Driven by the needs of applications both in sciences and in industry, the field of inverse problems has certainly been one of the fastest growing areas in applied mathematics recently. 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. Audience: This book, which can be read by students with a basic knowledge of functional analysis, should be useful both to mathematicians and to scientists and engineers who deal with inverse problems in their fields. It can be used as a text for a graduate course on inverse problems and will also be useful to specialists in the field as a reference work.
Author(s): Heinz Werner Engl, Martin Hanke, Andreas Neubauer
Series: Mathematics and Its Applications
Edition: 1
Publisher: Springer
Year: 1996
Language: English
Pages: 329
Contents......Page all_24762_to_00328.cpc0006.djvu
Preface......Page all_24762_to_00328.cpc0008.djvu
1. Introduction: Examples of Inverse Problems......Page all_24762_to_00328.cpc0010.djvu
1.1. Differentiation as an Inverse Problem......Page all_24762_to_00328.cpc0011.djvu
1.2. Radon Inversion (X-Ray Tomography)......Page all_24762_to_00328.cpc0014.djvu
1.3. Examples of Inverse Problems in Physics......Page all_24762_to_00328.cpc0017.djvu
1.4. Inverse Problems in Signal and Image Processing......Page all_24762_to_00328.cpc0019.djvu
1.5. Inverse Problems in Heat Conduction......Page all_24762_to_00328.cpc0025.djvu
1.6. Parameter Identification......Page all_24762_to_00328.cpc0030.djvu
1.7. Inverse Scattering......Page all_24762_to_00328.cpc0032.djvu
2. Ill-Posed Linear Operator Equations......Page all_24762_to_00328.cpc0038.djvu
2.1. The Moore-Penrose Generalized Inverse......Page all_24762_to_00328.cpc0039.djvu
2.2. Compact Linear Operators: Singular Value Expansion......Page all_24762_to_00328.cpc0043.djvu
2.3. Spectral Theory and Functional Calculus......Page all_24762_to_00328.cpc0049.djvu
3.1. Definition and Basic Results......Page all_24762_to_00328.cpc0056.djvu
3.2. Order Optimality......Page all_24762_to_00328.cpc0062.djvu
3.3. Regularization by Projection......Page all_24762_to_00328.cpc0070.djvu
4.1. A-priori Parameter Choice Rules......Page all_24762_to_00328.cpc0078.djvu
4.2. Saturation and Converse Results......Page all_24762_to_00328.cpc0087.djvu
4.3. The Discrepancy Principle......Page all_24762_to_00328.cpc0090.djvu
4.4. Improved A-posteriori Rules......Page all_24762_to_00328.cpc0096.djvu
4.5. Heuristic Parameter Choice Rules......Page all_24762_to_00328.cpc0107.djvu
4.6. Mollifier Methods......Page all_24762_to_00328.cpc0119.djvu
5.1. The Classical Theory......Page all_24762_to_00328.cpc0124.djvu
5.2. Regularization with Projection......Page all_24762_to_00328.cpc0133.djvu
5.3. Maximum Entropy Regularization......Page all_24762_to_00328.cpc0141.djvu
5.4. Convex Constraints......Page all_24762_to_00328.cpc0147.djvu
6.1. Landweber Iteration......Page all_24762_to_00328.cpc0161.djvu
6.2. Accelerated Landweber Methods......Page all_24762_to_00328.cpc0167.djvu
6.3. The \nu-Methods......Page all_24762_to_00328.cpc0173.djvu
7.1. Basic Properties......Page all_24762_to_00328.cpc0184.djvu
7.2. Stability and Convergence......Page all_24762_to_00328.cpc0188.djvu
7.3. The Discrepancy Principle......Page all_24762_to_00328.cpc0193.djvu
7.4. The Number of Iterations......Page all_24762_to_00328.cpc0198.djvu
8.1. Weighted Generalized Inverses......Page all_24762_to_00328.cpc0204.djvu
8.2. Regularization with Seminorms......Page all_24762_to_00328.cpc0209.djvu
8.3. Examples......Page all_24762_to_00328.cpc0214.djvu
8.4. Hilbert Scales......Page all_24762_to_00328.cpc0217.djvu
8.5. Regularization in Hilbert Scales......Page all_24762_to_00328.cpc0222.djvu
9.1. Derivation of the Discrete Problem......Page all_24762_to_00328.cpc0228.djvu
9.2. Reduction to Standard Form......Page all_24762_to_00328.cpc0231.djvu
9.3. Implementation of Tikhonov Regularization......Page all_24762_to_00328.cpc0235.djvu
9.4. Updating the Regularization Parameter......Page all_24762_to_00328.cpc0240.djvu
9.5. Implementation of Iterative Methods......Page all_24762_to_00328.cpc0244.djvu
10.1. Introduction......Page all_24762_to_00328.cpc0248.djvu
10.2. Convergence Analysis......Page all_24762_to_00328.cpc0250.djvu
10.3. A-posteriori Parameter Choice Rules......Page all_24762_to_00328.cpc0256.djvu
10.4. Regularization in Hilbert Scales......Page all_24762_to_00328.cpc0260.djvu
10.5. Applications......Page all_24762_to_00328.cpc0263.djvu
10.6. Convergence of Maximum Entropy Regularization......Page all_24762_to_00328.cpc0269.djvu
11.1. The Nonlinear Landweber Iteration......Page all_24762_to_00328.cpc0284.djvu
11.2. Newton Type Methods......Page all_24762_to_00328.cpc0292.djvu
A.1. Weighted Polynomial Minimization Problems......Page all_24762_to_00328.cpc0296.djvu
A.2. Orthogonal Polynomials......Page all_24762_to_00328.cpc0298.djvu
A.3. Christoffel Functions......Page all_24762_to_00328.cpc0302.djvu
Bibliography......Page all_24762_to_00328.cpc0306.djvu
Index......Page all_24762_to_00328.cpc0326.djvu
