Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg-Marquardt method

TL;DR

This paper introduces a new strategy for selecting Lagrange multipliers in the Levenberg-Marquardt method, improving convergence and stability for solving ill-posed nonlinear inverse problems, demonstrated through numerical experiments.

Contribution

A novel range-relaxed criterion for choosing Lagrange multipliers in the Levenberg-Marquardt method with proven convergence and stability properties.

Findings

Monotonically decreasing iteration error

Geometric decay of residuals

Effective performance on EIT inverse problem

Abstract

In this article we propose a novel strategy for choosing the Lagrange multipliers in the Levenberg-Marquardt method for solving ill-posed problems modeled by nonlinear operators acting between Hilbert spaces. Convergence analysis results are established for the proposed method, including: monotonicity of iteration error, geometrical decay of the residual, convergence for exact data, stability and semi-convergence for noisy data. Numerical experiments are presented for an elliptic parameter identification two-dimensional EIT problem. The performance of our strategy is compared with standard implementations of the Levenberg-Marquardt method (using a priori choice of the multipliers).