A doubly relaxed minimal-norm Gauss-Newton method for underdetermined nonlinear least-squares problems

TL;DR

This paper introduces a Gauss-Newton type method with two relaxation parameters for efficiently computing minimal-norm solutions in underdetermined nonlinear least-squares problems, with dynamic parameter estimation and convergence guarantees.

Contribution

It proposes a novel doubly relaxed minimal-norm Gauss-Newton method that adaptively estimates parameters and Jacobian rank to improve convergence in underdetermined problems.

Findings

Method ensures convergence for underdetermined nonlinear least-squares problems.

Numerical results demonstrate the effectiveness of the proposed approach.

Adaptive parameter estimation enhances solution accuracy.

Abstract

When a physical system is modeled by a nonlinear function, the unknown parameters can be estimated by fitting experimental observations by a least-squares approach. Newton's method and its variants are often used to solve problems of this type. In this paper, we are concerned with the computation of the minimal-norm solution of an underdetermined nonlinear least-squares problem. We present a Gauss-Newton type method, which relies on two relaxation parameters to ensure convergence, and which incorporates a procedure to dynamically estimate the two parameters, as well as the rank of the Jacobian matrix, along the iterations. Numerical results are presented.