Complexity Estimates for Severely Ill-posed Problems under A Posteriori Selection of Regularization Parameter

Abstract

In the article the authors developed two efficient algorithms for solving severely ill-posed problems such as Fredholm’s integral equations. The standard Tikhonov method is applied as a regularization. To select a regularization parameter we employ two different a posteriori rules, namely, discrepancy and balancing principles. It is established that proposed strategies not only achieved optimal order of accuracy on the class of problems under consideration, but also they are economical in the sense of used discrete information.

Keyword : severely ill-posed problem, complexity, Galerkin’s information, discrepancy principle, balancing principle