Convergence rates for Tikhonov regularization from different kinds of smoothness conditions

• Published in: Applicable analysis Vol. 85; no. 5; pp. 555 - 578
• Main Authors: Böttcher, Albrecht, Hofmann, Bernd, Tautenhahn, Ulrich, Yamamoto, Masahiro
• Format: Journal Article
• Language: English
• Published: Taylor & Francis Group 01.05.2006
• Subjects: Convergence rates; Index functions; Linear ill-posed problems; MSC 2000 Subject Classifications: 47A52; Operator monotone functions; Range inclusions; Smoothness conditions; Tikhonov regularization
• ISSN: 0003-6811; 1563-504X
• DOI: 10.1080/00036810500474838

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.