Bouligand-Levenberg-Marquardt iteration for a non-smooth ill-posed inverse problem

TL;DR

This paper introduces a modified Levenberg--Marquardt method tailored for non-smooth, ill-posed inverse problems, utilizing Bouligand subdifferentials to ensure convergence and stability.

Contribution

It develops a novel iterative regularization scheme for non-smooth inverse problems using Bouligand subdifferentials, extending classical methods to non-differentiable settings.

Findings

Proves convergence of the Bouligand-Levenberg-Marquardt method.

Demonstrates stability estimates for non-smooth inverse problems.

Numerical examples show improved performance over Bouligand-Landweber iteration.

Abstract

In this paper, we consider a modified Levenberg--Marquardt method for solving an ill-posed inverse problem where the forward mapping is not G\^ateaux differentiable. By relaxing the standard assumptions for the classical smooth setting, we derive asymptotic stability estimates that are then used to prove the convergence of the proposed method. This method can be applied to an inverse source problem for a non-smooth semilinear elliptic PDE where a Bouligand subdifferential can be used in place of the non-existing Fr\'echet derivative, and we show that the corresponding Bouligand-Levenberg-Marquardt iteration is an iterative regularization scheme. Numerical examples illustrate the advantage over the corresponding Bouligand-Landweber iteration.