Stable reconstruction of generalized impedance boundary conditions

TL;DR

This paper develops a stable method for reconstructing generalized impedance boundary conditions from far-field data, introducing new stability estimates and an optimization approach, with numerical validation in two dimensions.

Contribution

It provides the first global Lipschitz stability results for identifying second order surface operators in impedance boundary conditions, along with an effective optimization method.

Findings

Established Lipschitz stability for piecewise constant impedance functions.

Developed an optimization algorithm with $H^1$ regularization for boundary identification.

Numerical experiments demonstrate the method's effectiveness in 2D scenarios.

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 $\lambda$ and $\mu$ that define this boundary operator. We first derive a global Lipschitz stability result for the identification of $\ld$ or $\mu$ 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 $\lambda$ and $\mu$, 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.