A spectral projection method for transmission eigenvalues

Abstract

In this paper, we consider a nonlinear integral eigenvalue problem, which is a reformulation of the transmission eigenvalue problem arising in the inverse scattering theory. The boundary element method is employed for discretization, which leads to a generalized matrix eigenvalue problem. We propose a novel method based on the spectral projection. The method probes a given region on the complex plane using contour integrals and decides if the region contains eigenvalue(s) or not. It is particularly suitable to test if zero is an eigenvalue of the generalized eigenvalue problem, which in turn implies that the associated wavenumber is a transmission eigenvalue. Effectiveness and efficiency of the new method are demonstrated by numerical examples.Comment: The paper has been accepted for publication in SCIENCE CHINA Mathematic

The method of fundamental solutions for computing acoustic interior transmission eigenvalues

“…New methods such as the (generalized) linear sampling method, see [12,2,11], the factorization method, see [33], and the inside outside duality method, see [34,37], were developed which gave deeper insights into the inverse scattering problem. From a numerical point of view, techniques gained from these findings, but also the usual PDE solving tools like finite element methods, see [43,20,7,31,30,44,40,39,28,51,54,53,26,24,29,27,52,49,38,47,48], or boundary integral equations, see [21,22,36,55,35], are applicable and commonly used for ITEs. However, they either require the solution of regularized inverse problems, or they include the generation of a computational grid for the mathematical discretization of the scatterer followed by numerical integrations of singular kernels or of many test functions, respectively, which make these strategies in special cases too generic and numerically expensive.…”

Computation of transmission eigenvalues by the regularized Schur complement for the boundary integral operators

“…Let Ω be a disk with radius 1/2. In this case, the exact transmission eigenvalues are κ's such that [6,23] J 1 (κ/2)J 0 (2κ) − 4J 0 (κ/2)J 1 (2κ) = 0, and The numerical results are presented in Figures 1-3. We mark each location of the exact eigenvalues by a red line.…”

“…The integral based methods [1,9] were developed to compute the corresponding matrix eigenvalues. An approximation to the eigenvalue in a given simple closed curve in the complex plane is found by spectral projection using counter integral of the resolvent [9,23].…”

Maxwell Exterior Transmission Eigenvalue Problems and Their Applications to Electromagnetic Cloaking

“…In particular, the scattering of incident plane waves by a bounded non-absorbing medium is closely connected to the interior transmission eigenvalue problem and has been vigorously studied -cf. [8,33,34]. This problem is formulated in the form of two elliptic equations in a bounded domain with the same Cauchy data on the boundary.…”