Inverse Problems for Electrical Networks

by Curtis, Edward B.; Morrow, James A.

• ISBN13: 9789810241742
• ISBN10: 9810241747
• Format: Hardcover
• Copyright: 2000-05-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

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 graphtheory, 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.

Table of Contents

Preface	v
Introduction	1	(10)
Electrical Networks	1	(8)
Other Topics	9	(2)
Circular Planar Graphs	11	(16)
Connections	11	(3)
Y -- Δ Transformations	14	(2)
Edge Removal	16	(2)
Trivial Modifications	18	(2)
Well-connected Graphs	20	(7)
Resistor Networks	27	(32)
Conductivities on Graphs	27	(5)
The Response Matrix	32	(1)
The Kirchhoff Matrix	33	(2)
The Dirichlet Norm	35	(5)
The Schur Complement	40	(7)
Sub-matrices of the Response Matrix	47	(2)
Connections and Determinants	49	(6)
Recovery of Conductances I	55	(4)
Harmonic Functions	59	(24)
Harmonic Continuation	59	(3)
Recovering Conductances from Λ	62	(5)
Special Functions on Networks	67	(4)
Special Functions on G4m+3	71	(3)
Recovery of Conductances II	74	(3)
The Differential of L	77	(6)
Characterization I	83	(16)
Properties of Response Matrices	83	(2)
Some Matrix Algebra	85	(1)
Parametrizing Response Matrices	86	(4)
Principal Flow Paths	90	(3)
Proof of Theorem	93	(6)
Adjoining Edges	99	(10)
Adjoining a Boundary Edge	99	(4)
Adjoining a Boundary Pendant	103	(2)
Adjoining a Boundary Spike	105	(3)
Recovery of Conductances III	108	(1)
Characterization II	109	(12)
Totally Non-negative Matrices	109	(8)
Characterization of Response Matrices II	117	(4)
Medial Graphs	121	(28)
Constructing the Medial Graph	121	(3)
Coloring the Regions	124	(2)
Switching Arcs	126	(1)
Lenses	127	(3)
Uncrossing Arcs	130	(4)
Families of Chords	134	(6)
Standard Arrangements	140	(9)
Recovering a Graph	149	(24)
Connections	149	(3)
The Cut-point Lemma	152	(6)
Recovering a Medial Graph	158	(1)
Examples	159	(6)
Critical Graphs	165	(8)
Layered Networks	173	(8)
References	181	(2)
Index	183