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.

Keywords:

• severely ill-posed problem
• complexity
• Galerkin's information
• discrepancy principle
• balancing principle

AMS Subject Classification:

• 65R20
• 47A52

Additional information

Funding

This work was done while the authors were visiting the Johann Radon Institute for Computational and Applied Mathematics (RICAM). The authors gratefully acknowledge the partially support in the scope of joint European Project “Approximation Methods for Molecular Modelling and Diagnosis Tools” (“AMMODIT”), Grant no: 645672.