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Parallel Linear Systems Solvers: Sparse Iterative Methods

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Book cover High Performance Computing in Fluid Dynamics

Part of the book series: Series ((ERCO,volume 3))

Abstract

Iterative methods are quite popular for the approximate solution of large sparse linear systems. As we will see, the are very well-suited for parallel computing. From this point of view we will discuss a number of methods, representative for the class of so-called Krylov subspace methods.

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References

  1. O. Axelsson. Conjugate gradient type methods for unsymmetric and inconsistent systems of equations. Lin. Alg. and its Appl, 29:1–16, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  2. O. Axelsson and RS. Vassilevski. A black box generalized conjugate gradient solver with inner iterations and variable-step preconditioning. SI AMJ. Matrix Anal Appl, 12(4)625–644, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  3. Zhaojun Bai, Dan Hu, and Lothar Reichel. A Newton basis GMRES implementation. Technical Report 91-03, University of Kentucky, 1991.

    Google Scholar 

  4. R. Barrett, M. Berry, T. Chan, J. Demmel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, and H. van der Vorst.Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. SIAM, Philadelphia, PA, 1994.

    Google Scholar 

  5. E. Barszcz, R. Fatoohi, V. Venkatakrishnan, and S. Weeratunga. Triangular systems for CFD applications on parallel architectures. Technical report, NAS Applied Research Branch, NASA Ames Research Center, 1994.

    Google Scholar 

  6. H. Berryman, J. Saltz, W. Gropp, and R. Mirchandaney. Krylov methods preconditioned with incompletely factored matrices on the CM-2. Technical Report 89-54, NASA Langley Research Center, ICASE, Hampton, VA, 1989.

    Google Scholar 

  7. P.N. Brown. A theoretical comparison of the Arnoldi and GMRES algorithms. SIAMJ. Sci. Statist Comput 12:58–78, 1991.

    Article  MATH  Google Scholar 

  8. A.T. Chronopoulos and C.W. Gear. s-Step iterative methods for symmetric linear systems. J. on Comp. and Appl Math., 25:153–168, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  9. A.T. Chronopoulos and S.K. Kim. s-Step Orthomin and GMRES implemented on parallel computers. Technical Report 90/43R, UMSI, Minneapolis, 1990.

    Google Scholar 

  10. P. Concus and G.H. Golub. A generalized Conjugate Gradient method for nonsymmetric systems of linear equations. Technical Report STAN-CS-76-535, Stanford University, Stanford, CA, 1976.

    Google Scholar 

  11. G.C. (Lianne) Crone. The conjugate gradient method on the parsytec GCel-3/512. FGCS, 11:161-166, 1995.

    Google Scholar 

  12. L. Crone and H. van der Vorst. Communication aspects of the conjugate gradient method on distributed-memory machines. Supercomputer, X(6):4–9, 1993.

    Google Scholar 

  13. M.A. DeLong and J.M. Ortega. SOR as a Preconditioner. Appl Num. Math., 18:431–440, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  14. E. de Sturler. A parallel variant of GMRES(m). In R. Miller, editor, Proc. of the fifth Int.Symp. on Numer. Methods in Eng., 1991.

    Google Scholar 

  15. E. De Sturler. Iterative methods on distributed memory computers. PhD thesis, Delft University of Technology, Delft, the Netherlands, 1994.

    Google Scholar 

  16. E. De Sturler and D.R. Fokkema. Nested Krylov methods and preserving the orthogonality. In N. Duane Melson, T.A. ManteufFel, and S.F. McCormick, editors, Sixth Copper Mountain Conference on Multigrid Methods, volume P a r t 1 of NASA Conference Publication 3324, pages 111-126. NASA, 1993.

    Google Scholar 

  17. E. De Sturler and H.A. van der Vorst. Reducing the effect of global communication in GMRES(m) and CG on parallel distributed memory computers. J. Appl Num. Math., 1995.

    Google Scholar 

  18. J. Demmel, M. Heath, and H. van der Vorst.Parallel numerical linear algebra. In Acta Numerica 1993. Cambridge University Press, Cambridge, 1993.

    Google Scholar 

  19. S. Doi and A. Hoshi. Large numbered multicolor MILU preconditioning on SX-3/14. Int’l J. Computer Math., 44:143–152, 1992.

    Article  MATH  Google Scholar 

  20. J.J. Dongarra. Performance of various computers using standard linear equations software in a fortran environment. Technical Report CS-89-85, University of Tennessee, Knoxville, 1990.

    Google Scholar 

  21. J.J. Dongarra, I.S. Duff, D.C. Sorensen, and H.A. van der Vorst.Solving Linear Systems on Vector and Shared Memory Computers. SIAM, Philadelphia, PA, 1991.

    Google Scholar 

  22. J.J. Dongarra and H.A. van der Vorst. Performance of various computers using standard sparse linear equations solving techniques. Supercomputer, 9(5) 17–29, 1992.

    Google Scholar 

  23. P.F. Dubois, A. Greenbaum, and G.H. Rodrigue. Approximating the inverse of a matrix for use in iterative algorithms on vector processors. Computing, 22:257–268, 1979.

    Article  MathSciNet  MATH  Google Scholar 

  24. I.S. Duff and G.A. Meurant. The effect of ordering on preconditioned conjugate gradient. BIT, 29:635–657, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  25. H.C. Elman. Iterative methods for large sparse nonsymmetric systems of linear equations. PhD thesis, Yale University, New Haven, CT, 1982.

    Google Scholar 

  26. G.H. Golub and C.F. van Loan Matrix Computations. The Johns Hopkins University Press, Baltimore, 1989.

    Google Scholar 

  27. M.H. Gutknecht. Variants of BICGSTAB for matrices with complex spectrum. SIAMJ. Sci. Compute 14:1020–1033, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  28. M.R. Hestenes and E. Stiefel. Methods of conjugate gradients for solving linear systems. J. Res. Natl Bur. Stand., 49:409–436, 1954.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  30. J.C.C. Kuo and T.F. Chan. Two-color fourier analysis of iterative algorithms for elliptic problems with red/black ordering. SIAMJ. Sci. Stat. Comp., 11:767–793, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  31. 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., 31:148–162, 1977.

    MathSciNet  MATH  Google Scholar 

  32. G. Meurant. The block preconditioned conjugate gradient method on vector computers. BIT, 24:623–633, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  33. G. Meurant. Numerical experiments for the preconditioned conjugate gradient method on the CRAYX-MP/2. Technical Report LBL-18023, University of California, Berkeley, CA, 1984.

    Google Scholar 

  34. G. Meurant. Domain decomposition methods for partial differential equations on parallel computers. Int. J. Supercomputing Appls., 2:5–12, 1988.

    Article  Google Scholar 

  35. C.C. Paige and M.A. Saunders. Solution of sparse indefinite systems of linear equations. SIAMJ. Numer. Anal, 12:617–629, 1975.

    Article  MathSciNet  MATH  Google Scholar 

  36. Claude Pommerell. Solution of large unsymmetric systems of linear equations. PhD thesis, Swiss Federal Institute of Technology, Zurich, 1992.

    Google Scholar 

  37. G. Radicati di Brozolo and Y. Robert. Parallel conjugate gradient-like algorithms for solving sparse non-symmetric systems on a vector multiprocessor. Parallel Computing, 11:223–239, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  38. G. Radicati di Brozolo and M. Vitaletti. Sparse matrix-vector product and storage representations on the IBM 3090 with Vector Facility. Technical Report 513-4098, IBM-ECSEC, Rome, July 1986.

    Google Scholar 

  39. Y. Saad. Practical use of polynomial preconditionings for the conjugate gradient method. SIAMJ. Sci. Stat. Comput., 6:865–881, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  40. Y. Saad. Krylov subspace methods on supercomputers. Technical report, RIACS, Moffett Field, CA, September 1988.

    Google Scholar 

  41. Y. Saad. A flexible inner-outer preconditioned GMRES algorithm. SIAMI. Sci. Comput., 14:461–469, 1993.

    Article  MathSciNet  MATH  Google Scholar 

  42. Y. Saad and M.H. Schultz. Conjugate Gradient-like algorithms for solving nonsymmetric linear systems. Math, of Comp., 44:417–424, 1985.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  44. G.L.G. Sleijpen and H.A. Van der Vorst. Maintaining convergence properties of BICGSTAB methods in finite precision arithmetic. Numerical Algorithms, 10:203–223, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  45. G.L.G. Sleijpen, H.A. Van der Vorst, and D.R. Fokkema. Bi - CGSTAB and other hybrid Bi-CG methods. Numerical Algorithms, 7:75–109, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  46. G.L.G. Sleijpen and D.R. Fokkema. BICGSTAB(ℓ) for linear equations involving unsymmetric matrices with complex spectrum. ETNA, 1:11–32, 1993.

    MathSciNet  MATH  Google Scholar 

  47. P. Sonneveld. CGS: a fast Lanczos-type solver for nonsymmetric linear systems. SIAMJ. Sci. Statist Comput., 10:36–52, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  48. K.H. Tan. Local coupling in domain decomposition. PhD thesis, Utrecht University, Utrecht, the Netherlands, 1995.

    Google Scholar 

  49. H.A. van der Vorst. A vectorizable variant of some ICCG methods. SIAMJ. Sci. Stat Comput, 3:86–92, 1982.

    Google Scholar 

  50. H.A. van der Vorst. High performance preconditioning. SIAMJ. Sci. Statist. Comput., 10:1174–1185, 1989.

    Article  MATH  Google Scholar 

  51. H.A. van der Vorst. The convergence behaviour of preconditioned CG and CG-S in the presence of rounding errors. In O. Axelsson and L. Yu. Kolotilina, editors, Preconditioned Conjugate Gradient Methods, Berlin, 1990. Nijmegen 1989, Springer Verlag. Lecture Notes in Mathematics 1457.

    Google Scholar 

  52. H.A. van der Vorst. Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAMJ. Sci. Statist. Comput., 13:631–644, 1992.

    Article  MATH  Google Scholar 

  53. H.A. van der Vorst and C. Vuik. The superlinear convergence behaviour of GMRES. JCAM, 48:327–341, 1993.

    MATH  Google Scholar 

  54. H.A. van der Vorst and C. Vuik. GMRESR: A family of nested GMRES methods. Num. Lin. Alg. with Appl, 1:369–386, 1994.

    Article  MATH  Google Scholar 

  55. M.B. van Gijzen. Iterative solution methods for linear equations in finite element computations. PhD thesis, Delft University of Technology, Delft, the Netherlands, 1994.

    Google Scholar 

  56. P.K.W. Vinsome. ORTOMIN: an iterative method for solving sparse sets of simultaneous linear equations. In Proc.Fourth Symposium on Reservoir Simulation, pages 149-159. Society of Petroleum Engineers of AIME, 1976.

    Google Scholar 

  57. T. Washio and K. Hayami. Parallel block preconditioning based on SSOR and MILU. Numer. Lin. Alg. with Applic., 1:533–553, 1994.

    Article  MathSciNet  MATH  Google Scholar 

  58. O. Widlund. A Lanczos method for a class of nonsymmetric systems of linear equations. SIAMJ. Numer. Anal, 15:801–812, 1978.

    Article  MathSciNet  MATH  Google Scholar 

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© 1996 Kluwer Academic Publishers

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Van Der Vorst, H.A. (1996). Parallel Linear Systems Solvers: Sparse Iterative Methods. In: Wesseling, P. (eds) High Performance Computing in Fluid Dynamics. ERCOFTAC Series, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-0271-8_5

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  • DOI: https://doi.org/10.1007/978-94-009-0271-8_5

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-6606-8

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