The interior transmission problem

Colton, David; Päivärinta, Lassi; Sylvester, John

doi:10.3934/ipi.2007.1.13

Cited by 197 publications

(204 citation statements)

References 19 publications

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“…Since the transmission eigenvalues can be used to estimate the material properties of the scattering field, the transmission eigenvalue problems play a very important role in inverse scattering problems and have received much attention recently [2][3][4][5]9].…”

Section: Introductionmentioning

confidence: 99%

A Spectral-Element Method for Transmission Eigenvalue Problems

Shen

2013

J Sci Comput

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We develop an efficient spectral-element method for computing the transmission eigenvalues in two-dimensional radially stratified media. Our method is based on a dimension reduction approach which reduces the problem to a sequence of one-dimensional eigenvalue problems that can be efficiently solved by a spectral-element method. We provide an error analysis which shows that the convergence rate of the eigenvalues is twice that of the eigenfunctions in energy norm. We present ample numerical results to show that the method convergences exponentially fast for piecewise stratified media, and is very effective, particularly for computing the few smallest eigenvalues.

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Section: Introductionmentioning

confidence: 99%

A Spectral-Element Method for Transmission Eigenvalue Problems

Shen

2013

J Sci Comput

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“…The problem (25)- (27) resembles the interior transmission problem, which arises in the inverse scattering theory, and has initiated large research efforts during the last years, see [25]. The big difficulty here is that the eigenvalue is involved in the boundary condition (27).…”

Section: The Non-trivial Case (C3)mentioning

confidence: 99%

The Propagation Problem in a Bi-Isotropic Waveguide

Ioannidis¹,

Kristensson²,

Sjöberg³

2010

PIER B

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Abstract-We investigate the problem of defining propagating constants and modes in metallic waveguides of an arbitrary cross section, filled with a homogeneous bi-isotropic material. The approach follows the guidelines of the classical theory for the isotropic, homogeneous, lossless waveguide: starting with the Maxwell system, we formulate a spectral problem where the square of the propagation constant shows up as the eigenvalue and the corresponding mode as the eigenvector. The difficulty that arises, and this is a feature of chirality, is that the eigenvalue is involved in the boundary conditions. The main result is that the problem is solvable whenever the Dirichlet problem for the Helmholtz equation in the cross section is solvable and two technical hypotheses are fulfilled. Our method, inspired by the null-field method, is quite general and has a good potential to work in various geometries.

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“…In this section, we describe briefly the method introduced in [4,5,9] for estimation of the index of refraction of a non-absorbing inhomogeneous medium. We first show how to obtain transmission eigenvalues from far field data, and then describe an algorithm to reconstruct/estimate the index of refraction.…”

Section: Description Of the General Approachmentioning

confidence: 99%

“…The organization of the paper is as follows: In §2, we describe the general approach introduced in [4,5,9] for estimation of the index of refraction of a non-absorbing inhomogeneous medium. In §3, we derive a weak formulation, construct a Fourier-spectralelement method and derive error estimates for transmission eigenvalues in circular domains.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction

An¹,

Shen²

2014

JMS

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Abstract. An important step in estimating the index of refraction of electromagnetic scattering problems is to compute the associated transmission eigenvalue problem. We develop in this paper efficient and accurate spectral methods for computing the transmission eigenvalues associated to the electromagnetic scattering problems. We present ample numerical results to show that our methods are very effective for computing transmission eigenvalues (particularly for computing the smallest eigenvalue), and together with the linear sampling method, provide an efficient way to estimate the index of refraction of a non-absorbing inhomogeneous medium.

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The interior transmission problem

Cited by 197 publications

References 19 publications

A Spectral-Element Method for Transmission Eigenvalue Problems

A Spectral-Element Method for Transmission Eigenvalue Problems

The Propagation Problem in a Bi-Isotropic Waveguide

Efficient Spectral Methods for Transmission Eigenvalues and Estimation of the Index of Refraction

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