Bounds of Information Expenses in Constructing Projection Methods for Solving Ill-posed Problems

Abstract

An approach to constructing regularized projection methods to solve illposed problems is proposed. This approach is based on a modification of the Galerkin discretization scheme. It has been established that such a modification leads to a significant reduction of information expenses compared to other known methods.

On the conditions of convergence for one class of methods used for the solution of ill-posed problems

“…Then, within the framework of any method from A, the optimal (in order) rate of convergence to x † is guaranteed under conditions (19) and (20). The cardinality of the set of scalar products of the form (5) participating in the discretization of the coefficients A h and f δ has the following order: This statement is proved by analogy with Theorem 1 in [11]. Thus, according to Theorems 1 and 2, within the framework of the proposed approach to the solution of illposed problems in the case where the parameters of discretization n and regularization α satisfy only condition …”

“…Note that the class of these methods was constructed in [5] for the case where h = 0. Later, it was generalized in [11] to the case of sufficiently small h > 0.…”

“…Thus, we assume that, in the discretization scheme (4), the parameter n is chosen as the least natural number satisfying the condition [see relation (10) in [5]] …”

“…At the same time, an important aspect of substantiation of any approximate method, is connected with the analysis of conditions of its convergence to the exact solution of the original problem without any additional assumptions about this solution. In this sense, the present paper can be regarded as a continuation and generalization of [5]. Indeed, within the framework of investigations performed in what follows, we generalize the approach to the solution of ill-posed problems proposed in [5] to the case where the coefficients of the original equation are given with certain errors and establish sufficient conditions for the convergence of the constructed approximations to the required solution.…”

“…In this sense, the present paper can be regarded as a continuation and generalization of [5]. Indeed, within the framework of investigations performed in what follows, we generalize the approach to the solution of ill-posed problems proposed in [5] to the case where the coefficients of the original equation are given with certain errors and establish sufficient conditions for the convergence of the constructed approximations to the required solution.Consider an operator equation of the first kindin a separable Hilbert space ( , ) ⋅ ⋅ with scalar product ( , ) ⋅ ⋅ and norm x = ( , ) x x . Suppose that A ∈ L ( ) X , Rang( ) A ≠ Rang( ) A , and f A ∈Rang( ), where L ( ) X is a space of linear continuous operators acting in X.…”