On correction equations and domain decomposition for computing invariant subspaces

Philippe, Bernard; Saad, Yousef

doi:10.1016/j.cma.2006.03.026

Cited by 10 publications

(9 citation statements)

References 15 publications

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“…Equating (17) with (11) shows that X s X T s ≡  and X u X T s ≡ −, and thus, in contrast with the DD-FP scheme, we only need to compute the contour integrals − and  and ignore the block . As discussed in Section 4.3 and confirmed via experiments in Section 6, avoiding the computation of  can lead to considerable savings in some cases.…”

Section: Partial Integration Of the Matrix Resolventmentioning

confidence: 95%

“…Let the spectral projector , defined in (11), be expressed in the form  = XX T , X ∈ R n×r , where X is written as…”

Section: Partial Integration Of the Matrix Resolventmentioning

confidence: 99%

“…[1][2][3][4] We focus on distributed computing environments, possibly with a large number of processors, and study contour integration eigenvalue solvers (eigensolvers) from a domain decomposition viewpoint. 5,6 Domain decomposition techniques for the solution of eigenvalue problems have been studied in the past (see, e.g., other works [7][8][9][10][11] ), but to our knowledge, these methods have not yet been considered * within a contour integration framework. *An exception is the approach presented in the work of Kalantzis et al 12 that appeared while this paper was under review.…”

Section: Introductionmentioning

confidence: 99%

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Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

Kalantzis

Kestyn

Polizzi

et al. 2018

Numerical Linear Algebra App

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Summary This paper discusses techniques for computing a few selected eigenvalue–eigenvector pairs of large and sparse symmetric matrices. A recently developed class of techniques to solve this type of problems is based on integrating the matrix resolvent operator along a complex contour that encloses the interval containing the eigenvalues of interest. This paper considers such contour integration techniques from a domain decomposition viewpoint and proposes two schemes. The first scheme can be seen as an extension of domain decomposition linear system solvers in the framework of contour integration methods for eigenvalue problems, such as FEAST. The second scheme focuses on integrating the resolvent operator primarily along the interface region defined by adjacent subdomains. A parallel implementation of the proposed schemes is described, and results on distributed computing environments are reported. These results show that domain decomposition approaches can lead to reduced run times and improved scalability.

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Section: Partial Integration Of the Matrix Resolventmentioning

confidence: 95%

“…Let the spectral projector , defined in (11), be expressed in the form  = XX T , X ∈ R n×r , where X is written as…”

Section: Partial Integration Of the Matrix Resolventmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

Kalantzis

Kestyn

Polizzi

et al. 2018

Numerical Linear Algebra App

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“…Compared with directly using Kron's subspace iteration to improve the result of the first-order approximated Kron's method, as those have been done for the fixed-interface CMS approximation, [32][33][34] the proposed method does not need the modal transformation (the terms related to the high-order modal group) before the subspace iteration, and therefore, it (i) needs less operations and simpler programming and (ii) can employ enhancements such as the shifting and restarting throughout the iterations, especially for the initial guess.…”

Section: Initialization Of the Iterationsmentioning

confidence: 99%

An eigenvector‐based iterative procedure for the free‐interface component modal synthesis method

Cui

Wang

Xing

et al. 2018

Numerical Meth Engineering

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A novel eigenvector-based iteration procedure is developed for the free-interface component modal synthesis (CMS) method. To derive the iteration formula, Kron's substructuring is employed to distribute the computations of subspace iteration, and the free-interface component modes are chosen as the initial guess. Then, the modal transformation matrix of the first-order approximated Kron's CMS method is proved to be the free-interface component modes with one step of Kron's inverse iteration. The proposed CMS method has the advantages of both free-interface CMS approximation and subspace iteration: on one hand, the CMS approximation provides a high-quality initial guess and distributes the computational cost of the subspace iteration; on the other hand, the subspace iteration provides a more efficient way for truncation compensation and is compatible with using deflation, shifting, and restarting for further enhancements on the efficiency. Numerical examples show that the efficiency of the proposed method is higher than that of the conventional simultaneous iterative Kron's CMS method, especially for obtaining a large number of high-precision modes. Moreover, the proposed method is as efficient as the global Lanczos method for first-time analysis without parallelization while retaining the advantages of CMS methods for reanalysis tasks, parallelization, etc. KEYWORDScomponent modal synthesis, iterative, subspace iteration, substructure Int J Numer Methods Eng. 2018;116:723-740. wileyonlinelibrary.com/journal/nme

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“…Sylvester equations arise in various applications, such as the finite difference discretization of elliptic boundary value problems on a rectangular domain , problems in control theory and stability analysis , matrix differential and difference equations , block diagonalization of a block triangular matrix , and eigenproblems . The Lyapunov equations can be regarded as a special case of Sylvester equations, which arise in the stability and robust stability analysis, in determining the controllability and observability Grammians, and in H 2 ‐control.…”

Section: Introductionmentioning

confidence: 99%

Mixed, componentwise condition numbers and small sample statistical condition estimation of Sylvester equations

Diao

Xiang

Wei

2011

Numerical Linear Algebra App

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SUMMARY We present a componentwise perturbation analysis for the continuous‐time Sylvester equations. Componentwise, mixed condition numbers and new perturbation bounds are derived for the matrix equations. The small sample statistical method can also be applied for the condition estimation. These condition numbers and perturbation bounds are tested on numerical examples and compared with the normwise condition number. The numerical examples illustrate that the mixed condition number gives sharper bounds than the normwise one. Copyright © 2011 John Wiley & Sons, Ltd.

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On correction equations and domain decomposition for computing invariant subspaces

Cited by 10 publications

References 15 publications

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

Domain decomposition approaches for accelerating contour integration eigenvalue solvers for symmetric eigenvalue problems

An eigenvector‐based iterative procedure for the free‐interface component modal synthesis method

Mixed, componentwise condition numbers and small sample statistical condition estimation of Sylvester equations

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