Overlapping Domain Decomposition Methods for Linear Inverse Problems

TL;DR

This paper introduces efficient overlapping domain decomposition methods for linear inverse problems, enabling local problem solving with explicit solutions, demonstrating robustness and near-optimal convergence in numerical experiments.

Contribution

The paper develops novel overlapping domain decomposition algorithms specifically tailored for linear inverse problems, improving computational efficiency and convergence properties.

Findings

Methods are computationally efficient, requiring only local forward and adjoint problems.

Local minimizations have explicit solutions, simplifying implementation.

Numerical experiments show robustness and near-optimal convergence.

Abstract

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.