Inverse and Ill-posed Problems: Theory and Applications

Sergey I. Kabanikhin

Published by De Gruyter 2011

Inverse and Ill-posed Problems

Theory and Applications

Sergey I. Kabanikhin

Volume 55 in the series Inverse and Ill-Posed Problems Series

https://doi.org/10.1515/9783110224016

Overview
Contents

About this book

The theory of ill-posed problems originated in an unusual way. As a rule, a new concept is a subject in which its creator takes a keen interest. The concept of ill-posed problems was introduced by Hadamard with the comment that these problems are physically meaningless and not worthy of the attention of serious researchers. Despite Hadamard's pessimistic forecasts, however, his unloved "child" has turned into a powerful theory whose results are used in many fields of pure and applied mathematics. What is the secret of its success? The answer is clear. Ill-posed problems occur everywhere and it is unreasonable to ignore them.

Unlike ill-posed problems, inverse problems have no strict mathematical definition. In general, they can be described as the task of recovering a part of the data of a corresponding direct (well-posed) problem from information about its solution. Inverse problems were first encountered in practice and are mostly ill-posed. The urgent need for their solution, especially in geological exploration and medical diagnostics, has given powerful impetus to the development of the theory of ill-posed problems. Nowadays, the terms "inverse problem" and "ill-posed problem" are inextricably linked to each other.

Inverse and ill-posed problems are currently attracting great interest. A vast literature is devoted to these problems, making it necessary to systematize the accumulated material. This book is the first small step in that direction. We propose a classification of inverse problems according to the type of equation, unknowns and additional information. We consider specific problems from a single position and indicate relationships between them. The problems relate to different areas of mathematics, such as linear algebra, theory of integral equations, integral geometry, spectral theory and mathematical physics. We give examples of applied problems that can be studied using the techniques we describe.

This book was conceived as a textbook on the foundations of the theory of inverse and ill-posed problems for university students. The author's intention was to explain this complex material in the most accessible way possible. The monograph is aimed primarily at those who are just beginning to get to grips with inverse and ill-posed problems but we hope that it will be useful to anyone who is interested in the subject.

Covers a topic that is abundant in all areas of science and technology
Includes methods for incomplete input data
Enables readers to tackle real-world problems

Author / Editor information

Sergey I. Kabanikhin, Sobolev Institute of Mathematics, Novosibirsk, Russia.

Supplementary Materials

Contents

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Frontmatter Requires Authentication Unlicensed Licensed Download PDF	i
Preface Sergey I. Kabanikhin Requires Authentication Unlicensed Licensed Download PDF	v
Denotations Requires Authentication Unlicensed Licensed Download PDF	ix
Contents Requires Authentication Unlicensed Licensed Download PDF	xiii
Chapter 1. Basic concepts and examples Requires Authentication Unlicensed Licensed Download PDF	1
Chapter 2. Ill-posed problems Requires Authentication Unlicensed Licensed Download PDF	22
Chapter 3. Ill-posed problems of linear algebra Requires Authentication Unlicensed Licensed Download PDF	68
Chapter 4. Integral equations Requires Authentication Unlicensed Licensed Download PDF	98
Chapter 5. Integral geometry Requires Authentication Unlicensed Licensed Download PDF	129
Chapter 6. Inverse spectral and scattering problems Requires Authentication Unlicensed Licensed Download PDF	154
Chapter 7. Linear problems for hyperbolic equations Requires Authentication Unlicensed Licensed Download PDF	187
Chapter 8. Linear problems for parabolic equations Requires Authentication Unlicensed Licensed Download PDF	209
Chapter 9. Linear problems for elliptic equations Requires Authentication Unlicensed Licensed Download PDF	233
Chapter 10. Inverse coefficient problems for hyperbolic equations Requires Authentication Unlicensed Licensed Download PDF	249
Chapter 11. Inverse coefficient problems for parabolic and elliptic equations Requires Authentication Unlicensed Licensed Download PDF	319
Appendix A Requires Authentication Unlicensed Licensed Download PDF	331
Appendix B Requires Authentication Unlicensed Licensed Download PDF	411
Epilogue Requires Authentication Unlicensed Licensed Download PDF	431
Bibliography Requires Authentication Unlicensed Licensed Download PDF	433
Index Requires Authentication Unlicensed Licensed Download PDF	457

(Deutsch)

Language: English
Publisher: De Gruyter
Audience: Senior Undergraduate Students, Graduate and PhD Students, Recent Graduates getting Practical Experience; Engineers; Researchers in Physics, Geophysics, Medicine, Financial Mathematics, Acoustics, Electrodynamics, and so on; Academic Libraries
Pages
- Front matter: 16
- Main content: 459
Illustrations
- Tables: 2
Keywords: Ill-posed Problems; Inverse Problem; Regularization; Integral Equation; Differential Equation

eBook

Published: December 23, 2011
ISBN: 9783110224016

Hardcover

Published: December 23, 2011
ISBN: 9783110224009