Aplicação do método das soluções fundamentais em problemas inversos geométricos

Abstract

The aim of this work is to explore some applications of the Method of Fundametal Solutions (MFS) to solve geometric problems. Initially, we study the reconstructions of a subdomain contained in a bounded set for the modified Helmhotz and Laplace equations. Finally, two ways of using the Fundamental Solutions Method for a problem of Electrical Impedance Tomography (EIT) are presented. Such methods are aimed at reconstructing the wet area of water from voltage measurements obtained from a system of electrodes fed with a certain electrical current and placed on the walls containing that liquid. The basic idea of the methodology used to solve both of the above problems is to minimize a cost function, which relates the measurements and the numerical data obtained through the MFS, the last data being dependent on the boundary to be reconstructed. Several numerical examples are presented to check the validity of the results. For a simulation more realistic, we incorporate White Gaussian Noise (WGN) into the data.