Generalized Arcangeli’s discrepancy principles for a class of regularization methods for solving ill-posed problems

Abstract

- While applying regularization procedures for obtaining stable solutions of ill-posed problems, one of the crucial step is the choice of the regularization parameter. Among the well considered discrepancy principles in the literature, Morozov’s method and Arcangeli’s method are widely used because of their simplicity for the purpose of applications. Although Morozov’s method and their variations have been considered extensively in the literature for general class of regularization methods, the Arcangeli’s method is known to have applied only for Tikhonov regularization. The reason could be the belief that it can never yield a rate better than Morozov’s procedure, under any smoothness assumption on the solution. However, this belief was misplaced as it has been showed by Nair (1992) that Arcangeli’s method do provide the best rate O(δ^2/3) for Tikhonov regularization under sufficient smoothness assumption on the solution, while Morozov’s method gives the rate only up to O(δ^1/2).

The purpose of this paper is to consider a generalized form of Arcangeli’s method for a general class of regularization methods for the case when there is no error on the modeling, and then extend the procedure which allow error in the modeling as well.