Levenberg-Marquardt method with Singular Scaling and applications

TL;DR

This paper explores a modified Levenberg-Marquardt method with singular scaling matrices, proving convergence properties and demonstrating its effectiveness in parameter identification problems in heat conduction.

Contribution

It introduces a new Levenberg-Marquardt variant with singular scaling, establishing convergence and applicability to inverse heat conduction problems.

Findings

The method is well-defined under a completeness condition.

It achieves local quadratic convergence under an error bound.

It improves solution quality in heat conduction parameter identification.

Abstract

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.