Stable determination of sound-soft polyhedral scatterers by a single measurement

Abstract

We prove optimal stability estimates for the determination of a finite number of sound-soft polyhedral scatterers in R^3 by a single far-field measurement. The admissible multiple polyhedral scatterers satisfy minimal a priori assumptions of Lipschitz type and may include at the same time obstacles, screens and even more complicated scatterers. We characterize any multiple\ud polyhedral scatterer by a size parameter h which is related to the minimal size of the cells of its boundary. In a first step we show that, provided the error epsilon on the far-field measurement is small enough with respect to h, then the corresponding error, in the Hausdorff distance, on the multiple polyhedral scatterer can be controlled by an explicit function of epsilon which approaches zero,\ud as epsilon goes to 0, in an essentially optimal, although logarithmic, way. Then, we show how to improve this stability estimate, provided we restrict our attention to multiple polyhedral obstacles and epsilon is even smaller with respect to h. In this case we obtain an explicit estimate essentially of Hoelder type

Mosco convergence for $H$ (curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems

“…with C 1 and C 2 depending on the classD obst only. The estimates obtained in Theorem 5.5, in particular the uniform bound (5.4) for R = R 0 + 3 and and the uniform decay (5.5), are the crucial preliminary results that are required to extend the stability results obtained in the acoustic case, in [34] for sound-soft scatterers and in [20] for soundhard scatterers, to the electromagnetic case.…”

“…The corresponding stability issue has been treated first in dimension 3 for general sound-soft polyhedral scatterers, [34], with a single scattering measurement and under minimal regularity assumptions. This result has been extended to any dimension and for a more general and versatile class of polyhedral scatterers in [20].…”

“…In all these stability results, a crucial preliminary result is to establish bounds for the solutions of the direct scattering problems which are uniform for the widest possible class of admissible scatterers. For the sound-soft case this was done in [34], whereas for the sound-hard case such a result follows from [25].…”

Reconstruction of shapes and impedance functions using few far-field measurements

“…Through (18) we obtain a velocity W defined only on @D nÀ1 . We extend the velocity W by a linear transport equation described by (25) in Section 4.3 and plug it in (24), then we obtain an updated level set function / n ðxÞ. 4.…”

“…Moreover, if we know in advance that the obstacle D is polygonal, then two incident directions are enough for detecting ð@D; rÞ, see [21] and the references there, while for the Dirchlet case one incident direction is enough, see [1]. For this particular form of the obstacle, stability estimates are also provided in [25]. For general forms of obstacles and for Robin boundary conditions, the local uniqueness question is still an open issue.…”