Abstract
We discuss the solution of large sparse systems using Gaussian elimination on both local and shared memory parallel computers.
There is a natural parallelism to Gaussian elimination that has been frequently exploited. We can take advantage of this parallelism in addition to that provided by the sparsity itself. We discuss this latter parallelism in some detail.
We discuss an approach that exploits the parallelism due to the sparsity and that can automatically benefit also from the parallelism of Gaussian elimination. This approach, which is applicable to quite general systems, is based on a multifrontal technique.
We look at the implementation of the multifrontal approach on shared memory machines and discuss its implementation on a hypercube.
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Bhatt, S.N. and Ipsen, I.C.F. (1985). How to embed trees in hypercubes. Report YALEU/DCS/RR-443, Department of Computer Science, Yale University, Connecticut.
Duff, I.S. (1985). Parallel implementation of multifrontal schemes. Report CSS 174, Computer Science and Systems Division, AERE Harwell. Parallel Computing (To appear).
Duff, I.S. and Johnsson, S.L. (1986). Node orderings and concurrency in sparse problems: an experimental investigation. Proceedings International Conference on Vector and Parallel Computing, Loen, Norway, June 2–6, 1986. (To appear).
Duff, I. S. and Reid, J. K. (1983). The multifrontal solution of indefinite sparse symmetric linear systems. ACM Trans. Math. Softw. 9, 302–325.
Duff, I.S., Erisman, A.M., and Reid, J.K. (1986). Direct methods for sparse matrices. Oxford University Press, London.
Geist, G.A. (1985). Efficient parallel LU factorization with pivoting on a hypercube processor. Report ORNL-6211, Engineering Physics and Mathematics Division, Oak Ridge National Laboratory, Tennessee.
Geist, G.A. and Heath, M.T. (1985). Parallel Cholesky factorization on a hypercube multiprocessor. Report ORNL-6190, Engineering Physics and Mathematics Division, Oak Ridge National Laboratory, Tennessee.
George, A. (1973). Nested dissection of a regular finite-element mesh. SIAM J. Numer. Anal. 10, 345–363.
George, A., Heath, M., Liu, J., and Ng, E. (1986a). Sparse Cholesky factorization on a local-memory multiprocessor. Report CS-86-01. Department of Computer Science, York University, Ontario, Canada.
George, A., Heath, M., Ng, E., and Liu, J. (1986b). Symbolic Cholesky factorization on a local-memory multiprocessor. Proceedings International Conference on Vector and Parallel Computing, Loen, Norway, June 2–6, 1986. Parallel Computing (To appear).
Heller, D. (1978). A survey of parallel algorithms in numerical linear algebra. SIAM Review 20, 740–777.
Kung, H., Sproull, R., and Steele, G. (Eds.) (1981). VLSI systems and computations. Computer Science Press, Rockville, Maryland.
Kung, S.-Y., Arun, K., Bhuskerio, D., and Ho, Y. (1981). A matrix data flow language/architecture for parallel matrix operations based on computational wave concept. In Kung, Sproull, and Steele (1981).
Liu, J.W.H. (1985). Computational models and task scheduling for parallel sparse Cholesky factorization. Report CS-85-01. Department of Computer Science, York University, Ontario, Canada.
O'Leary, D.P. and Stewart, G.W. (1985). Data-flow algorithms for parallel matrix computations. Communications ACM 28, 620–632.
Saad, Y. (1986). Gaussian elimination on hypercubes. Report YALEU/DCS/RR-462, Department of Computer Science, Yale University, Connecticut.
Sameh, A.H. (1983). An overview of parallel algorithms in numerical linear algebra. Bulletin de la Direction des Etudes et Recherches, EDF, France. Serie C., 129–134.
Tinney, W.F. and Walker, J.W. (1967). Direct solutions of sparse network equations by optimally ordered triangular factorization. Proc. IEEE 55, 1801–1809.
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Duff, I.S. (1986). The parallel solution of sparse linear equations. In: Händler, W., Haupt, D., Jeltsch, R., Juling, W., Lange, O. (eds) CONPAR 86. CONPAR 1986. Lecture Notes in Computer Science, vol 237. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-16811-7_149
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DOI: https://doi.org/10.1007/3-540-16811-7_149
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