Parametrizing Response Matrices 5.4 Principal Flow Paths 5.5 Proof of Theorem 5.1

83 83 85 86 90 93

6

Adjoining E d g e s 6.1 Adjoining a Boundary Edge 6.2 Adjoining a Boundary Pendant 6.3 Adjoining a Boundary Spike 6.4 Recovery of Conductances III

99 99 103 105 108

7

C h a r a c t e r i z a t i o n II 7.1 Totally Non-negative Matrices 7.2 Characterization of Response Matrices II

109 109 117

8

Medial Graphs 8.1 Constructing the Medial Graph 8.2 Coloring the Regions 8.3 Switching Arcs 8.4 Lenses 8.5 Uncrossing Arcs 8.6 Families of Chords 8.7 Standard Arrangements

121 121 124 126 127 130 134 140

9

Recovering a Graph 9.1 Connections 9.2 The Cut-point Lemma 9.3 Recovering a Medial Graph 9.4 Examples 9.5 Critical Graphs

149 149 152 158 159 165

10 Layered Networks

173

CONTENTS

XI

References

181

Index

183

Chapter 1

Introduction 1.1

Electrical Networks

Suppose an electrical network is inside a black box as in Figure 1-1. The interior of the box consists of nodes joined by conductors. The nodes are the vertices, and the conductors are the edges of a graph G. • The inverse problem is to find the conductance of each edge in G from measurements of voltages and currents at the boundary nodes. The forward problem assumes that the graph G and the conductance j(pq) of each edge pq in G are known. If a voltage is imposed at the boundary nodes, there is a function u defined throughout the network which agrees with / at the boundary nodes, and which satisfies Kirchhoff's Law at each interior node. LAW: At each interior node p, the sum of the currents from p to its neighboring nodes is 0. KIRCHHOFF'S

This function u is called the potential due to / . The resulting current at the boundary nodes is called the network response. The linear map A = A 7 which takes the boundary voltage / to the boundary current cj> is called the response map. A is sometimes called the voltage-to-current map because it gives the current (i.e., the response) to any voltage imposed at the boundary nodes. The response map will be known when the potential is found for each boundary function / , and the resulting boundary current <j> is calculated. If the standard basis is used to represent the boundary function / and the 1

CHAPTER

2

1.

INTRODUCTION

?

Figure 1-1: Black Box

boundary current <j>, the response map is represented by an n by n matrix also denoted A, called the response matrix. On the other hand, if the response map A 7 is given, but the conductivity function 7 is unknown, the inverse problem is to use A 7 to calculate the conductance of each edge in G. If the graph G is unknown, then that too must be deduced from the response matrix A 7 . The inverse problem as articulated (for a continuous conducting medium) by Calderon in [1], can be broken into four questions. ( Q l ) Uniqueness: Is the map 7 —► A 7 one-to-one? (Q2) Characterization: Which linear maps A are response maps? (Q3) Algorithm: Is there a procedure for calculating 7 from the response map A 7 ? (Q4) Continuity: If 7 is near fi, does it follow that A 7 is near A^,? To these we add a fifth question. (Q5) Can the graph G be deduced from the response matrix?

1.1. ELECTRICAL

NETWORKS

3

Figure 1-2: Circular planar graph G

Example 1.1 Suppose given a resistor network with five boundary nodes, one interior node and seven edges as in Figure 1-2. Measurements of voltages and currents are made at the boundary nodes v\, V2, v%, V4, v$, located on the dashed circle (not part of the network). The response matrix A is: 2.4 - 1 . 2 - 0 . 8 0 -0.4 -1.2 7.1 - 5 . 6 0 -0.3 -0.8 -5.6 7.6 - 1 . 0 - 0 . 2 4.0 - 3 . 0 0 ' 0 -1.0 3.9 -0.4 -0.3 -0.2 -3.0

(1.1)

The inverse problem is to calculate the conductances of each of the seven edges in G from A. Returning to the general situation, the solution to the forward problem reveals some facts about the response matrix. For any resistor network T = (G,7) with n boundary nodes, the response matrix A is an n by n matrix which has the following three properties. (1) A is symmetric: Ajj = Aj^ (2) The sum of the entries in each row is 0.

CHAPTER

4

1.

INTRODUCTION

(3)Fovi^j,Ai}j2, • • • , vn, as follows. For each pair (i,j) with Aij 7^ 0, place an edge joining V{ to Vj and assign the conductance of this edge to be jiviVj) = — Aij. The response matrix for this network will be A 7 = A. The inverse problem for resistor networks becomes interesting only if a restriction is placed on the type of graph allowed in the interior of the box. A circular planar resistor network consists of a graph, embedded in the disc in the plane, with the boundary nodes on the boundary circle, and with a conductance assigned to each of the edges. • This text is concerned with circular planar resistor networks. The surprising outcome is that, for circular planar resistor networks, there is a positive answer to the five questions ( Q l ) - (Q5). The answers involve three main techniques, which turn out to be closely related. (I) SCHUR COMPLEMENTS. (II) HARMONIC CONTINUATION. (III)

MEDIAL GRAPHS.

The first use of Schur complements is to obtain the response matrix from the Kirchoff matrix. More subtle is the use of Schur complements to obtain formulas (3.14) and (3.16) for calculating conductances of boundary edges and boundary spikes from the response matrix A. These same formulas can also be arrived at by a process called harmonic continuation. The medial graphs in Chapters 8 and 9 give even more insight into the same formulas. These three techniques are discussed briefly in the remainder of this chapter, and will be dealt with in much greater detail in the succeeding chapters. Some important concepts concerning circular planar graphs are path, connection, critical, and well-connected. A pathp