On the determination of Dirichlet or transmission eigenvalues from far field data

Cakoni, Fioralba; Colton, David; Haddar, Houssem

doi:10.1016/j.crma.2010.02.003

Cited by 142 publications

(164 citation statements)

References 8 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 following theorem formalizes previous considerations (the formulation is an adaptation of the results in [5]). …”

Section: Transmission Eigenvalues and Their Identification Using Farfmentioning

confidence: 94%

“…(Ω) for several choices of z, respectively for (n e , n i ) = (8,8), (11,5) Tables 2, 3, describe the results achievable when the two layered inclusion of previous experiment is now embedded in an inhomogeneous background. The inhomogeneous background consists into a circular domain of radius 0.75 and with refractive index n b embedded in the vacuum.…”

Section: The Case Of Circular Geometriesmentioning

confidence: 99%

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Computing estimates of material properties from transmission eigenvalues

Giorgi

Haddar

2012

Inverse Problems

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This work is motivated by inverse scattering problems, those problems where one is interested in reconstructing the shape and the material properties of an inclusion from electromagnetic farfields measurements. More precisely we are interested in complementing the so called sampling methods, (those methods that enables one to reconstruct just the geometry of the scatterer), by providing estimates on the material properties. We shall use for that purpose the so-called transmission eigenvalues. Our method is based on reformulating the so-called interior transmission eigenvalue problem into an eigenvalue problem for the material coefficients. We shall restrict ourselves to the two dimensional setting of the problem and treat the cases of both TE and TM polarizations. We present a number of numerical experiments that validate our methodology for homogeneous and inhomogeneous inclusions and backgrounds. We also treat the case of a background with absorption and the case of scatterers with multiple connected components of different refractive indexes.

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“…First, the support of the scattering obstacle can be recovered by using the measured scattering data and the linear sampling method [12], and the transmission eigenvalues can be identified from the far field data. Then, the bounds for smallest and largest eigenvalues of the (matrix) index of refraction can be obtained in terms of the support of the scattering obstacle and the first transmission eigenvalue of the anisotropic media [3]. Finally, reconstructions of the electric permittivity (if it is a scalar constant) or an estimate of the eigenvalues of the matrix in the case of anisotropic permittivity can be obtained [5].…”

Section: Introductionmentioning

confidence: 99%

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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On the determination of Dirichlet or transmission eigenvalues from far field data

Cited by 142 publications

References 8 publications

A Spectral-Element Method for Transmission Eigenvalue Problems

A Spectral-Element Method for Transmission Eigenvalue Problems

Computing estimates of material properties from transmission eigenvalues

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

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