Foschiatti Sonia

Department: 
Department of Mathematics, Informatics and Geosciences
Year of the Programme: 
3
Field: 
Mathematical Analysis
Supervisor: 
Research Title: 
Stability issue in inverse problems and its application to Geophysics
Project description: 

My research project deals with  the stability issue for the Calderon’s problem that consists in the recovery of the conductivity from known data measurements. Among the many applications of this problem, there is the Electrical Impedance Tomography (EIT). EIT is a medical imaging technique based on the reconstruction of the conductivity in the tissues by means of electrostatic measurements of voltages and current flux taken on the surface of the skin. Another application is geophysical exploration. In this framework, the electric current is injected into the ground through a pair of electrodes at the boundary while the voltage is measured with another pair of electrodes. This process is repeated various times at different locations, thus the data measurements are pairs voltage-current that can be modelled as a pseudodifferential operator, known as Neumann-to-Dirichlet (NtoD) map. It is well known that the inverse boundary value problem (IBVP) of determining the conductivity from the NtoD map or, equivalently, from its inverse, the Dirichlet-to-Naumann (DtoN) map is severely ill-posed. Several results in the literature suggest introducing special a-priori assumptions on the structure of the unknown conductivity to mitigate the ill-posedness of the problem. An assumption that seems to suit the physical problem in question is the one to confine the conductivity to a finite dimensional space, by assuming for example that it is piecewise constant on a finite given partition of the conductor. Previous literature along this line of research shows that, in this setting, the conductivity depends upon the data with a modulus of continuity of Lipschitz type. Our project aims at generalizing such results to include certain type of anisotropy in the model proving Lipschitz stability estimates, which are keys ingredients for the set up of efficient and reliable reconstruction procedures. 

Final thesis title: 
Stability estimates for some anisotropic inverse problems
Last update: 04-12-2024 - 01:40