One-iteration reconstruction algorithm for geometric inverse source problem

Abstract

In this paper, we address the reconstruction of characteristic source functions \(\delta g^*\) in the elliptic partial differential equations \(-\Delta u+u=\delta g^*\) from the knowledge of the boundary measurements. We will detect the shape and location of the unknown source term from additional boundary conditions. We propose a new reconstruction method based on the Kohn–Vogelius formulation and the topological gradient method. The inverse problem is formulated as a topological optimization one. An asymptotic expansion for an energy function is derived with respect to a small topological perturbation of the source term. The unknown source is reconstructed using a level-set curve of the topological gradient. A non-iterative reconstruction procedure based on the topological sensitivity is implemented. The efficiency and accuracy of the reconstruction algorithm are illustrated by some numerical results.