TY - JOUR

T1 - On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method

AU - Li, Tiexiang

AU - Huang, Tsung Ming

AU - Lin, Wen Wei

AU - Wang, Jenn Nan

N1 - Funding Information:
2010 Mathematics Subject Classification. Primary: 78A46, 35R30; Secondary: 78M10. Key words and phrases. Two-dimensional transmission eigenvalue problem, pseudo-chiral model, transverse magnetic mode, linear sampling method, singular value decomposition. Li is supported in parts by the NSFC 11471074. Huang was partially supported by the Ministry of Science and Technology (MOST) 105-2115-M-003-009-MY3, National Center of Theoretical Sciences (NCTS) in Taiwan. Lin was partially supported by MOST, NCTS and ST Yau Center in Taiwan. Wang was partially supported by MOST 105-2115-M-002-014-MY3. ∗ Corresponding author: Jenn-Nan Wang.

PY - 2018/8

Y1 - 2018/8

N2 - 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.

AB - 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.

KW - Linear sampling method

KW - Pseudo-chiral model

KW - Singular value decomposition

KW - Transverse magnetic mode

KW - Two-dimensional transmission eigenvalue problem

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U2 - 10.3934/ipi.2018043

DO - 10.3934/ipi.2018043

M3 - Article

AN - SCOPUS:85051704877

VL - 12

SP - 1033

EP - 1054

JO - Inverse Problems and Imaging

JF - Inverse Problems and Imaging

SN - 1930-8337

IS - 4

ER -