On Minimization Strategies for Choice of the Regularization Parameter in Ill-Posed Problems

Abstract

We consider solving of linear ill-posed problems using the Tikhonov method (in self-adjoint case the Lavrentiev method), its iterated variant, Landweber method and conjugate gradient type methods. Several rules for a posteriori choice of the regularization parameter are proposed. In case of known noise level of data we propose to compute in Tikhonov method certain 2 parameters and take for regularization parameter minimal of them. In case of unknown noise level we consider family of rules where a certain function is minimized. The quasioptimality criterion and Hanke-Raus rule are included, error estimates are given. Extensive numerical experiments show an advantage of proposed rules over known rules.

Keywords:

• Conjugate gradients
• Heuristic rules
• Ill-posed problem
• Iterated Lavrentiev method
• Iterated Tikhonov method
• Landweber method
• Quasioptimality criterion
• Regularization parameter
• Stopping rule

AMS Subject Classification:

• 65J20
• 47A52

ACKNOWLEDGMENT

This work was supported by the Estonian Science Foundation Research Grant No. 7489.