arXiv subjects

Zou Jun

Publications and source records attributed to Zou Jun.

3 recordsLinked to original sources

arXiv2016

Local Lipschitz Stability for Inverse Robin Problems in Some Elliptic and Parabolic Systems

In this work, we shall study the nonlinear inverse problems of recovering the Robin coefficients in elliptic and parabolic systems of second order, and establish their local Lipschitz stabilities. Some local Lipschitz stability was derived for an elliptic inverse Robin problem. We shall first restructure the arguments in \cite{chou04} for the local Lipschitz stability so that the stability follows from three basic conditions for the elliptic inverse Robin problem. The new arguments are then generalized to help establish a novel local Lipschitz stability for parabolic inverse Robin problems.

math.AP↗

arXiv2015

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 and efficiency of the methods.

math.NA↗

arXiv2013

Overlapping Domain Decomposition Methods for Linear Inverse Problems

We shall derive and propose several efficient overlapping domain decomposition methods for solving some typical linear inverse problems, including the identiffication of the flux, the source strength and the initial temperature in second order elliptic and parabolic systems. The methods are iterative, and computationally very efficient: only local forward and adjoint problems need to be solved in each subdomain, and the local minimizations have explicit solutions. Numerical experiments are provided to demonstrate the robustness and efficiency of the methods, in particular, the convergences seem nearly optimal, i.e., they do not deteriorate or deteriorate only slightly when the mesh size reduces.

math.NA↗