arXiv subjects

Everton Boos

Publications and source records attributed to Everton Boos.

3 recordsLinked to original sources

arXiv2025

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

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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arXiv2024

Convergence analysis of Levenberg-Marquardt method with Singular Scaling for nonzero residue nonlinear least-squares problems

Recently, a Levenberg-Marquardt method with Singular Scaling matrix, called LMMSS, was proposed and successfully applied in parameter estimation in heat conduction problems, where the choice of suitable singular scaling matrix resulted in better quality approximate solutions than those of the classical Levenberg-Marquardt. In this paper, we study convergence properties of LMMSS when applied to nonzero residual nonlinear least-squares problems. We show that the local convergence of the iterates depends both on the control of the gradient linearization error and on a suitable choice of the regularization parameter. Incidentally, we show that the rate of convergence is dictated by a measure of nonlinearity and residual size, so that if such a measure goes to zero quickly enough, the convergence can be superlinear, otherwise, in general, we show that not even linear convergence can be expected if such a measure is not small enough. Additionally, we propose a globalized version of the method and prove that any limit point of the generated sequence is stationary for the least-squares function. Some examples are provided to illustrate our theoretical results.

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arXiv2023

Levenberg-Marquardt method with Singular Scaling and applications

Inspired by certain regularization techniques for linear inverse problems, in this work we investigate the convergence properties of the Levenberg-Marquardt method using singular scaling matrices. Under a completeness condition, we show that the method is well-defined and establish its local quadratic convergence under an error bound assumption. We also prove that the search directions are gradient-related allowing us to show that limit points of the sequence generated by a line-search version of the method are stationary for the sum-of-squares function. The usefulness of the method is illustrated with some examples of parameter identification in heat conduction problems for which specific singular scaling matrices can be used to improve the quality of approximate solutions.

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