Alternatives for parallel Krylov subspace basis computation
Numerical methods related to Krylov subspaces are widely used in large sparse numerical linear algebra. Vectors in these subspaces are manipulated via their representation onto orthonormal bases. Nowadays, on serial computers, the method of Arnoldi is considered as a reliable technique for construct...
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| Vydáno v: | Numerical linear algebra with applications Ročník 4; číslo 4; s. 305 - 331 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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West Sussex
John Wiley & Sons, Ltd
01.07.1997
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| ISSN: | 1070-5325, 1099-1506 |
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| Abstract | Numerical methods related to Krylov subspaces are widely used in large sparse numerical linear algebra. Vectors in these subspaces are manipulated via their representation onto orthonormal bases. Nowadays, on serial computers, the method of Arnoldi is considered as a reliable technique for constructing such bases. However, although easily parallelizable, this technique is not as scalable as expected for communications. In this work we examine alternative methods aimed at overcoming this drawback. Since they retrieve upon completion the same information as Arnoldi's algorithm does, they enable us to design a wide family of stable and scalable Krylov approximation methods for various parallel environments. We present timing results obtained from their implementation on two distributed‐memory multiprocessor supercomputers: the Intel Paragon and the IBM Scalable POWERparallel SP2. © 1997 John Wiley & Sons, Ltd. |
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| AbstractList | Numerical methods related to Krylov subspaces are widely used in large sparse numerical linear algebra. Vectors in these subspaces are manipulated via their representation onto orthonormal bases. Nowadays, on serial computers, the method of Arnoldi is considered as a reliable technique for constructing such bases. However, although easily parallelizable, this technique is not as scalable as expected for communications. In this work we examine alternative methods aimed at overcoming this drawback. Since they retrieve upon completion the same information as Arnoldi's algorithm does, they enable us to design a wide family of stable and scalable Krylov approximation methods for various parallel environments. We present timing results obtained from their implementation on two distributed‐memory multiprocessor supercomputers: the Intel Paragon and the IBM Scalable POWERparallel SP2. © 1997 John Wiley & Sons, Ltd. |
| Author | Sidje, Roger B. |
| Author_xml | – sequence: 1 givenname: Roger B. surname: Sidje fullname: Sidje, Roger B. organization: Department of Mathematics, University of Queensland, Brisbane QLD 4072, Australia |
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| Cites_doi | 10.1137/0729014 10.1137/0910067 10.1137/0913071 10.1142/S0129183193000082 10.1137/0915030 10.1016/0167-8191(86)90004-9 10.1016/0167-8191(89)90003-3 10.1007/BF01462266 10.1016/0010-4655(85)90148-1 10.1002/nla.1680010605 10.1093/imanum/14.4.563 10.1145/62038.62043 10.1137/0911057 10.1137/0912056 10.1016/0024-3795(94)90493-6 10.1016/0167-8191(93)90076-W 10.1007/BF02017352 10.1016/0167-8191(89)90063-X 10.1016/0167-8191(90)90163-4 |
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Parallel Comput, 16, 99-112, 1990. S. Petiton, Y. Saad, K. Wu and Ferng W. Basic sparse matrix computations on the CM-5. International Journal of Modern Physics, 4 (1), 65-83, 1994. E. Chu and A. George QR factorization of a dense matrix on a shared-memory multiprocessor. Parallel. Comput., 11, 55-71, 1989. A. H. Sameh On some parallel algorithms on a ring of processors. Comput. Phys. Comm., 37, 159-166, 1985. G. W. Stewart The economical storage of plane rotations. Numer. Math., 25, 137-138, 1976. I. S. Duff, R. G. Grimes and J. G. Lewis Sparse matrix test problems. ACM Trans. Math. Softw., 15, 1-14, 1989. G. H. Golub and C. F. Van Loan Matrix Computations, second edition. The Johns Hopkins University Press, Baltimore, 1989. A. Pothen and P. Raghavan Distributed orthogonal factorization: Givens and Householder algorithms. SIAM J. Sci. Stat. Comput., 10, 1113-1135, 1989. Y. Saad Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal, 29 (1), 208-227, 1992. E. Gallopoulos and Y. Saad Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. Comput., 13 (5), 1236-1264, 1992. E. Chu and A. George QR factorization of a dense matrix on a hypercube multiprocessor. SIAM J. Sci. Stat. Comput, 11, 990-1028, 1990. J. N. Shadid and R. S. Tuminaro A comparison of preconditioned nonsymmetric Krylov methods on a large-scale MIMD machine. SIAM J. Sci. Stat. Comput, 15 (2), 440-459, 1994. J. Zhu QR factorization for the regularized least squares problem on hypercubes. Parallel Comput, 19, 939-948, 1993. 1990; 30 1990; 11 1976; 25 1991; 12 1990; 16 1994; 197 1994 1992; 13 1993 1992 1991 1995; 3 1989; 11 1993; 19 1989; 12 1989; 10 1986; 3 1986 1992; 29 1994; 14 1985 1994; 15 1989; 15 1994; 1 1985; 37 1994; 4 1989 Duff (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB11) 1989; 15 Sameh (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB26) 1985; 37 Bjorck (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB4) 1994; 197 Erhel (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB12) 1995; 3 Bai (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB1) 1992 Hindmarsh (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB16) 1986 De Sturler (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB9) 1994 Chu (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB8) 1990; 11 Golub (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB15) 1989 Pothen (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB23) 1989; 10 Gallopoulos (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB14) 1992; 13 Reichel (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB24) 1990; 30 Dongarra (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB10) 1986; 3 Philippe (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB22) 1995; 3 Petiton (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB21) 1994; 4 Sameh (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB27) 1985 Barnard (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB3) 1992 Sidje (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB29) 1994 Calvetti (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB6) 1993 O'Leary (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB20) 1990; 16 Flatt (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB13) 1989; 12 Jalby (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB17) 1991; 12 Saad (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB25) 1992; 29 Stewart (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB31) 1976; 25 Zhu (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB34) 1993; 19 Bodin (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB5) 1993 Shadid (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB28) 1994; 15 Walker (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB33) 1994; 1 Kim (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB18) 1991 Sidje (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB30) 1994 Temam (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB32) 1992 Bai (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB2) 1994; 14 Chu (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB7) 1989; 11 Lewis (10.1002/(SICI)1099-1506(199707/08)4:4<305::AID-NLA104>3.0.CO;2-D-BIB19) 1993 |
| References_xml | – reference: H. P. Flatt and K. Kennedy Performance of parallel processors. Parallel Comput., 12, 1-20, 1989. – reference: J. Erhel A parallel GMRES version for general sparse matrices. ETNA, 3, 160-176, 1995. – reference: A. H. Sameh On some parallel algorithms on a ring of processors. Comput. Phys. Comm., 37, 159-166, 1985. – reference: H. Walker and L. Zhou A simpler GMRES. Num. Lin. Alg. Appl., 1 (16), 571-581, 1994. – reference: G. W. Stewart The economical storage of plane rotations. Numer. Math., 25, 137-138, 1976. – reference: W. Jalby and B. Philippe Stability analysis and improvement of the block Gram-Schmidt algorithm. SIAM J. Sci. Stat. Comput, 12, 1058-1073, 1991. – reference: D. P. O'Leary and P. Whitman Parallel QR factorization by Householder and modified Gram- Schmidt algorithms. Parallel Comput, 16, 99-112, 1990. – reference: J. J. Dongarra, A. H. Sameh and D. C. Sorensen Implementation of some concurrent algorithms for matrix factorization. Parallel Comput., 3, 25-34, 1986. – reference: Y. Saad Analysis of some Krylov subspace approximations to the matrix exponential operator. SIAM J. Numer. Anal, 29 (1), 208-227, 1992. – reference: J. Zhu QR factorization for the regularized least squares problem on hypercubes. Parallel Comput, 19, 939-948, 1993. – reference: G. H. Golub and C. F. Van Loan Matrix Computations, second edition. The Johns Hopkins University Press, Baltimore, 1989. – reference: E. Chu and A. George QR factorization of a dense matrix on a hypercube multiprocessor. SIAM J. Sci. Stat. Comput, 11, 990-1028, 1990. – reference: S. Petiton, Y. Saad, K. Wu and Ferng W. Basic sparse matrix computations on the CM-5. International Journal of Modern Physics, 4 (1), 65-83, 1994. – reference: L. Reichel Newton interpolation at Leja points. BIT, 30, 332-346, 1990. – reference: A. Pothen and P. Raghavan Distributed orthogonal factorization: Givens and Householder algorithms. SIAM J. Sci. Stat. Comput., 10, 1113-1135, 1989. – reference: Z. Bai, D. Hu and L. Reichel A Newton basis GMRES implementation. IMA J. Numer. Anal., 14, 563-581, 1994. – reference: I. S. Duff, R. G. Grimes and J. G. Lewis Sparse matrix test problems. ACM Trans. Math. Softw., 15, 1-14, 1989. – reference: E. Gallopoulos and Y. Saad Efficient solution of parabolic equations by Krylov approximation methods. SIAM J. Sci. Stat. Comput., 13 (5), 1236-1264, 1992. – reference: E. Chu and A. George QR factorization of a dense matrix on a shared-memory multiprocessor. Parallel. Comput., 11, 55-71, 1989. – reference: Å. Bjorck Numerics of Gram-Schmidt orthogonalization. Linear Algebra and its Applications, 197, 297-316, 1994. – reference: J. N. Shadid and R. S. Tuminaro A comparison of preconditioned nonsymmetric Krylov methods on a large-scale MIMD machine. SIAM J. Sci. Stat. Comput, 15 (2), 440-459, 1994. – volume: 3 start-page: 156 year: 1995 end-page: 165 – volume: 37 start-page: 159 year: 1985 end-page: 166 article-title: On some parallel algorithms on a ring of processors publication-title: Comput. Phys. Comm. – start-page: 191 year: 1985 end-page: 200 – volume: 19 start-page: 939 year: 1993 end-page: 948 article-title: QR factorization for the regularized least squares problem on hypercubes publication-title: Parallel Comput – volume: 4 start-page: 65 issue: 1 year: 1994 end-page: 83 article-title: Basic sparse matrix computations on the CM‐5 publication-title: International Journal of Modern Physics – volume: 3 start-page: 160 year: 1995 end-page: 176 article-title: A parallel GMRES version for general sparse matrices publication-title: ETNA – year: 1989 – volume: 25 start-page: 137 year: 1976 end-page: 138 article-title: The economical storage of plane rotations publication-title: Numer. Math. – volume: 10 start-page: 1113 year: 1989 end-page: 1135 article-title: Distributed orthogonal factorization: Givens and Householder algorithms publication-title: SIAM J. Sci. Stat. Comput. – volume: 1 start-page: 571 issue: 16 year: 1994 end-page: 581 article-title: A simpler GMRES publication-title: Num. Lin. Alg. Appl. – volume: 15 start-page: 1 year: 1989 end-page: 14 article-title: Sparse matrix test problems publication-title: ACM Trans. Math. Softw. – volume: 12 start-page: 1 year: 1989 end-page: 20 article-title: Performance of parallel processors publication-title: Parallel Comput. – volume: 29 start-page: 208 issue: 1 year: 1992 end-page: 227 article-title: Analysis of some Krylov subspace approximations to the matrix exponential operator publication-title: SIAM J. Numer. Anal – volume: 3 start-page: 25 year: 1986 end-page: 34 article-title: Implementation of some concurrent algorithms for matrix factorization publication-title: Parallel Comput. – year: 1992 – year: 1994 – start-page: 484 year: 1993 end-page: 492 – volume: 13 start-page: 1236 issue: 5 year: 1992 end-page: 1264 article-title: Efficient solution of parabolic equations by Krylov approximation methods publication-title: SIAM J. Sci. Stat. Comput. – year: 1986 – start-page: 167 year: 1991 end-page: 170 – volume: 30 start-page: 332 year: 1990 end-page: 346 article-title: Newton interpolation at Leja points publication-title: BIT – volume: 11 start-page: 55 year: 1989 end-page: 71 article-title: QR factorization of a dense matrix on a shared‐memory multiprocessor publication-title: Parallel. Comput. – volume: 16 start-page: 99 year: 1990 end-page: 112 article-title: Parallel QR factorization by Householder and modified Gram‐ Schmidt algorithms publication-title: Parallel Comput – volume: 14 start-page: 563 year: 1994 end-page: 581 article-title: A Newton basis GMRES implementation publication-title: IMA J. Numer. Anal. – volume: 11 start-page: 990 year: 1990 end-page: 1028 article-title: QR factorization of a dense matrix on a hypercube multiprocessor publication-title: SIAM J. Sci. Stat. Comput – year: 1993 – start-page: 31 year: 1993 end-page: 46 – volume: 197 start-page: 297 year: 1994 end-page: 316 article-title: Numerics of Gram‐Schmidt orthogonalization publication-title: Linear Algebra and its Applications – volume: 12 start-page: 1058 year: 1991 end-page: 1073 article-title: Stability analysis and improvement of the block Gram‐Schmidt algorithm publication-title: SIAM J. Sci. Stat. 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| Snippet | Numerical methods related to Krylov subspaces are widely used in large sparse numerical linear algebra. Vectors in these subspaces are manipulated via their... |
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| SubjectTerms | IBM SP2 Intel Paragon Krylov subspace Leja points parallel QR factorization sparse matrix |
| Title | Alternatives for parallel Krylov subspace basis computation |
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