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

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(10 2 ). 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.

A mixed element scheme for the Helmholtz transmission eigenvalue problem for anisotropic media

“…where I is the identity matrix. To solve the transmission eigenvalue problem for isotropic media, theories and algorithms have been developed in [9,12,16,17,21,27], etc. Among these studies, the fourth-order problem of the transmission eigenvalue problem was also constructed in [9,16,17,27], but based on an auxiliary variable w − v rather than A∇w − ∇v in the anisotropic case.…”

An inverse medium problem using Stekloff eigenvalues and a Bayesian approach

“…Given reconstructed eigenvalues, a legitimate question is what information about the scatterer can be obtained. For inhomogeneous non-absorbing media, transmission eigenvalues have been used to reconstruct the shape of the obstacle [31] and estimate the index of refraction [1,3,6,13,20,24,30]. However, the use of transmission eigenvalues has two drawbacks: (1) multi-frequency data are necessary; and (2) only real transmission eigenvalues can be determined from the scattering data so far.…”