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

## Palavras-chave:

Discrete Inverse Problems, Helmholtz Equation, Finite Elements Method, Regularization Schemes## 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.