Stable determination of sound-hard polyhedral scatterers by a minimal number of scattering measurements

Abstract

The aim of the paper is to establish optimal stability estimates for the determination of sound-hard polyhedral scatterers in $\mathbb{R}^N$, $N \geq 2$, by a minimal number of far-field measurements. This work is a significant and highly nontrivial extension of the stability estimates for the determination of sound-soft polyhedral scatterers by far-field measurements, proved by one of the authors, to the much more challenging sound-hard case. The admissible polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time solid obstacles and screen-type components. In this case we obtain a stability estimate with $N$ far-field measurements. Important features of such an estimate are that we have an explicit dependence on the parameter $h$ representing the minimal size of the cells forming the boundaries of the admissible polyhedral scatterers, and that the modulus of continuity, provided the error is small enough with respect to $h$, does not depend on $h$. If we restrict to $N=2,3$ and to polyhedral obstacles, that is to polyhedra, then we obtain stability estimates with fewer measurements, namely first with $N-1$ measurements and then with a single measurement. In this case the dependence on $h$ is not explicit anymore and the modulus of continuity depends on $h$ as well.