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