In this textbook the basics of linear ill-posed problems in normed spaces are introduced. In the first chapter of this monograph, the general concepts for well-posedness are introduced and some sufficient conditions for well-posedness are given. This chapter also contains examples of linear ill-problems as well as stability results.
In Chapter 2 the general concepts for the regularization of linear ill-posed problems with perturbed right-hand sides and operators are presented. In Chapter 3 special regularization methods like Tikhonov and Lavrentiev regularization and the quasisolution method are introduced and their relations are investigated. Additionally, the truncated singular value decomposition and simple iteration methods are considered. For most of the considered regularization methods, parameter choices and stopping rules are considered.
Chapter 4 deals with the optimality of parameter choices and stopping rules for regularization methods, and additionally error estimates for some regularization methods are presented. In Chapter 5 methods for the evaluation of unbounded operators are presented, and, finally, in Chapter 6 finite-dimensional approximations for Tikhonov’s regularization method are considered, and several numerical experiments are provided.
This monograph is a translation of the book published by Nauka in 1978 (see Zbl 0391.35059) but the considered topics are still of interest, in particular due to the consideration of operator perturbations. The presentation of the material is clear and self-contained, and the monograph can be recommended to researchers as well as to students. As an extension to the Nauka edition, recent developments have been included at the end of each chapter, and some recent publications have been added to the list of references.