On numerical realization of quasioptimal parameter choices in (iterated) Tikhonov and Lavrentiev regularization

Abstract

We consider linear ill‐posed problems in Hilbert spaces with noisy right hand side and given noise level. For approximation of the solution the Tikhonov method or the iterated variant of this method may be used. In self‐adjoint problems the Lavrentiev method or its iterated variant are used. For a posteriori choice of the regularization parameter often quasioptimal rules are used which require computing of additionally iterated approximations. In this paper we propose for parameter choice alternative numerical schemes, using instead of additional iterations linear combinations of approximations with different parameters.

Key words:

• ill‐posed problem
• regularization
• (iterated) Tikhonov method
• (iterated) Lavrentiev method
• quasioptimal rules
• parameter choice
• numerical schemes

Notes

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