11 Maggio, 2016 16:15 in punto

Sezione di Analisi

Some inverse boundary value problems for PDEs: theory and applications

Matteo Santacesaria, Politecnico di Milano

Aula seminari III piano

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. In particular we will review some classical strategy to attack these problems, based on the so-called complex geometrical optics solutions and inverse scattering theory. Then we will present a new reconstruction method able to detect singularities of a conductivity from the Dirichlet-to-Neumann map: this is based on some microlocal properties of our PDE. Numerical results will be presented as well. The latter is an ongoing project in collaboration with A. Greenleaf, M. Lassas, S. Siltanen and G. Uhlmann.