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 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.\ud The admissible polyhedral scatterers satisfy minimal apriori 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, N being the space dimension. 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

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.…”

“…Obviously, we haveB scat (r, L, R, r 1 ,C, ω, δ) ⊂B scat (r, L, R, ω, δ). We notice that any scatterer Σ ∈D obst is indeed an obstacle, that is, Σ is the closure of its interior which is a bounded open set with Lipschitz boundary, with constants r and L. By Corollary 2.3 and Proposition 2.1 in [20], for some constants and functions depending on r, L, and R only, we havê D obst (r, L, R) ⊂B scat (r,L, R, r 1 ,C, ω, δ).…”

“…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]. More importantly, in [20] the stability result has been extended to the sound-hard case, for the same class of polyhedral scatterers as in the sound-soft case, and with the minimal amount of scattering measurements, that is, N measurements for general polyhedral scatterers and a single measurement for polyhedra, at least for N = 2, 3 in this last case.…”

“…Also concerning the domains where the Maxwell system is defined, as pointed out above with respect to scatterers, these are always as general as possible, with boundaries which may have a rather complex structure. The classes of domains we use, described in Subsection 3.2, were introduced in [25] and further developed in [20]. We believe that, even if improvements may be obtained in this direction, their generality and versatility put them at the top of the art in this moment.…”

“…In Subsection 3.2, we discuss suitable classes of admissible scatterers. These kinds of classes were introduced in [25] and further developed in [20].…”