MSRI - Approximation of inverse boundary value problems by phase-field methods

Abstract: We consider the inverse problem of determining insulating
defects, such as cracks or cavities, by performing few electrostatic
measurements on the boundary. A possible method for solving in a
convenient way the direct problem is to approximate the defects by using
suitable phase-field functions. We address the question whether such an
approximation may be useful for the inverse problem as well. With the help
of a regularization of perimeter-penalization type, we show by a
convergence analysis that this is indeed the case. However, there are
important differences between the crack and cavity cases that we wish to
illustrate. We conclude with some numerical experiments, done in
collaboration with Wolfgang Ring (University of Graz, Austria), showing
the performance of these methods.