Regularization of Inverse Problems

Inbunden, Engelska, 1996

AvHeinz Werner Engl,Martin Hanke,A. Neubauer

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Driven by the needs of applications both in sciences and in industry, the field of inverse problems has become a growing area in applied mathematics. This book starts with an overview over some classes of inverse problems of practical interest. Inverse problems typically lead to mathematical models that are ill-posed in the sense of Hadamard. Especially, their solution is unstable under data perturbations, so that special numerical methods that can cope with these instabilities, so-called regularization methods, have to be developed. This book is devoted to the mathematical theory of regularization methods and is intended to give an up-to-date account of the currently available results about regularization methods both for linear and for nonlinear ill-posed problems. Both continuous and iterative regularization methods are considered in detail with special emphasis on the development of parameter choice and stopping rules which lead to optimal convergence rates.

Produktinformation

Utgivningsdatum1996-07-31
Mått156 x 234 x 24 mm
Vikt666 g
FormatInbunden
SpråkEngelska
SerieMathematics and Its Applications
Antal sidor322
Upplaga1996
FörlagKluwer Academic Publishers
ISBN9780792341574

Tillhör följande kategorier

1. Introduction: Examples of Inverse Problems.- 1.1. Differentiation as an Inverse Problem.- 1.2. Radon Inversion (X-Ray Tomography).- 1.3. Examples of Inverse Problems in Physics.- 1.4. Inverse Problems in Signal and Image Processing.- 1.5. Inverse Problems in Heat Conduction.- 1.6. Parameter Identification.- 1.7. Inverse Scattering.- 2. Ill-Posed Linear Operator Equations.- 2.1. The Moore-Penrose Generalized Inverse.- 2.2. Compact Linear Operators: Singular Value Expansion.- 2.3. Spectral Theory and Functional Calculus.- 3. Regularization Operators.- 3.1. Definition and Basic Results.- 3.2. Order Optimality.- 3.3. Regularization by Projection.- 4. Continuous Regularization Methods.- 4.1. A-priori Parameter Choice Rules.- 4.2. Saturation and Converse Results.- 4.3. The Discrepancy Principle.- 4.4. Improved A-posteriori Rules.- 4.5. Heuristic Parameter Choice Rules.- 4.6. Mollifier Methods.- 5. Tikhonov Regularization.- 5.1. The Classical Theory.- 5.2. Regularization with Projection.- 5.3. Maximum Entropy Regularization.- 5.4. Convex Constraints.- 6. Iterative Regularization Methods.- 6.1. Landweber Iteration.- 6.2. Accelerated Landweber Methods.- 6.3. The ?-Methods.- 7. The Conjugate Gradient Method.- 7.1. Basic Properties.- 7.2. Stability and Convergence.- 7.3. The Discrepancy Principle.- 7.4. The Number of Iterations.- 8. Regularization With Differential Operators.- 8.1. Weighted Generalized Inverses.- 8.2. Regularization with Seminorms.- 8.3. Examples.- 8.4. Hilbert Scales.- 8.5. Regularization in Hilbert Scales.- 9. Numerical Realization.- 9.1. Derivation of the Discrete Problem.- 9.2. Reduction to Standard Form.- 9.3. Implementation of Tikhonov Regularization.- 9.4. Updating the Regularization Parameter.- 9.5. Implementation of Iterative Methods.- 10. TikhonovRegularization of Nonlinear Problems.- 10.1. Introduction.- 10.2. Convergence Analysis.- 10.3. A-posteriori Parameter Choice Rules.- 10.4. Regularization in Hilbert Scales.- 10.5. Applications.- 10.6. Convergence of Maximum Entropy Regularization.- 11. Iterative Methods for Nonlinear Problems.- 11.1. The Nonlinear Landweber Iteration.- 11.2. Newton Type Methods.- A. Appendix.- A.1. Weighted Polynomial Minimization Problems.- A.2. Orthogonal Polynomials.- A.3. Christoffel Functions.

'It is written in a very clear style, the material is well organized, and there is an extensive bibliography with 290 items. There is no doubt that this book belongs to the modern standard references on ill-posed and inverse problems. It can be recommended not only to mathematicians interested in this, but to students with a basic knowledge of functional analysis, and to scientists and engineers working in this field.' Mathematical Reviews Clippings, 97k '... it will be an extremely valuable tool for researchers in the field, who will find under the same cover and with unified notation material that is otherwise scattered in extremely diverse publications.' SIAM Review, 41:2 (1999)