A topological derivative-based method for an inverse problem modeled by the Helmholtz equation
Lucas dos Santos Fernandez1; Antonio André Novotny1; Ravi Prakash2
1 Laboratório Nacional de Computação Científica; 2 Universidad de Concepción
doi:10.20906/CPS/CILAMCE2017-0277
Resumo
This paper deals with an inverse problem whose forward model is governed by Helmholtz equation. The inverse problem consists in the reconstruction of a set of geometrical subdomains contained in a reference domain from partial measurements of a scalar field of interest. 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. We are using the concepts of topological derivatives, hence the shape functional measuring the misfit between the known target data and the calculated data is minimized with respect to a set of ball-shaped inclusions. It leads to a non-iterative reconstruction algorithm which is independent of any initial guess. As a result, the reconstruction process becomes very robust with respect to the noisy data. A numerical example is presented in order to demonstrate the effectiveness of the proposed algorithm.
Palavras-chave: Inverse problem; Helmholtz equation; higher-order topological derivatives; Topology optimization; Reconstruction method