New sets of eigenvalues in inverse scattering for inhomogeneous media and their determination from scattering data

In this paper we develop a general mathematical framework to determine interior eigenvalues from a knowledge of the modified far field operator associated with an unknown (anisotropic) inhomogeneity. The modified far field operator is obtained by subtracting from the measured far field operator the...

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Bibliographic Details
Published in:Inverse problems 2017-12, Vol.33 (12), p.125011-30
Main Authors: Audibert, Lorenzo, Cakoni, Fioralba, Haddar, Houssem
Format: Article
Language:English
Subjects:
Computer Science
generalized linear sampling method
inhomogenous media
inverse scattering
Numerical Analysis
Steklov eigenvalues
transmission eigenvalues
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Summary:In this paper we develop a general mathematical framework to determine interior eigenvalues from a knowledge of the modified far field operator associated with an unknown (anisotropic) inhomogeneity. The modified far field operator is obtained by subtracting from the measured far field operator the computed far field operator corresponding to a well-posed scattering problem depending on one (possibly complex) parameter. Injectivity of this modified far field operator is related to an appropriate eigenvalue problem whose eigenvalues can be determined from the scattering data, and thus can be used to obtain information about material properties of the unknown inhomogeneity. We discuss here two examples of such modification leading to a Steklov eigenvalue problem, and a new type of the transmission eigenvalue problem. We present some numerical examples demonstrating the viability of our method for determining the interior eigenvalues form far field data.
ISSN:0266-5611
1361-6420
DOI:10.1088/1361-6420/aa982f