Modified iterated Tikhonov methods for solving systems of nonlinear ill-posed equations

TL;DR

This paper introduces modified iterated Tikhonov methods combined with a Kaczmarz strategy to stabilize solutions of nonlinear ill-posed equations, including a new approach for noisy data with convergence guarantees.

Contribution

The paper proposes a novel regularization method that integrates iterated Tikhonov regularization with a Kaczmarz strategy and introduces a modified approach for noisy data with proven convergence.

Findings

The proposed method is a convergent regularization technique.

A modified loping iterated Tikhonov-Kaczmarz method effectively handles noisy data.

Convergence analysis confirms the method's stability and reliability.

Abstract

We investigate iterated Tikhonov methods coupled with a Kaczmarz strategy for obtaining stable solutions of nonlinear systems of ill-posed operator equations. We show that the proposed method is a convergent regularization method. In the case of noisy data we propose a modification, the so called loping iterated Tikhonov-Kaczmarz method, where a sequence of relaxation parameters is introduced and a different stopping rule is used. Convergence analysis for this method is also provided.