Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Plestenjak, Bor; Gheorghiu, Călin-Ioan; Hochstenbach, Michiel E.

doi:10.1016/j.jcp.2015.06.015

Cited by 17 publications

(35 citation statements)

References 34 publications

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“…Mathieu's equation is a well known second-order ordinary differential equation (ODE), endowing with periodic coefficient (McLachlan 1947;Bateman & Erdelyi, 1955;Meixner, Schäfke & Wolf, 1980). A number of physical phenomena and engineering problems can be described by Mathieu's equation, which also appears in the solution of the Helmholtz equation of an elliptic membrane by using the method of separation of variables (Plestenjak, Gheorghiu & Hochstenbach, 2015). Mathieu's equation comes into the realm of mathematical physics for wave propagation, electromagnetic, elastic membrane and heat conduction, when there is some elliptic symmetry in the problems.…”

Section: Introductionmentioning

confidence: 99%

Analytic Solutions of the Eigenvalues of Mathieu’s Equation

Liu¹

2019

JMR

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Mathieu's eigenvalue problem −y (x) + 2e 0 cos(2x)y(x) = λy(x), 0 < x < is symmetric if cos(2x) = cos(2 − 2x) for = k 0 π, k 0 ∈ N, and skew-symmetric if cos(2x) = − cos(2 − 2x) for = π/2. Two typical boundary conditions are considered. When the eigenfunctions are expanded by the orthonormal bases of sine functions or cosine functions, we can derive an n-dimensional matrix eigenvalue problem, endowing with a special structure of the symmetric coefficient matrix A := [a i j ], a i j = 0 if i + j is an odd integer. Based on it, we can obtain the eigenvalues easily and analytically. When = k 0 π, k 0 ∈ N, we have a i j = 0 if |i − j| > 2k 0 . Besides the diagonal band, A has two off-diagonal bands, and furthermore, a cross band appears when k 0 ≥ 2. The product formula, the recursion formulas of characteristic functions and a fictitious time integration method (FTIM) are developed to find the eigenvalues of Mathieu's equation.

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Section: Introductionmentioning

confidence: 99%

Analytic Solutions of the Eigenvalues of Mathieu’s Equation

Liu¹

2019

JMR

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“…This example will be solved numerically in Section 5.1 more accurately than in [20] as the new methods can deal with larger matrices coming from finer discretizations.…”

Section: Ellipsoidal Wave Equationsmentioning

confidence: 99%

“…Although we cannot write (20) as a Sylvester equation in the 3-parameter setting, we can borrow some ideas from the Krylov method for the 2-parameter case that is based on the solutions of Sylvester equations by the low-rank approximation approach due to Hu-Reichel. In particular, suppose that we are looking for a low-rank approximation of the solution of (19).…”

Section: Subspace Iterationmentioning

confidence: 99%

“…Specifically, spectral collocation is used in the aforementioned work for the discretization, which gives rise to relatively small matrices and accurate results. While several suitable numerical methods for the case k = 2 exist (see, e.g., the work of Plestenjak et al and the references therein), available feasible numerical methods for k ≥ 3 are limited to problems with very small matrices, which means that, even by using spectral collocation, we cannot obtain many accurate eigenvalues of . We introduce new variants of numerical methods for three‐parameter eigenvalue problems that exceed the above limitations and can be applied to problems with larger matrices.…”

Section: Introductionmentioning

confidence: 99%

“…When eigenvalues ( λ 1 ,…, λ k ) with the smallest | λ k | are sought, subspace iteration or an Arnoldi iteration operating directly on Δ k z = λ k Δ 0 z appears more appropriate. Such ideas have been explored well in the two‐parameter eigenvalue setting and applied for the solution of various separable boundary value problems . This success is mostly due to the fact that linear systems of the form Δ 2 w = Δ 0 v for a given v can be expressed as Sylvester equations involving the matricizations of the vectors v and w and, thus, can be solved efficiently at a cost of

O false(n_{1}^{3} + n_{2}^{3} false)

.…”

Section: Introductionmentioning

confidence: 99%

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Subspace methods for three‐parameter eigenvalue problems

Hochstenbach

Meerbergen

Mengi

et al. 2019

Numerical Linear Algebra App

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Summary We propose subspace methods for three‐parameter eigenvalue problems. Such problems arise when separation of variables is applied to separable boundary value problems; a particular example is the Helmholtz equation in ellipsoidal and paraboloidal coordinates. While several subspace methods for two‐parameter eigenvalue problems exist, their extensions to a three‐parameter setting seem challenging. An inherent difficulty is that, while for two‐parameter eigenvalue problems, we can exploit a relation to Sylvester equations to obtain a fast Arnoldi‐type method, such a relation does not seem to exist when there are three or more parameters. Instead, we introduce a subspace iteration method with projections onto generalized Krylov subspaces that are constructed from scratch at every iteration using certain Ritz vectors as the initial vectors. Another possibility is a Jacobi–Davidson‐type method for three or more parameters, which we generalize from its two‐parameter counterpart. For both approaches, we introduce a selection criterion for deflation that is based on the angles between left and right eigenvectors. The Jacobi–Davidson approach is devised to locate eigenvalues close to a prescribed target; yet, it often also performs well when eigenvalues are sought based on the proximity of one of the components to a prescribed target. The subspace iteration method is devised specifically for the latter task. The proposed approaches are suitable especially for problems where the computation of several eigenvalues is required with high accuracy. MATLAB implementations of both methods have been made available in the package MultiParEig (see http://www.mathworks.com/matlabcentral/fileexchange/47844-multipareig).

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Subspace method for multiparameter‐eigenvalue problems based on tensor‐train representations

Ruymbeek

Meerbergen²,

Michiels³

2022

Numerical Linear Algebra App

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In this article, we solve m‐parameter eigenvalue problems (mEPs), with m any natural number by representing the problem using tensor‐trains (TTs) and designing a method based on this format. mEPs typically arise when separation of variables is applied to separable boundary value problems. Often, methods for solving mEP are restricted to m=3, due to the fact that, to the best of our knowledge, no available solvers exist for m>3 and reasonable size of the involved matrices. In this article, we prove that computing the eigenvalues of a mEP can be recast into computing the eigenvalues of TT‐operators. We adapted the algorithm in9 for symmetric eigenvalue problems in TT‐format to an algorithm for solving generic mEPs. This leads to a subspace method whose subspace dimension does not depend on m, in contrast to other subspace methods for mEPs. This allows us to tackle mEPs with m>3 and reasonable size of the matrices. We provide theoretical results and report numerical experiments. The MATLAB code is publicly available.

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Spectral collocation for multiparameter eigenvalue problems arising from separable boundary value problems

Cited by 17 publications

References 34 publications

Analytic Solutions of the Eigenvalues of Mathieu’s Equation

Analytic Solutions of the Eigenvalues of Mathieu’s Equation

Subspace methods for three‐parameter eigenvalue problems

Subspace method for multiparameter‐eigenvalue problems based on tensor‐train representations

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