Convergence rates for the iteratively regularized Gauss–Newton method in Banach spaces**Research has been partly conducted during the Mini Special Semester on Inverse Problems, 18 May–15 July 2009, organized by RICAM (Austrian Academy of Sciences), Linz, Austria, and moreover supported by Deutsche Forschungsgemeinschaft (DFG) under Grant HO1454/7-2 as well as within the Cluster of Excellence SimTech, University of Stuttgart.

Abstract

In this paper we consider the iteratively regularized Gauss–Newton method (IRGNM) in a Banach space setting and prove optimal convergence rates under approximate source conditions. These are related to the classical concept of source conditions that is available only in Hilbert space. We provide results in the framework of general index functions, which include, e.g. Hölder and logarithmic rates. Concerning the regularization parameters in each Newton step as well as the stopping index, we provide both a priori and a posteriori strategies, the latter being based on the discrepancy principle.