Slim Chaabane

Logarithmic stability estimates for a Robin coefficient in two-dimensional Laplace inverse problems

We establish some global stability results together with logarithmic estimates in Sobolev norms for the inverse problem of recovering a Robin coefficient on part of the boundary of a smooth 2D domain from overdetermined measurements on the complementary part of a solution to the Laplace equation in the domain, using tools from analytic function theory.

Differentiability properties of the L1-tracking functional and application to the Robin inverse problem

We investigate an optimization problem (OP) in a non-standard form: the cost functional measures the L1 distance between the solution u of the direct Robin problem and a function f L1(M). After proving positivity, monotonicity and control properties of the state u with respect to , we prove the existence of an optimal control ? to the problem (OP) and establish Newton differentiability of the functional . As an application to this optimization problem the inverse problem of determining a Robin parameter inv by measuring the data f on M is considered. In that case f is assumed to be the trace on M of . In spite of the fact that we work with the L1-norm we prove differentiability of the cost functional by using complex analysis techniques. The proof is strongly related to positivity and monotonicity of the derivative of the state with respect to . An identifiability result is also proved for the set of admissible parameters ?ad consisting of positive functions in L?.

Parameter identification for Laplace equation and approximation in Hardy classes

We consider the inverse problem of identifying a Robin coefficient on some part of the boundary of a smooth 2D domain from overdetermined data available on the other part of the boundary, for Laplace equation in the domain. Using tools from complex analysis and analytic functions theory, we provide a constructive and convergent identification scheme for this inverse problem, together with numerical experiments.