Inverse Problems for Partial Differential Equations (V. Bacchelli, E. Beretta, C. Cerutti, M. Di Cristo, D. Pierotti)

Inverse problems require to determine the cause from a set of observations. Such problems appear for example  in  medical imaging, non destructive testing of materials, computerized tomography, source reconstructions in acoustics,  computer vision, geophysics and seismology, to mention but a few, and their mathematical solutions represent breakthroughs in applications. In many situations the mathematical modelling of these problems leads to the study of inverse boundary value problems for linear and nonlinear partial differential equations and systems where one wants to determine coefficients appearing in the equation or system form observations of the solution at the boundary. Analysis of such problems involves different subjects of mathematics such as  for example complex analysis, harmonic analysis, microlocal analysis, numerical analysis, optimization, operator theory, probability, statistics etc. and  represents an area of interaction of pure and applied mathematics.

This type of problems are generally highly non-linear and  ill-posed in the sense of Hadamard; small errors in the data may cause uncontrollable errors in the solution.

An illuminating example is the celebrated inverse conductivity problem, modeling an imaging technique, also known as Electrical Impedance Tomography, which aims to detect the conductivity inside an object from boundary measurements (encoded by the so-called Dirichlet to Neumann map).

The problem was introduced the first time by Calderon in the early 80's motivated by an application in geophysical prospection. Since then a lot of beautiful mathematics has been developed in order to investigate the wellposedness of this inverse problem. The conductivity problem is severely ill-posed. Actually, despite the a-priori smoothness assumptions on the unknown conductivity, a conditional stability estimate of logarithmic type of the conductivity in terms of the data is the best possible. Precisely because of this feature, the analysis of these instabilities and their regularization towards a successful reconstruction are crucial.

Our research is mainly focused on investigating stability, uniqueness and reconstruction of inverse boundary value problems and, in the case of severe ill-posedness (log-type stability) using  regularization procedures that restore wellposedness of the problems.

Politecnico di Milano - Dipartimento di Matematica