Recursive integral method for transmission eigenvalues

Huang, Ruihao; Struthers, Allan; Sun, Jiguang; Zhang, Ruming

doi:10.1016/j.jcp.2016.10.001

Cited by 47 publications

(56 citation statements)

References 36 publications

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“…Another critical problem of RIM is how to define the indicator δ S . As seen above, the indicator in our other work defined by is to project a random vector twice. One needs to solve linear systems with different right‐hand sides, that is, f and P f /| P f |.…”

Section: An Efficient Indicatormentioning

confidence: 99%

“…In a recent work, we have developed an eigenvalue solver RIM (recursive integral method). RIM is based on spectral projection and uses a technique similar to bisection.…”

Section: Introductionmentioning

confidence: 99%

“…Since f is random, | P f | could be very small. The solution in our other work is to normalize P f and project again. The indicator δ S is set to be

δ_{S} : = (||, P ((), \frac{P bold-italicf}{false| P bold-italicf false|})) .

Analytically, P 2 f = P f .…”

Section: Introductionmentioning

confidence: 99%

“…According to , δ S = 1 if there exists at least one eigenvalue in S and δ S = 0 if there is no eigenvalue in S . Since only “yes” ( δ S = 1) or “no” ( δ S = 0) is needed in the algorithm and quadrature is used to evaluate P f , it is natural to use a threshold δ 0 to differentiate “yes” and “no.” The strategy of RIM in our other work selects a small threshold δ 0 = 0.1, that is, S contains eigenvalue(s) whenever δ S ≥ δ 0 . This choice of threshold for selecting a region systematically leans toward further investigation of regions that may potentially contain eigenvalues.…”

Section: Introductionmentioning

confidence: 99%

“…To compute δ S , one needs to solve the linear system

false(A - z_{j} B false) x_{j} = bold-italicf

at each quadrature point z j . In the first version of RIM , MATLAB ‘\’ is used to solve . Thus, the number of linear solves is related to the size of the region S , the required precision d 0 , and the density of the eigenvalues in S , which is certainly inefficient.…”

Section: Introductionmentioning

confidence: 99%

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Recursive integral method with Cayley transformation

Huang

Sun

Yang

2018

Numerical Linear Algebra App

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Summary The recently developed RIM (recursive integral method) finds eigenvalues in a region of the complex plane. It computes an indicator to test if the region contains eigenvalues using an approximate spectral projection. If the indicator shows that the region contains eigenvalues, it is subdivided into smaller regions, which are then recursively tested and subdivided until any eigenvalues are isolated to a specified precision. We propose an enhancement to RIM that uses Cayley transformations and Arnoldi's method to greatly decrease the number of factorizations required to solve the linear systems that arise from a discretized contour integral to compute the indicator, which substantially reduces the computational cost. We demonstrate the efficacy of our method with numerical examples and compare with the implicitly restarted Arnoldi method, as implemented in eigs in MATLAB.

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Section: An Efficient Indicatormentioning

confidence: 99%

“…In a recent work, we have developed an eigenvalue solver RIM (recursive integral method). RIM is based on spectral projection and uses a technique similar to bisection.…”

Section: Introductionmentioning

confidence: 99%

“…Since f is random, | P f | could be very small. The solution in our other work is to normalize P f and project again. The indicator δ S is set to be

δ_{S} : = (||, P ((), \frac{P bold-italicf}{false| P bold-italicf false|})) .

Analytically, P 2 f = P f .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…To compute δ S , one needs to solve the linear system

false(A - z_{j} B false) x_{j} = bold-italicf

Section: Introductionmentioning

confidence: 99%

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Recursive integral method with Cayley transformation

Huang

Sun

Yang

2018

Numerical Linear Algebra App

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Spectral Indicator Method for a Non-selfadjoint Steklov Eigenvalue Problem

2019

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We propose an efficient numerical method for a non-selfadjoint Steklov eigenvalue problem. The Lagrange finite element is used for discretization. The convergence is proved using the spectral perturbation theory for compact operators. The non-sefadjointness of the problem leads to non-Hermitian matrix eigenvalue problem. Due to the existence of complex eigenvalues and lack of a priori spectral information, we propose a modified version of the recently developed spectral indicator method to compute (complex) eigenvalues in a given region on the complex plane. In particular, to reduce computational cost, the problem is transformed into a much smaller matrix eigenvalue problem involving the unknowns only on the boundary of the domain. Numerical examples are presented to validate the effectiveness of the proposed method.1. Introduction. Steklov eigenvalue problems arise in mathematical physics with spectral parameters in the boundary conditions [24]. Applications of Steklov eigenvalues include surface waves, mechanical oscillators immersed in a viscous fluid, the vibration modes of a structure in contact with an incompressible fluid, etc [24,13,14]. Recently, Steklov eigenvalues have been used in the inverse scattering theory to reconstruct the index of refraction of an inhomogeneous media [12]. Note that most Steklov eigenvalue problems considered in the literature are related to partial differential equations of second order. However, Steklov eigenvalue problems of higher order were also studied, e.g., the fourth order Steklov eigenvalue problem [2].In contrast to the theoretical study of the Steklov eigenvalue problem, numerical methods, in particular, finite element methods have attracted some researchers rather recently [3,6,7,25,14,1,23,16,21]. Various methods have been proposed, including the isoparametric finite element method [3], the virtual element method [25], non-conforming finite element methods [16,21], the spectral-Galerkin method [2], adaptive methods [6], multilevel methods [32], etc. All of the above works consider the selfadjoint cases. In this paper, we consider a non-selfadjoint Steklov eigenvalue problem arising in the study of non-homogeneous absorbing medium in inverse scattering theory [12]. There seems to exist only one paper by Bramble and Osborn [9], which considered the non-selfadjoint case. However, the second order non-selfadjoint operator is assumed to be uniformly elliptic and no numerical results were reported in [9]. In this sense, the current paper is the first paper contains both finite element theory and numerical examples for a non-selfadjoint Steklov eigenvalue problem, to the authors' knowledge. For the general theory and examples of finite element methods for eigenvalue problems, we refer the readers to the book chapter by Babuška and Osborn [4], the review paper by Boffi [8], and the recently published book by Sun and Zhou [31].There are two major challenges to develop effective finite element methods for non-selfadjoint eigenvalue problems [26,4,31]. The first one is the l...

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