The Ivanov regularized Gauss-Newton method in Banach space with an a posteriori choice of the regularization radius

TL;DR

This paper develops an Ivanov regularized Gauss-Newton method with an a posteriori rule for selecting the regularization radius, proving convergence in Banach spaces and demonstrating effectiveness through numerical experiments.

Contribution

It introduces a new a posteriori rule for the Ivanov regularization radius within the Gauss-Newton method in Banach spaces, with proven convergence and error analysis.

Findings

Convergence and rates are established under variational source conditions.

The method is applicable in general Banach spaces, including nonreflexive ones.

Numerical experiments validate the theoretical results.

Abstract

In this paper we consider the iteratively regularized Gauss-Newton method, where regularization is achieved by Ivanov regularization, i.e., by imposing a priori constraints on the solution. We propose an a posteriori choice of the regularization radius, based on an inexact Newton / discrepancy principle approach, prove convergence and convergence rates under a variational source condition as the noise level tends to zero, and provide an analysis of the discretization error. Our results are valid in general, possibly nonreflexive Banach spaces, including, e.g., $L^\infty$ as a preimage space. The theoretical findings are illustrated by numerical experiments.