A latency tolerant hybrid sparse solver using incomplete Cholesky factorization

Numerical Linear Algebra with Applications

Volumn 10, Issue 5-6, 2003, Pages 541-560

  Raghavan, Padma ^a   Teranishi, Keita ^a   Ng, Esmond G ^b  

^a PENNSYLVANIA STATE UNIVERSITY   (United States)

^b LAWRENCE BERKELEY NATIONAL LABORATORY   (United States)

Author keywords

Conjugate gradient; Incomplete Cholesky; Preconditioners; Selective inversion; Sparse matrix factorization

Indexed keywords

EID: 0042843372     PISSN: 10705325     EISSN: None     Source Type: Journal    
DOI: 10.1002/nla.327     Document Type: Article

References (36)

1. Hestenes MR, Stiefel E. Methods of conjugate gradients for solving linear systems. Journal of Research of the National Bureau of Standards 1952; 49:409-436.
2. Golub G, O'Leary D. Some history of the conjugate gradient and Lanczos methods. SIAM Review 1989; 31: 50-102.
3. Meijerink JA, Van der Vorst, HA. An iterative solution methods for which the coefficient matrix is a symmetric M-matrix. Mathematics of Computation 1977; 31:148-162.
4. Ng E, Peyton B, Raghavan P. A blocked incomplete cholesky preconditioner for hierarchical-memory computers. In Iterative Methods in Scientific Computation IV, D. R. Kincaid, A. C. Elster (eds). IMACS: Rutgers University, NJ 08903, 1999; 211-222.
5. Benzi M, Cullum J, Tuma M. Robust approximate inverse preconditioning for the conjugate gradient method. Technical Report LA-UR-99-2899. Los Alamos National Laboratory, Los Alamos, MM, 1999.
6. Heath M, Raghavan P. Parallel sparse triangular solution. In IMA Volumes in Mathematics and its Applications, vol. 105, R. Gulliver, M. Heath, R. Schreiber, P. Bjorstad (eds). Springer-Verlag, Berlin, 1998; 289-306.
7. Grote M, Huckle T. Parallel preconditioning with sparse approximate inverses. SIAM Journal on Scientific Computing 1997; 18:838-853.
8. Kolotolina LY, Yeremin AY. Factorized sparse approximate inverse preconditionings. SILM Journal on Matrix Analysis and Applications 1993; 14:45-58.
9. Chow E. Parallel implementation and performance characteristics of least squares sparse approximate inverse preconditioners. Technical Report UCRL-JC-138883. Lawrence Livermore National Laboratory, Livermore, CA, 2000.
10. Raghavan, P. Efficient parallel triangular solution with selective inversion. Parallel Processing Letters 1998; 8(1):29-40.
11. Chow E. ParaSails: Parallel sparse approximate inverse (least-squares) preconditioner. Software distributed by the author [2000].
12. Campbell Y, Davis TA. Incomplete LU factorization: a multifrontal approach. Technical Report TR-95-024. Computer and Information Science Department, University of Florida, Gainesville FL 32611, October 1994.
13. Chan T. Fourier analysis of relaxed incomplete factorization preconditioners. SIAM Journal on Statistical and Scientific Computing 1991; 12:668-680.
14. Chan TF, Vassilevski PS. A framework for block ILU factorizations using block-size reduction. Mathematics of Computation 1995; 64:129-156.
15. Gustafsson I. A class of first order factorization methods. BIT 1978; 18:142-156.
16. Jones MT, Plassman PE. An improved incomplete Cholesky factorization. ACM Transactions on Mathematical Software 1995; 21:5-17.
17. Munksgaard N. Solving sparse symmetric sets of linear equations by preconditioned conjugate gradients. ACM Transactions an Mathematical Software 1980; 6:206-219.
18. Saad Y. Preconditioning techniques for indefinite and nonsymmetric linear systems. Journal of Computational and Applied Mathematics 1988; 24:89-105.
19. Ashcraft C. SParse Object Oriented Linear Equations Solver, http://netlib.ccpl4.ac.uk/linalg/spooles/spooles. 2.2.html [1999].
20. Gupta A, Kumar V. A scalable parallel algorithm for sparse matrix factorization. Technical Report 94-19. Department of Computer Science, University of Minnesota, Minneapolis, MN, 1994. A short version of the paper appears in Proceedings of Supercomputing '94, November 1994, Washington DC.
21. Gupta A, Rothberg E, Ng E, Peyton BW. Parallel sparse Cholesky factorization algorithms for shared-memory multiprocessor systems. In Advances in Computer Methods for Partial Differential Equations-VII (1992), R. Vichnevetsky, D. Knight, G. Richter (eds). International Association for Mathematics and Computers in Simulation (IMACS); New Brunswick. NJ, 1992; 622-628.
22. Heath MT, Raghavan P. A Cartesian nested dissection algorithm. SIAM Journal on Matrix Analysis and Applications 1995; 16(1):235-253.
23. Raghavan P. Distributed sparse Gaussian elimination and orthogonal factorization. SIAM Journal on Scientific Computing 1995: 16(6): 1462-1477.
24. Rothberg E. Performance of panel and block approaches to sparse Cholesky factorization on the iPSC/860 and Paragon multiprocessors. Technical Report Intel Supercomputer Systems Division, 14924 N. W. Greenbrier Parkway, Beaverton, OR 97006, September 1993.
25. Ng E, Raghavan P. Towards a scalable hybrid sparse solver. Concurrency: Practice and Experience 2000; 12:1-16.
26. George JA, Liu J, Ng E. Communication results for parallel sparse Cholesky factorization on a hypercube. Parallel Computing 1989; 10:287-298.
27. Ashcraft C, Eisenstat S, Liu JW-H, Peyton B, Sherman A. A compute-ahead implementation of the fan-in sparse distributed factorization scheme. Technical Report ORNL/TM-11496. Oak Ridge National Laboratory, Oak Ridge, TN, 1990.
28. Duff IS, Reid JK. The multifrontal solution of indefinite sparse symmetric linear equations. ACM Transactions on Mathematical Software 1983; 9:302-325.
29. Heath M, Romine C. Parallel solution of triangular systems on distributed-meraory multiprocessors. SIAM Journal on Statistical and Scientific Computing 1988; 9:558-588.
30. George A. Nested dissection of a regular finite element mesh. SIAM Journal an Numerical Analysis 1973; 10:345-363.
31. Raghavan P. DSCPACK: A domain-separator Cholesky package.
32. Balay S, Gropp W, McInnes LC, Smith B. PETSc 2.0 Users Manual. Technical Report ANL-95-11. Mathematics and Computer Science Division, Argonne National Laboratory, April 1999.
33. Forum MPI. MPI: A message-passing interface standard. Technical Report CS-94-230. Computer Science Department, University of Tennessee, Knoxville, April 1994.
34. Lewis JG. Cruising (approximately) at 41,000 feet-iterative methods at Boeing. Presentation at Seventh SIAM Conference on Applied Linear Algebra, Raleigh, NC, 2000.
35. Anderson E, Bai Z, Bischof C, Demmel J, Dongarra JJ, DuCroz J, Greenbaum A, Hammarling s, McKenney A, Ostrouchov S, Sorensen D. LAPACK Users' Guide (2nd edn). SIAM Publications: Philadelphia. PA, 1995.
36. Dongarra JJ, Croz J, Hammarling S, Duff IS. An extended set of basic linear algebra subprograms. ACM Transactions on Mathematical Software 1988; 14:1-17.