On the regularization property of Levenberg-Marquardt method with Singular Scaling for nonlinear inverse problems

TL;DR

This paper introduces a regularization framework for the Levenberg-Marquardt method with Singular Scaling (LMMSS) applied to nonlinear inverse problems, demonstrating convergence and stability under noise and a new tangent cone condition.

Contribution

It extends the analysis of LMM with Singular Scaling by establishing convergence and regularization properties for noisy data in nonlinear inverse problems.

Findings

LMMSS iterates converge to true solutions as noise decreases.

The discrepancy principle effectively determines stopping criteria.

The new tangent cone condition supports convergence analysis.

Abstract

Recently, in Applied Mathematics and Computation 474 (2024) 128688, a Levenberg-Marquardt method (LMM) with Singular Scaling was analyzed and successfully applied in parameter estimation problems in heat conduction where the use of a particular singular scaling matrix (semi-norm regularizer) provided approximate solutions of better quality than those of the classic LMM. Here we propose a regularization framework for the Levenberg-Marquardt method with Singular Scaling (LMMSS) applied to nonlinear inverse problems with noisy data. Assuming that the noise-free problem admits exact solutions (zero-residual case), we consider the LMMSS iteration where the regularization effect is induced by the choice of a possibly singular scaling matrix and an implicit control of the regularization parameter. The discrepancy principle is used to define a stopping index that ensures stability of the computed solutions with respect to data perturbations. Under a new Tangent Cone Condition, we prove that the iterates obtained with noisy data converge to a solution of the unperturbed problem as the noise level tends to zero. This work represents a first step toward the analysis of regularizing properties of the LMMSS method and extends previous results in the literature on regularizing LM-type methods.