A fast reconstruction algorithm for geometric inverse problems using topological sensitivity analysis and Dirichlet-Neumann cost functional approach

TL;DR

This paper introduces a fast, accurate algorithm for reconstructing objects in anisotropic media from boundary data, utilizing topological sensitivity analysis and a level-set method within a topology optimization framework.

Contribution

It develops a novel topological asymptotic expansion for anisotropic Laplace operators and integrates it into a level-set based reconstruction algorithm.

Findings

Algorithm demonstrates high accuracy in numerical tests

Reconstruction is efficient and robust in anisotropic settings

Topological sensitivity provides detailed object detection

Abstract

This paper is concerned with the detection of objects immersed in anisotropic media from boundary measurements. We propose an accurate approach based on the Kohn-Vogelius formulation and the topological sensitivity analysis method. The inverse problem is formulated as a topology optimization one minimizing an energy like functional. A topological asymptotic expansion is derived for the anisotropic Laplace operator. The unknown object is reconstructed using a level-set curve of the topological gradient. The efficiency and accuracy of the proposed algorithm are illustrated by some numerical results. MOTS-CL\'ES : Probl\`eme inverse g\'eom\'etrique, Laplace anisotrope, formulation de Kohn-Vogelius, analyse de sensibilit\'e, optimisation topologique.