Abstract
Massively parallel SIMD computers, in principle, should be good platforms for performing direct factorization of large, sparse matrices. However, the high arithmetic speed of these machines can easily be overcome by overhead in intra- and inter-processor data motion. Furthermore, load balancing is difficult for an “unstructured” sparsity pattern that cannot be dissected conveniently into equal-size domains. Nevertheless, some progress has been made recently in LU and QR factorization of unstructured sparse matrices, using some familiar concepts from vector-supercomputer implementations (elimination trees, supernodes, etc.) and some new ideas for distributing the computations across many processors. This paper describes programs based on the standard data-parallel computing model, as well as those using a SIMD machine to implement a dataflow paradigm
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References
O. Mcbryan, The Connection Machine: PDE Solution on 65,536 Processors, Thinking Machines Corp. Technical Report CS86–1, 1986.
A. Dave AND I. Duff, Sparse Matrix Calculations on the Cray-2, Parallel Comput., 5 (1987), pp. 55–64.
C. Yang, A Vector/parallel Implementation of the Multifrontal Method for Sparse Symmetric Positive Definite Linear Systems on the Cray Y/MP, Cray Research Inc. Technical Report 1990.
E. Rotherberg AND A. Gupta, Techniques for Improving the Performance of Sparse Matrix Factorization on Multiprocessor Workstations, Stanford Univ. Report CSL-TR-90–430, 1990.
A. George, M. Heath AND J. Liu, Parallel Cholesky Factorization on a Shared-Memory Multiprocessor, Lin. Alg. Appl., 77 (1986), pp. 165–187.
R. Lucas, W. Blank AND J. Tieman, A Parallel Solution Method for Large Sparse Systems of Equations, IEEE Trans. Computer Aided Design, CAD-6 (1987), pp. 981–991.
P. Worley AND R. Schreiber, Nested Dissection on a Mesh-Connected Processor Array, in New Computing Environment: Parallel Vector and Systolic, ed. by A. Wouk, SIAM, 1986.
J. Liu, The Role of Elimination Trees in Sparse Factorization, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 134–172.
R. Schreiber, A New Implementation of Sparse Gaussian Elimination, ACM Trans. Math. Software, 8 (1982) pp. 256–276.
A. George AND M. Heath, Solution of Sparse Linear Least Squares Problems Using Givens Rotations, Lin. Alg. Appl., 34 (1980), pp. 69–83.
J. Liu, On General Row Merging Schemes for Sparse Givens Transformations, SIAM J. Sci. Stat. Comp., 7 (1986), pp. 1190–1211.
A. George AND J. Liu, Householder Reflections versus Givens Rotations in Sparse Orthogonal Decomposition, Lin. Alg. Appl., 88 (1987), pp. 223–238.
J. Gilbert AND R. Schreiber, Highly Parallel Sparse Cholesky Factorization, SIAM J. Scientific and Statistical Computing, 13 (1992) pp. 1151–1172.
S. Kratzer, Sparse LU Factorization on Massively Parallel SIMD Computers, Technical Report SRC-TR-92–072, Supercomputing Research Center, April, 1992.
S. Kratzer, Massively Parallel Sparse Matrix Computations, Technical Report SRC-TR-90–008, Supercomputing Research Center, February, 1990.
M. Heath, E. Ng AND B. Peyton, Parallel Algorithms for Sparse Linear Systems, SIAM Review, 33 (1991), pp. 420–460.
C. Ashcraft, S. Eisenstat, J. Liu, AND A. Sherman, A Comparison of Three Column-based Distributed Sparse Factorization Schemes, Technical Report, Dept. of Computer Science, York Univ., 1990.
A. Cleary, A Comparison of Algorithms for Cholesky Factorization on a Massively Parallel MIMD Computer, Proc. 5th SIAM Conf. on Parallel Processing, March, 1991.
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© 1993 Springer-Verlag New York, Inc.
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Kratzer, S.G., Cleary, A.J. (1993). Sparse Matrix Factorization on SIMD Parallel Computers. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds) Graph Theory and Sparse Matrix Computation. The IMA Volumes in Mathematics and its Applications, vol 56. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-8369-7_10
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DOI: https://doi.org/10.1007/978-1-4613-8369-7_10
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