Partitioning sparse matrices for parallel preconditioned iterative methods

SIAM Journal on Scientific Computing

Volumn 29, Issue 4, 2007, Pages 1683-1709

  Uçar, Bora ^a   Aykanat, Cevdet ^b  

^a CERFACS   (France)

^b BILKENT UNIVERSITY   (Turkey)

Author keywords

Iterative method; Matrix partitioning; Parallel computing; Preconditioning

Indexed keywords

ITERATIVE METHODS;

APPROXIMATE INVERSE; MATRIX; MATRIX PARTITIONING; PARALLEL COM- PUTING; PARALLELIZATIONS; PARALLELIZING; PRECONDITIONED ITERATIVE METHODS; PRECONDITIONERS; PRECONDITIONING; SPARSE MATRICES;

MATRIX ALGEBRA;

EID: 41449101917     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/040617431     Document Type: Article

References (64)

1. C. ABABEI, N. SELVAKKUMARAN, K. BAZARGAN, AND G. KARYPIS, Multi-objective circuit partitioning for cutsize and path-based delay minimization, in Proceedings of the 2002 IEEE/ACM International Conference on Computer-aided Design (ICCAD '02), San Jose, CA, 2002, pp. 181-185.
2. C. AYKANAT, A. PINAR, AND Ü. V. ÇATALYÜREK, Permuting sparse rectangular matrices into block-diagonal form, SIAM J. Sci. Comput., 25 (2004), pp. 1860-1879.
3. R. BARRETT, M. W. BERRY, T. F. CHAN, J. DEMMEL, J. DONATO, J. DONGARRA, V. EIJKHOUT, R. POZO, C. ROMINE, AND H. A. VAN DER VORST, Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM, Philadelphia, 1994.
4. M. BENZI, Preconditioning techniques for large linear systems: A survey, J. Comput. Phys., 182 (2002), pp. 418-477.
5. M. BENZI, J. K. CULLUM, AND M. TŮMA, Robust approximate inverse preconditioning for the conjugate gradient method, SIAM J. Sci. Comput., 22 (2000), pp. 1318-1332.
6. M. BENZI, J. C. HAWS, AND M. TŮMA, Preconditioning highly indefinite and nonsymmetric matrices, SIAM J. Sci. Comput., 22 (2000), pp. 1333-1353.
7. M. BENZI, C. D. MEYER, AND M. TŮMA, A sparse approximate inverse preconditioner for the conjugate gradient method, SIAM J. Sci. Comput., 17 (1996), pp. 1135-1149.
8. M. BENZI AND M. TŮMA, A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput., 19 (1998), pp. 968-994.
9. M. BENZI AND M. TŮMA, A comparative study of sparse approximate inverse preconditioners, Appl. Numer. Math., 30 (1999), pp. 305-340.
10. M. BENZI AND M. TŮMA, A parallel solver for large-scale Markov chains, Appl. Numer. Math., 41 (2002), pp. 135-153.
11. M. BENZI AND M. TŮMA, Private communication, 2003.
12. L. BERGAMASCHI, G. PINI, AND F. SARTORETTO, Approximate inverse preconditioning in the solution of sparse eigenproblems, Numer. Linear Algebra Appl., 7 (2000), pp. 99-116.
13. R. H. BISSELING AND W. MEESEN, Communication balancing in parallel sparse matrix-vector multiplication, Electron. Trans. Numer. Anal., 21 (2005), pp. 47-65.
14. K. BROWN, S. ATTAWAY, S. J. PLIMPTON, AND B. HENDRICKSON, Parallel strategies for crash and impact simulations, Comput. Methods Appl. Mech. Engrg., 184 (2000), pp. 373-390.
15. G. BURNS, R. DAOUD, AND J. VAIGL, LAM: An open cluster environment for MPI, in Proceedings of Supercomputing Symposium '94, J. W. Ross, ed., University of Toronto, 1994, pp. 379-386.
16. U. V. ÇATALYÜREK, Hypergraph Models for Sparse Matrix Partitioning and Reordering, Ph.D. thesis, Computer Engineering and Information Science, Bilkent University, Ankara, Turkey, 1999.
17. U. V. ÇATALYÜREK AND C. AYKANAT, Decomposing irregularly sparse matrices for parallel matrix-vector multiplications, in Parallel Algorithms for Irregularly Structured Problems, Lecture Notes in Comput. Sci. 1117, Springer, Berlin, 1996, pp. 75-86.
18. U. V. ÇATALYÜREK AND C. AYKANAT, Hypergraph-partitioning based decomposition for parallel sparse-matrix vector multiplication, IEEE Trans. Parallel Distrib. Systems, 10 (1999), pp. 673-693.
19. Ü. V. ÇATALYÜREK AND C. AYKANAT, PaToH: A Multilevel Hypergraph Partitioning Tool, Version 3.0, Tech. Report BU-CE-9915, Computer Engineering Department, Bilkent University, Ankara, Turkey, 1999.
20. U. V. ÇATALYÜREK AND C. AYKANAT, A hypergraph-partitioning approach for coarse-grain decomposition, in Proceedings of Scientific Computing 2001 (SC2001), Denver, CO, 2001, pp. 10-16.
21. T. F. CHAN, E. CHOW, Y. SAAD, AND M. C. YEUNG, Preserving symmetry in preconditioned Krylov subspace methods, SIAM J. Sci. Comput., 20 (1998), pp. 568-581.
22. C. CHANG, T. KURC, A. SUSSMAN, Ü. V. ÇATALYÜREK, AND J. SALTZ, A hypergraph-based workload partitioning strategy for parallel data aggregation, in Proceedings of the Tenth SIAM Conference on Parallel Processing for Scientific Computing, Portsmouth, VA, CD-ROM, SIAM, Philadelphia, 2001.
23. E. CHOW, A priori sparsity patterns for parallel sparse approximate inverse preconditioners, SIAM J. Sci. Comput., 21 (2000), pp. 1804-1822.
24. E. CHOW, Parallel implementation and practical use of sparse approximate inverse preconditioners with a priori sparsity patterns, Internat. J. High Perform. Comput. Appl., 15 (2001), pp. 56-74.
25. E. CHOW AND Y. SAAD, Approximate inverse preconditioners via sparse-sparse iterations, SIAM J. Sci. Comput., 19 (1998), pp. 995-1023.
26. J. K. CULLUM, K. JOHNSON, AND M. TŮMA, Effects of problem decomposition on the convergence behavior of parallel numerical algorithms, Numer. Linear Algebra Appl., 10 (2003), pp. 445-465.
27. T. DAVIS, University of Florida Sparse Matrix Collection, http://www.cise.ufl.edu/research/sparse/matrices/.
28. R. W. FREUND, A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems, SIAM J. Sci. Comput., 14 (1993), pp. 470-482.
29. R. W. FREUND AND N. M. NACHTIGAL, A New Krylov-Subspace Method for Symmetric Indefinite Linear Systems, Tech. Report ORNL/TM-12754, Oak Ridge National Laboratories, 1994.
30. G. H. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., Johns Hopkins University Press, Baltimore, MD, 1996.
31. N. I. M. GOULD AND J. A. SCOTT, Sparse approximate-inverse preconditioners using normminimization techniques, SIAM J. Sci. Comput., 19 (1998), pp. 605-625.
32. M. J. GROTE, SPAI: SParse Approximate inverse Preconditioner, http://www.sam.math.ethz.ch/~grote/spai/.
33. M. J. GROTE AND T. HUCKLE, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838-853.
34. B. HENDRICKSON, Graph partitioning and parallel solvers: Has the emperor no clothes?, in Workshop on Parallel Algorithms lor Irregularly Structured Problems, Lecture Notes in Comput. Sci. 1457, Springer, Berlin, 1998, pp. 218-225.
35. B. HENDRICKSON AND K. DEVINE, Dynamic load balancing in computational mechanics, Comput. Methods Appl. Mech. Engrg., 184 (2000), pp. 485-500.
36. B. HENDRICKSON AND T. G. KOLDA, Graph partitioning models for parallel computing, Parallel Comput., 26 (2000), pp. 1519-1534.
37. B. HENDRICKSON AND T. G. KOLDA, Partitioning rectangular and structurally unsymmetric sparse matrices for parallel processing, SIAM J. Sci. Comput., 21 (2000), pp. 2048-2072.
38. C. HOOVER, A. DEGROOT, J. MALTBY, AND R. PROCASSINI, Paradyn: Dyna3d for Massively Parallel Computers, Presentation at Tri-Laboratory Engineering Conference on Computational Modeling, Pleasanton, CA, 1995.
39. T. HUCKLE, Approximate Sparsity Patterns for the Inverse of a Matrix and Preconditioning, Tech. Report 342/12/98 A, Technische Universität München, Germany, 1998.
40. G. KARYPIS, Multi-constraint Mesh Partitioning for Contact/Impact Computations, Tech. Report 03-022, Department of Computer Science and Engineering/Army HPC Research Center, University of Minnesota, Minneapolis, MN, 2003.
41. G. KARYPIS AND V. KUMAR, Multilevel Algorithms for Multi-constraint Graph Partitioning, Tech. Report 98-019, Department of Computer Science/Army HPC Research Center, University of Minnesota, Minneapolis, MN, 1998.
42. G. KARYPIS AND V. KUMAR, Multilevel Algorithms for Multi-constraint Hypergraph Partitioning, Tech. Report 99-034, Department of Computer Science/Army HPC Research Center, University of Minnesota, Minneapolis, MN, 1998.
43. T. G. KOLDA, Partitioning sparse rectangular matrices for parallel processing, in Solving Irregularly Structured Problems in Parallel, Lecture Notes in Comput. Sci. 1457, Springer, Berlin, 1998, pp. 68-79.
44. L. YU. KOLOTILINA AND A. YU. YEREMIN, Factorized sparse approximate inverse preconditionings I. Theory, SIAM J. Matrix Anal. Appl., 14 (1993), pp. 45-58.
45. L. YU. KOLOTILINA AND A. YU. YEREMIN, Factorized sparse approximate inverse preconditioning II: Solution of 3D FE systems on massively parallel computers, Internat. J. High Speed Comput., 7 (1995), pp. 191-216.
46. T. LENGAUER, Combinatorial Algorithms for Integrated Circuit Layout, Wiley-Teubner, Chichester, UK, 1990.
47. K. MCMANUS, M. CROSS, C. WALSHAW, S. JOHNSON, AND P. LEGGETT, A scalable strategy for the parallelization of multiphysics unstructured mesh-iterative codes on distributed-memory systems, Internat. J. High Perform. Comput. Appl., 14 (2000), pp. 137-174.
48. G. MEURANT, Computer Solution of Large Linear Systems, Elsevier, Amsterdam, The Netherlands, 1999.
49. A. PINAR, Ü. V. ÇATALYÜREK, C. AYKANAT, AND M. PINAR, Decomposing linear programs for parallel solution, in Proceedings of the Second International Workshop on Applied Parallel Computing in Physics, Chemistry and Engineering Science, Lecture Notes in Comput. Sci. 1041, Springer, Berlin, 1996, pp. 473-482.
50. S. PLIMPTON, S. ATTAWAY, B. HENDRICKSON, J. SWEGLE, C. VAUGHAN, AND D. GARDNER, Transient dynamics simulations: Parallel algorithms for contact detection and smoothed particle hydrodynamics, J. Parallel Distrib. Comput., 50 (1998), pp. 104-122.
51. S. PLIMPTON, B. HENDRICKSON, AND J. STEWART, A parallel rendezvous algorithm for interpolation between multiple grids, in Proceedings of the 1998 ACM/IEEE Conference on Super-computing, San Jose, CA, 1998, CD-ROM, pp. 1-8.
52. Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS Publishing, Boston, MA, 1996.
53. K. SCHLOEGEL, G. KARYPIS, AND V. KUMAR, A New Algorithm for Multi-objective Graph Partitioning, Tech. Report 99-033, Department of Computer Science and Engineering, University of Minnesota, Minneapolis, MN, 1999.
54. K. SCHLOEGEL, G. KARYPIS, AND V. KUMAR, Graph partitioning for high-performance scientific simulations, in CRPC Parallel Computing Handbook, J. Dongarra, I. Foster, G. Fox, W. Gropp, K. Kennedy, L. Torczon, and A. White, eds., Morgan Kaufmann, San Francisco, CA, 2002, Chap. 18, pp. 491-541.
55. N. SELVAKKUMARAN AND G. KARYPIS, Multi-objective hypergraph partitioning algorithms for cut and maximum subdomain degree minimization, in Proceedings of the 2003 IEEE/ACM International Conference on Computer-Aided Design (ICCAD '03), San Jose, CA, 2003, pp. 726-733.
56. T. STERLING, D. SAVARESE, D. J. BECKER, J. E. DORBAND, U. A. RANAWEKE, AND C. V. PACKER, BEOWULF: A parallel workstation for scientific computation, in Proceedings of the 24th International Conference on Parallel Processing, Oconomowoc, WI, 1995, pp. 11-34.
57. R. S. TUMINARO, J. N. SHADID, AND S. A. HUTCHINSON, Parallel sparse matrix vector multiply software for matrices with data locality, Concurrency: Practice and Experience, 10 (1998), pp. 229-247.
58. B. UÇAR AND C. AYKANAT, Minimizing communication cost in fine-grain partitioning of sparse matrices, in Proceedings of the 18th International Symposium on Computer and Information Sciences (ISCIS XVIII), A. Yazici and C. Şener, eds., Antalya, Turkey, 2003.
59. B. UÇAR AND C. AYKANAT, Encapsulating multiple communication-cost metrics in partitioning sparse rectangular matrices for parallel matrix-vector multiplies, SIAM J. Sci. Comput., 25 (2004), pp. 1837-1859.
60. B. UÇAR AND C. AYKANAT, Experiments on Hypergraph Models for Parallelizing Preconditioned Iterative Methods, Tech. Report BU-CE-0413, Department of Computer Engineering, Bilkent University, Ankara, Turkey, 2004.
61. B. UÇAR AND C. AYKANAT, A Library for Parallel Sparse Matrix-Vector Multiplies, Tech. Report BU-CE-0506, Department of Computer Engineering, Bilkent University, Ankara, Turkey, 2005.
62. H. A. VAN DER VORST, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Statist. Comput., 13 (1992), pp. 631-644.
63. B. VASTENHOUW AND R. H. BISSELING, A two-dimensional data distribution method for parallel sparse matrix-vector multiplication, SIAM Rev., 47 (2005), pp. 67-95.
64. C. WALSHAW, M. CROSS, AND K. MCMANUS, Multiphase mesh partitioning, Appl. Math. Modeling, 25 (2000), pp. 123-140.