Iterative processes of Fejér type in ill-posed problems with a priori information

Abstract

In the latter thirty years, the solution of ill-posed problems with a priori information formed a separate field of research in the theory of ill-posed problems. We mean the class of problems, where along with the basic equation one has some additional data on the desired solution. Namely, one states some relations and constraints which contain important information on the object under consideration. As a rule, taking into account these data in a solution algorithm, one can essentially increase its accuracy for solving ill-posed (unstable) problems. It is especially important in the solution of applied problems in the case when a solution is not unique, because this approach allows one to choose a solution that meets the reality. In this paper we survey the methods for solving such problems. We briefly describe all relevant approaches (known to us), but we pay the main attention to the method proposed by us. This technique is based on the application of iterative processes of Fejer type which admit a flexible and effective realization for a wide class of a priori constraints.