Convergence rates for Tikhonov regularization from different kinds of smoothness conditions

Abstract

The article is concerned with ill-posed operator equations Ax = y where A:X →Y is an injective bounded linear operator with non-closed range and X and Y are Hilbert spaces. The solution x=x ^† is assumed to be in the range of some selfadjoint strictly positive bounded linear operator G:X →X. Under several assumptions on G, such as or more generally , inequalities of the form , or range inclusions , convergence rates for the regularization error of Tikhonov regularization are established. We also show that part of our assumptions automatically imply so-called source conditions. The article contains a series of new results but also intends to uncover cross-connections between the different kinds of smoothness conditions that have been discussed in the literature on convergence rates for Tikhonov regularization.

Keywords:

• Linear ill-posed problems
• Tikhonov regularization
• Convergence rates
• Smoothness conditions
• Index functions
• Operator monotone functions
• Range inclusions

Keywords:

• MSC 2000 Subject Classifications: 47A52
• 65J20
• 65R30

Acknowledgments

The authors express their sincere thanks to Dr Dana Düvelmeyer (TU Chemnitz) for a couple of helpful comments on the paper, which were made during her stay in Tokyo in July 2005. The fourth named author is partly supported by Grant 15340027 from the Japan Society for the Promotion of Science and Grant 17654019 from the Ministry of Education, Culture, Sports and Technology.