Some questions related to geometric inverse problems

Abstract

We will consider geometric inverse problems of determining by external measurements a portion of the domain in which certain partial differential equations are satisfied. We will consider real-world applications problems and will explore two crucial aspects: uniqueness and numerical reconstruction based on certain optimization problems. We will present results that have been obtained in collaboration with different authors. This paper will consist of two parts. First, we will present some numerical domain reconstruction techniques related to optimization problems. We will focus on meshless technique based on the Method of Fundamental Solutions which will be introduced in the context of an elliptic equation. The second part of the paper will be devoted to analyzing the sensitivity of the inverse problems to the boundary and initial data which we will present in the context of the variable density Burguers equation.