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We present an algorithm for the computation of scattering poles for an impenetrable obstacle with Dirichlet or Robin boundary conditions in acoustic scattering. This paper builds upon the previous work of Cakoni et al. (2020) titled ‘A duality between scattering poles and transmission eigenvalues in scattering theory’ (Cakoni et al. 2020 Proc. A. 476, 20200612 (doi:10.1098/rspa.2020.0612)), where the authors developed a conceptually unified approach for characterizing the scattering poles and interior eigenvalues corresponding to a scattering problem. This approach views scattering poles as dual to interior eigenvalues by interchanging the roles of incident and scattered fields. In this framework, both sets are linked to the kernel of the relative scattering operator that maps incident fields to scattered fields. This mapping corresponds to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Leveraging this dual characterization and inspired by the generalized linear sampling method for computing the interior eigenvalues, we present a novel numerical algorithm for computing scattering poles without relying on an iterative scheme. Preliminary numerical examples showcase the effectiveness of this computational approach. 

We discuss a novel approach for imaging local faults inside an infinite bi-periodic layered medium in ℝ3 using acoustic measurements of scattered fields at the bottom or the top of the layer. The faulted area is represented by compactly supported perturbations with erroneous material properties. Our method reconstructs the support of perturbations without knowing or reconstructing the constitutive material parameters of healthy or faulty bi-period layer; only the size of the period is needed. This approach falls under the class of non-iterative imaging methods, known as the generalized linear sampling method with differential measurements, first introduced in [2] and adapted to periodic layers in [25]. The advantage of applying differential measurements to our inverse problem is that instead of comparing the measured data against measurements due to healthy structures, one makes use of periodicity of the layer where the data operator restricted to single Floquet-Bloch modes plays the role of the one corresponding to healthy material. This leads to a computationally efficient and mathematically rigorous reconstruction algorithm. We present numerical experiments that confirm the viability of the approach for various configurations of defects 

This paper concerns the analysis of a passive, broadband approximate cloaking scheme for the Helmholtz equation in Rd for d = 2 or d = 3. Using ideas from transformation optics, we construct an approximate cloak by “blowing up” a small ball of radius ϵ > 0 to one of radius 1. In the anisotropic cloaking layer resulting from the “blow-up” change of variables, we incorporate a Drude-Lorentz- type model for the index of refraction, and we assume that the cloaked object is a soft (perfectly conducting) obstacle. We first show that (for any fixed ϵ) there are no real transmission eigenvalues associated with the inhomogeneity representing the cloak, which implies that the cloaking devices we have created will not yield perfect cloaking at any frequency, even for a single incident time harmonic wave. Secondly, we establish estimates on the scattered field due to an arbitrary time harmonic incident wave. These estimates show that, as ϵ approaches 0, the L2 -norm of the scattered field outside the cloak, and its far field pattern, approach 0 uniformly over any bounded band of frequencies. In other words: our scheme leads to broadband approximate cloaking for arbitrary incident time harmonic waves. 

In this paper we examine necessary conditions for an inhomogeneity to be non‐scattering, or equivalently, by negation, sufficient conditions for it to be scattering. These conditions are formulated in terms of the regularity of the boundary of the inhomogeneity. We examine broad classes of incident waves in both two and three dimensions. Our analysis is greatly influenced by the analysis carried out by Williams in order to establish that a domain, which does not possess the Pompeiu Property, has a real analytic boundary. That analysis, as well as ours, relies crucially on classical free boundary regularity results due to Kinderlehrer and Nirenberg, and Caffarelli. 

Abstract We investigate an inverse scattering problem for a thin inhomogeneous scatterer in R m , m = 2, 3, which we model as an m − 1 dimensional open surface. The scatterer is referred to as a screen. The goal is to design target signatures that are computable from scattering data in order to detect changes in the material properties of the screen. This target signature is characterized by a mixed Steklov eigenvalue problem for a domain whose boundary contains the screen. We show that the corresponding eigenvalues can be determined from appropriately modified scattering data by using the generalized linear sampling method. A weaker justification is provided for the classical linear sampling method. Numerical experiments are presented to support our theoretical results. 

In recent years, a new approach has been proposed in the study of the inverse scattering problem for electromagnetic waves. In particular, a study is made of the analytic properties of the scattering operator, and the results of this study are used to design target signatures that respond to changes in the electromagnetic parameters of the scattering medium. These target signatures are characterized by novel eigenvalue problems such that the eigenvalues can be determined from measured scattering data. Changes in the structural properties of the material or the presence of flaws cause changes in the measured eigenvalues. In this article, we provide a general framework for developing target signatures and numerical evidence of the efficacy of new target signatures based on recently introduced eigenvalue problems arising in electromagnetic scattering theory for anisotropic media. 

In this paper we consider the inverse problem of determining structural properties of a thin anisotropic and dissipative inhomogeneity in \begin{document}$$ {\mathbb R}^m $$\end{document}, \begin{document}$ m = 2, 3 $$\end{document} from scattering data. In the asymptotic limit as the thickness goes to zero, the thin inhomogeneity is modeled by an open \begin{document}$$ m-1 $$\end{document}$ dimensional manifold (here referred to as screen), and the field inside is replaced by jump conditions on the total field involving a second order surface differential operator. We show that all the surface coefficients (possibly matrix valued and complex) are uniquely determined from far field patterns of the scattered fields due to infinitely many incident plane waves at a fixed frequency. Then we introduce a target signature characterized by a novel eigenvalue problem such that the eigenvalues can be determined from measured scattering data, adapting the approach in [20]. Changes in the measured eigenvalues are used to identified changes in the coefficients without making use of the governing equations that model the healthy screen. In our investigation the shape of the screen is known, since it represents the object being evaluated. We present some preliminary numerical results indicating the validity of our inversion approach