Polyhedral Scatterers Research Articles

Inverse Time-Harmonic Electromagnetic Scattering from Coated Polyhedral Scatterers With a Single Far-Field Pattern

It is proved that a convex polyhedral scatterer of impedance type can be uniquely determined by the electric far-field pattern of a non-vanishing incident field. The incoming wave is allowed to bean electromagnetic plane wave, a vector Herglotz wave function or a point source wave incited by some magnetic dipole. Our proof relies on the reflection principle for Maxwell's equations with the impedance (or Leontovich) boundary condition enforcing on a hyper-plane. We prove that it is impossible to analytically extend the total field across any vertex of the scatterer. This leads to a data-driven inversion scheme for imaging an arbitrary convex polyhedron.

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

This paper is concerned with the scattering problem for time-harmonic electromagnetic waves, due to the presence of scatterers and of inhomogeneities in the medium. We prove a sharp stability result for the solutions to the direct electromagnetic scattering problem, with respect to variations of the scatterer and of the inhomogeneity, under minimal regularity assumptions for both of them. The stability result leads to bounds on solutions to the scattering problems which are uniform for an extremely general class of admissible scatterers and inhomogeneities. These uniform bounds are a key step in tackling the challenging stability issue for the corresponding inverse electromagnetic scattering problem. In this paper we establish two optimal stability results of logarithmic type for the determination of polyhedral scatterers by a minimal number of electromagnetic scattering measurements. In order to prove the stability result for the direct electromagnetic scattering problem, we study two fundamental issues in the theory of Maxwell equations: Mosco convergence for H (curl) spaces and higher integrability properties of solutions to Maxwell equations in nonsmooth domains.

Decoupling elastic waves and its applications

In this paper, we consider time-harmonic elastic wave scattering governed by the Lamé system. It is known that the elastic wave field can be decomposed into the shear and compressional parts, namely, the pressure and shear waves that are generally coexisting, but propagating at different speeds. We consider the third or fourth kind impenetrable scatterer and derive two geometric conditions, respectively, related to the mean and Gaussian curvatures of the boundary surface of the scatterer that can ensure the decoupling of the shear and pressure waves. The decoupling results are new to the literature and are of significant interest for their own sake. As an interesting application, we apply the decoupling results to the uniqueness and stability analysis for inverse elastic scattering problems in determining polyhedral scatterers by a minimal number of far-field measurements.

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

The aim of the paper is to establish optimal stability estimates for the determination of sound-hard polyhedral scatterers in RN, N≥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.

Unique determination of balls and polyhedral scatterers with a single point source wave

In this paper, we prove uniqueness in determining a sound-soft ball or polyhedral scatterer in the inverse acoustic scattering problem with a single incident point source wave in (N = 2,3). Our proofs rely on the reflection principle for the Helmholtz equation with respect to a Dirichlet hyperplane or sphere, which is essentially a ‘point-to-point’ extension formula. The method has been adapted to proving uniqueness in inverse scattering from sound-soft cavities with interior measurement data deriving from a single point source. The corresponding uniqueness for sound-hard balls or polyhedral scatterers has also been discussed.

Recovery of polyhedral scatterers by a single electromagnetic far-field measurement

We prove that a polyhedral obstacle in R3 consisting of finitely many polyhedra with mixed perfect electric conductor and perfect magnetic conductor boundary conditions can be uniquely determined by a single electric or magnetic far-field measurement, namely, the far-field pattern corresponding to a single incident wave. A unique novelty of our new technique for proving the uniqueness is to realize that the existence of an “unbounded” perfect plane implies certain symmetries of the underlying scatterer.

The No Response Test for the reconstruction of polyhedral objects in electromagnetics

We develop a No Response Test for the reconstruction of a polyhedral obstacle from two or few time-harmonic electromagnetic incident waves in electromagnetics. The basic idea of the test is to probe some region in space with waves which are small on some test domain and, thus, do not generate a response when the scatterer is inside of this test domain. We will prove that the No Response Test checks analytic continuability of a time-harmonic field from the far field pattern into the domain B e ≔ R 3 ∖ B ¯ for a non-vibrating test domain B . We show that two incident waves, defined by one incident direction and two appropriately chosen directions of polarization, are enough to recover the convex hull of polyhedrals. Based on this uniqueness result, we build up the No Response Test and we prove convergence in the sense that it fully reconstructs a convex polyhedral scatterer D or the convex hull of an arbitrary polyhedral scatterer. Further, we will describe the algorithmic realization of the No Response Test and show the feasibility of the method by reconstruction of convex polyhedral objects in three dimensions. This is the first formulation of the No Response Test for electromagnetics.

On recovering polyhedral scatterers with acoustic far-field measurements

We prove that an acoustic sound-hard scatterer, consisting of finitely many solid polyhedra in R n (n ≥ 2), is uniquely determined by the far-field patterns corresponding to n - 1 different incident waves. By suitable modifications, the method can also be used to show a similar uniqueness result in the setting without knowing the a priori physical properties of the underlying scatterer.

On uniqueness in inverse acoustic and electromagnetic obstacle scattering problems

We review some of the uniqueness results in inverse acoustic and electromagnetic obstacle scattering problems. The emphasis is given to the recent progress on unique determination of general polyhedral scatterers by the far field data corresponding to a single or several incident fields.

Uniqueness in determining polyhedral sound-hard obstacles with a single incoming wave

We consider the inverse acoustic scattering problem of determining a sound-hard obstacle by far-field measurements. It is proved that a polyhedral scatterer in , consisting of finitely many solid polyhedra, is uniquely determined by a single incoming plane wave.

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

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 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 e 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 e which approaches zero, as e → 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 e is even smaller with respect to h. In this case we obtain an explicit estimate essentially of Holder type.

On unique determination of partially coated polyhedral scatterers with far field measurements

This work is a continuation of our early study in Liu and Zou (2006 Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers Inverse Problems 22 515–24; 2006 Uniqueness in determining multiple polygonal or polyhedral scatterers of mixed type Technical Report 2006-03(337) The Chinese University of Hong Kong) and addresses the unique determination of partially coated polyhedral scatterers in (N ⩾ 2) along with their surface impedance from far field data. Two global uniqueness results are established for this inverse problem with a scatterer consisting of multiple solid polyhedra: the first one is to determine such a scatterer of mixed sound-soft and impedance type by a single incident plane wave and the other is to determine such a scatterer of mixed sound-soft, sound-hard and impedance type by N different incident waves in the N-dimensional case with N ⩾ 3 and by only one incident wave for the two-dimensional case. Then we present some examples to show that as long as a scatterer admits the presence of (sound-hard) crack-type obstacles, then one cannot determine the scatterer uniquely by any less than N different incident plane waves. These examples also reveal that the uniqueness results achieved earlier in [15, 16] for polyhedral scatterers are optimal. Finally, the uniqueness results that have been solved or are still unsolved for the polyhedral-type scatterers with both solid and crack components are summarized in the conclusion.

Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers

This paper addresses the uniqueness for an inverse acoustic obstacle scattering problem. It is proved that a general sound-hard polyhedral scatterer in , possibly consisting of finitely many solid polyhedra and subsets of (N − 1)-dimensional hyperplanes, is uniquely determined by N far-field measurements corresponding to N incident plane waves given by a fixed wave number and N linearly independent incident directions. A simple proof, which is quite different from that in Alessandrini and Rondi (2005 Proc. Am. Math. Soc. 6 1685–91), is also provided for the unique determination of a general sound-soft polyhedral scatterer by a single incoming wave.

Determining a sound-soft polyhedral scatterer by a single far-field measurement

We prove that a sound-soft polyhedral scatterer is uniquely determined by the far-field pattern corresponding to an incident plane wave at one given wavenumber and one given incident direction.