Interior Transmission Problem Research Articles

The interior transmission spectrum in one dimension

This paper seeks conditions on acoustic profiles c(x) and b(x) which imply the existence of intervals free of transmission eigenvalues. The existence of intervals free of transmission eigenvalues has important implications in the question of unique recovery of the acoustic profile for inverse acoustic scattering theory using the linear sampling method and for the inverse source problem in thermoacoustic tomography. We examine the interior transmission spectrum relative to two acoustic profiles on the unit interval. This research includes the case when the difference of the two acoustic profiles changes sign and is, to the author's knowledge, the only work to date on the interior transmission spectrum allowing this behavior. It is shown that for a large class of acoustic profiles the transmission spectrum is asymptotically sparse. The spectrum exhibits three distinct types of behavior dependent on the relation of the two acoustic profiles. Numerical examples of each type of behavior are given which show how the transmission eigenvalues are distributed depending on the relation between c(x) and b(x). Our method of study is to reduce the interior transmission problem on [0, 1] to a problem of finding roots of a determinant of fundamental solutions to a Sturm–Liouville problem. The asymptotic expansion of solutions of the Sturm–Liouville problem then allows us to analyze properties of the transmission spectrum for large values of the wavenumber.

Bounds on positive interior transmission eigenvalues

This paper contains lower bounds on the counting function of the positive eigenvalues of the interior transmission problem when the latter is elliptic. In particular, these bounds justify the existence of an infinite set of interior transmission eigenvalues and provide asymptotic estimates from above on the counting function for the large values of the wave number. They also lead to certain important upper estimates on the first few interior transmission eigenvalues. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle.

Interior transmission eigenvalue problem for Maxwell's equations: the T-coercivity as an alternative approach

In this paper, we examine the interior transmission problem for Maxwell’s equations in the case where both ε and μ, the physical parameters of the scattering medium, differ from ε0 and μ0 modelling the background medium. Using the T-coercivity method, we propose an alternative approach to the classical techniques to prove that this problem is of Fredholm type and that the so-called transmission eigenvalues form at most a discrete set. The T-coercivity approach allows us to deal with cases where ε − ε0 and μ − μ0 can change sign. We also provide results of localization and Faber–Krahn-type inequalities for the transmission eigenvalues.

Remarks on interior transmission eigenvalues, Weyl formula and branching billiards

This paper contains the Weyl formula for the counting function of the interior transmission problem when the latter is parameter elliptic. Branching billiard trajectories are constructed, and the second term of the Weyl asymptotics is estimated from above under some conditions on the set of periodic billiard trajectories.

Nature of the transmission eigenvalue spectrum for elastic bodies

This study develops a spectral theory of the interior transmission problem (ITP) for heterogeneous and anisotropic elastic solids. The subject is central to the so-called qualitative methods for inverse scattering involving penetrable obstacles. Although simply stated as a coupled pair of elastodynamic wave equations, the ITP for elastic bodies is neither self-adjoint nor elliptic. To help deal with such impediments, earlier studies have established the well-posedness of an elastodynamic ITP under notably restrictive assumptions on the contrast in elastic and mass density parameters between the scatterer and the background solid. Due to lack of self-adjointness of the problem, these analyses were further successful in substantiating the discreteness of the relevant eigenvalue spectrum but not its existence. The aim of this work is to provide a systematic treatment of the ITP for elastic bodies that transcends the limitations of earlier analyses. Considering a broad range of material-contrast configurations, this paper investigates the questions of the solvability of the ITP, the discreteness of its eigenvalues and, for the first time, of the existence of such eigenvalue spectrum. Necessitated by the breadth of material configurations studied, the relevant claims are established via a suite of variational formulations, each customized to meet the needs of a particular subclass of eigenvalue problems.

A multistep reciprocity gap functional method for the inverse problem in a multilayered medium

We introduce a multistep reciprocity gap functional method to reconstruct the support of a target embedded in a multi-layered medium from near field measurements at a fixed frequency. The approach is based on the use of the standard reciprocity gap functional method for homogeneous background to strip the layers and recover the data at each interface by means of the so-called interior transmission problem and then, at the last step, to reconstruct the support of the scatterer. The advantage of this approach is that it avoids computing the Green's function for multi-layered medium. Numerical examples are given, showing the performance of our reconstruction algorithm.

The Green function of the interior transmission problem and its applications

The interior transmission problem appears naturally in thescattering theory. In this paper, we construct theGreen function associated to this problem. In addition, we providepoint-wise estimates of this Green function similar to those knownfor the Green function related to the classical transmissionproblems. These estimates are, in particular, useful to the study ofvarious inverse scattering problems. Here, we apply them tojustify some asymptotic formulas already used for detectingpartially coated dielectric mediums from far field measurements.

Ellipticity in the Interior Transmission Problem in Anisotropic Media

The paper concerns the discreteness of the eigenvalues and the solvability of the interior transmission problem for anisotropic media. Conditions for the ellipticity of the problem are written explicitly, and it is shown that they do not guarantee the discreteness of the eigenvalues. Some simple sufficient conditions for the discreteness and solvability are found. They are expressed in terms of the values of the anisotropy matrix and the refraction index at the boundary of the domain. The discreteness of the eigenvalues and the solvability of the interior transmission problem are shown if a small perturbation is applied to the refraction index.

Transmission eigenvalues for inhomogeneous media containing obstacles

We consider the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The presence of Dirichlet obstacle brings new difficulties to already complicated situation dealing with a non-selfadjoint eigenvalue problem. In this paper, we employ a variety of variational techniques under various assumptions on the index of refraction as well as the size of the Dirichlet obstacle.

Sampling type methods for an inverse waveguide problem

We consider the problem of locating a penetrable obstacle in an acoustic waveguide from measurements of pressure waves due to point sources inside the waveguide. More precisely, we assume that we are given the scattered field and its normal derivative for any source point and receiver placed on a pair of surfaces known as the source and the measurement surfaces, respectively. A novel feature of this work is that the obstacle is allowed to touch the boundary of the pipe. &nbsp We first analyze the associated interior transmission problem. Then, we adapt and analyze the Reciprocity Gap Method (RGM) and the Linear Sampling Method (LSM) to deal with the inverse problem. We also study the relationship between these two methods and provide numerical results.

On the Nonexistence of Transmission Eigenvalues for Regions with Cavities

We consider the interior transmission problem when the inhomogeneous medium has a cavity region. In this case we establish the Fredholm property for this problem and show that there does not exist a transmission eigenvalue under a new condition.

On the use of T-coercivity to study the interior transmission eigenvalue problem

In this Note, we investigate the so-called interior transmission problem using the T-coercivity approach. In particular, we prove that this problem, which appears when one is interested in the reconstruction of the support of an inclusion embedded in a homogeneous medium, is of Fredholm type and that so-called transmission eigenvalues form at most a discrete set. Our approach treats cases where the difference between the inclusion index and the background index can change sign, which are not covered by other techniques that can be found in the literature. We also provide Faber–Krahn type inequalities associated with this general case.

A coupled BEM and FEM for the interior transmission problem in acoustics

The interior transmission problem (ITP) is a boundary value problem arising in inverse scattering theory, and it has important applications in qualitative methods. In this paper, we propose a coupled boundary element method (BEM) and a finite element method (FEM) for the ITP in two dimensions. The coupling procedure is realized by applying the direct boundary integral equation method to define the so-called Dirichlet-to-Neumann (DtN) mappings. We show the existence of the solution to the ITP for the anisotropic medium. Numerical results are provided to illustrate the accuracy of the coupling method.

The Electromagnetic Interior Transmission Problem for Regions with Cavities

We consider the electromagnetic interior transmission problem in the case when the medium has cavities, i.e. regions in which the index of refraction is the same as in the host medium. We address the configuration where the electromagnetic permeability is constant while the electric permittivity is variable and may be anisotropic. In this case, using appropriate reformulation of the problem into a fourth order pde, we establish the Fredholm property for this problem and show that transmission eigenvalues exist and form a discrete set. Monotonicity properties of the first eigenvalue in terms of the permittivity and the size of the cavity are established.

On the multi-frequency obstacle reconstruction via the linear sampling method

This paper investigates the possibility of multi-frequency reconstruction of sound-soft and penetrable obstacles via the linear sampling method involving either far-field or near-field observations of the scattered field. On establishing a suitable approximate solution to the linear sampling equation and making an assumption of continuous frequency sweep, two possible choices for a cumulative multi-frequency indicator function of the scatterer's support are proposed. The first alternative, termed the ‘serial’ indicator, is taken as a natural extension of its monochromatic companion in the sense that its computation entails space-frequency (as opposed to space) L2-norm of a solution to the linear sampling equation. Under a set of assumptions that include experimental observations down to zero frequency and compact frequency support of the wavelet used to illuminate the obstacle, this indicator function is further related to its time-domain counterpart. As a second possibility, the so-called parallel indicator is alternatively proposed as an L2-norm, in the frequency domain, of the monochromatic indicator function. On the basis of a perturbation analysis which demonstrates that the monochromatic solution of the linear sampling equation behaves as O(|k2 − k2∗|−m), m ⩾ 1 in the neighborhood of an isolated eigenvalue, k2∗, of the associated interior (Dirichlet or transmission) problem, it is found that the ‘serial’ indicator is unable to distinguish the interior from the exterior of a scatterer in situations when the prescribed frequency band traverses at least one such eigenvalue. In contrast the ‘parallel’ indicator is, due to its particular structure, shown to be insensitive to the presence of pertinent interior eigenvalues (unknown beforehand and typically belonging to a countable set), and thus to be robust in a generic scattering configuration. A set of numerical results, including both ‘fine’ and ‘coarse’ frequency sampling, is included to illustrate the performance of the competing (multi-frequency) indicator functions, demonstrating behavior that is consistent with the theoretical results.

Analytical and computational methods for transmission eigenvalues

The interior transmission problem is a boundary value problem that arises in the scattering of time-harmonic waves by an inhomogeneous medium of compact support. The associated transmission eigenvalue problem has important applications in qualitative methods in inverse scattering theory. In this paper, we first establish optimal conditions for the existence of transmission eigenvalues for a spherically stratified medium and give numerical examples of the existence of both real and complex transmission eigenvalues in this case. We then propose three finite element methods for the computation of the transmission eigenvalues for the cases of a general non-stratified medium and use these methods to investigate the accuracy of recently established inequalities for transmission eigenvalues.

On the Existence and Uniqueness of a Solution to the Interior Transmission Problem for Piecewise-Homogeneous Solids

The interior transmission problem (ITP), which plays a fundamental role in inverse scattering theories involving penetrable defects, is investigated within the framework of mechanical waves scattered by piecewise-homogeneous, elastic or viscoelastic obstacles in a likewise heterogeneous background solid. For generality, the obstacle is allowed to be multiply connected, having both penetrable components (inclusions) and impenetrable parts (cavities). A variational formulation is employed to establish sufficient conditions for the existence and uniqueness of a solution to the ITP, provided that the excitation frequency does not belong to (at most) countable spectrum of transmission eigenvalues. The featured sufficient conditions, expressed in terms of the mass density and elasticity parameters of the problem, represent an advancement over earlier works on the subject in that (i) they pose a precise, previously unavailable provision for the well-posedness of the ITP in situations when both the obstacle and the background solid are heterogeneous, and (ii) they are dimensionally consistent, i.e., invariant under the choice of physical units. For the case of a viscoelastic scatterer in an elastic solid it is further shown, consistent with earlier studies in acoustics, electromagnetism, and elasticity that the uniqueness of a solution to the ITP is maintained irrespective of the vibration frequency. When applied to the situation where both the scatterer and the background medium are viscoelastic, i.e., dissipative, on the other hand, the same type of analysis shows that the analogous claim of uniqueness does not hold. Physically, such anomalous behavior of the “viscoelastic-viscoelastic” case (that has eluded previous studies) has its origins in a lesser known fact that the homogeneous ITP is not mechanically insulated from its surroundings—a feature that is particularly cloaked in situations when either the background medium or the scatterer are dissipative. A set of numerical results, computed for ITP configurations that meet the sufficient conditions for the existence of a solution, is included to illustrate the problem. Consistent with the preceding analysis, the results indicate that the set of transmission values is indeed empty in the “elastic-viscoelastic” case, and countable for “elastic-elastic” and “viscoelastic-viscoelastic” configurations.

The Interior Transmission Problem for Regions with Cavities

We consider the interior transmission problem in the case when the inhomogeneous medium has cavities, i.e., regions in which the index of refraction is the same as the host medium. In this case we establish the Fredholm property for this problem and show that transmission eigenvalues exist and form a discrete set. We also derive Faber–Krahn-type inequalities for the transmission eigenvalues.

The Existence of an Infinite Discrete Set of Transmission Eigenvalues

We prove the existence of an infinite discrete set of transmission eigenvalues corresponding to the scattering problem for isotropic and anisotropic inhomogeneous media for both the Helmholtz and Maxwell's equations. Our discussion includes the case of the interior transmission problem for an inhomogeneous medium with cavities, i.e., subregions with contrast zero.

Transmission Eigenvalues for Operators with Constant Coefficients

In this paper we study the interior transmission problem and transmission eigenvalues for multiplicative perturbations of linear partial differential operator of order $\geq2$ with constant real coefficients. Under suitable growth conditions on the symbol of the operator and the perturbation, we show the discreteness of the set of transmission eigenvalues and derive sufficient conditions on the existence of transmission eigenvalues. We apply these techniques to the case of the biharmonic operator and the Dirac system. In the hypoelliptic case we present a connection to scattering theory.