On projective Landweber-Kaczmarz methods for solving systems of nonlinear ill-posed equations

TL;DR

This paper introduces a combined projective Landweber-Kaczmarz method for solving large-scale nonlinear ill-posed equations, demonstrating convergence and superior performance in inverse problems.

Contribution

The authors develop and analyze a new iterative regularization method combining projective Landweber and Kaczmarz techniques under the tangential cone condition.

Findings

Proven convergence of the proposed method as a regularization technique.

Numerical tests show improved stability and accuracy over existing methods.

Effective for large-scale nonlinear inverse problems.

Abstract

In this article we combine the projective Landweber method, recently proposed by the authors, with Kaczmarz's method for solving systems of non-linear ill-posed equations. The underlying assumption used in this work is the tangential cone condition. We show that the proposed iteration is a convergent regularization method. Numerical tests are presented for a non-linear inverse problem related to the Dirichlet-to-Neumann map, indicating a superior performance of the proposed method when compared with other well established iterations. Our preliminary investigation indicates that the resulting iteration is a promising alternative for computing stable solutions of large scale systems of nonlinear ill-posed equations.