Stable Solution of Inverse Problems
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Advanced Lectures in Mathematics Edited by Gerd Fischer
Jochen Werner Optimization. Theory and Applications Manfred Denker Asymptotic Distribution Theory in Nonparametric Statistics Klaus Lamotke Regular Solids and Isolated Singularities Francesco Guaraldo, Patrizia Macri, Alessandro Tancredi Topics on Real Analytic Spaces Ernst Kunz Kahler Differentials Johann Baumeister Stable Solution of Inverse Prob.lems
Johann Baumeister
Stable Solution of Inverse Problems
Friedr. Vieweg & Sohn
Braunschweig/Wiesbaden
AMS Subject Classification: 35R25, 35R30, 45A05, 45L05, 65F20
1987 All rights reserved © Friedr. Vieweg & Sohn Verlagsgesellschaft mbH, Braunschweig 1987
No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior permission of the copyright holder.
Produced by Lengericher Handelsdruckerei, Lengerich
ISBN-13: 978-3-528-08961-0 e-ISBN-13: 978-3-322-83967-1 DOl: 10.1007/978-3-322-83967-1
v
PREFACE
These notes are intended to describe the basic concepts of solving inverse problems in a stable way. Since almost all inverse problems are ill-posed in its original formulation the discussion of methods to overcome difficulties which result from this fact is the main subject of this book. Over the past fifteen years, the number of publications on inverse problems has grown rapidly. Therefore, these notes can be neither a comprehensive introduction nor a complete monograph on the topics considered; it is designed to provide the main ideas and methods. Throughout, we have not striven for the most general statement, but the clearest one which would cover the most situations. The presentation is intended to be accessible to students whose mathematical background includes basic courses in advanced calculus, linear algebra and functional analysis. Each chapter contains bibliographical comments. At the end of Chapter 1 references are given which refer to topics which are not studied in this book. I am very grateful to Mrs. B. Brodt for typing and to W. Scondo and
u.
Schuch for inspecting the manuscript.
Frankfurt/Main, November 1986
Johann Baumeister
VI TABLE OF CONTENTS PART
I
Chapter
BASIC CONCEPTS Introduction 1.1 Inverse problems 1.2 Some examples of inverse problems 1.3 Analysis of inverse problems
Chapter
Chapter
4
14
2 Ill-posed problems
16
2.1 General properties
16
2.2 Restoration of continuity in the linear case
18
2.3 Stability estimates
23
3 Regularization
27
3.1 Reconstruction from non-exact data
27
3.2 Preliminary results on Tikhonov's method
34
3.3 Regularizing schemes
37
3.4 A tutorial example: The reconstruction of a derivative 3.5 Optimal reconstruction of linear functionals PART II Chapter
Chapter
41 43
REGULARIZATION METHODS 4 The singular value decomposition
49 49
4.1 Compact operators 4.2 The spectrum of compact selfadjoint operators 4.3 The singular value decomposition
53 57
4.4 The min-max principle
64
4.5 The asymptot
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