Numerical Methods for the Solution of Ill-Posed Problems

Many problems in science, technology and engineering are posed in the form of operator equations of the first kind, with the operator and RHS approximately known. But such problems often turn out to be ill-posed, having no solution, or a non-unique solution, and/or an unstable solution. Non-existence and non-uniqueness can usually be overcome by settling for `generalised' solutions, leading to the need to develop regularising algorithms.
The theory of ill-posed problems has advanced greatly since A. N. Tikhonov laid its foundations, the Russian original of this book (1990) rapidly becoming a classical monograph on the topic. The present edition has been completely updated to consider linear ill-posed problems with or without a priori constraints (non-negativity, monotonicity, convexity, etc.).
Besides the theoretical material, the book also contains a FORTRAN program library.
Audience: Postgraduate students of physics, mathematics, chemistry, economics, engineering. Engineers and scientists interested in data processing and the theory of ill-posed problems.

Inhalt

1. Regularization methods.- 2. Numerical methods for the approximate solution of ill-posed problems on compact sets.- 3. Algorithms for the approximate solution of ill-posed problems on special sets.- 4. Algorithms and programs for solving linear ill-posed problems.- Appendix: Program listings.- I. Program for solving Fredholm integral equations of the first kind, using Tikhonov's method with transformation of the Euler equation to tridiagonal form.- II. Program for solving Fredholm integral equations of the first kind by Tikhonov's method, using the conjugate gradient method.- III. Program for solving Fredholm integral equations of the first kind on the set of nonnegative functions, using the regularization method.- IV. Program for solving one-dimensional integral equations of convolution type.- V. Program for solving two-dimensional integral equations of convolution type.- VI. Program for solving Fredholm integral equations of the first kind on the sets of monotone and (or) convex functions. The method of the conditional gradient.- VII. Program for solving Fredholm integral equations of the first kind on the sets of monotone and (or) convex functions. The method of projection of conjugate gradients.- VIII. Program for solving Fredholm integral equations of the first kind on the sets of monotone and (or) convex functions. The method of projection of conjugate gradients onto the set of vectors with nonnegative coordinates.- IX. General programs.- Postscript.- 1. Variational methods.- 2. Iterative methods.- 3. Statistical methods.- 4. Textbooks.- 5. Handbooks and Conference Proceedings.