Regularization of Inverse Problems

'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)

• 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.