Stable reconstruction of generalized impedance boundary conditions

Abstract

We are interested in the identification of a Generalized Impedance Boundary Condition from the far--fields created by one or several incident plane waves at a fixed frequency. We focus on the particular case where this boundary condition is expressed with the help of a second order surface operator: the inverse problem then amounts to retrieve the two functions $λ$ and $μ$ that define this boundary operator. We first derive a global Lipschitz stability result for the identification of $\ld$ or $μ$ from the far--field for bounded piecewise constant impedance coefficients and we give a new type of stability estimate when inexact knowledge of the boundary is assumed. We then introduce an optimization method to identify $λ$ and $μ$, using in particular a $H^1$-type regularization of the gradient. We lastly show some numerical results in two dimensions, including a study of the impact of some various parameters, and by assuming either an exact knowledge of the shape of the obstacle or an approximate one.