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Published in: Journal of Scientific Computing 1/2015

01-07-2015

On Spectral Analysis and a Novel Algorithm for Transmission Eigenvalue Problems

Authors: Tiexiang Li, Wei-Qiang Huang, Wen-Wei Lin, Jijun Liu

Published in: Journal of Scientific Computing | Issue 1/2015

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Abstract

The transmission eigenvalue problem, besides its critical role in inverse scattering problems, deserves special interest of its own due to the fact that the corresponding differential operator is neither elliptic nor self-adjoint. In this paper, we provide a spectral analysis and propose a novel iterative algorithm for the computation of a few positive real eigenvalues and the corresponding eigenfunctions of the transmission eigenvalue problem. Based on approximation using continuous finite elements, we first derive an associated symmetric quadratic eigenvalue problem (QEP) for the transmission eigenvalue problem to eliminate the nonphysical zero eigenvalues while preserve all nonzero ones. In addition, the derived QEP enables us to consider more refined discretization to overcome the limitation on the number of degrees of freedom. We then transform the QEP to a parameterized symmetric definite generalized eigenvalue problem (GEP) and develop a secant-type iteration for solving the resulting GEPs. Moreover, we carry out spectral analysis for various existence intervals of desired positive real eigenvalues, since a few lowest positive real transmission eigenvalues are of practical interest in the estimation and the reconstruction of the index of refraction. Numerical experiments show that the proposed method can find those desired smallest positive real transmission eigenvalues accurately, efficiently, and robustly.

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Footnotes
1
Here \(\mathcal {O}\) denotes the “big O”.
 
2
Since we are only interested in finding the positive real eigenvalues of the QEP (10), we restrict our discussion to the case \(\tau \ge 0\).
 
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Metadata
Title
On Spectral Analysis and a Novel Algorithm for Transmission Eigenvalue Problems
Authors
Tiexiang Li
Wei-Qiang Huang
Wen-Wei Lin
Jijun Liu
Publication date
01-07-2015
Publisher
Springer US
Published in
Journal of Scientific Computing / Issue 1/2015
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI
https://doi.org/10.1007/s10915-014-9923-0

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