On Spectral Analysis and A Novel Algorithm for Transmission Eigenvalue Problems

Tiexiang Li Southeast University Wei-Qiang Huang National Chiao Tung University Wen-Wei Lin National Chiao Tung University Jijun Liu Southeast University

Numerical Analysis and Scientific Computing mathscidoc:1608.25002

Distinguished Paper Award in 2019

Journal of Scientific Computing, 64, (1), 2015
Thetransmissioneigenvalueproblem,besidesitscriticalroleininversescattering problems, deserves special interest of its own due to the fact that the corresponding differen- tial 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 eigen- values 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 symmet- ric 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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@inproceedings{tiexiang2015on,
  title={On Spectral Analysis and A Novel Algorithm for Transmission Eigenvalue Problems},
  author={Tiexiang Li, Wei-Qiang Huang, Wen-Wei Lin, and Jijun Liu},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818151951605013244},
  booktitle={Journal of Scientific Computing},
  volume={64},
  number={1},
  year={2015},
}
Tiexiang Li, Wei-Qiang Huang, Wen-Wei Lin, and Jijun Liu. On Spectral Analysis and A Novel Algorithm for Transmission Eigenvalue Problems. 2015. Vol. 64. In Journal of Scientific Computing. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160818151951605013244.
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