A derivative-free Levenberg-Marquardt method for sparse nonlinear least squares problems

TL;DR

This paper introduces a derivative-free Levenberg-Marquardt algorithm tailored for sparse nonlinear least squares problems, utilizing Jacobian models built via $\ell_1$ minimization and interpolation, with proven convergence and demonstrated efficiency.

Contribution

It presents a novel derivative-free method that constructs Jacobian models through $\ell_1$ minimization and interpolation, offering a new approach for sparse nonlinear least squares problems.

Findings

Jacobian models are probabilistically first-order accurate.

The algorithm converges globally almost surely.

Numerical experiments confirm the method's efficiency.

Abstract

This paper studies sparse nonlinear least squares problems, where the Jacobian matrices are unavailable or expensive to compute, yet have some underlying sparse structures. We construct the Jacobian models by the $ \ell_1 $ minimization subject to a small number of interpolation constraints with interpolation points generated from some certain distributions,and propose a derivative-free Levenberg-Marquardt algorithm based on such Jacobian models.It is proved that the Jacobian models are probabilistically first-order accurate and the algorithm converges globally almost surely.Numerical experiments are presented to show the efficiency of the proposed algorithm for sparse nonlinear least squares problems.