Abstract
Consider the solution of a large sparse linear system Ax = b on multiprocessors. A parallel sparse matrix factorization is required in a direct solver. Alternatively, if Krylov subspace iterative methods are used, then incomplete forms of parallel sparse factorization are required for preconditioning. In such schemes, the underlying parallel computation is tree-structured, utilizing task-parallelism at lower levels of the tree and data-parallelism at higher levels. The proportional heuristic has typically been used to map the data and computation to processors. However, for sparse systems from finite-element methods on complex domains, the resulting assignments can exhibit significant load-imbalances. In this paper, we develop a multi-pass mapping scheme to reduce such load imbalances and we demonstrate its effectiveness for a test suite of large sparse matrices. Our scheme can also be used to generate improved mappings for tree-structured applications beyond those considered in this paper.
The work was supported in part by the National Science Foundation through grants NSF ACI-0102537 and NSF CCF-0444345, and by the Director, Office of Science, Division of Mathematical, Information, and Computational Sciences of the U.S. Department of Energy under contract number DE-AC03-76SF00098.
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Demmel, J., Eisenstat, S.C., Gilbert, J.R., Li, X.S., Liu, J.W.H.: A supernodal approach to sparse partial pivoting. Technical Report CSL–94–14, Xerox Palo Alto Research Center (1995)
Duff, I.S.: Parallel implementation of multifrontal schemes. Parallel Computing 3, 193–204 (1986)
George, A., Liu, J.W.H.: An automatic nested dissection algorithm for irregular finite element problems. SIAM J. Numer. Anal. 15, 1053–1069 (1978)
George, J.A., Liu, J.W.H.: Computer Solution of Large Sparse Positive Definite Systems. Prentice-Hall Inc., Englewood (1981)
Grigori, L., Li, X.S.: A new scheduling algorithm for parallel sparse LU factorization with static pivoting. In: Proceedings of the 2002 ACM/IEEE conference on Supercomputing, pp. 1–18. IEEE Computer Society Press, Los Alamitos (2002)
Gupta, A., Gustavson, F., Joshi, M., Karypis, G., Kumar, V.: PSPASES: An efficient and scalable parallel sparse direct solver (1999), See http://www-users.cs.umn.edu/~mjoshi/pspases
Gupta, A., Kumar, V., Sameh, A.: Performance and scalability of preconditioned conjugate gradient methods on the CM-5. In: Sincovec, R.F., Keyes, D.E., Leuze, M.R., Petzold, L.R., Reed, D.A. (eds.) Proceedings of the Sixth SIAM Conference on Parallel Processing for Scientific Computing, Philadephia, PA, pp. 664–674. SIAM Publications, Philadelphia (1993)
Heath, M.T., Ng, E., Peyton, B.W.: Parallel algorithms for sparse linear systems. SIAM Review 33, 420–460 (1991)
Heath, M.T., Raghavan, P.: Performance of a fully parallel sparse solver. Int. J. Supercomputing Appl. 11, 49–64 (1997)
Jones, M.T., Plassman, P.E.: An improved incomplete Cholesky factorization. ACM Trans. Math. Software 21, 5–17 (1995)
Karypis, G., Kumar, V.: METIS: Unstructured graph partitioning and sparse matrix ordering system. Technical report, Department of Computer Science, University of Minnesota, Minneapolis, MN (1995)
Liu, J.W.H.: The role of elimination trees in sparse factorization. SIAM J. Matrix Anal. Appl. 11, 134–172 (1990)
Liu, J.W.H.: The multifrontal method for sparse matrix solution: theory and practice. SIAM Review 34, 82–109 (1992)
Ng, E., Peyton, B.W.: A supernodal Cholesky factorization algorithm for shared-memory multiprocessors. SIAM J. Sci. Comput. 14, 761–769 (1993)
Pothen, A., Sun, C.: A mapping algorithm for parallel sparse Cholesky factorization. SIAM J. Sci. Comput. 14(5), 1253–1257 (1993)
Raghavan, P.: DSCPACK: Domain-Separator Codes for the parallel solution of sparse linear systems. Software for solving sparse linear systems on multiprocessors and NOWs using C and MPI. Package has a two stage parallel nested dissection, higher-level BLAS, fast repeated solves and an easy to use parallel interface (2002), See http://www.cse.psu.edu/Dscpack
Raghavan, P., Teranishi, K., Ng, E.: Scalable parallel preconditioning with incomplete factors. In: Proceedings of the Seventh SIAM Conference on Applied Linear Algebra (2000)
Raghavan, P., Teranishi, K., Ng, E.: A latency tolerant hybrid sparse solver using incomplete Cholesky factorization. Numerical Linear Algebra 10, 541–560 (2003)
Saad, Y.: Iterative Methods for Sparse Linears Systems. PWS Publishing Co., Boston (1996)
Yang, C., Raghavan, P., Arrowood, L., Sumpter, B., Noid, D.: Large-scale normal coordinate analysis on distributed memory mu ltiprocessors and nows. Int. J. Supercomputing Appl. 1(4), 409–424 (2002)
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Malkowski, K., Raghavan, P. (2005). Multi-pass Mapping Schemes for Parallel Sparse Matrix Computations. In: Sunderam, V.S., van Albada, G.D., Sloot, P.M.A., Dongarra, J.J. (eds) Computational Science – ICCS 2005. ICCS 2005. Lecture Notes in Computer Science, vol 3514. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11428831_31
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DOI: https://doi.org/10.1007/11428831_31
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