Quadratic Convergence of Levenberg-Marquardt Method for Elliptic and Parabolic Inverse Robin Problems

TL;DR

This paper proves the quadratic convergence of the Levenberg-Marquardt method for solving highly nonlinear and ill-posed inverse Robin problems in elliptic and parabolic systems, introducing an adaptive regularization strategy and surrogate functional approach.

Contribution

It establishes the first rigorous proof of quadratic convergence for the L-M method in these inverse problems, with an innovative adaptive regularization and explicit solution approach.

Findings

Quadratic convergence is rigorously proven for the L-M method.

The adaptive regularization strategy effectively ensures convergence.

Numerical experiments confirm the method's accuracy and efficiency.

Abstract

We study the Levenberg-Marquardt (L-M) method for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The L-M method transforms the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the L-M method is rigorously established for the nonlinear elliptic and parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the L-M iteration. Then the surrogate functional approach is adopted to solve the strongly ill-conditioned convex minimizations, resulting in an explicit solution of the minimisation at each L-M iteration for both the elliptic and parabolic cases. Numerical experiments are provided to demonstrate the accuracy and efficiency of the methods.