Solution of generalized shifted linear systems with complex symmetric matrices

Sogabe, Tomohiro; Hoshi, Takeo; Zhang, Shao‐Liang; Fujiwara, Takeo

doi:10.1016/j.jcp.2012.04.046

Cited by 13 publications

(18 citation statements)

References 35 publications

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“…And refer to the works of Liu and Zhong, and Zhong et al for block variants of SGMRES for solving linear systems with multiple right hand sides. For other related methods, refer some studies …”

Section: Introductionmentioning

confidence: 98%

“…For other related methods, refer some studies. 4,[12][13][14][15][16][17][18][19] This paper focuses on the development of a new variant of SGMRES for shifted system (referred to as SGMRES-Sh). Just like the non-collinearity of the GMRES residuals r m and r m , respectively, of the seed and the add systems, 20 similar situation may occur when restarted SGMRES is used to solve simultaneously the seed and the add systems.…”

Section: Introductionmentioning

confidence: 99%

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A simpler GMRES and its adaptive variant for shifted linear systems

Jing

Yuan

Huang

2016

Numerical Linear Algebra App

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Summary A variant of the simpler GMRES method is developed for solving shifted linear systems (SGMRES‐Sh), exhibiting almost the same advantage of the simpler GMRES method over the regular GMRES method. Because the remedy adapted by GMRES‐Sh is no longer feasible for SGMRES‐Sh due to the differences between simpler GMRES and GMRES for constructing the residual vectors of linear systems, we take an alternative strategy to force the residual vectors of the add system also be orthogonal to the subspaces, to which the residual vectors of the seed system are orthogonal when the seed system is solved with the simpler GMRES method. In addition, a seed selection strategy is also employed for solving the rest non‐converged linear systems. Furthermore, an adaptive version of SGMRES‐Sh is presented for the purpose of improving the stability of SGMRES‐Sh based on the technique of the adaptive choice of the Krylov subspace basis developed for the adaptive simpler GMRES. Numerical experiments demonstrate the benefits of the presented methods.

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Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

A simpler GMRES and its adaptive variant for shifted linear systems

Jing

Yuan

Huang

2016

Numerical Linear Algebra App

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“…conjugate gradients for shifted linear systems [12,13], stochastic trace estimation [14,15], and stochastic Lanczos quadrature [16][17][18] are suggested in the bibliography.…”

Section: Preliminariesmentioning

confidence: 99%

Stochastic Lanczos estimation of genomic variance components for linear mixed-effects models

Border

Becker

2019

Preprint

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Background: Linear mixed-effects models (LMM) are a leading method in conducting genome-wide association studies (GWAS) but require residual maximum likelihood (REML) estimation of variance components, which is computationally demanding. Previous work has reduced the computational burden of variance component estimation by replacing direct matrix operations with iterative and stochastic methods and by employing loose tolerances to limit the number of iterations in the REML optimization procedure. Here, we introduce two novel algorithms, stochastic Lanczos derivative-free REML (SLDF_REML) and Lanczos first-order Monte Carlo REML (L_FOMC_REML), that exploit problem structure via the principle of Krylov subspace shift-invariance to speed computation beyond existing methods. Both novel algorithms only require a single round of computation involving iterative matrix operations, after which their respective objectives can be repeatedly evaluated using vector operations. Further, in contrast to existing stochastic methods, SLDF_REML can exploit precomputed genomic relatedness matrices (GRMs), when available, to further speed computation.Results: Results of numerical experiments are congruent with theory and demonstrate that interpreted-language implementations of both algorithms match or exceed existing compiled-language software packages in speed, accuracy, and flexibility. Conclusions:Both the SLDF_REML and L_FOMC_REML algorithms outperform existing methods for REML estimation of variance components for LMM and are suitable for incorporation into existing GWAS LMM software implementations.

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“…One has considerable freedom in choosing the approach to solve Equation 32 based on the properties of the matrices in Equation 32, such as the Krylov subspace based methods. [40][41][42] Because Equation 32 is generalized shifted systems with multiple right-hand sides, the direct methods, such as the sparse Gaussian LU factorization, are also highly recommended. Notice that once we obtain the LU factors, they can be reused when we solve Equation 32 in the subsequent iterations.…”

Section: Endmentioning

confidence: 99%

A FEAST algorithm with oblique projection for generalized eigenvalue problems

Yin

Chan

Yeung

2017

Numer. Linear Algebra Appl.

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Abstract. The contour-integral based eigensolvers are the recent efforts for computing the eigenvalues inside a given region in the complex plane. The best-known members are the Sakurai-Sugiura (SS) method, its stable version CIRR, and the FEAST algorithm. An attractive computational advantage of these methods is that they are easily parallelizable. The FEAST algorithm was developed for the generalized Hermitian eigenvalue problems. It is stable and accurate. However, it may fail when applied to non-Hermitian problems. In this paper, we extend the FEAST algorithm to non-Hermitian problems. The approach can be summarized as follows: (i) to construct a particular contour integral to form a subspace containing the desired eigenspace, and (ii) to use the oblique projection technique to extract desired eigenpairs with appropriately chosen test subspace. The related mathematical framework is established. We also address some implementation issues such as how to choose a suitable starting matrix and design good stopping criteria. Numerical experiments are provided to illustrate that our method is stable and efficient.

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Solution of generalized shifted linear systems with complex symmetric matrices

Cited by 13 publications

References 35 publications

A simpler GMRES and its adaptive variant for shifted linear systems

A simpler GMRES and its adaptive variant for shifted linear systems

Stochastic Lanczos estimation of genomic variance components for linear mixed-effects models

A FEAST algorithm with oblique projection for generalized eigenvalue problems

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