Reconstruction algorithm of the potential in a wave equation
This talk aims to present some recent works in collaboration with Maya de Buhan and Sylvain Ervedoza regarding an inverse problem for the wave equation. More specifically, we study the determination of the potential in a wave equation with given Dirichlet boundary data from a measurement of the flux of the solution on a part of the boundary. Several uniqueness and stability results are available in the literature about this inverse problem. In particular, we can mention a Lipschitz stability result under a classical geometric condition obtained by Imanuvilov and Yamamoto. We will present in this talk a new globally convergent reconstruction algorithm of the potential. The design and convergence of the algorithm are based on a global Carleman estimate for the wave operator, traditionally used to prove the Lipschitz stability of the inverse problem. We will finally give some simple illustrative numerical simulations for 1-d problems.