Solving Inverse Problems with Spectral Data
We consider a two dimensional membrane. The goal is to find properties of the membrane or properties of a force on the membrane. The data is natural frequencies or mode shape measurements. As a result, the functional relationship between the data and the
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SpringerWienN ewYork
Dr. David Colton Department of Mathematical Sciences, University of Delaware, Newark, Delaware, U.S.A.
Dr. Heinz W. Engl Institut fur Mathematik, Johannes-Kepler-UniversiUit, Linz, Austria
Dr. Alfred K. Louis Fachbereich Mathematik, Universitat des Saarlandes, Saarbriicken, Federal Republic of Germany
Dr. Joyce R. McLaughlin Department of Mathematical Sciences, Rensselaer Polytechnic Institute Troy, New York, U.S.A.
Dr. William Rundell Department of Mathematics, Texas A & M University, College Station, Texas, U.S.A.
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machines or similar means, and storage in data banks. © 2000 Springer-VerlaglWien Typesetting: Camera ready by authors Printing: Novographic Druck G.m.b.H., A-1238 Wien Graphic design: Ecke Bonk SPIN 10732324
With 41 Figures
ISBN 3-211-83470-2 Springer-Verlag Wien New York
Contents
D. Colton, H. W. Engl, A. K. Louis, J. R. McLaughlin and W. Rundell Introduction
1
H. W. Engl and O. Scherzer Convergence Rates Results for Iterative Methods for Solving Nonlinear Ill-Posed Problems
7
M. Hanke Iterative Regularization Techniques in Image Reconstruction
35
P. K. Lamm A Survey of Regularization Methods for First-Kind Volterra Equations
53
J. Sylvester
Layer Stripping
83
D. Colton, P. Monk and A. Kirsch The Linear Sampling Method in Inverse Scattering Theory
107
M. V. Klibanov Carleman Estimates and Inverse Problems in the Last Two Decades
119
A. K. Louis and E. T. Quinto Local Tomographic Methods in Sonar
147
T. Kohler, P. Maass and P. Wust Efficient Methods in Hyperthermia Treatment Planning
155
J. R. McLaughlin Solving Inverse Problems with Spectral Data
169
L. Borcea and G. C. Papanicolaou Low Frequency Electromagnetic Fields in High Contrast Media G. Uhlmann Inverse Scattering in Anisotropic Media
235
P. B. Stark Inverse Problems as Statistics
253
195
Introd uction D. Colton, H.W. Engl, A.K. Louis, J.R. McLaughlin and W. Rundell It has only been since the mid-1960s that inverse problems has been identified as a proper subfield of mathematics. Prior to this conventional wisdom held it was not an area appropriate for mathematical analysis. This historical prejudice dates back to Hadamard who claimed that the only problems of physical interest were those that had a unique solution depending continuously on the given data. Such problems were well-posed and problems that were not well-posed were labeled ill-posed. In particular, ill-posed problems connected with partial differential equations of mathematical physics were considered to be of purely academic interest and not worthy of serious study. In the meantime, the success of radar and sonar during the Second World War caused scientists to ask the question if more could be determined about a scattering object than simply its location. Such problems are in the category of in
