Shape optimization methods for the Inverse Obstacle Problem with generalized impedance boundary conditions

Fabien Caubet (1) , Marc Dambrine (2) , Djalil Kateb (1)

1 LMAC - Laboratoire de Mathématiques Appliquées de Compiègne
2 LMAP - Laboratoire de Mathématiques et de leurs Applications [Pau]

We aim to reconstruct an inclusion ω immersed in a perfect fluid flowing in a larger bounded domain Ω via boundary measurements on ∂Ω. The obstacle ω is assumed to have a thin layer and is then modeled using generalized boundary conditions (precisely Ventcel boundary conditions). We first obtain an identifiability result (i.e. the uniqueness of the solution of the inverse problem) for annular configurations through explicit computations. Then, this inverse problem of reconstructing ω is studied thanks to the tools of shape optimization by minimizing a least squares type cost functional. We prove the existence of the shape derivatives with respect to the domain ω and characterize the gradient of this cost functional in order to make a numerical resolution. We also characterize the shape Hessian and prove that this inverse obstacle problem is unstable in the following sense: the functional is degenerated for highly oscillating perturbations. Finally, we present some numerical simulations in order to confirm and extend our theoretical results.

Mots clés: geometric inverse problem; shape calculus; generalized impedance boundary conditions; Ventcel boundary conditions; identifiability result

Domaines: Optimisation et contrôle [math.OC]

Fabien Caubet, Marc Dambrine, Djalil Kateb. Shape optimization methods for the Inverse Obstacle Problem with generalized impedance boundary conditions. Inverse Problems, 2013, 29 (11), ⟨10.1088/0266-5611/29/11/115011⟩. ⟨hal-00780735⟩