Bouligand-Landweber iteration for a non-smooth ill-posed problem

TL;DR

This paper develops a new convergence analysis for a Bouligand-Landweber iterative method applied to a non-smooth, ill-posed inverse problem, demonstrating strong convergence under certain conditions.

Contribution

It introduces a novel convergence analysis for a Bouligand-Landweber method applicable to non-smooth problems without Gâteaux differentiability, using asymptotic stability and a generalized tangential cone condition.

Findings

Convergence is proven for the Bouligand-Landweber iteration with exact and noisy data.

The method is verified on an inverse source problem for an elliptic PDE with non-smooth nonlinearity.

Strong convergence results are established under the discrepancy principle.

Abstract

This work is concerned with the iterative regularization of a non-smooth nonlinear ill-posed problem where the forward mapping is merely directionally but not G\^ateaux differentiable. Using a Bouligand subderivative of the forward mapping, a modified Landweber method can be applied; however, the standard analysis is not applicable since the Bouligand subderivative mapping is not continuous unless the forward mapping is G\^ateaux differentiable. We therefore provide a novel convergence analysis of the modified Landweber method that is based on the concept of asymptotic stability and merely requires a generalized tangential cone condition. These conditions are verified for an inverse source problem for an elliptic PDE with a non-smooth Lipschitz continuous nonlinearity, showing that the corresponding Bouligand--Landweber iteration converges strongly for exact data as well as in the limit of vanishing data if the iteration is stopped according to the discrepancy principle. This is illustrated