Abstract
In this paper we survey a recent approach for solving sparse triangular systems of equations on highly parallel computers. This approach employs a partitioned representation of the inverse of the triangular matrix so that the solution can be computed by matrix-vector multiplication. The number of factors in the partitioned inverse is proportional to the number of general communication steps (router steps on a CM-2) required in a highly parallel algorithm. We describe partitioning algorithms that minimize the number of factors in the partitioned inverse over all symmetric permutations of the triangular matrix such that the permuted matrix continues to be triangular. For a Cholesky factor we describe an O(n) time and space algorithm to solve the partitioning problem above, where n is the order of the matrix. Our computational results on a CM-2 demonstrate the potential superiority of the partitioned inverse approach over the conventional substitution algorithm for highly parallel sparse triangular solution. Finally we describe current and future extensions of these results.
A part of this work was done while the authors were visiting the Institute for Mathematics and its Applications (IMA) at the University of Minnesota. We thank the IMA for its support.
Electrical and Computer Engineering Department, 1425 Johnson Drive, The University of Wisconsin, Madison, WI 53706 (alvarado@ece.wisc.edu). This author was supported under NSF Contracts ECS-8822654 and ECS-8907391.
Department of Computer Science, University of Waterloo, Waterloo, Ontario Canada N2L 3G1 (apothen@narnia.uwaterloo.ca, na.pothen@na-net.ornl.gov). This author was supported by NSF grant CCR-9024954 and by U. S. Department of Energy grant DE-FG02–91ER25095 at the Pennsylvania State University and by the Canadian Natural Sciences and Engineering Research Council under grant OGP0008111 at the University of Waterloo.
RIACS, MS T045–1, NASA Ames Research Center, Moffett Field, CA 94035 (schreiber@riacs.edu). This author was supported by the NAS Systems Division under Cooperative Agreement NCC 2–387 between NASA and the University Space Research Association (USRA).
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© 1993 Springer-Verlag New York, Inc.
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Alvarado, F.L., Pothen, A., Schreiber, R. (1993). Highly Parallel Sparse Triangular Solution. 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_7
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DOI: https://doi.org/10.1007/978-1-4613-8369-7_7
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