Local Analysis of Inverse Problems: H\"{o}lder Stability and Iterative Reconstruction

TL;DR

This paper investigates the local convergence of nonlinear Landweber iterations for inverse problems, linking convergence conditions to Hölder stability of the inverse map in Banach spaces.

Contribution

It provides new conditions for local convergence of iterative methods based on Hölder stability in the context of nonlinear inverse problems.

Findings

Established local convergence criteria for Landweber iteration.

Connected Hölder stability to iterative convergence.

Applicable to inverse problems with Banach space data.

Abstract

We consider a class of inverse problems defined by a nonlinear map from parameter or model functions to the data. We assume that solutions exist. The space of model functions is a Banach space which is smooth and uniformly convex; however, the data space can be an arbitrary Banach space. We study sequences of parameter functions generated by a nonlinear Landweber iteration and conditions under which these strongly converge, locally, to the solutions within an appropriate distance. We express the conditions for convergence in terms of H\"{o}lder stability of the inverse maps, which ties naturally to the analysis of inverse problems.