Error Estimation in Ill-Posed Problems in Special Cases

Abstract

In this review we consider ill-posed inverse problems for linear operator equations Az = u with an operator A acting between two normed spaces. It is well known that, in general, no error estimate can be provided for approximate solution of an ill-posed problem. But in some special cases when we are aware of some a priori information about the unknown exact solution, error estimation can be done. In this paper we review inverse problems on compact sets, as well as inverse problems with source-wise represented solutions. We also touch inverse problems in partially ordered spaces. A posteriori error estimates for regularized solutions with special regularization properties are also discussed here. This work was supported by the Swedish institute (Visby program) and the RFBR grants 11-01-0040_a and 12-01-00524_a.