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

Jiang, Daijun; Feng, Hui; Zou, Jun

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 3, pp. 1085-1107.

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.

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