The linear sampling method for anisotropic media

Cakoni, Fioralba; Colton, David; Haddar, Houssem

doi:10.1016/s0377-0427(02)00361-8

Cited by 100 publications

(126 citation statements)

References 9 publications

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“…[5]. Assumption (A2a) seems to be a bit unnatural and differs from the canonical form as, e.g., in [6].…”

Section: Theorem 22 the Operatormentioning

confidence: 99%

“…In the case of γ 0 = id and q 0 ≡ 1 the transmission problem (3.15), (3.16) and its non-homogeneous counterpart has recently been studied in [5,6]. It has been shown under essentially the same assumptions as above that for real γ 1 and q 1 there exist at most a countable number of eigenvalues and each eigenspace is finite dimensional.…”

Section: (B) We Consider Only the Operator T − T And Note That (T − Tmentioning

confidence: 99%

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The factorization method for a class of inverse elliptic problems

Kirsch

2005

Mathematische Nachrichten

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Key words Elliptic partial differential equations, inverse problem, factorization method, anisotropic media MSC (2000) Primary: 35R25; Secondary: 35R25In this paper the factorization method from inverse scattering theory and impedance tomography is extended to a class of general elliptic differential equations in divergence form. The inverse problem is to determine the interface ∂Ω of an interior change of the material parameters from the Neumann-Dirichlet map. Since absorption is allowed a suitable combination of the real and imaginary part of the Neumann-Dirichlet map is needed to explicitely characterize Ω by the data.

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“…[5]. Assumption (A2a) seems to be a bit unnatural and differs from the canonical form as, e.g., in [6].…”

Section: Theorem 22 the Operatormentioning

confidence: 99%

Section: (B) We Consider Only the Operator T − T And Note That (T − Tmentioning

confidence: 99%

The factorization method for a class of inverse elliptic problems

Kirsch

2005

Mathematische Nachrichten

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“…Roughly speaking, two main approaches are available in this direction, namely integral equation and projection methods [8], [11], and variational methods typically applied to a fourth order equivalent boundary value problem [3], [5], [6], [10], [17]. On the other hand, except for the case of spherically stratified medium [7], [9], until recently little was known about the existence and properties of transmission eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

On the existence of transmission eigenvalues in an inhomogeneous medium

Cakoni

Haddar

2009

Applicable Analysis

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Abstract:We prove the existence of transmission eigenvalues corresponding to the inverse scattering problem for isotropic and anisotropic media for both scalar Helmholtz equation and Maxwell's equations. Considering a generalized abstract eigenvalue problem, we are able to extend the ideas of Päivärinta and Sylvester [15] to prove the existence of transmission eigenvalues for a larger class of interior transmission problems. Our analysis includes both the case of a medium with positive contrast and of a medium with negative contrast provided that this contrast is large enough.

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“…It gives for example a powerful tool for uniqueness issues in the case of anisotropic media [1], and plays also a fundamental role in the justiÿcation of the linear sampling method [2] to reconstruct the support of penetrable objects [3][4][5]. The present paper completes these existing results by studying the case of inhomogeneous, anisotropic Maxwell's equations.…”

Section: Introductionmentioning

confidence: 55%