On the interplay of source conditions and variational inequalities for nonlinear ill-posed problems

Abstract

In the past few years, convergence rates results for Tikhonov regularization of nonlinear ill-posed problems in Banach spaces have been published, where the classical concept of source conditions was replaced with variational inequalities holding on some level sets. This essentially advanced the analysis of non-smooth situations with respect to forward operators and solutions. In fact, such variational inequalities combine both structural conditions on the nonlinearity of the operator and smoothness properties of the solution. Varying exponents in the variational inequalities correspond to different levels of convergence rates. In this article, we discuss the range of occurring exponents in the Banach space setting. To lighten the cross-connections between generalized source conditions, degree of nonlinearity of the forward operator and associated variational inequalities we study the Hilbert space situation and even prove some converse result for linear operators. Finally, we outline some aspects for the interplay of variational regularization and conditional stability estimates for partial differential equations. As an example, we apply the theory to a specific parameter identification problem for a parabolic equation.

Keywords:

• nonlinear ill-posed problems
• regularization
• variational inequalities
• source conditions
• degree of nonlinearity
• convergence rates
• conditional stability
• inverse PDE problems

AMS Subject Classifications::

• 47J06
• 65J20
• 47J20
• 47A52

Acknowledgements

This article was started during a research stay of B. Hofmann at the Graduate School of Mathematical Sciences of the University of Tokyo in February/March 2009 and has been completed in Linz/Austria during the Mini Special Semester on Inverse Problems, 18 May – 15 July, 2009, organized by RICAM, Austrian Academy of Sciences. The stay at the University of Tokyo was supported by Global COE Program ‘The Research and Training Center for New Development in Mathematics’. B. Hofmann thanks both hosts for kind hospitality and allowance. The research of B. Hofmann was also supported by Deutsche Forschungsgemeinschaft (DFG) under Grant HO1454/7-2. M. Yamamoto was partly supported by Grants 20654011 and 21340021 from Japan Society for the Promotion of Science. Moreover, both the authors thank Radu Ioan Boţ and Jens Geissler (TU Chemnitz) for fruitful discussions.