The Stability of Block Variants of Classical Gram--Schmidt

Carson, Erin; Lund, Kathryn; Rozložńık, Miroslav

doi:10.1137/21m1394424

Cited by 15 publications

(22 citation statements)

References 15 publications

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“…At present, we can see that efficient parallelization of GMRES represents the frontier for linear systems. Rigorous stability studies of parallel and block algorithms are still lacking; we refer to [52] as a recent work in this direction. Furthermore, for the acceleration strategies mentioned above, preconditioning may offer significant advantages in a variety of contexts, and thus should play a role in solving challenging problems.…”

Section: Discussionmentioning

confidence: 99%

“…Here, BGS amounts to a block version of CGS. For more details on BGS, we refer the reader to [52,53] and references therein. BGS and TSQR are both regarded as kernels (see [187,141]).…”

Section: S-step Gmresmentioning

confidence: 99%

“…The CGS process involving (30) could be denoted by CGS-P (see, e.g., [246,52]), in which one finds that only one reduction is required, whereas two are required in the traditional CGS-Arnoldi algorithm. This results in the single-reduction GMRES algorithm.…”

Section: Pipelined Gmresmentioning

confidence: 99%

“…Numerical results reported promising performance of the low-sync Gram-Schmidt and GMRES algorithms. A good survey of Gram-Schmidt up to 2010 was given by Leon et al [162] in 2013; we also refer the interested reader to [23,22,52,53] and their references for the discussion of block Gram-Schmidt algorithms. On the other hand, a low-sync GMRES variant based on Householder QR was derived by Walker [277] in 1988.…”

Section: Low-sync Gmresmentioning

confidence: 99%

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GMRES algorithms over 35 years

Zou

2021

Preprint

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Section: Discussionmentioning

confidence: 99%

“…Here, BGS amounts to a block version of CGS. For more details on BGS, we refer the reader to [52,53] and references therein. BGS and TSQR are both regarded as kernels (see [187,141]).…”

Section: S-step Gmresmentioning

confidence: 99%

Section: Pipelined Gmresmentioning

confidence: 99%

Section: Low-sync Gmresmentioning

confidence: 99%

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GMRES algorithms over 35 years

Zou

2021

Preprint

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“…Especially the classical Gram-Schmidt procedure may produce a significant orthogonalization error. The stability properties of various block Gram-Schmidt procedures are analyzed in the paper of Carson et al [4] and an excellent overview of existing block Gram-Schmidt methods is given in [5], taking performance and stability into account.…”

mentioning

confidence: 99%

A Hardware-aware and Stable Orthogonalization Framework

Dreier¹,

Engwer²

2022

Preprint

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The orthogonalization process is an essential building block in Krylov space methods, which takes up a large portion of the computational time. Commonly used methods, like the Gram-Schmidt method, consider the projection and normalization separately and store the orthogonal base explicitly. We consider the problem of orthogonalization and normalization as a QR decomposition problem on which we apply known algorithms, namely CholeskyQR and TSQR. This leads to methods that solve the orthogonlization problem with reduced communication costs, while maintaining stability and stores the orthogonal base in a locally orthogonal representation. Furthermore, we discuss the novel method as a framework which allows us to combine different orthogonalization algorithms and use the best algorithm for each part of the hardware. After the formulation of the methods, we show their advantageous performance properties based on a performance model that takes data transfers within compute nodes as well as message passing between compute nodes into account. The theoretic results are validated by numerical experiments.

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A flexible block classical Gram–Schmidt skeleton with reorthogonalization

Zou

2023

Numerical Linear Algebra App

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We investigate a variant of the reorthogonalized block classical Gram–Schmidt method for computing the QR factorization of a full column rank matrix. Our aim is to bound the loss of orthogonality even when the first local QR algorithm is only conditionally stable. In particular, this allows the use of modified Gram–Schmidt instead of Householder transformations as the first local QR algorithm. Numerical experiments confirm the stable behavior of the new variant. We also examine the use of non‐QR local factorization and show by example that the resulting variants, although less stable, may also be applied to ill‐conditioned problems.

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The Stability of Block Variants of Classical Gram--Schmidt

Cited by 15 publications

References 15 publications

GMRES algorithms over 35 years

GMRES algorithms over 35 years

A Hardware-aware and Stable Orthogonalization Framework

A flexible block classical Gram–Schmidt skeleton with reorthogonalization

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