Range Inclusions and Approximate Source Conditions with General Benchmark Functions

Abstract

The paper is devoted to the analysis of ill-posed operator equations Ax = y with injective linear operator A and solution x ₀ in a Hilbert space setting. We present some new ideas and results for finding convergence rates in Tikhonov regularization based on the concept of approximate source conditions by means of using distance functions with a general benchmark. For the case of compact operator A and benchmark functions of power-type, we can show that there is a one-to-one correspondence between the maximal power-type decay rate of the distance function and the best possible Hölder exponent for the noise-free convergence rate in Tikhonov regularization. As is well-known, this exponent coincides with the supremum of exponents in power-type source conditions. The main theorem of this paper is devoted to the impact of range inclusions under the smoothness assumption that x ₀ is in the range of some positive self-adjoint operator G. It generalizes a convergence rate result proven for compact G in Hofmann and Yamamoto (Inverse Problems 2005; 21:805–820) to the case of general operators G with nonclosed range.

Keywords:

• Convergence rates
• Distance function
• Hölder exponent
• Index function
• Linear ill-posed problems
• Qualification
• Range inclusion
• Source condition
• Tikhonov regularization

AMS Subject Classification:

• 47A52
• 65J20

ACKNOWLEDGMENTS

The main part of this paper was written during the stays of the first and of the second named authors in Tokyo as research fellows of the Japan Society for the Promotion of Science (JSPS) in summer 2005 and spring 2006, respectively. Both authors express their sincere thanks to JSPS for the generous financial support. The second named author is also supported by Deutsche Forschungsgemeinschaft (DFG) under grant HO1454/7-1. The third named author was 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.