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 ${\mathbb{R}}^N$, $N\ge 2$, 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.

39 pages

References in corpus (2)

• Stable determination of a scattered wave from its far-field pattern: the high frequency asymptotics
• A stability result for Neumann problems in dimension $N\ge 3$

Cited by in corpus (19)

• Mosco convergence for H(curl) spaces, higher integrability for Maxwell's equations, and stability in direct and inverse EM scattering problems
• Infinite-dimensional inverse problems with finite measurements
• Scattering by curvatures, radiationless sources, transmission eigenfunctions and inverse scattering problems
• On a novel inverse scattering scheme using resonant modes with enhanced imaging resolution
• A novel quantitative inverse scattering scheme using interior resonant modes
• Inverse problems on low-dimensional manifolds
• Stable determination of a rigid scatterer in elastodynamics
• Unique determination of the shape of a scattering screen from a passive measurement
• Determining a piecewise conductive medium body by a single far-field measurement
• On corners scattering stably and stable shape determination by a single far-field pattern
• On nodal and generalized singular structures of Laplacian eigenfunctions and applications in ${\mathbb{R}}^3$
• Local geometric properties of conductive transmission eigenfunctions and applications
• On local and global structures of transmission eigenfunctions and beyond
• On the geometric structures of transmission eigenfunctions with a conductive boundary condition and applications
• On generalized Holmgren's principle to the Lamé operator with applications to inverse elastic problems
• Machine learning based data retrieval for inverse scattering problems with incomplete data
• Two unique Identifiability results for inverse scattering problems within polyhedral geometries
• Mathematical design of a novel gesture-based instruction/input device using wave detection
• Decoupling Elastic Waves and Its Applications