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Supercomputer Implementations of Preconditioned Krylov Subspace Methods

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Algorithmic Trends in Computational Fluid Dynamics

Part of the book series: ICASE/NASA LaRC Series ((ICASE/NASA))

Abstract

Preconditioned Krylov subspace methods are among the preferred iterative techniques for solving large sparse linear systems of equations. As computer architectures are evolving and problems are becoming more complex, iterative techniques are undergoing several mutations. In particular, there has been much recent work on new preconditioned that yield higher parallelism or on new implementations of the standard ones. In the past, a conservative and well understood approach has consisted of porting standard preconditioners to the new computers. However, in addition to their limited parallelism, such preconditioners have other drawbacks. The simple ILU(O) pre-conditioner may fail to converge for realistic problems arising from applications such as Computational Fluid Dynamics. The more robust analogues ILU(k) [30] or ILUT(k) [40] that allow more fill-in are not always a good alternative since they tend to be sequential in nature. In this paper we discuss these issues and give an overview of the standard approaches. Then we will propose a number of alternatives. It will be argued that some approaches based on multi-coloring can offer good compromises between generality and efficiency.

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References

  1. L. Adams and H. Jordan. Is SOR color-blind? SIAM J. Sci. Statist. Comp, 6: 490–506, 1985.

    MathSciNet  Google Scholar 

  2. L. Adams and J. Ortega. A multi-color SOR Method for Parallel Computers. In Proceedings 1982 Int. Conf. Par. Proc., pages 53–56, 1982.

    Google Scholar 

  3. L. M. Adams. Iterative algorithms for large sparse linear systems on parallel computers. PhD thesis, Applied Mathematics, University of Virginia, Charlottsville, VA, 22904, 1982. Also NASA Contractor Report 166027.

    Google Scholar 

  4. E. C. Anderson. Parallel implementation of preconditioned conjugate gradient methods for solving sparse systems of linear equations. Technical Report 805, CSRD, University of Illinois, Urbana, IL, 1988. MS Thesis.

    Google Scholar 

  5. E. C. Anderson and Y. Saad. Solving sparse triangular systems on parallel computers. Technical Report 794, University of Illinois, CSRD, Urbana, IL, 1988.

    Google Scholar 

  6. S. F. Ashby, T. A. Manteuffel, and P. E. Saylor. Adaptive polynomial preconditioning for Hermitian indefinite linear systems. BIT, 29: 583–609, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  7. C. C. Ashcraft and R. G. Grimes. On vectorizing incomplete factorization and SSOR preconditioners. SIAM J. Sci. Statist. Comput., 9: 122–151, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  8. O. Axelsson. A generalized conjugate gradient, least squares method. Num. Math., 51: 209–227, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  9. D. Baxter, J. Saltz, M. H. Schultz, and S. C. Eisenstat. Preconditioned Krylov solvers and methods for runtime loop paral-lelization. Technical Report 655, Computer Science, Yale University, New Haven, CT, 1988.

    Google Scholar 

  10. D. Baxter, J. Saltz, M. H. Schultz, S. C. Eisenstat, and K. Crowley. An experimental study of methods for parallel preconditioned Krylov methods. Technical Report 629, Computer Science, Yale University, New Haven, CT, 1988.

    Google Scholar 

  11. M. Benantar and J. E. Flaherty. A Six color procedure for the parallel solution of Elliptic systems using the finite Quadtree structure. In J. Dongarra, P. Messina, D. C. Sorenson, and R. G. Voigt, editors, Proceedings of the fourth SIAM conference on parallel processing for scientific computing, pages 230–236, 1990.

    Google Scholar 

  12. I. S. Duff, A. M. Erisman, and J. K. Reid. Direct Methods for Sparse Matrices. Clarendon Press, Oxford, 1986.

    MATH  Google Scholar 

  13. I. S. Duff and G. A. Meurant. The effect of reordering on preconditioned conjugate gradients. BIT, 29: 635–657, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  14. H. C. Elman. Iterative Methods for Large Sparse Nonsymmetric Systems of Linear Equations. PhD thesis, Yale University, Computer Science Dept., New Haven, CT., 1982.

    Google Scholar 

  15. H. C. Elman and E. Agron. Ordering techniques for the preconditioning conjugate gradient method on parallel computers. Technical Report UMIACS-TR-88-53, UMIACS, University of Maryland, College Park, MD, 1988.

    Google Scholar 

  16. H. C. Elman and G. H. Golub. Iterative methods for cyclically reduced non-self-adjoint linear systems. Technical Report CS-TR-2145, Dept. of Computer Science, University of Maryland, College Park, MD, 1988.

    Google Scholar 

  17. R. M. Ferencz. Element-by-element preconditioning techniques for large scale vectorized finite element analysis in nonlinear solid and structural mechanics. PhD thesis, Applied Mathematics, Stanford, CA, 1989.

    Google Scholar 

  18. R. Freund, M. H. Gutknecht, and N. M. Nachtigal. An implementation of the Look-Ahead Lanczos algorithm for non-Hermitian matrices, Part I. Technical Report 90-11, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1990.

    Google Scholar 

  19. R. Freund and N. M. Nachtigal. An implementation of the look-ahead lanczos algorithm for non-Hermitian matrices, Part II. Technical Report 90-11, Massachusetts Institute of Technology, Cambridge, Massachusetts, 1990.

    Google Scholar 

  20. R. S. Varga G. H. Golub. Chebyshev semi iterative methods successive overrelaxation iterative methods and second order Richardson iterative methods. Numer. Math., 3: 147–168, 1961.

    Article  MathSciNet  Google Scholar 

  21. A. Greenbaum. Solving triangular linear systems using fortran with parallel extensions on the nyu ultracomputer prototype. Technical Report 99, Courant Institute, New York University, New York, NY, 1986.

    Google Scholar 

  22. A. L. Hageman and D. M. Young. Applied Iterative Methods. Academic Press, New York, 1981.

    MATH  Google Scholar 

  23. M. A. Heroux, P. Vu, and C. Yang. A parallel preconditioned conjugate gradient package for solving sparse linear systems on a cray Y-MP. Technical report, Cray Research, Eagan, MN, 1990. Proc. of the 1990 Cray Tech. Symp.

    Google Scholar 

  24. T. J. R. Hughes, R. M. Ferencz, and J. O. Hallquist. Large-scale vectorized implicit calculations in solid mechanics on a cray x-mp/48 utilizing ebe preconditioning conjugate gradients. Computer Methods in Applied Mechanics and Engineering, 61: 215–248, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  25. K. C. Jea and D. M. Young. Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods. Linear Algebra Appl., 34: 159–194, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  26. M. T. Jones and P. E. Plassmann. Parallel iterative solution of sparse linear systems using ordering from graph coloring heuristics. Technical Report MCS-P198-1290, Argonne National Lab., Argonne, IL, 1990.

    Book  Google Scholar 

  27. T. I. Karush, N. K. Madsen, and G. H. Rodrigue. Matrix multiplication by diagonals on vector/parallel processors. Technical Report UCUD, Lawrence Livermore National Lab., Livermore, CA, 1975.

    Google Scholar 

  28. T. A. Manteuffel. The Tchebychev iteration for nonsymmetric linear systems. Numer. Math., 28: 307–327, 1977.

    Article  MathSciNet  MATH  Google Scholar 

  29. T. A. Manteuffel. Adaptive procedure for estimation of parameter for the nonsymmetric Tchebychev iteration. Numer. Math., 28: 187–208, 1978.

    Google Scholar 

  30. J. A. Meijerink and H. A. van der Vorst. An iterative solution method for linear systems of which the coefficient matrix is a symmetric M-matrix. Math. Comp., 31 (137): 148–162, 1977.

    MathSciNet  MATH  Google Scholar 

  31. R. Melhem. Solution of linear systems with striped sparse matrices. Parallel Comput., 6: 165–184, 1988.

    Article  MathSciNet  MATH  Google Scholar 

  32. T. C. Oppe, W. Joubert, and D. R. Kincaid. Nspcg user’s guide, a package for solving large linear systems by various iterative methods. Technical report, The University of Texas at Austin, 1988.

    Google Scholar 

  33. T. C. Oppe and D. R. Kincaid. The performance of ITPACK on vector computers for solving large sparse linear systems arising in sample oil reservoir simulation problems. Communications in applied numerical methods, 2: 1–7, 1986.

    Article  Google Scholar 

  34. J. M. Ortega. Introduction to Parallel and Vector Solution of Linear Systems. Plenum Press, New York, 1988.

    MATH  Google Scholar 

  35. G. A. Paolini and G. Radicati di Brozólo. Data structures to vectorize CG algorithms for general sparsity patterns. BIT, 29: 4, 1989.

    Article  Google Scholar 

  36. E. L Poole and J. M. Ortega. Mullticolor ICCG methods for vector computers. SIAM J. Numer. Anal., 24: 1394–1418, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  37. Y. Saad. Least squares polynomials in the complex plane and their use for solving sparse nonsymmetric linear systems. SIAM J. Num. Anal., 24: 155–169, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  38. Y. Saad. SPARSKIT: A basic tool kit for sparse matrix computations. Technical Report 90-20, Research Institute for Advanced Computer Science, NASA Ames Research Center, Moffet Field, CA, 1990.

    Google Scholar 

  39. Y. Saad. A flexible inner-outer preconditioned GMRES algorithm. Technical Report 91-279, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota, 1991.

    Google Scholar 

  40. Y. Saad. ILUT: a dual strategy accurate incomplete ilu factorization. Technical Report -, Minnesota Supercomputer Institute, University of Minnesota, Minneapolis, 1991. in preparation.

    Google Scholar 

  41. Y. Saad. Massively parallel preconditioned Krylov subspace methods. Technical Report -, Minnesota Supercomputer Institute, Minneapolis, Minnesota, 1991. In preparation.

    Google Scholar 

  42. Y. Saad and M. H. Schultz. Conjugate gradient-like algorithms for solving nonsymmetric linear systems. Mathematics of Computation, 44 (170): 417–424, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  43. Y. Saad and M. H. Schultz. Parallel implementations of preconditioned conjugate gradient methods. Research report 425, Dept Computer Science, Yale University, 1985.

    Google Scholar 

  44. Y. Saad and M. H. Schultz. GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Statist. Comput., 7: 856–869, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  45. J. H. Saltz. Automated problem scheduling and reduction of synchronization delay effects. Technical Report 87-22, ICASE, Hampton, VA, 1987.

    Google Scholar 

  46. F. Shakib. Finite element analysis of the compressible Euler and Navier Stokes Equations. PhD thesis, Aeronautics Dept., Stanford, CA, 1989.

    Google Scholar 

  47. T. E. Tezduyar, M. Behr, S. K. A. Abadi, and S. E. Ray. A mixed CEBE/CC preconditionning for finite element computations. Technical Report UMSI 91/160, University of Minnesota, Minnesota Supercomputing Institute, Mineapolis, Minnesota, June 1991.

    Google Scholar 

  48. H. A. van der Vorst. The performance of FORTRAN implementations for preconditioned conjugate gradient methods on vector computers. Parallel Comput., 3: 49–58, 1986.

    Article  MATH  Google Scholar 

  49. H. A. van der Vorst. Large tridiagonal and block tridiagonal linear systems on vector and parallel computers. Par. Comp., 5: 303–311, 1987.

    Article  MATH  Google Scholar 

  50. H. A. van der Vorst. High performance preconditioning. SIAM j. Scient. Stat. Comput., 10: 1174–1185, 1989.

    Article  MATH  Google Scholar 

  51. R. S. Varga. Matrix Iterative Analysis. Prentice Hall, Englewood Cliffs, NJ, 1962.

    Google Scholar 

  52. V. Venkatakrishnan. Preconditioned Conjugate Gradient methods for the compressible Navier Stokes equations. AIAA Journal, 29: 1092–1100, 1991.

    Article  Google Scholar 

  53. V. Venkatakrishnan and D. J. Mavriplis. Implicit solvers for unstructured grids. In Proceedings of the AIAA 10th CFD Conference, June, 1991, HI., 1991.

    Google Scholar 

  54. V. Venkatakrishnan, H. D. Simon, and T. J. Barth. A MIMD Implementation of a Parallel Euler Solver for Unstructured Grids. Technical Report RNR-91-024, NASA Ames research center, Moffett Field, CA, 1991.

    Google Scholar 

  55. P. K. W. Vinsome. Orthomin, an iterative method for solving sparse sets of simultaneous linear equations. In Proceedings of the Fourth Symposium on Resevoir Simulation, pages 149–159. Society of Petroleum Engineers of AIME, 1976.

    Google Scholar 

  56. O. Wing and J. W. Huang. A computation model of parallel solution of linear equations. IEEE Transactions on Computers, C-29: 632–638, 1980.

    Article  MathSciNet  Google Scholar 

  57. C. H. Wu. A multicolour SOR method for the finite-element method. J. of Comput and App. Math., 30: 283–294, 1990.

    Article  MATH  Google Scholar 

  58. D.M. Young. Iterative solution of large linear systems. Academic Press, New-York, 1971.

    MATH  Google Scholar 

  59. D. M. Young, T. C. Oppe, D. R. Kincaid, and L. J. Hayes. On the use of vector computers for solving large sparse linear systems. Technical Report CNA-199, Center for Numerical Analysis, University of Texas at Austin, Austin, Texas, 1985.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Computer Science Department, University of Minnesota, 4-192 EE/CSci Building, 200 Union Street S.E, Minneapolis, Minnesota, 55455, USA

    Youcef Saad

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  1. Youcef Saad
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Editors and Affiliations

  1. Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Mail Stop 132-C, 23681-0001, Hampton, VA, USA

    M. Y. Hussaini

  2. NASA Langley Research Center, Mail Stop 156, 23681-0001, Hampton, VA, USA

    A. Kumar  & M. D. Salas  & 

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© 1993 Springer-Verlag New York, Inc.

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Saad, Y. (1993). Supercomputer Implementations of Preconditioned Krylov Subspace Methods. In: Hussaini, M.Y., Kumar, A., Salas, M.D. (eds) Algorithmic Trends in Computational Fluid Dynamics. ICASE/NASA LaRC Series. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-2708-3_8

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  • DOI: https://doi.org/10.1007/978-1-4612-2708-3_8

  • Publisher Name: Springer, New York, NY

  • Print ISBN: 978-1-4612-7638-8

  • Online ISBN: 978-1-4612-2708-3

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