SlabLU: a two-level sparse direct solver for elliptic PDEs
The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The sche...
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| Vydáno v: | Advances in computational mathematics Ročník 50; číslo 4; s. 90 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Springer US
01.08.2024
Springer Nature B.V Springer |
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| ISSN: | 1019-7168, 1572-9044 |
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| Abstract | The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of
H
2
-matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of
H
or
H
2
matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses
H
2
-matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multidomain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size
1000
λ
×
1000
λ
(for which
N
=
100
M
) is solved in 15 min to 6 correct digits on a high-powered desktop with GPU acceleration. |
|---|---|
| AbstractList | The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of
H
2
-matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of
H
or
H
2
matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses
H
2
-matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multidomain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size
1000
λ
×
1000
λ
(for which
N
=
100
M
) is solved in 15 min to 6 correct digits on a high-powered desktop with GPU acceleration. Not provided. The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme decomposes the domain into thin subdomains, or “slabs” and uses a two-level approach that is designed with parallelization in mind. The scheme takes advantage of H2-matrix structure emerging during factorization and utilizes randomized algorithms to efficiently recover this structure. As opposed to multi-level nested dissection schemes that incorporate the use of H or H2 matrices for a hierarchy of front sizes, SlabLU is a two-level scheme which only uses H2-matrix algebra for fronts of roughly the same size. The simplicity allows the scheme to be easily tuned for performance on modern architectures and GPUs. The solver described is compatible with a range of different local discretizations, and numerical experiments demonstrate its performance for regular discretizations of rectangular and curved geometries. The technique becomes particularly efficient when combined with very high-order accurate multidomain spectral collocation schemes. With this discretization, a Helmholtz problem on a domain of size 1000λ×1000λ (for which N=100M) is solved in 15 min to 6 correct digits on a high-powered desktop with GPU acceleration. |
| ArticleNumber | 90 |
| Author | Yesypenko, Anna Martinsson, Per-Gunnar |
| Author_xml | – sequence: 1 givenname: Anna orcidid: 0009-0008-1409-4075 surname: Yesypenko fullname: Yesypenko, Anna email: annayesy@utexas.edu organization: Oden Institute, University of Texas at Austin – sequence: 2 givenname: Per-Gunnar orcidid: 0000-0002-1048-5270 surname: Martinsson fullname: Martinsson, Per-Gunnar organization: Oden Institute, University of Texas at Austin |
| BackLink | https://www.osti.gov/biblio/2576229$$D View this record in Osti.gov |
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| Keywords | 65N35 Direct solver Sparse direct solver Randomized linear algebra Helmholtz equation High-order discretization GPU 65N22 65F05 Multifrontal solver |
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| Snippet | The paper describes a sparse direct solver for the linear systems that arise from the discretization of an elliptic PDE on a two-dimensional domain. The scheme... Not provided. |
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| SubjectTerms | Computational mathematics Computational Mathematics and Numerical Analysis Computational Science and Engineering Discretization Domains Elliptic differential equations Graphics processing units Linear algebra Linear systems Mathematical and Computational Biology Mathematical Modeling and Industrial Mathematics Mathematics Mathematics and Statistics Matrix algebra Parabolic differential equations Partial differential equations Solvers Sparsity Visualization |
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| Title | SlabLU: a two-level sparse direct solver for elliptic PDEs |
| URI | https://link.springer.com/article/10.1007/s10444-024-10176-x https://www.proquest.com/docview/3254225525 https://www.osti.gov/biblio/2576229 |
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