|Abstract:|| In this talk we will focus on two inverse boundary value problems, the Calderón problem and the Gelfand-Calderon problem. The first concerns the reconstruction of an electrical conductivity from voltage and current measurements on the boundary of an object; its related imaging method is called Electrical Impedance Tomography and has applications from medical imaging to non destructive testing. In the Gelfand-Calderon problem one wants to reconstruct a potential in the Schrödinger equation from some information of its solutions at the boundary of a domain (Dirichlet to Neumann map). This problem can be seen as a model for acoustic tomography, namely with applications in geophysical prospecting.
We will first discuss theoretical properties of these problems, in particular their ill-posedness and stability estimates. Then we will present some recent numerical reconstruction results. First a numerical implementation for the reconstruction of a potential in the Schrödinger equation from high energy data; this new algorithm clearly shows how the resolution increases with the energy. Then we will present a new imaging technique based on electrical measurements which allows the reconstruction of inclusions inside a body. Its main feature is the ability to identify inclusions inside inclusions and the main application is a new method for early detection of brain strokes