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Solving Unsymmetric Sparse Systems of Linear Equations with PARDISO

  • Olaf Schenk
  • Klaus Gärtner
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2330)

Abstract

Supernode pivoting for unsymmetric matrices coupled with supernode partitioning and asynchronous computation can achieve high gigaflop rates for parallel sparse LU factorization on shared memory parallel computers. The progress in weighted graph matching algorithms helps to extend these concepts further and prepermutation of rows is used to place large matrix entries on the diagonal. Supernode pivoting allows dynamical interchanges of columns and rows during the factorization process. The BLAS-3 level efficiency is retained. An enhanced left—right looking scheduling scheme is uneffected and results in good speedup on SMP machines without increasing the operation count. These algorithms have been integrated into the recent unsymmetric version of the PARDISO solver. Experiments demonstrate that a wide set of unsymmetric linear systems can be solved and high performance is consistently achieved for large sparse unsymmetric matrices from real world applications.

Keywords

Sparse Matrice Sparse Linear System Operation Count Partial Pivoting Nest Dissection 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    P. R. Amestoy, I. S. Duff, J.-Y. L’Excellent, and X. S. Li. Analysis and comparison of two general sparse solvers for distributed memory computers. Technical Report TR/PA/00/90, CERFACS, Toulouse, France, December 2000. Submitted to ACM Trans. Math. Softw. Google Scholar
  2. 2.
    Patrick R. Amestoy, Iain S. Duff, Jean-Yves L’Excellent, and Jacko Koster. A fully asynchronous multifrontal solver using distributed dynamic scheduling. SIAM J. Matrix Analysis and Applications, 23(1):15–41, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Mario Arioli, James W. Demmel, and Iain S. Duff. Solving sparse linear systems with sparse backward error. SIAMJ. Matrix Analysis and Applications, 10:165–190, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R.E. Bank, D.J. Rose, and W. Fichtner. Numerical methods for semiconductor device simulation. SIAM Journal on Scientific and Statistical Computing, 4(3):416–435, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    T. A. Davis and I. S. Duff. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Analysis and Applications, 18(1):140–158, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Timothy A. Davis. UMFPACK. Software for unsymmetric multifrontal method. In NA Digest, 01(11), March 18, 2001., http://www.cise.ufl.edu/research/sparse/umfpack.
  7. 7.
    J. Demmel, J. Gilbert, and X. Li. An asynchronous parallel supernodal algorithm to sparse partial pivoting. SIAM Journal on Matrix Analysis and Applications, 20(4):915–952, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    J. W. Demmel, S. C. Eisenstat, J. R. Gilbert, X. S. Li, and J. W.-H. Liu. A supernodal approach to sparse partial pivoting. SIAM J. Matrix Analysis and Applications, 20(3):720–755, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    I. S. Duff and J. Koster. The design and use of algorithms for permuting large entries to the diagonal of sparse matrices. Technical Report TR/PA/97/45, CERFACS, Toulouse, France, 1997. Also appeared as Report RAL-TR-97-059, Rutherford Appleton Laboratories, Oxfordshire.Google Scholar
  10. 10.
    I. S. Duff and J. Koster. The design and use of algorithms for permuting large entries to the diagonal of sparse matrices. SIAM J. Matrix Analysis and Applications, 20(4):889–901, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    A. Gupta. Fast and effective algorithms for solving graph partitioning and sparse matrix ordering. IBM Journal of Research and Development, 41(1/2):171–183, January/March 1997.Google Scholar
  12. 12.
    A. Gupta. Improved symbolic and numerical factorization algorithms for unsymmetric sparse matrices. Technical Report RC 22137 (99131), IBM T. J. Watson Research Center, Yorktown Heights, NY, August 1, 2001.Google Scholar
  13. 13.
    A. Gupta. Recent advances in direct methods for solving unsymmetric sparse systems of linear equations. Technical Report RC 22039 (98933), IBM T. J. Watson Research Center, Yorktown Heights, NY, April 20, 2001.Google Scholar
  14. 14.
    G. Karypis and V. Kumar. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM Journal on Scientific Computing, 20(1):359–392, 1998.CrossRefMathSciNetGoogle Scholar
  15. 15.
    X.S. Li and J.W Demmel. A scalable sparse direct solver using static pivoting. In Proceeding of the 9th SIAM conference on Parallel Processing for Scientic Computing, San Antonio, Texas, March 22-34, 1999.Google Scholar
  16. 16.
    E.G. Ng and B.W. Peyton. Block sparse Cholesky algorithms on advanced uniprocessor computers. SIAM Journal on Scientific Computing, 14:1034–1056, 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    O. Schenk. Scalable Parallel Sparse LU Factorization Methods on Shared Memory Multiprocessors. PhD thesis, ETH Zürich, 2000.Google Scholar
  18. 18.
    O. Schenk and K. Gärtner. Two-level scheduling in PARDISO: Improved scalability on shared memory multiprocessing systems. Accepted for publication in Parallel Computing.Google Scholar
  19. 19.
    O. Schenk and K. Gärtner. PARDISO: a high performance serial and parallel sparse linear solver in semiconductor device simulation. Future Generation Computer Systems, 789(1):1–9, 2001.Google Scholar
  20. 20.
    O. Schenk, K. Gärtner, and W Fichtner. Efficient sparse LU factorization with left-right looking strategy on shared memory multiprocessors. BIT, 40(1):158–176, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    P. Sonneveld. CGS, a fast Lanczos-type solver for nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing, 10:36–52, 1989.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Olaf Schenk
    • 1
  • Klaus Gärtner
    • 2
  1. 1.Department of Computer ScienceUniversity of BaselBaselSwitzerland
  2. 2.Weierstrass Institute for Applied Analysis and StochasticsBerlinGermany

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