Preconditioning correction equations in Jacobi--Davidson type methods for computing partial singular value decompositions of large matrices Article Swipe
Jinzhi Huang
,
Zhongxiao Jia
·
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2404.12568
YOU?
·
· 2024
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.2404.12568
In a Jacobi--Davidson (JD) type method for singular value decomposition (SVD) problems, called JDSVD, a large symmetric and generally indefinite correction equation is solved iteratively at each outer iteration, which constitutes the inner iterations and dominates the overall efficiency of JDSVD. In this paper, a convergence analysis is made on the minimal residual (MINRES) method for the correction equation. Motivated by the results obtained, at each outer iteration a new correction equation is derived that extracts useful information from current subspaces to construct effective preconditioners for the correction equation and is proven to retain the same convergence of outer iterations of JDSVD.The resulting method is called inner preconditioned JDSVD (IPJDSVD) method; it is also a new JDSVD method, and any viable preconditioner for the correction equations in JDSVD is straightforwardly applicable to those in IPJDSVD. Convergence results show that MINRES for the new correction equation can converge much faster when there is a cluster of singular values closest to a given target. A new thick-restart IPJDSVD algorithm with deflation and purgation is proposed that simultaneously accelerates the outer and inner convergence of the standard thick-restart JDSVD and computes several singular triplets. Numerical experiments justify the theory and illustrate the considerable superiority of IPJDSVD to JDSVD, and demonstrate that a similar two-stage IPJDSVD algorithm substantially outperforms the most advanced PRIMME\_SVDS software nowadays for computing the smallest singular triplets.
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- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/2404.12568
- https://arxiv.org/pdf/2404.12568
- OA Status
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- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4395022015
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https://openalex.org/W4395022015Canonical identifier for this work in OpenAlex
- DOI
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https://doi.org/10.48550/arxiv.2404.12568Digital Object Identifier
- Title
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Preconditioning correction equations in Jacobi--Davidson type methods for computing partial singular value decompositions of large matricesWork title
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preprintOpenAlex work type
- Language
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enPrimary language
- Publication year
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2024Year of publication
- Publication date
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2024-04-19Full publication date if available
- Authors
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Jinzhi Huang, Zhongxiao JiaList of authors in order
- Landing page
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https://arxiv.org/abs/2404.12568Publisher landing page
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https://arxiv.org/pdf/2404.12568Direct link to full text PDF
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YesWhether a free full text is available
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greenOpen access status per OpenAlex
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https://arxiv.org/pdf/2404.12568Direct OA link when available
- Concepts
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Type (biology), Mathematics, Value (mathematics), Applied mathematics, Singular value, Singular value decomposition, Algebra over a field, Pure mathematics, Algorithm, Statistics, Physics, Eigenvalues and eigenvectors, Geology, Paleontology, Quantum mechanicsTop concepts (fields/topics) attached by OpenAlex
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10Other works algorithmically related by OpenAlex
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| abstract_inverted_index.information | 77 |
| abstract_inverted_index.iteratively | 24 |
| abstract_inverted_index.outperforms | 214 |
| abstract_inverted_index.superiority | 200 |
| abstract_inverted_index.PRIMME\_SVDS | 218 |
| abstract_inverted_index.considerable | 199 |
| abstract_inverted_index.decomposition | 9 |
| abstract_inverted_index.substantially | 213 |
| abstract_inverted_index.thick-restart | 164, 184 |
| abstract_inverted_index.preconditioned | 107 |
| abstract_inverted_index.preconditioner | 121 |
| abstract_inverted_index.simultaneously | 174 |
| abstract_inverted_index.preconditioners | 84 |
| abstract_inverted_index.Jacobi--Davidson | 2 |
| abstract_inverted_index.straightforwardly | 129 |
| cited_by_percentile_year | |
| countries_distinct_count | 0 |
| institutions_distinct_count | 2 |
| citation_normalized_percentile |