ILUM: A multi-elimination ILU preconditioner for general sparse matrices

SIAM Journal on Scientific Computing

Volumn 17, Issue 4, 1996, Pages 830-847

  Saad, Y ^a  

^a University of Minnesota   (United States)

Author keywords

Graph coloring; Incomplete LU factorizations; Independent set orderings; Multicoloring; Preconditioned Krylov subspace methods; Sparse linear systems; Threshold dropping strategies

Indexed keywords

EID: 0001085784     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/0917054     Document Type: Article

References (53)

1. L. ADAMS AND J. ORTEGA, A multi-color SOR Method for Parallel Computers, in Proceedings of the 1982 International Conference on Pararallel Processing, 1982, pp. 53-56.
2. L. M. ADAMS, Iterative Algorithms for Large Sparse Linear Systems on Parallel Computers, Ph.D. thesis, Applied Mathematics, University of Virginia, Charlottesville, VA, 1982. Also NASA Contractor Report 166027. NASA Langley Research Center, Hampton, VA.
3. E. C. ANDERSON AND Y. SAAD, Solving sparse triangular systems on parallel computers, Internat. J. High Speed Comput., 1 (1989), pp, 73-96.
4. O. AXELSSON, V. EIJKOUT, B. POLMAN, AND P. VASSILEVSKI, Incomplete block-matrix factorization iterative methods for convection-diffusion problems, BIT. 29 (1989), pp. 867-889.
5. O. AXELSSON AND B. POLMAN, On approximate factorization methods for block matrices suitable for vector and parallel processors. Linear Algebra Appl., 77 (1986) pp. 3-26.
6. M. BENANTAR AND J. E. FLAHERTY, A six color procedure for the parallel solution of elliptic systems using the finite quadtree structure, in Proceedings of the Fourth SIAM Conference on Parallel Processing for Scientific Computing, J. Dongarra, P. Messina, D. C. Sorenson, and R. G. Voigt, eds., Society for Industrial and Applied Mathematics, Philadelphia, PA, 1990, pp. 230-236.
7. T. F. COLEMAN AND J. J. MORE, Estimation of sparse Jacobian matrices and graph coloring problems, SIAM J. Numer. Anal., 20 (1983), pp. 187-209.
8. P. CONCUS, G. H. GOLUB, AND G. MEURANT, Block preconditioning for the conjugate gradient method, SIAM J. Sci. Stat. Comput., 6 (1985), pp. 309-332.
9. T. H. CORMEN, C. E. LEISERSON, AND R. L. RIVEST, Introduction to Algorithms, McGraw-Hill, New York, 1990.
10. P. J. DAVIS, Interpolation and Approximation, Blaisdell, Waltham, MA, 1963.
11. T. A. DAVIS, A Parallel Algorithm for Sparse Unsymmetric LU Factorizations, Ph.D. thesis, University of Illinois at Urbana Champaign, Urbana, IL, 1989.
12. E. F. D'AZEVEDO, F. A. FORSYTH, AND W. P. TANG, Ordering methods for preconditioned conjugate gradient methods applied to unstructured grid problems, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 944-961.
13. _, Towards a cost effective ILU preconditioner with high level fill, BIT, 31 (1992), pp. 442-463.
14. J. J. DONGARRA, I. S. DUFF, D. SORENSEN, AND H. A. VAN DER VORST, Solving Linear Systems on Vector and Shared Memory Computers. Society for Industrial and Applied Mathematics, Philadelphia, PA, 1991.
15. I. S. DUFF, A. M. ERISMAN, AND J. K. REID, Direct Methods for Sparse Matrices, Clarendon Press, Oxford, UK, 1986.
16. I. S. DUFF, R. G. GRIMES, AND J. G. LEWIS, Sparse matrix test problems, ACM Trans. Math. Software, 15 (1989), pp. 1-14.
17. _, User's Guide for the Harwell-Boeing Sparse Matrix Collection, Tech. Rep. TR/PA/92/86, CERFACS, ! Toulouse, France, 1992.
18. I. S. DUFF AND G.A. MEURANT, The effect of reordering on preconditioned conjugate gradients, BIT, 29 (1989), pp. 635-657.
19. L. C. DUTTO, The effect of reordering on the preconditioned GMRES algorithm for solving the compressible Navier-Stokes equations, Internat. J. Numer. Methods Engrg., 36 (1993), pp. 457-497.
20. H. C. ELMAN AND E. AGRON, Ordering techniquesfor the preconditioned conjugate gradientmethod on parallel computers, Comput. Phys. Comm., 53 (1989), pp. 253-269.
21. H. C. ELMAN AND G. H. GOLUB, Iterative methods for cyclically reduced non-self-adjoint linear systems. Math. Comp., 54 (1990), pp. 671-700.
22. R. M. FERENCZ, Element-by-Element Preconditioning Techniques for Large Scale Vectorized Finite Element Analysis in Nonlinear Solid and Structural Mechanics, Ph.D. thesis, Applied Mathematics, Stanford University, CA, 1989.
23. R. S. VARGA AND G. H. GOLUB, Chebyshev semi iterative methods successive overrelaxation terative methods and second order Richardson iterative methods, Numer. Math., 3 (1961), pp. 147-168.
24. K. GALLIVAN, A. SAMEH, AND Z. ZLATEV, A parallel hybrid sparse linear system solver, Comput. Systems Engrg., 1 (1990), pp. 183-195.
25. J. A. GEORGE AND J. W. LIU, Computer Solution of Large Sparse Positive Definite Systems, Prentice-Hall, Englewood Cliffs, NJ, 1981.
26. 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 preconditioned conjugate gradients, Comput. Methods Appl. Mech. Engrg., 61 (1987), pp. 215-248.
27. O. G. JOHNSON, C. A. MICCHELLI, AND G. PAUL, Polynomial preconditionings for conjugate gradient calculations, SIAM J. Numer. Anal., 20 (1983), pp. 362-376.
28. M. T. JONES AND P. E. PLASSMANN, Parallel Iterative Solution of Sparse Linear Systems Using Ordering from Graph Coloring Heuristics, Tech. Rep. MCS-P198-1290, Argonne National Lab., Argonne, IL, 1990.
29. M. T. JONES AND P. E. PLASSMANN, A parallel graph coloring heuristic, Tech. Rep. MCS-P246-0691, Argonne National Lab., Argonne, IL, 1991.
30. _, The effect of many-color orderings on the convergence of methods. Tech. Rep. MCS-P292-0292, Argonne National Lab., Argonne, IL, 1992.
31. _, Scalable iterative solution of sparse linear systems. Tech. Rep. MCS-P277-1191, Argonne National Lab., Argonne, IL, 1992.
32. R. M. KARP, Reducibility among combinatorial problems, in Complexity of Computer Computations, Plenum Press, New York, 1972, pp. 85-104.
33. R. LEUZE, Independent set orderings for parallel matrix factorizations by Gaussian elimination, Parallel Comput., 10 (1989), pp. 177-191.
34. J. G. LEWIS, B. W. PEYTON, AND A. POTHEN, A fast algorithm for reordering sparse matrices for parallel factorizations, SIAM J. Sci. Stat. Comput., 6 (1989), pp. 1146-1173.
35. S. MA AND Y. SAAD, Distributed ILU(0) and SOR Preconditioned for Unstructured Sparse Linear Systems, Tech. Rep. 94-027, Army High Performance Computing Research Center, University of Minnesota, Minneapolis, MN, 1994.
36. J. A. MEIJERINK AND H. A. VAN DER VORST, An iterative solution method for linear systems of which the coefficient matrix is asymmetric M-matrix, Math. Comp., 31 (1977), pp. 148-162.
37. J. M. ORTEGA, Introduction to Parallel and Vector Solution of Linear Systems, Plenum Press, New York, 1988.
38. O. OSTERBY AND Z. ZLATEV, Direct Methods for Sparse Matrices, Springer-Verlag, New York, 1983.
39. E. L. POOLE AND J. M. ORTEGA, Multicolor ICCG methods for vector computers, SIAM J. Numer. Anal., 24 (1987), pp. 1394-1418.
40. I. S. DUFF AND J. REID, The multifrontal solution of unsymmetric sets of linear equations, SIAM J. Sci. Stat. Comput., 5 (1984), pp. 633-641.
41. G. RODRIGUE AND D. WOUTZER, Preconditioning by incomplete block cyclic reduction, Math. Comp., 42 (1984), pp. 549-565.
42. Y. SAAD, Krylov subspace methods on supercomputers, SIAM J. Sci. Stat. Comput., 10 (1989), pp. 1200-1232.
43. _, A flexible inner-outer preconditioned GURES algorithm, SIAM J. Sci. Comput., 14 (1993), pp. 461-469.
44. _, Highly parallel preconditioners for general sparse matrices, in Recent Advances in Iterative Methods, IMA Volumes in Mathematics and its Applications, Vol. 60, G. Golub, M. Luskin, and A. Greenbaum, eds., Springer-Verlag, Berlin, 1994, pp. 165-199.
45. _, ILUT: A dual threshold incomplete ILU factorization, Numer. Linear Algebra Appl., 1 (1994), pp. 387-402.
46. F. SHAKIB, Finite element analysis of the compressible Euler and Navier Stokes Equations, Ph.D. thesis, Aeronautics Dept., Stanford University, CA, 1989.
47. H. A. VAN DER VORST, A vectorizable version of some ICCG methods, SIAM J. Sci. Stat. Comput., 3 (1982), pp. 350-356.
48. H. A. VAN DER VORST AND C. VUIK, GMRESR: A family of nested GMRES methods, Numer. Linear Algebra Appl., 1 (1994), pp. 369-386.
49. R. S. VARGA, Matrix Iterative Analysis, Prentice Hall, Englewood Cliffs, NJ, 1962.
50. V. VENKATAKRISHNAN, H. D. SIMON, AND T. J. BARTH, A MIMD Implementation of a Parallel Euler Solver for Unstructured Grids, Tech. Rep. RNR-91-024, NASA Ames Research Center, Moffett Field, CA, 1991.
51. D. M. YOUNG, Iterative Solution of Large Linear Systems, Academic Press, New York, 1971.
52. D. P. YOUNG, R. G. MELVIN, F. T. JOHNSON, J. E. BUSSOLETTI, L. B. WIGTON, AND S. S. SAMANT, Application of sparse matrix solvers as effective preconditioners, SIAM J. Sci. Stat. Comput., 10 (1989), pp. 1186-1199.
53. Z. ZLATEV, Use of iterative refinement in the solution of sparse linear systems, SIAM J. Numer. Anal., 19 (1982), pp. 381-399.