Analysis Seminar

On some inverse boundary value problems: Ill-posedness and remedies

Speaker: Elena Beretta, Politecnico di Milano

Location: Warren Weaver Hall 1302

Date: Thursday, May 16, 2019, 11 a.m.


Inverse boundary value problems consist of recovering unknown parameters of a partial differential equation (PDE) or of a system of PDE’s from boundary data. In applications, this corresponds to reconstructing internal properties of a medium (e.g. conduction, stiffness, density) from observations made at its boundary; such problems lie at the heart of scientific and technological development and have applications, for example, in medical imaging, nondestructive testing of materials, seismic imaging and underground prospection.

In general, parameter estimation problems are highly nonlinear and ill-posed in the sense of Hadamard: small errors in the data may cause incontrollable errors in the unknowns. In view of the many applications, this leads to the search of appropriate methods to contain such instability.

In the first part of my talk I will introduce Calderon’s problem, a prototypical example of ill-posed nonlinear inverse problem at the basis of Electrical Impedence Tomography (EIT). In the second part, I will show that by introducing, mathematically suitable but physically relevant, a-priori assumptions on the unknown parameters one can mitigate the ill-posedeness. I will show that for Calder ́on’s problem one can obtain the best possibly stability (i.e. Lipschitz) of the unknowns from the data. The regularizing effect of additional assumptions can be used to implement efficient reconstruction methods and is then briefly discussed for other models, together with some open issues.