Orateur
Description
In this lecture, we will present approaches based on Carleman estimates of several questions related to inverse problems for evolution partial differential equations. The focus will be the recovery of parameters in the wave equation posed in a bounded domain with Dirichlet boundary conditions.
Carleman estimates are a type of weighted energy inequalities. They are particularly useful in proving for instance unique continuation properties, controlability results, or stability of inverse problems. Here, we will mainly consider the determination of the potential (a zeroth order term) in the whole domain from measurements of the Neumann boundary value of the solution of the wave equation. We will first present the proof of the uniqueness and stability of the inverse problem, that indeed came first historically, more than 25 years ago, using local and then global Carleman estimates. Nevertheless, our goal will be the proposition of a reconstruction algorithm, quite more recent, that uses the same technical tools.
This algorithm has the particularity of being designed using the strategy of the proof of stability of the inverse problem. It is indeed relying on the minimization of a functional based on the corresponding Carleman weight function. Conventional methods can encounter local minima, which is not the case with this approach, that is globally convergent.
I will outline the milestones of demonstrating the convergence of this algorithm in various cases of reconstructing a time-independent potential. The difficulties we encountered and some numerical simulations will be presented. We will conclude with a brief discussion of the extension of our approach to the reconstruction of the speed of wave propagation, a problem of more directly applied interest.
