Extensions of certain graph-based algorithms for preconditioning

SIAM Journal on Scientific Computing

Volumn 29, Issue 5, 2007, Pages 2144-2161

  Fritzsche, David ^a   Frommer, Andreas ^b   Szyld, Daniel B ^a  

^a Temple University   (United States)

^b UNIVERSITY OF WUPPERTAL   (Germany)

Author keywords

Block Gauss Seidel; Block partitioning; ILU decomposition; PABLO; Preconditioning; TPABLO

Indexed keywords

GAUSSIAN DISTRIBUTION; GRAPHIC METHODS; JACOBIAN MATRICES;

BLOCK DIAGONAL; BLOCK GAUSS-SEIDEL; BLOCK PARTITIONING; GAUSS-SEIDEL; GRAPH-BASED ALGORITHMS; ILU DECOMPOSITION; PABLO; PRECONDITIONERS; PRECONDITIONING; TPABLO;

LINEAR SYSTEMS;

EID: 52649142658     PISSN: 10648275     EISSN: None     Source Type: Journal    
DOI: 10.1137/060661284     Document Type: Conference Paper

References (37)

1. G. ALEFELD, Über die Durchführbarkeit des Gaufischen Algorithmus bei Gleichungen mit Intervallen als Koeffizienten (On the feasibility of the Gaussian algorithm for equations with interval coefficients), Computing Supplementum, 1 (1977), pp. 15-19 (in German).
2. E. ANDERSON, Z. BAI, C. BISCHOF, S. BLACKFORD, J. DEMMEL, J. DONGARRA, J. Du CROZ, A. GREENBAUM, S. HAMMARLING, A. MCKENNEY, AND D. SORENSEN, LAPACK Users' Guide, 3rd ed., Software Environ. Tools 9, SIAM, Philadelphia, 1999.
3. M. ARIOLI, I. DUFF, AND D. RUIZ, Stopping criteria for iterative solvers, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 138-144.
4. M. BENZI, Preconditioning techniques for large linear systems: A survey, J. Comput. Phys., 182 (2002), pp. 418-477.
5. M. BENZI, H. CHOI, AND D. B. SZYLD, Threshold ordering for preconditioning nonsymmetric problems, in Scientific Computing, Proceedings of the Workshop, (Hong Kong, 1997), G. Golub, S.-H. Lui, F. Luk, and R. Plemmons, eds., Springer-Verlag, Singapore, 1997, pp. 159-165.
6. M. BENZ I, J. C. HAWS, AND M. TŮMA, Preconditioning highly indefinite and nonsymmetric matrices, SIAM J. Sci. Comput., 22 (2000), pp. 1333-1353.
7. M. BENZI, W. JOUBERT, AND G. MATEESCU, Numerical experiments with parallel orderings for ILU preconditioners, Electron. Trans. Numer. Anal., 8 (1999), pp. 88-114.
8. M. BENZI, D. B. SZYLD, AND A. van DUIN, Orderings for incomplete factorization preconditioning of nonsymmetric problems, SIAM J. Sci. Comput., 20 (1999), pp. 1652-1670.
9. M. BENZI AND M. TUMA, Orderings for factorized sparse approximate inverse preconditioners, SIAM J. Sci. Comput., 21 (2000), pp. 1851-1868.
10. A. BERMAN AND R. J. PLEMMONS, Nonnegative Matrices in the Mathematical Sciences, Classics Appl. Math., SIAM, Philadelphia, 1994. Originally published by Academic Press, New York, 1979.
11. M. BOLLHÖFER AND Y. SAAD, ILU PACK-preconditioning software package, release ul. 0, May 14, 2004, available online at http://www.tu-berlin.de/ilupack/.
12. A. BUNSE-GERSTNER AND R. STÖVER, On a conjugate gradient-type method for solving complex symmetric linear systems, Linear Algebra Appl., 287 (1999), pp. 105-123.
13. H. CHOI AND D. B. SZYLD, Application of threshold partitioning of sparse matrices to Markov chains, in IEEE International Computer Performance and Dependability Symposium IPDS'96, Urbana-Champaign, I L, 1996, pp. 158-165.
14. H. CHOI AND D. B. SZYLD, Threshold Ordering for Preconditioning Nonsymmetric Problems with Highly Varying Coefficients, Tech. report 96-51, Department of Mathematics, Temple University, Philadelphia, PA, 1996.
15. T. H. GORMEN C. E. LEISERSON, R. L. RIVEST, AND C. STEIN, Introduction to Algorithms, 2nd ed., MIT Press, Cambridge, MA, 2001.
16. T. A. DAVIS, Algorithm 832: UMFPACK V4.3: An unsymmetric-pattern multifrontal method, ACM Trans. Math. Software, 30 (2004), pp. 196-199. Also available online at http://www.cise.ufl.edu/research/ sparse/urnfpack/.
17. T. A. DAVIS, University of Florida Sparse Matrix Collection. Available online at http://www.cise.ufi.edu/research/sparse/ matrices/.
18. I. S. DUFF, MA57-A code for the solution of sparse symmetric definite and indefinite systems, ACM Trans. Math. Software, 30 (2004), pp. 118-144.
19. I. S. DUFF, A. M. ERISMAN, AND J. K. REID, Direct Methods for Sparse Matrices, Monographs on Numerical Analysis, Clarendon Press, Oxford, UK, 1986.
20. I. S. DUFF AND J. KOSTER, The design and use of algorithms for permuting large entries to the diagonal of sparse matrices, SIAM J. Matrix Anal. Appl., 20 (1999), pp. 889-901.
21. I. S. DUFF AND J. KOSTER, On algorithms for permuting large entries to the diagonal of a sparse matrix, SIAM J. Matrix Anal. Appl., 22 (2001), pp. 973-996.
22. I. S. DUFF AND J. A. SCOTT, A parallel direct solver for large sparse highly unsymmetric linear systems, ACM Trans. Math. Software, 30 (2004), pp. 95-117.
23. L. C. DUTTO, W. G. HABASHI, AND M. FORTIN, Parallelizable block diagonal preconditioners for the compressible Navier-Stokes equations, Comput. Methods Appl. Mech. Engrg., 117 (1994), pp. 15-47.
24. L. C. DUTTO, The effect of ordering on preconditioned GMRES algorithm, for solving the compressible Navier-Stokes equations, Internat. J. Numer. Methods Engrg., 36 (1993), pp. 457-497.
25. S. C. EISENSTAT, Efficient implementation of a class of preconditioned conjugate gradient methods, SIAM J. Sci. Statist. Comput., 2 (1981), pp. 1-4.
26. S. FcISCHER, Uber auf Takagi-Faktorisierungen beruhende Präkondilionierer für CSYM (On Takagi-factorization based preconditioners for CSYM), Ph.D. thesis, Department of Mathematics and Science, University of Wuppertal, Wuppertal, Germany, 2006 (in German).
27. S. FISCHER AND A. FROMMER, Preconditioning CSYM, in preparation.
28. D. FRITZSCHE, Graph theoretical methods for preconditioners, Master's thesis, Department of Mathematics and Science, University of Wuppertal, Wuppertal, Germany, 2004.
29. A. GREENBAUM, Iterative Methods for Solving Linear Systems, Frontiers Appl. Math. 17, SIAM, Philadelphia, 1997.
30. J. O'NEIL AND D. B. SZYLD, A block ordering method for sparse matrices, SIAM J. Sci. Statist. Comput., 11 (1990), pp. 811-823.
31. J. M. ORTEGA, Efficient implementations of certain iterative methods, SIAM J. Sci. Statist. Comput., 9 (1988), pp. 882-891.
32. J. M. ORTEGA, Introduction to Parallel and Vector Solution of Linear Systems, Plenum Press, New York, 1988.
33. C. C. PAIGE AND Z. STRAKOŠ, Residual and backward error bounds in minimum residual Krylov subspace methods, SIAM J. Sci. Comput., 23 (2002), pp. 1898-1923.
34. Y. SAAD, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003.
35. O. SCHENK, S. RÖLLIN, AND A. GUPTA, The effects of unsymmetric matrix permutations and scalings in semiconductor device and circuit simulation, IEEE Transactions on CAD of Integrated Circuits and Systems, 23 (2004), pp. 400-411.
36. V. SIMONCINI AND D. B. SZYLD, Recent computational developments in Krylov subspace methods for linear systems, Numer. Linear Algebra Appl., 14 (2007), pp. 1-59.
37. R. S. TUMINARO, M. HEROUX, S. A. HUTCHINSON, AND J. N. SHADID, Official Aztec User Guide, version 2.1, Tech. report SAND99-8801J, Sandia National Laboratory, Albuquerque, NM, 1999. Available online at http://www.cs.sandia.gov/CRF/aztecl.html.