Stable reconstruction of generalized impedance boundary conditions

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 λ or μ from the far field for bounded piecewise constant impedance coefficients and 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 an H¹-type regularization of the gradient. Finally, we show some numerical results in two dimensions, including a study of the impact of various parameters, and by assuming either an exact knowledge of the shape of the obstacle or an approximate one.