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
Hlavní autor: Sidje, Roger B.
Médium: Journal Article
Jazyk:angličtina
Vydáno: 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.
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.
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References_xml – reference: H. P. Flatt and K. Kennedy Performance of parallel processors. Parallel Comput., 12, 1-20, 1989.
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– 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.
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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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