Supernodal sparse Cholesky factorization on graphics processing units
SUMMARY Sparse Cholesky factorization is the most computationally intensive component in solving large sparse linear systems and is the core algorithm of numerous scientific computing applications. A large number of sparse Cholesky factorization algorithms have previously emerged, exploiting archite...
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| Vydáno v: | Concurrency and computation Ročník 26; číslo 16; s. 2713 - 2726 |
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| Jazyk: | angličtina |
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Blackwell Publishing Ltd
01.11.2014
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| ISSN: | 1532-0626, 1532-0634 |
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| Abstract | SUMMARY
Sparse Cholesky factorization is the most computationally intensive component in solving large sparse linear systems and is the core algorithm of numerous scientific computing applications. A large number of sparse Cholesky factorization algorithms have previously emerged, exploiting architectural features for various computing platforms. The recent use of graphics processing units (GPUs) to accelerate structured parallel applications shows the potential to achieve significant acceleration relative to desktop performance. However, sparse Cholesky factorization has not been explored sufficiently because of the complexity involved in its efficient implementation and the concerns of low GPU utilization.
In this paper, we present a new approach for sparse Cholesky factorization on GPUs. We present the organization of the sparse matrix supernode data structure for GPU and propose a queue‐based approach for the generation and scheduling of GPU tasks with dense linear algebraic operations. We also design a subtree‐based parallel method for multi‐GPU system. These approaches increase GPU utilization, thus resulting in substantial computational time reduction.
Comparisons are made with the existing parallel solvers by using problems arising from practical applications. The experiment results show that the proposed approaches can substantially improve sparse Cholesky factorization performance on GPUs. Relative to a highly optimized parallel algorithm on a 12‐core node, we were able to obtain speedups in the range 1.59× to 2.31× by using one GPU and 1.80× to 3.21× by using two GPUs. Relative to a state‐of‐the‐art solver based on supernodal method for CPU‐GPU heterogeneous platform, we were able to obtain speedups in the range 1.52× to 2.30× by using one GPU and 2.15× to 2.76× by using two GPUs. Concurrency and Computation: Practice and Experience, 2013. Copyright © 2013 John Wiley & Sons, Ltd. |
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| AbstractList | Sparse Cholesky factorization is the most computationally intensive component in solving large sparse linear systems and is the core algorithm of numerous scientific computing applications. A large number of sparse Cholesky factorization algorithms have previously emerged, exploiting architectural features for various computing platforms. The recent use of graphics processing units (GPUs) to accelerate structured parallel applications shows the potential to achieve significant acceleration relative to desktop performance. However, sparse Cholesky factorization has not been explored sufficiently because of the complexity involved in its efficient implementation and the concerns of low GPU utilization. In this paper, we present a new approach for sparse Cholesky factorization on GPUs. We present the organization of the sparse matrix supernode data structure for GPU and propose a queue-based approach for the generation and scheduling of GPU tasks with dense linear algebraic operations. We also design a subtree-based parallel method for multi-GPU system. These approaches increase GPU utilization, thus resulting in substantial computational time reduction. Comparisons are made with the existing parallel solvers by using problems arising from practical applications. The experiment results show that the proposed approaches can substantially improve sparse Cholesky factorization performance on GPUs. Relative to a highly optimized parallel algorithm on a 12-core node, we were able to obtain speedups in the range 1.59 to 2.31 by using one GPU and 1.80 to 3.21 by using two GPUs. Relative to a state-of-the-art solver based on supernodal method for CPU-GPU heterogeneous platform, we were able to obtain speedups in the range 1.52 to 2.30 by using one GPU and 2.15 to 2.76 by using two GPUs. Concurrency and Computation: Practice and Experience, 2013. Copyright copyright 2013 John Wiley & Sons, Ltd. SUMMARY Sparse Cholesky factorization is the most computationally intensive component in solving large sparse linear systems and is the core algorithm of numerous scientific computing applications. A large number of sparse Cholesky factorization algorithms have previously emerged, exploiting architectural features for various computing platforms. The recent use of graphics processing units (GPUs) to accelerate structured parallel applications shows the potential to achieve significant acceleration relative to desktop performance. However, sparse Cholesky factorization has not been explored sufficiently because of the complexity involved in its efficient implementation and the concerns of low GPU utilization. In this paper, we present a new approach for sparse Cholesky factorization on GPUs. We present the organization of the sparse matrix supernode data structure for GPU and propose a queue‐based approach for the generation and scheduling of GPU tasks with dense linear algebraic operations. We also design a subtree‐based parallel method for multi‐GPU system. These approaches increase GPU utilization, thus resulting in substantial computational time reduction. Comparisons are made with the existing parallel solvers by using problems arising from practical applications. The experiment results show that the proposed approaches can substantially improve sparse Cholesky factorization performance on GPUs. Relative to a highly optimized parallel algorithm on a 12‐core node, we were able to obtain speedups in the range 1.59× to 2.31× by using one GPU and 1.80× to 3.21× by using two GPUs. Relative to a state‐of‐the‐art solver based on supernodal method for CPU‐GPU heterogeneous platform, we were able to obtain speedups in the range 1.52× to 2.30× by using one GPU and 2.15× to 2.76× by using two GPUs. Concurrency and Computation: Practice and Experience, 2013. Copyright © 2013 John Wiley & Sons, Ltd. Sparse Cholesky factorization is the most computationally intensive component in solving large sparse linear systems and is the core algorithm of numerous scientific computing applications. A large number of sparse Cholesky factorization algorithms have previously emerged, exploiting architectural features for various computing platforms. The recent use of graphics processing units (GPUs) to accelerate structured parallel applications shows the potential to achieve significant acceleration relative to desktop performance. However, sparse Cholesky factorization has not been explored sufficiently because of the complexity involved in its efficient implementation and the concerns of low GPU utilization. In this paper, we present a new approach for sparse Cholesky factorization on GPUs. We present the organization of the sparse matrix supernode data structure for GPU and propose a queue‐based approach for the generation and scheduling of GPU tasks with dense linear algebraic operations. We also design a subtree‐based parallel method for multi‐GPU system. These approaches increase GPU utilization, thus resulting in substantial computational time reduction. Comparisons are made with the existing parallel solvers by using problems arising from practical applications. The experiment results show that the proposed approaches can substantially improve sparse Cholesky factorization performance on GPUs. Relative to a highly optimized parallel algorithm on a 12‐core node, we were able to obtain speedups in the range 1.59× to 2.31× by using one GPU and 1.80× to 3.21× by using two GPUs. Relative to a state‐of‐the‐art solver based on supernodal method for CPU‐GPU heterogeneous platform, we were able to obtain speedups in the range 1.52× to 2.30× by using one GPU and 2.15× to 2.76× by using two GPUs. Concurrency and Computation: Practice and Experience, 2013. Copyright © 2013 John Wiley & Sons, Ltd. |
| Author | Deng, Lin Zou, Dan Guo, Song Dou, Yong Li, Rongchun |
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| References | Yuen DA, Wang L, Chi X, Johnsson L, Ge W, Shi Y. GPU Solutions to Multi-scale Problems in Science and Engineering. Springer: New York, USA, 2013. Pissanetsky S. Sparse Matrix Technology. Academic Press: New York, 1984. Dongarra JJ, Du Croz J, Hammarling S, Duff IS. A set of level 3 basic linear algebra subprograms. ACM Transactions on Mathematical Software 1990; 16:1-17. Hogg JD, Reid JK, Scott JA. Design of a multi-core sparse Cholesky factorization using DAGs. Proc. SIAM Journal on Scientific Computing 2010; 32(6):3627-3649. Dongarra JJ, Bader DA, Kurzak J. Scientific Computing with Multi-Core and Accelerators. CRC Press Inc: Boca Raton, Florida, USA, 2010. Chen Y, Davis TA, Hager WW, Rajamanickam S. Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Transactions on Mathematical Software 2008; 35(3):1-14. Liu JWH. The role of elimination trees in sparse factorization. SIAM Journal on Matrix Analysis and Applications 11:134-172, 1990. Fred G Gustavson. Two fast algorithms for sparse matrices: multiplication and permuted transposition. ACM Transactions on Mathematical Software 1978; 4:250-269. Duff IS, Reid JK. The multifrontal solution of indefinite sparse symmetric linear. ACM Transactions on Mathematical Software 1983; 9:302-325. Li XS, Demmel JW, Eisenstat SC, Gilbert JR, Liu JWH. A supernodal approach to sparse partial pivoting. SIAM Journal on Matrix Analysis and Applications 1999; 20:720-755. Liu JWH. The multifrontal method for sparse matrix solution: theory and practice. SIAM Review 1992; 34:82-109. JWH Liu, Ng EG, Peyton BW. On finding supernodes for sparse matrix computations. SIAM Journal on Matrix Analysis and Applications 1993; 14:242-252. Lawson, CL, Hanson, RJ, Kincaid, DR, Krogh, FT. Basic linear algebra subprograms for Fortran usage. ACM Transactions on Mathematical Software 1979; 5:308-323. 1993; 14 2010; 32 1990; 11 2012 2011 1990; 16 2010 1978; 4 1979; 5 2008 2007 2006 1984 1983; 9 2008; 35 1999; 20 2003 2013 1992; 34 1999 e_1_2_7_5_1 e_1_2_7_4_1 e_1_2_7_3_1 e_1_2_7_9_1 e_1_2_7_8_1 e_1_2_7_7_1 e_1_2_7_19_1 e_1_2_7_17_1 e_1_2_7_16_1 e_1_2_7_2_1 e_1_2_7_15_1 e_1_2_7_14_1 e_1_2_7_13_1 e_1_2_7_12_1 e_1_2_7_11_1 e_1_2_7_10_1 Dongarra JJ (e_1_2_7_24_1) 2010 Pissanetsky S (e_1_2_7_6_1) 1984 Saule E (e_1_2_7_26_1) 2012 Dongarra JJ (e_1_2_7_18_1) 1990; 16 e_1_2_7_25_1 e_1_2_7_23_1 e_1_2_7_22_1 e_1_2_7_21_1 e_1_2_7_20_1 |
| References_xml | – reference: JWH Liu, Ng EG, Peyton BW. On finding supernodes for sparse matrix computations. SIAM Journal on Matrix Analysis and Applications 1993; 14:242-252. – reference: Hogg JD, Reid JK, Scott JA. Design of a multi-core sparse Cholesky factorization using DAGs. Proc. SIAM Journal on Scientific Computing 2010; 32(6):3627-3649. – reference: Duff IS, Reid JK. The multifrontal solution of indefinite sparse symmetric linear. ACM Transactions on Mathematical Software 1983; 9:302-325. – reference: Liu JWH. The role of elimination trees in sparse factorization. SIAM Journal on Matrix Analysis and Applications 11:134-172, 1990. – reference: Li XS, Demmel JW, Eisenstat SC, Gilbert JR, Liu JWH. A supernodal approach to sparse partial pivoting. SIAM Journal on Matrix Analysis and Applications 1999; 20:720-755. – reference: Dongarra JJ, Bader DA, Kurzak J. Scientific Computing with Multi-Core and Accelerators. CRC Press Inc: Boca Raton, Florida, USA, 2010. – reference: Dongarra JJ, Du Croz J, Hammarling S, Duff IS. A set of level 3 basic linear algebra subprograms. ACM Transactions on Mathematical Software 1990; 16:1-17. – reference: Liu JWH. The multifrontal method for sparse matrix solution: theory and practice. SIAM Review 1992; 34:82-109. – reference: Chen Y, Davis TA, Hager WW, Rajamanickam S. Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate. ACM Transactions on Mathematical Software 2008; 35(3):1-14. – reference: Fred G Gustavson. Two fast algorithms for sparse matrices: multiplication and permuted transposition. ACM Transactions on Mathematical Software 1978; 4:250-269. – reference: Yuen DA, Wang L, Chi X, Johnsson L, Ge W, Shi Y. GPU Solutions to Multi-scale Problems in Science and Engineering. Springer: New York, USA, 2013. – reference: Pissanetsky S. Sparse Matrix Technology. Academic Press: New York, 1984. – reference: Lawson, CL, Hanson, RJ, Kincaid, DR, Krogh, FT. Basic linear algebra subprograms for Fortran usage. ACM Transactions on Mathematical Software 1979; 5:308-323. – volume: 34 start-page: 82 year: 1992 end-page: 109 article-title: The multifrontal method for sparse matrix solution: theory and practice publication-title: SIAM Review – year: 2011 – year: 1984 – volume: 4 start-page: 250 year: 1978 end-page: 269 article-title: Two fast algorithms for sparse matrices: multiplication and permuted transposition publication-title: ACM Transactions on Mathematical Software – start-page: 1629 year: 2012 end-page: 1639 – volume: 35 start-page: 1 issue: 3 year: 2008 end-page: 14 article-title: Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate publication-title: ACM Transactions on Mathematical Software – year: 2008 – year: 2007 – year: 2006 – year: 2003 – volume: 20 start-page: 720 year: 1999 end-page: 755 article-title: A supernodal approach to sparse partial pivoting publication-title: SIAM Journal on Matrix Analysis and Applications – volume: 11 start-page: 134 year: 1990 end-page: 172 article-title: The role of elimination trees in sparse factorization publication-title: SIAM Journal on Matrix Analysis and Applications – volume: 14 start-page: 242 year: 1993 end-page: 252 article-title: On finding supernodes for sparse matrix computations publication-title: SIAM Journal on Matrix Analysis and Applications – volume: 16 start-page: 1 year: 1990 end-page: 17 article-title: A set of level 3 basic linear algebra subprograms publication-title: ACM Transactions on Mathematical Software – volume: 9 start-page: 302 year: 1983 end-page: 325 article-title: The multifrontal solution of indefinite sparse symmetric linear publication-title: ACM Transactions on Mathematical Software – volume: 32 start-page: 3627 issue: 6 year: 2010 end-page: 3649 article-title: Design of a multi‐core sparse Cholesky factorization using DAGs publication-title: Proc. SIAM Journal on Scientific Computing – volume: 5 start-page: 308 year: 1979 end-page: 323 article-title: Basic linear algebra subprograms for Fortran usage publication-title: ACM Transactions on Mathematical Software – year: 2010 – year: 2012 – year: 1999 – year: 2013 – ident: e_1_2_7_20_1 – ident: e_1_2_7_4_1 doi: 10.1137/0611010 – start-page: 1629 volume-title: Proceedings IEEE International Parallel & Distributed Processing Symposium's Workshop on Multithreaded Architectures and Applications year: 2012 ident: e_1_2_7_26_1 – ident: e_1_2_7_15_1 doi: 10.1007/978-3-642-19328-6_9 – ident: e_1_2_7_25_1 doi: 10.1007/978-3-642-16405-7 – ident: e_1_2_7_5_1 doi: 10.1145/355791.355796 – ident: e_1_2_7_17_1 – ident: e_1_2_7_12_1 doi: 10.1137/S0895479895291765 – volume-title: Sparse Matrix Technology year: 1984 ident: e_1_2_7_6_1 – ident: e_1_2_7_23_1 doi: 10.1145/1391989.1391995 – volume-title: Scientific Computing with Multi‐Core and Accelerators year: 2010 ident: e_1_2_7_24_1 – ident: e_1_2_7_8_1 – ident: e_1_2_7_9_1 doi: 10.1145/356044.356047 – ident: e_1_2_7_16_1 – ident: e_1_2_7_2_1 – ident: e_1_2_7_10_1 doi: 10.1137/1034004 – ident: e_1_2_7_13_1 – ident: e_1_2_7_21_1 – volume: 16 start-page: 1 year: 1990 ident: e_1_2_7_18_1 article-title: A set of level 3 basic linear algebra subprograms publication-title: ACM Transactions on Mathematical Software doi: 10.1145/77626.79170 – ident: e_1_2_7_11_1 doi: 10.1137/0614019 – ident: e_1_2_7_14_1 doi: 10.1109/IPDPS.2011.44 – ident: e_1_2_7_19_1 – ident: e_1_2_7_7_1 doi: 10.1145/355841.355847 – ident: e_1_2_7_22_1 doi: 10.1137/090757216 – ident: e_1_2_7_3_1 doi: 10.1137/1.9780898718881 |
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Sparse Cholesky factorization is the most computationally intensive component in solving large sparse linear systems and is the core algorithm of... Sparse Cholesky factorization is the most computationally intensive component in solving large sparse linear systems and is the core algorithm of numerous... |
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| SubjectTerms | Algorithms Cholesky factorization Computation Concurrency GPU Graphics processing units Platforms Solvers sparse Cholesky factorization supernodal method Utilization |
| Title | Supernodal sparse Cholesky factorization on graphics processing units |
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