arXiv subjects

Rafaela Filippozzi

Publications and source records attributed to Rafaela Filippozzi.

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.

math.NA↗

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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arXiv2022

First-order methods for the convex hull membership problem

The convex hull membership problem (CHMP) consists in deciding whether a certain point belongs to the convex hull of a finite set of points, a decision problem with important applications in computational geometry and in foundations of linear programming. In this study, we review, compare and analyze first-order methods for CHMP, namely, Frank-Wolfe type methods, Projected Gradient methods and a recently introduced geometric algorithm, called Triangle Algorithm (TA). We discuss the connections between this algorithm and Frank-Wolfe, showing that TA can be interpreted as an inexact Frank-Wolfe. Despite this similarity, TA is strongly based on a theorem of alternatives known as distance duality. By using this theorem, we propose suitable stopping criteria for CHMP to be integrated into Frank-Wolfe type and Projected Gradient, specializing these methods to the membership decision problem. Interestingly, Frank-Wolfe integrated with such stopping criteria coincides with a greedy version of the Triangle Algorithm which is, in its turn, equivalent to an algorithm due to von Neumann. We report numerical experiments on random instances of CHMP, carefully designed to cover different scenarios, that indicate which algorithm is preferable according to the geometry of the convex hull and the relative position of the query point. Concerning potential applications, we present two illustrative examples, one related to linear programming feasibility problems and another related to image classification problems.

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