Convergence and adaptive discretization of the IRGNM Tikhonov and the IRGNM Ivanov method under a tangential cone condition in Banach space

TL;DR

This paper analyzes the convergence of IRGNM methods in Banach spaces under a tangential cone condition, introducing discretization error control and demonstrating applications to nonlinear elliptic inverse problems.

Contribution

It extends convergence analysis of IRGNM Tikhonov and Ivanov methods to Banach spaces under weaker conditions and includes discretization error control via goal-oriented estimators.

Findings

Convergence proven without source conditions under tangential cone condition.

Discretization error bounds achieved using goal-oriented weighted dual residual estimators.

Numerical results demonstrate effectiveness on nonlinear elliptic inverse problems.

Abstract

In this paper we consider the Iteratively Regularized Gauss-Newton Method (IRGNM) in its classical Tikhonov version and in an Ivanov type version, where regularization is achieved by imposing bounds on the solution. We do so in a general Banach space setting and under a tangential cone condition, while convergence (without source conditions, thus without rates) has so far only been proven under stronger restrictions on the nonlinearity of the operator and/or on the spaces. Moreover, we provide a convergence result for the discretized problem with an appropriate control on the error and show how to provide the required error bounds by goal oriented weighted dual residual estimators. The results are illustrated for an inverse source problem for a nonlinear elliptic boundary value problem, for the cases of a measure valued and of an $L^\infty$ source. For the latter, we also provide numerical results with the Ivanov type IRGNM.