Regularization of systems of nonlinear ill-posed equations: I. Convergence Analysis

TL;DR

This paper introduces new iterative regularization methods for solving systems of nonlinear ill-posed equations, focusing on convergence analysis and stability, using Kaczmarz-type approaches with novel stopping criteria.

Contribution

It develops two novel Kaczmarz-type regularization methods with specific strategies and proves their well-posedness, stability, and convergence for nonlinear ill-posed systems.

Findings

Both methods are proven to be well-posed and stable.

Convergence of the methods is rigorously established.

The techniques effectively handle nonlinear ill-posed problems.

Abstract

In this article we develop and analyze novel iterative regularization techniques for the solution of systems of nonlinear ill--posed operator equations. The basic idea consists in considering separately each equation of this system and incorporating a loping strategy. The first technique is a Kaczmarz-type method, equipped with a novel stopping criteria. The second method is obtained using an embedding strategy, and again a Kaczmarz-type approach. We prove well-posedness, stability and convergence of both methods.