We consider a problem in nondestructive testing in which small changes in the (possibly complex valued) refractive index $n(x)$ of an inhomogeneous medium of compact support are to be determined from changes in measured far field data due to incident plane waves. The problem is studied by considering a modified far field operator ${\cal F}$ whose kernel is the difference of the measured far field pattern due to the scattering object and the far field pattern of an auxiliary scattering problem with the Stekloff boundary condition imposed on the boundary of a domain $B$, where $B$ is either the support of the scattering object or a ball containing the scattering object in its interior. It is shown that ${\cal F}$ can be used to determine the Stekloff eigenvalues corresponding to $B$, where, if $B\not=D$, the refractive index is set equal to one in $B\setminus\overline{D}$. A formula is obtained relating changes in $n(x)$ to changes in the Stekloff eigenvalues and numerical examples are given showing the effectiveness of determining changes to the refractive index in this way.

Stekloff Eigenvalues in Inverse Scattering

We introduce an inside-outside duality approach for the determination of interior transmission eigenvalues of a possibly anisotropic dielectric electromagnetic scattering object using time-harmonic electromagnetic far field data. To this end, we exploit a self-adjoint factorization of the far field operator to link the electromagnetic interior transmission eigenvalues to the maximal or minimal phase of the eigenvalues of the corresponding far field operator, depending on whether the sign of the contrast function is positive or negative.

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Inside-Outside Duality and the Determination of Electromagnetic Interior Transmission Eigenvalues

In this article we extend the inside-outside duality for acoustic transmission eigenvalue problems by allowing scattering objects that may contain cavities. In this context we provide the functional analytical framework necessary to transfer the techniques that have been used in Kirsch and Lechleiter (2013 Inverse Problems, 29 104011) to derive the inside-outside duality. Additionally, extensive numerical results are presented to show that we are able to successfully detect interior transmission eigenvalues with the inside-outside duality approach for a variety of obstacles with and without cavities in three dimensions. In this context, we also discuss the advantages and disadvantages of the inside-outside duality approach from a numerical point of view. Furthermore we derive the integral equations necessary to extend the algorithm in Kleefeld (2013 Inverse Problems, 29 104012) to compute highly accurate interior transmission eigenvalues for scattering objects with cavities, which we will then use as reference values to examine the accuracy of the inside-outside duality algorithm.

https://iopscience.iop.org/article/10.1088/0266-5611/32/4/045001/pdf

Numerical computations of interior transmission eigenvalues for scattering objects with cavities

In this thesis, we extend the theory of non-scattering energies on two fronts. First, we shall consider the discreteness of non-scattering energies corresponding to non-compactly supported potentials using the approach via transmission eigenvalues and fourth-order operators. The method requires the support of the potential to exhibit certain compact Sobolev embedding and to be contained in a half-space and the potential to have controlled polynomial or exponential decay at infinity. Also, in order to connect the non-scattering energies to the fourth-order operators, a generalization of the classical Rellich theorem to unbounded domains is required. This is of independent interest, and we obtain several such results, including a discrete analogue. Our second contribution (joint work with L. Paivarinta and M. Salo) is extending a recent result on non-existence of non-scattering energies for potentials with rectangular corners to arbitrary corners of angle smaller than ⇡ in two dimensions, and to prove in three dimensions that the set of strictly convex circular conical corners for which non-scattering energies might exist is at most countable. This thesis consists of the papers I. Vesalainen, E. V.: Transmission eigenvalues for non-compactly supported potentials, Inverse Problems, 29 (2013), 104006, 1–11. II. Vesalainen, E. V.: Rellich type theorems for unbounded domains, to appear in Inverse Problems and Imaging. Preprint available at arXiv:1401. 4531 [math.AP]. III. Paivarinta, L., M. Salo, and E. V. Vesalainen: Strictly convex corners scatter, submitted. Preprint available at arXiv:1404.2513 [math.AP]. The author of this thesis had an equal role in the research and writing of the joint article.

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On Non-Scattering Energies

The interior transmission problem arises in inverse scattering theory for inhomogeneous media. It is a boundary value problem for a set of equations defined in a bounded domain coinciding with the support of the scattering object. Of particular interest is the spectrum associated with this boundary value problem, more specifically the existence of eigenvalues known as transmission eigenvalues. Indeed, on one hand, in the context of sampling methods for reconstructing the support of the scatterer one needs to avoid those frequencies that correspond to transmission eigenvalues, and hence it is important to know that the transmission eigenvalues form a discrete set. On the other hand, one can use transmission eigenvalues to obtain information about physical properties of the scattering medium [1], [3] and therefore it is important to know whether they exist and to understand their connection with the index of refraction. The latter application is based on the recent results in [2] which justify the numerical observation that the transmission eigenvalues can be computed from the far field data. Either way, the investigation of the spectral properties of the interior transmission problem has become an interesting mathematical question in inverse scattering theory. We present here the most recent developments on transmission eigenvalues, in particular we show the existence of infinitely many transmission eigenvalues and provide lower and upper bounds for the first transmission eigenvalue in terms of the geometry and physical properties of the scattering object [4], [5]. Then, we show how to use these bounds to obtain information on the index of refraction of a general anisotropic scattering medium, as well on the presence of defects inside the scattering medium.

D. Colton

Papers

Transmission Eigenvalues and their Application to Inverse Scattering Problems