Quadratic convergence of Levenberg-Marquardt method for elliptic and parabolic inverse robin problems

¹ School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, P.R. China
² School of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China
³ Department of Mathematics, The Chinese University of Hong Kong, Shatin, P.R. China.

^**** Corresponding author: zou@math.cuhk.edu.hk

Received: 30 September 2016
Accepted: 9 February 2018

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, efficiency and quadratic convergence of the methods.