Topological derivative method for an inverse problem modeled by the Schr¨odinger equation

Abstract

This paper deals with an inverse potential problem whose forward problem is governed by Schr¨odinger equation. The inverse problem consists in the reconstruction of a set of anomalies embedded into a fluid from partial measurements of the substance concentration. Since the inverse problem, we are dealing with, is written in the form of an ill-posed boundary value problem, the idea is to rewrite it as a topology optimization problem. In particular, a shape functional is defined to measure the misfit of the solution obtained from the model and the data taken from the partial measurements. This shape functional is minimized with respect to a set of ball-shaped anomalies by using the concept of topological derivatives. It means that the shape functional is expanded asymptotically and then truncated up to the desired order term. The resulting expression is trivially minimized with respect to the parameters under consideration, leading to a non-iterative second order reconstruction algorithm. Finally, a numerical example is presented to show the effectiveness of the proposed reconstruction algorithm.