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

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.

Domain Decomposition Methods for Recovering Robin Coefficients in Elliptic and Parabolic Systems

We shall derive and propose several efficient domain decomposition methods for solving the nonlinear inverse problem of identifying the Robin coefficients in elliptic and parabolic systems. The highly ill-posed inverse problems are transformed into output least-squares nonlinear and non-convex minimizations with classical Tikhonov regularization. The Levenberg–Marquardt method is applied to transform the non-convex minimizations into convex minimizations, which will be solved by several efficient domain decomposition methods. The methods are completely local and the local minimizers have explicit expressions within the subdomains. Several numerical experiments are presented to show the accuracy and efficiency of the methods; in particular, the convergence seems nearly optimal in the sense that the iteration number of the methods is nearly independent of mesh sizes.