An indefinite variant of LOBPCG for definite matrix pencils

Kreßner, Daniel; Pandur, Marija Miloloža; Shao, Meiyue

doi:10.1007/s11075-013-9754-3

Cited by 24 publications

(53 citation statements)

References 27 publications

(57 reference statements)

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“…To study this issue, in the following two sections, we will introduce an efficient and robust eigensolver, called LOBPCG [22,23], that can exclude the disturbance of infinite eigenvalues when solving the μ-SDGEP (3.1). In fact, for any fixed μ > 0, one can see that the desired positive eigenvalues of (3.1) suffer from the disturbance of a cluster of infinite eigenvalues.…”

Section: A Secant-type Methods For Computing Positive Transmission Eigmentioning

confidence: 99%

“…However, we can immediately note that applying the SILM to solve (3.1) has some drawbacks. For the case in which B is an indefinite matrix, two variants of LOBPCG were recently studied in [23]. (ii) When the desired eigenpairs of (3.1) are convergent, it is natural to use the associated eigenvectors as the initial vectors for the next μ-SDGEP to accelerate the convergence.…”

Section: Lobpcg Methodmentioning

confidence: 99%

“…Based on the results in Theorem 4.3, Kressner, Pandur, and Shao [23] obtained a corresponding Ky Fan-type theorem (trace minimization principle) and used it to develop two indefinite variants of the LOBPCG method. Algorithm 2 is the indefinite LOBPCG method with one preconditioner for computing the smallest positive eigenvalues of a positive definite pencil A − λB.…”

Section: Lobpcg Methodmentioning

confidence: 99%

“…Now, we have Algorithm 3, which summarizes the practical procedure for solving a few positive eigenvalues of the QEP (2.12) by SecTypIt [25] combined with the indefinite LOBPCG with one preconditioner [23]. All computations in this section are carried out in MATLAB 2014b.…”

Section: Stopping Tolerance For Thementioning

confidence: 99%

See 3 more Smart Citations

A Robust Numerical Algorithm for Computing Maxwell's Transmission Eigenvalue Problems

Huang¹,

Wei²,

Lin³

2015

SIAM J. Sci. Comput.

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We study a robust and efficient eigensolver for computing a few smallest positive eigenvalues of the three-dimensional Maxwell's transmission eigenvalue problem. The discretized governing equations by the Nédélec edge element result in a large-scale quadratic eigenvalue problem (QEP) for which the spectrum contains many zero eigenvalues and the coefficient matrices consist of patterns in the matrix form XY −1 Z, both of which prevent existing eigenvalue solvers from being efficient. To remedy these difficulties, we rewrite the QEP as a particular nonlinear eigenvalue problem and develop a secant-type iteration, together with an indefinite locally optimal block preconditioned conjugate gradient (LOBPCG) method, to sequentially compute the desired positive eigenvalues. Furthermore, we propose a novel method to solve the linear systems in each iteration of LOBPCG.Intensive numerical experiments show that our proposed method is robust, although the desired real eigenvalues are surrounded by complex eigenvalues.Recently, there have been some papers [12,15,18,19,21,25,28,32,33] addressing numerical computations in transmission eigenvalue problems. In [12], three finite element methods (FEMs) were proposed for solving the two-dimensional (2D) transmission eigenvalue problem. A coupled boundary element method and FEM was introduced for the interior transmission problem in [15]. Then, Sun [32] proposed two iterative methods together with convergence analysis based on the existence theory of the fourth-order reformulation for the transmission eigenvalues [9,30]. A mixed FEM for 2D transmission eigenvalue problems was proposed in [18] and the corresponding non-Hermitian quadratic eigenvalue problem (QEP) was solved by the classical secant iteration with an adaptive Arnoldi method. In [19], Ji, Sun, and Xie used the multilevel correction method to transform the solution of the transmission problem into a series of solutions corresponding to linear boundary value problems and solved them by the multigrid method. Li et al. in [25] rewrote the QEP as a particular parameterized generalized eigenvalue problem (GEP) for which the eigenvalue curves are arranged in a monotonic order so that the desired curves can be sequentially solved with a new secant-type iteration.For a three-dimensional (3D) Maxwell's transmission eigenvalue problem, two FEMs with an adaptive Arnoldi method were proposed in [28]. The resulting GEPs are large, sparse, and non-Hermitian. The numerical challenges for solving the corresponding GEPs are (i) a few of the smallest positive eigenvalues, which may be surrounded by complex eigenvalues, are of interest; (ii) the number of zero eigenvalues of the GEP is huge because the nullity of the discrete double curl operator equals the number of edges in the spanning tree of a finite element mesh [2]; (iii) efficient solution of the associated large sparse linear system in each iteration of the eigensolver. To tackle drawbacks (i) and (ii), in [33], a mixed FEM was applied to an equivalent quadcurl eigenvalue proble...

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Section: A Secant-type Methods For Computing Positive Transmission Eigmentioning

confidence: 99%

Section: Lobpcg Methodmentioning

confidence: 99%

Section: Lobpcg Methodmentioning

confidence: 99%

Section: Stopping Tolerance For Thementioning

confidence: 99%

See 2 more Smart Citations

A Robust Numerical Algorithm for Computing Maxwell's Transmission Eigenvalue Problems

Huang¹,

Wei²,

Lin³

2015

SIAM J. Sci. Comput.

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“…It is also easy to see that if the factorization B = QR exists, it can be regarded as an implicit Cholesky-like factorization of the symmetric indefinite matrix M = B T AB = R T ΩR (without its explicit computation), delivering the same upper triangular factor R. Conversely, given the Cholesky-like factorization of M , the (A, Ω)-orthogonal factor Q can be then recovered as Q = BR −1 . Such problems appear explicitly [15] or implicitly in many applications such as eigenvalue problems, matrix pencils and structure-preserving algorithms [21,25], saddle-point problems, and optimization with interior-point methods [13,36,29] or indefinite least squares problems [4,9,23,24].…”

Section: Introductionmentioning

confidence: 99%

Cholesky-Like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms

Rozložńık¹,

Okulicka-Dłużewska²,

Smoktunowicz³

2015

SIAM J. Matrix Anal. & Appl.

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It is well known that orthogonalization of column vectors in a rectangular matrix B with respect to the bilinear form induced by a nonsingular symmetric indefinite matrix A can be eventually seen as its factorization B = QR that is equivalent to the Cholesky-like factorization in the form B T AB = R T ΩR, where R is upper triangular and Ω is a signature matrix. Under the assumption of nonzero principal minors of the matrix M = B T AB we give bounds for the conditioning of the triangular factor R in terms of extremal singular values of M and of only those principal submatrices of M where there is a change of sign in Ω. Using these results we study the numerical behavior of two types of orthogonalization schemes and we give the worst-case bounds for quantities computed in finite precision arithmetic. In particular, we analyze the implementation based on the Cholesky-like factorization of M and the Gram-Schmidt process with respect to the bilinear form induced by the matrix A. To improve the accuracy of computed results we consider also the Gram-Schmidt process with reorthogonalization and show that its behavior is similar to the scheme based on the Cholesky-like factorization with one step of iterative refinement. Introduction.For a real symmetric (in general indefinite) nonsingular matrix A ∈ R m,m and for a full-column rank matrix B ∈ R m,n (m ≥ n) we look for a factorization B = QR, where Q ∈ R m,n is so-called (A, Ω)-orthogonal, i.e., its columns are mutually orthogonal with respect to the bilinear form induced by the matrix A, with Q T AQ = Ω ∈ R n,n being a signature matrix Ω ∈ diag(±1), and where R ∈ R n,n is upper triangular with positive diagonal elements. Note that the full-column rank condition of the matrix B is not enough for the existence of the factors Q and R such that Q is (A, Ω)-orthogonal and R is upper triangular with positive diagonal entries. It is also easy to see that if the factorization B = QR exists, it can be regarded as an implicit Cholesky-like factorization of the symmetric indefinite matrix M = B T AB = R T ΩR (without its explicit computation), delivering the same upper triangular factor R. Conversely, given the Cholesky-like factorization of M , the (A, Ω)-orthogonal factor Q can be then recovered as Q = BR −1 . Such problems appear explicitly [15] or implicitly in many applications such as eigenvalue problems, matrix pencils and structure-preserving algorithms [21,25], saddle-point problems, and optimization with interior-point methods [13,36,29] or indefinite least squares problems [4,9,23,24].

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Convergence analysis of vector extended locally optimal block preconditioned extended conjugate gradient method for computing extreme eigenvalues

Benner

Liang

2022

Numerical Linear Algebra App

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This article is concerned with the convergence analysis of an extension of the locally optimal preconditioned conjugate gradient method for the extreme eigenvalue of a Hermitian matrix polynomial that possesses some extended form of Rayleigh quotient. This work is a generalization of the analysis by Ovtchinnikov (SIAM J Numer Anal. 2008;46(5):2567-92.). As instances, the algorithms for definite matrix pairs and hyperbolic quadratic matrix polynomials are shown to be globally convergent and to have an asymptotically local convergence rate. Also, numerical examples are given to illustrate the convergence behavior.

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An indefinite variant of LOBPCG for definite matrix pencils

Cited by 24 publications

References 27 publications

A Robust Numerical Algorithm for Computing Maxwell's Transmission Eigenvalue Problems

A Robust Numerical Algorithm for Computing Maxwell's Transmission Eigenvalue Problems

Cholesky-Like Factorization of Symmetric Indefinite Matrices and Orthogonalization with Respect to Bilinear Forms

Convergence analysis of vector extended locally optimal block preconditioned extended conjugate gradient method for computing extreme eigenvalues

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