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

Rafaela Filippozzi; Everton Boos; Douglas S. Gonçalves; Fermin S. V. Bazán

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arXiv · 2506.00190

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

Rafaela Filippozzi·Everton Boos·Douglas S. Gonçalves·Fermin S. V. Bazán

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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.

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Rafaela Filippozzi, Everton Boos, Douglas S. Gonçalves, Fermin S. V. Bazán. 2025-05-30. On the regularization property of Levenberg-Marquardt method with Singular Scaling for nonlinear inverse problems. https://arxiv.org/abs/2506.00190

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