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World Scientific Publishing Company, Incorporated
Inverse Problems for Electrical Networks

Inverse Problems for Electrical Networks


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This book is a very timely exposition of part of an important subject which goes under the general name of “inverse problems”. The analogous problem for continuous media has been very much studied, with a great deal of difficult mathematics involved, especially partial differential equations. Some of the researchers working on the inverse conductivity problem for continuous media (the problem of recovering the conductivity inside from measurements on the outside) have taken an interest in the authors' analysis of this similar problem for resistor networks.The authors' treatment of inverse problems for electrical networks is at a fairly elementary level. It is accessible to advanced undergraduates, and mathematics students at the graduate level. The topics are of interest to mathematicians working on inverse problems, and possibly to electrical engineers. A few techniques from other areas of mathematics have been brought together in the treatment. It is this amalgamation of such topics as graph theory, medial graphs and matrix algebra, as well as the analogy to inverse problems for partial differential equations, that makes the book both original and interesting.

Product Details

ISBN-13: 9789810241742
Publisher: World Scientific Publishing Company, Incorporated
Publication date: 03/28/2000
Series: Series On Applied Mathematics Series , #13
Pages: 196
Product dimensions: 6.30(w) x 8.60(h) x 0.80(d)

Table of Contents

1.1Electrical Networks1
1.2Other Topics9
2Circular Planar Graphs11
2.2Y - [Delta] Transformations14
2.3Edge Removal16
2.4Trivial Modifications18
2.5Well-connected Graphs20
3Resistor Networks27
3.1Conductivities on Graphs27
3.2The Response Matrix32
3.3The Kirchhoff Matrix33
3.4The Dirichlet Norm35
3.5The Schur Complement40
3.6Sub-matrices of the Response Matrix47
3.7Connections and Determinants49
3.8Recovery of Conductances I55
4Harmonic Functions59
4.1Harmonic Continuation59
4.2Recovering Conductances from [Lambda]62
4.3Special Functions on Networks67
4.4Special Functions on G[subscript 4m+3]71
4.5Recovery of Conductances II74
4.6The Differential of L77
5Characterization I83
5.1Properties of Response Matrices83
5.2Some Matrix Algebra85
5.3Parametrizing Response Matrices86
5.4Principal Flow Paths90
5.5Proof of Theorem 5.193
6Adjoining Edges99
6.1Adjoining a Boundary Edge99
6.2Adjoining a Boundary Pendant103
6.3Adjoining a Boundary Spike105
6.4Recovery of Conductances III108
7Characterization II109
7.1Totally Non-negative Matrices109
7.2Characterization of Response Matrices II117
8Medial Graphs121
8.1Constructing the Medial Graph121
8.2Coloring the Regions124
8.3Switching Arcs126
8.5Uncrossing Arcs130
8.6Families of Chords134
8.7Standard Arrangements140
9Recovering a Graph149
9.2The Cut-point Lemma152
9.3Recovering a Medial Graph158
9.5Critical Graphs165
10Layered Networks173

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