Distance-two interpolation for parallel algebraic multigrid
Algebraic multigrid (AMG) is one of the most efficient and scalable parallel algorithms for solving sparse linear systems on unstructured grids. However, for large 3D problems, the coarse grids that are normally used in AMG often lead to growing complexity in terms of memory use and execution time p...
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| Veröffentlicht in: | Numerical linear algebra with applications Jg. 15; H. 2-3; S. 115 - 139 |
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Chichester, UK
John Wiley & Sons, Ltd
01.03.2008
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| Abstract | Algebraic multigrid (AMG) is one of the most efficient and scalable parallel algorithms for solving sparse linear systems on unstructured grids. However, for large 3D problems, the coarse grids that are normally used in AMG often lead to growing complexity in terms of memory use and execution time per AMG V‐cycle. Sparser coarse grids, such as those obtained by the parallel modified independent set (PMIS) coarsening algorithm, remedy this complexity growth but lead to nonscalable AMG convergence factors when traditional distance‐one interpolation methods are used. In this paper, we study the scalability of AMG methods that combine PMIS coarse grids with long‐distance interpolation methods. AMG performance and scalability are compared for previously introduced interpolation methods as well as new variants of them for a variety of relevant test problems on parallel computers. It is shown that the increased interpolation accuracy largely restores the scalability of AMG convergence factors for PMIS‐coarsened grids, and in combination with complexity reducing methods, such as interpolation truncation, one obtains a class of parallel AMG methods that enjoy excellent scalability properties on large parallel computers. Copyright © 2007 John Wiley & Sons, Ltd. |
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| AbstractList | Algebraic multigrid (AMG) is one of the most efficient and scalable parallel algorithms for solving sparse linear systems on unstructured grids. However, for large 3D problems, the coarse grids that are normally used in AMG often lead to growing complexity in terms of memory use and execution time per AMG V‐cycle. Sparser coarse grids, such as those obtained by the parallel modified independent set (PMIS) coarsening algorithm, remedy this complexity growth but lead to nonscalable AMG convergence factors when traditional distance‐one interpolation methods are used. In this paper, we study the scalability of AMG methods that combine PMIS coarse grids with long‐distance interpolation methods. AMG performance and scalability are compared for previously introduced interpolation methods as well as new variants of them for a variety of relevant test problems on parallel computers. It is shown that the increased interpolation accuracy largely restores the scalability of AMG convergence factors for PMIS‐coarsened grids, and in combination with complexity reducing methods, such as interpolation truncation, one obtains a class of parallel AMG methods that enjoy excellent scalability properties on large parallel computers. Copyright © 2007 John Wiley & Sons, Ltd. Algebraic multigrid (AMG) is one of the most efficient and scalable parallel algorithms for solving sparse linear systems on unstructured grids. However, for large 3D problems, the coarse grids that are normally used in AMG often lead to growing complexity in terms of memory use and execution time per AMG V-cycle. Sparser coarse grids, such as those obtained by the parallel modified independent set (PMIS) coarsening algorithm, remedy this complexity growth but lead to nonscalable AMG convergence factors when traditional distance-one interpolation methods are used. In this paper, we study the scalability of AMG methods that combine PMIS coarse grids with long-distance interpolation methods. AMG performance and scalability are compared for previously introduced interpolation methods as well as new variants of them for a variety of relevant test problems on parallel computers. It is shown that the increased interpolation accuracy largely restores the scalability of AMG convergence factors for PMIS-coarsened grids, and in combination with complexity reducing methods, such as interpolation truncation, one obtains a class of parallel AMG methods that enjoy excellent scalability properties on large parallel computers. |
| Author | De Sterck, Hans Yang, Ulrike Meier Nolting, Joshua W. Falgout, Robert D. |
| Author_xml | – sequence: 1 givenname: Hans surname: De Sterck fullname: De Sterck, Hans organization: Department of Applied Mathematics, University of Waterloo, Waterloo, Ont., Canada N2L 3G1 – sequence: 2 givenname: Robert D. surname: Falgout fullname: Falgout, Robert D. organization: Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, U.S.A – sequence: 3 givenname: Joshua W. surname: Nolting fullname: Nolting, Joshua W. organization: Department of Applied Mathematics, University of Colorado at Boulder, Campus Box 526, Boulder, CO 80302, U.S.A – sequence: 4 givenname: Ulrike Meier surname: Yang fullname: Yang, Ulrike Meier email: umyang@llnl.gov organization: Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94551, U.S.A |
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| Cites_doi | 10.1007/BFb0018531 10.1137/1.9780898719505 10.1137/040615729 10.1016/S0168-9274(01)00115-5 10.1145/1089014.1089018 10.1137/0215074 10.1137/1.9781611971057.ch4 10.1016/j.parco.2006.06.009 10.1137/S1064827598339402 |
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| References | De Sterck H, Yang UM, Heys JJ. Reducing complexity in parallel algebraic multigrid preconditioners. SIAM Journal on Matrix Analysis and Applications 2006; 27:1019-1039. Luby M. A simple parallel algorithm for the maximal independent set problem. SIAM Journal on Computing 1986; 15:1036-1053. Cleary AJ, Falgout RD, Henson VE, Jones JE, Manteuffel TA, McCormick SF, Miranda GN, Ruge JW. Robustness and scalability of algebraic multigrid. SIAM Journal on Scientific Computing 2000; 21:1886-1908. Baker A, Falgout RD, Yang UM. An assumed partition algorithm for determining processor inter-communication. Parallel Computing 2006; 32:394-414. Henson VE, Yang UM. BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Applied Numerical Mathematics 2002; 41:155-177. Briggs WL, Henson VE, McCormick SF. A Multigrid Tutorial (2nd edn). SIAM: Philadelphia, PA, 2000. Falgout RD, Jones JE, Yang UM. Pursuing scalability for hypre's conceptual interfaces. ACM Transactions on Mathematical Software 2005; 31:326-350. 1987 1998 2005; 31 1984 2006 2002; 41 2000 2006; 32 2000; 21 2006; 27 1986; 15 e_1_2_1_6_2 e_1_2_1_7_2 e_1_2_1_5_2 e_1_2_1_11_2 e_1_2_1_3_2 e_1_2_1_12_2 e_1_2_1_10_2 Brandt A (e_1_2_1_2_2) 1984 e_1_2_1_13_2 Stüben K (e_1_2_1_4_2) 2000 e_1_2_1_8_2 e_1_2_1_9_2 |
| References_xml | – reference: Luby M. A simple parallel algorithm for the maximal independent set problem. SIAM Journal on Computing 1986; 15:1036-1053. – reference: Cleary AJ, Falgout RD, Henson VE, Jones JE, Manteuffel TA, McCormick SF, Miranda GN, Ruge JW. Robustness and scalability of algebraic multigrid. SIAM Journal on Scientific Computing 2000; 21:1886-1908. – reference: Falgout RD, Jones JE, Yang UM. Pursuing scalability for hypre's conceptual interfaces. ACM Transactions on Mathematical Software 2005; 31:326-350. – reference: Baker A, Falgout RD, Yang UM. An assumed partition algorithm for determining processor inter-communication. Parallel Computing 2006; 32:394-414. – reference: De Sterck H, Yang UM, Heys JJ. Reducing complexity in parallel algebraic multigrid preconditioners. SIAM Journal on Matrix Analysis and Applications 2006; 27:1019-1039. – reference: Briggs WL, Henson VE, McCormick SF. A Multigrid Tutorial (2nd edn). SIAM: Philadelphia, PA, 2000. – reference: Henson VE, Yang UM. BoomerAMG: a parallel algebraic multigrid solver and preconditioner. Applied Numerical Mathematics 2002; 41:155-177. – year: 2000 – year: 1984 – volume: 27 start-page: 1019 year: 2006 end-page: 1039 article-title: Reducing complexity in parallel algebraic multigrid preconditioners publication-title: SIAM Journal on Matrix Analysis and Applications – start-page: 73 year: 1987 end-page: 130 – volume: 41 start-page: 155 year: 2002 end-page: 177 article-title: BoomerAMG: a parallel algebraic multigrid solver and preconditioner publication-title: Applied Numerical Mathematics – volume: 32 start-page: 394 year: 2006 end-page: 414 article-title: An assumed partition algorithm for determining processor inter‐communication publication-title: Parallel Computing – volume: 21 start-page: 1886 year: 2000 end-page: 1908 article-title: Robustness and scalability of algebraic multigrid publication-title: SIAM Journal on Scientific Computing – volume: 15 start-page: 1036 year: 1986 end-page: 1053 article-title: A simple parallel algorithm for the maximal independent set problem publication-title: SIAM Journal on Computing – year: 2006 – year: 1998 – volume: 31 start-page: 326 year: 2005 end-page: 350 article-title: Pursuing scalability for hypre's conceptual interfaces publication-title: ACM Transactions on Mathematical Software – volume-title: Multigrid year: 2000 ident: e_1_2_1_4_2 – ident: e_1_2_1_13_2 doi: 10.1007/BFb0018531 – ident: e_1_2_1_8_2 doi: 10.1137/1.9780898719505 – ident: e_1_2_1_6_2 doi: 10.1137/040615729 – ident: e_1_2_1_9_2 doi: 10.1016/S0168-9274(01)00115-5 – ident: e_1_2_1_11_2 doi: 10.1145/1089014.1089018 – volume-title: Sparsity and its Applications year: 1984 ident: e_1_2_1_2_2 – ident: e_1_2_1_7_2 doi: 10.1137/0215074 – ident: e_1_2_1_3_2 doi: 10.1137/1.9781611971057.ch4 – ident: e_1_2_1_10_2 – ident: e_1_2_1_12_2 doi: 10.1016/j.parco.2006.06.009 – ident: e_1_2_1_5_2 doi: 10.1137/S1064827598339402 |
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| Title | Distance-two interpolation for parallel algebraic multigrid |
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