AN INVERSE PROBLEM APPROACH FOR THE IDENTIFICATION OF THE REFRACTIVE INDEX IN THE HELMHOLTZ EQUATION

Resumo

The Identification of parameters for differential equations is one of the most common types of discrete
inverse problems. Formulations like these can be used to calculate the thermal conductivity of a material, the drag
coefficient in a body during freefall and the diffusion of matter through a certain media. Following this trend, this
paper shows how the refraction index of a pre-established domain can be calculated applying inverse problem
techniques to the Helmholtz equation. As the main endeavor in inverse problems arise from the necessity of solving
matrices with large condition number, this work contains an overall review and a numerical comparison between
classical and more recent regularization schemes to solve these ill-posed matrices. In order to validate the results,
the problem was also solved in a direct manner using the Finite Elements Method. Results include numerical
examples for uniform and non-uniform mesh grids.