Regularization of Inverse Problems

Heinz W. Engl, Martin Hanke, and Andreas Neubauer, Regularization of Inverse Problems, Kluwer, Dordrecht, 1996.
Hardcover: ISBN 0-7923-4157-0
Paperback: ISBN 0-7923-6140-7.

Table of Contents

Preface
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 $\nu $-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. Tikhonov Regularization 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
Bibliography
Index