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Parallel sparse linear algebra and application to structural mechanics

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Abstract

The framework of this paper is the parallelization of a plasticity algorithm that uses an implicit method and an incremental approach. More precisely, we will focus on some specific parallel sparse linear algebra algorithms which are the most time-consuming steps to solve efficiently such an engineering application. First, we present a general algorithm which computes an efficient static scheduling of block computations for parallel sparse linear factorization. The associated solver, based on a supernodal fan-in approach, is fully driven by this scheduling. Second, we describe a scalable parallel assembly algorithm based on a distribution of elements induced by the previous distribution for the blocks of the sparse matrix. We give an overview of these algorithms and present performance results on an IBM SP2 for a collection of grid and irregular problems.

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References

  1. P. Amestoy, T. Davis and I. Duff, An approximate minimum degree ordering algorithm, SIAM J. Matrix Anal. Appl. 17 (1996) 886-905.

    Article  MATH  MathSciNet  Google Scholar 

  2. C. Ashcraft, The Fan-Both family of column-based distributed Cholesky factorization algorithms, in: Graph Theory and Sparse Matrix Computation, The IMA Volumes in Mathematics and its Applications, Vol. 56 (Springer, New York, 1993) pp. 159-190.

    Google Scholar 

  3. C. Ashcraft, S.C. Eisenstat and J.W.-H. Liu, A fan-in algorithm for distributed sparse numerical factorization, SIAM J. Sci. Statist. Comput. 11(3) (1990) 593-599.

    Article  MATH  MathSciNet  Google Scholar 

  4. C. Ashcraft, S.C. Eisenstat, J.W.-H. Liu and A. Sherman, A comparison of three column based distributed sparse factorization schemes, in: Proc. of 5th SIAM Conf. on Parallel Processing for Scientific Computing (1991).

  5. H. Bramble, J.E. Pasciak and J. Xu, Parallel multilevel preconditioner, in: Proc. of 3rd SIAM Internat. Symp. on Domain Decomposition Method for Partial Differential Equations (1999) pp. 341-357.

  6. V.E. Bulgakov and G. Kuhn, High performance multilevel iterative aggregation solver for large finite element structural analysis problems, Internat. J. Numer. Methods Engrg. 38 (1995) 3529-3544.

    Article  MATH  Google Scholar 

  7. P. Charrier and J. Roman, Algorithmique et calculs de complexité pour un solveur de type dissections emboî tées, Numer. Math. 55 (1989) 463-476.

    Article  MathSciNet  Google Scholar 

  8. J.K. Dickinson and P.A. Forsyth, Preconditioned conjugate gradient methods for 3d linear elasticity, Internat. J. Numer. Methods Engrg. 37 (1994) 2211-2234.

    Article  MATH  MathSciNet  Google Scholar 

  9. M.C. Dracopoulos and M.A. Crisfield, Coarse/fine mesh preconditioners for the iterative solution of finite element problems, Internat. J. Numer. Methods Engrg. 38 (1995) 3297-3313.

    Article  MATH  Google Scholar 

  10. I.S. Duff, Sparse numerical linear algebra: Direct methods and preconditioning, Technical Report TR/PA/96/22, CERFACS (1996).

  11. G.A. Geist and E. Ng, Task scheduling for parallel sparse Cholesky factorization, Internat. J. Parallel Programming 18(4) (1989) 291-314.

    Article  MATH  MathSciNet  Google Scholar 

  12. A. George, M.T. Heath, J.W.-H. Liu and E.G.-Y. Ng, Sparse Cholesky factorization on a local memory multiprocessor, SIAM J. Sci. Statist. Comput. 9 (1988) 327-340.

    Article  MATH  MathSciNet  Google Scholar 

  13. A. George and J.W.-H. Liu, Computer Solution of Large Sparse Positive Definite Systems (Prentice-Hall, Englewood Cliffs, NJ, 1981).

    MATH  Google Scholar 

  14. E. Graham and P.A. Forsyth, Preconditioning methods for very ill conditioned 3d linear elasticity problems, Internat. J. Numer. Methods Engrg. 44 (1999) 77-99.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Gupta, G. Karypis and V. Kumar, Scalable parallel algorithms for sparse linear systems, in: Proc. of Stratagem'96, Sophia-Antipolis (July 1996) pp. 97-110.

  16. A. Gupta, G. Karypis and V. Kumar, Highly scalable parallel algorithms for sparse matrix factorization, IEEE Trans. Parallel Distributed Systems 8(5) (1997) 502-520.

    Article  Google Scholar 

  17. A. Gupta and V. Kumar, WSSMP: A high-performance serial and parallel symmetric sparse linear solver, in: PARA'98 Workshop on Applied Parallel Computing in Large Scale Scientific and Industrial Problems, Umeå , Sweden (June 1998).

  18. P. Hénon, P. Ramet and J. Roman, A mapping and scheduling algorithm for parallel sparse fan-in numerical factorization, in: Proc. of EuroPAR'99, Lecture Notes in Computer Science, Vol. 1685 (Springer, New York, 1999) pp. 1059-1067.

    Google Scholar 

  19. P. Hénon, P. Ramet and J. Roman, PaStiX: A parallel sparse direct solver based on a static scheduling for mixed 1D/2D block distributions, in: Proc. of IRREGULAR 2000, Lecture Notes in Computer Science (Springer, New York, 2000) pp. 519-525.

    Google Scholar 

  20. P. Laborde, B. Toson and J.-J. Pesqué, On the consistent tangent operator algorithm for thermo-pastic problems, Comput. Methods Appl. Mech. Engrg. 146 (1997) 215-230.

    Article  MATH  MathSciNet  Google Scholar 

  21. F.J. Lingen, A generalized conjugate residual method for the solution of non-symmetric systems of equations with multiple right hand sides, Internat. J. Numer. Methods Engrg. 43 (1999) 641-656.

    Article  MathSciNet  Google Scholar 

  22. T.A. Manteuffel, The shifted incomplete cholesky factorization, Technical Report, Sandia Laboratories, Livermore.

  23. J.A. Mitchell and J.N. Reddy, A multilevel hierarchical preconditioner for thin elastic solids, Internat. J. Numer. Methods Engrg. 43 (1998) 1383-1400.

    Article  MATH  Google Scholar 

  24. F. Pellegrini and J. Roman, Sparse matrix ordering with SCOTCH, in: Proc. of HPCN'97, Vienna (April 1997), Lecture Notes in Computer Science, Vol. 1225 (Springer, New York) pp. 370-378.

    Google Scholar 

  25. F. Pellegrini, J. Roman and P. Amestoy, Hybridizing nested dissection and halo approximate minimum degree for efficient sparse matrix ordering, in: Proc. of IRREGULAR'99, Puerto Rico (April 1999), Lecture Notes in Computer Science, Vol. 1586 (Springer, New York) pp. 986-995.

    Google Scholar 

  26. A. Pothen and C. Sun, A mapping algorithm for parallel sparse Cholesky factorization, SIAM J. Sci. Comput. 14(5) (1993) 1253-1257.

    Article  MATH  MathSciNet  Google Scholar 

  27. E. Rothberg, Performance of panel and block approaches to sparse Cholesky factorization on the iPSC/860 and Paragon multicomputers, SIAM J. Sci. Comput. 17(3) (1996) 699-713.

    Article  MATH  MathSciNet  Google Scholar 

  28. E. Rothberg and A. Gupta, An efficient block-oriented approach to parallel sparse Cholesky factorization, SIAM J. Sci. Comput. 15(6) (1994) 1413-1439.

    Article  MATH  MathSciNet  Google Scholar 

  29. E. Rothberg and R. Schreiber, Improved load distribution in parallel sparse Cholesky factorization, in: Proc. of Supercomputing'94 (IEEE Press, New York, 1994) pp. 783-792.

    Google Scholar 

  30. Y. Saad, Iterative Methods for Sparse Linear Systems (PWS, 1996).

  31. R. Schreiber, Scalability of sparse direct solvers, Technical Report TR 92.13, RIACS, NASA Ames Research Center (May 1992).

  32. P. Schulze et al., The Cologne Challenge, Finite Element News (4).

  33. M. Suarjana and K. Law, A robust incomplete factorization based on value and space constraints, Internat. J. Numer. Methods Engrg. 38 (1995) 1703-1719.

    Article  MATH  MathSciNet  Google Scholar 

  34. W.F. Tinney and J.W. Walker, Direct solutions of sparse network equations by optimally ordered triangular factorization, J. Proc. IEEE 55 (1967) 1801-1809.

    Article  Google Scholar 

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Authors and Affiliations

  1. LaBRI, UMR CNRS 5800, ENSERB et Université Bordeaux 1, 351, cours de la Libération, F-33405, Talence, France

    David Goudin, Pascal Hénon, François Pellegrini, Pierre Ramet & Jean Roman

  2. CEA/CESTA, BP 2, F-33114, Le Barp, France

    Jean-Jacques Pesqué

Authors
  1. David Goudin
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  2. Pascal Hénon
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  3. François Pellegrini
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  4. Pierre Ramet
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  5. Jean Roman
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  6. Jean-Jacques Pesqué
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Goudin, D., Hénon, P., Pellegrini, F. et al. Parallel sparse linear algebra and application to structural mechanics. Numerical Algorithms 24, 371–391 (2000). https://doi.org/10.1023/A:1019118015711

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