On inertial Levenberg-Marquardt type methods for solving nonlinear ill-posed operator equations

TL;DR

This paper introduces an inertial Levenberg-Marquardt method for stable solutions of nonlinear ill-posed operator equations, demonstrating convergence, stability, and efficiency through numerical experiments in PDEs and machine learning.

Contribution

It proposes a novel inertial LM method with proven convergence and stability properties for ill-posed problems, enhancing computational efficiency over traditional methods.

Findings

Monotonicity and convergence for exact data

Stability and semi-convergence for noisy data

Improved computational efficiency in numerical experiments

Abstract

In these notes we propose and analyze an inertial type method for obtaining stable approximate solutions to nonlinear ill-posed operator equations. The method is based on the Levenberg-Marquardt (LM) iteration. The main obtained results are: monotonicity and convergence for exact data, stability and semi-convergence for noisy data. Regarding numerical experiments we consider: i) a parameter identification problem in elliptic PDEs, ii) a parameter identification problem in machine learning; the computational efficiency of the proposed method is compared with canonical implementations of the LM method.