Iteratively regularized landweber iteration method: Convergence analysis via H\"older Stability

TL;DR

This paper analyzes the local convergence of the iteratively regularized Landweber method for nonlinear inverse problems in Banach spaces, establishing convergence rates based on H"older stability assumptions for both noisy and noise-free data.

Contribution

It provides a new convergence analysis framework for the Landweber iteration under H"older stability, including convergence rates for noisy and noise-free data.

Findings

Convergence of iterates within a specific ball around the solution.

Establishment of convergence rates under H"older stability.

Validation of convergence for both noisy and non-noisy data cases.

Abstract

In this paper, the local convergence of Iteratively regularized Landweber iteration method is investigated for solving non-linear inverse problems in Banach spaces. Our analysis mainly relies on the assumption that the inverse mapping satisfies the H\"older stability estimate locally. We consider both noisy as well as non-noisy data in our analysis. Under the a-priori choice of stopping index for noisy data, we show that the iterates remain in a certain ball around exact solution and obtain the convergence rates. The convergence of the Iteratively regularized Landweber iterates to the exact solution is shown under certain assumptions in the case of non-noisy data and as a by-product, under different conditions, two different convergence rates are obtained.