On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method

Tiexiang Li, Tsung Ming Huang, Wen-Wei Lin, Jenn Nan Wang*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

In this paper, we consider the two-dimensional Maxwell’s equations with the TM mode in pseudo-chiral media. The system can be reduced to the acoustic equation with a negative index of refraction. We first study the transmission eigenvalue problem (TEP) for this equation. By the continuous finite element method, we discretize the reduced equation and transform the study of TEP to a quadratic eigenvalue problem by deflating all nonphysical zeros. We then estimate half of the eigenvalues are negative with order of O(1) and the other half of eigenvalues are positive with order of O(102). In the second part of the paper, we present a practical numerical method to reconstruct the support of the inhomogeneity by the near-field measurements, i.e., Cauchy data. Based on the linear sampling method, we propose the truncated singular value decomposition to solve the ill-posed near-field integral equation, at one wave number which is not a transmission eigenvalue. By carefully chosen an indicator function, this method produce different jumps for the sampling points inside and outside the support. Numerical results show that our method is able to reconstruct the support reliably.

Original languageEnglish
Pages (from-to)1033-1054
Number of pages22
JournalInverse Problems and Imaging
Volume12
Issue number4
DOIs
StatePublished - 1 Aug 2018

Keywords

  • Linear sampling method
  • Pseudo-chiral model
  • Singular value decomposition
  • Transverse magnetic mode
  • Two-dimensional transmission eigenvalue problem

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