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A column pre-ordering strategy for the unsymmetric-pattern multifrontal method

Author:

Timothy A. DavisAuthors Info & Claims

ACM Transactions on Mathematical Software (TOMS), Volume 30, Issue 2

Pages 165 - 195

https://doi.org/10.1145/992200.992205

Published: 01 June 2004 Publication History

Abstract

A new method for sparse LU factorization is presented that combines a column pre-ordering strategy with a right-looking unsymmetric-pattern multifrontal numerical factorization. The column ordering is selected to give a good a priori upper bound on fill-in and then refined during numerical factorization (while preserving the bound). Pivot rows are selected to maintain numerical stability and to preserve sparsity. The method analyzes the matrix and automatically selects one of three pre-ordering and pivoting strategies. The number of nonzeros in the LU factors computed by the method is typically less than or equal to those found by a wide range of unsymmetric sparse LU factorization methods, including left-looking methods and prior multifrontal methods.

References

[1]

Amestoy, P. R., Davis, T. A., and Duff, I. S. 1996a. An approximate minimum degree ordering algorithm. SIAM J. Matrix Anal. Applic. 17, 4, 886--905.

[2]

Amestoy, P. R., Davis, T. A., and Duff, I. S. 2004. Algorithm: AMD, an approximate minimum degree ordering algorithm. ACM Trans. Math. Softw., to appear.

[3]

Amestoy, P. R., Davis, T. A., Duff, I. S., and Hadfield, S. M. 1996b. Parallelism in multifrontal methods for matices with unsymmetric structures. In Abstracts of the Second SIAM Conference on Sparse Matrices. Snowbird, Utah.

[4]

Amestoy, P. R. and Duff, I. S. 1989. Vectorization of a multiprocessor multifrontal code. Intl. J. Supercomputer Appl. 3, 3, 41--59.

[5]

Amestoy, P. R., Duff, I. S., L'Excellent, J.-Y., and Koster, J. 2001a. A fully asynchronous multifrontal solver using distributed dynmamic scheduling. SIAM J. Matrix Anal. Applic. 23, 1, 15--41.

[6]

Amestoy, P. R., Duff, I. S., L'Excellent, J.-Y., and Li, X. S. 2001b. Analysis and comparison of two general sparse solvers for distributed memory computers. ACM Trans. Math. Softw. 27, 4, 388--421.

[7]

Amestoy, P. R., Duff, I. S., and Puglisi, C. 1996a. Multifrontal QR factorization in a multiprocessor environment. Numer. Linear Algebra Appl. 3, 4, 275--300.

[8]

Amestoy, P. R. and Puglisi, C. 2002. An unsymmetrized multifrontal LU factorization. SIAM J. Matrix Anal. Applic. 24, 553--569.

[9]

Arioli, M., Demmel, J. W., and Duff, I. S. 1989. Solving sparse linear systems with sparse backward error. SIAM J. Matrix Anal. Applic. 10, 2, 165--190.

[10]

Ashcraft, C. C. and Grimes, R. 1989. The influence of relaxed supernode partitions on the multifrontal method. ACM Trans. Math. Softw. 15, 4, 291--309.

[11]

Ashcraft, C. C. and Liu, J. W. H. 1998. Robust ordering of sparse matrices using multisection. SIAM J. Matrix Anal. Applic. 19, 3, 816--832.

[12]

Berger, A. J., Mulvey, J. M., Rothberg, E., and Vanderbei, R. J. 1995. Solving multistage stochastic programs using tree dissection. Tech. Rep. SOR-95-07, Dept. of Civil Eng. and Operations Research, Princeton Univ., Princeton, NJ. June.

[13]

Brainman, I. and Toledo, S. 2002. Nested-dissection orderings for sparse LU with partial pivoting. SIAM J. Matrix Anal. Applic. 23, 998--1012.

[14]

Bunch, J. R. and Parlett, B. N. 1971. Direct methods for solving symmetric indefinite systems of linear equations. SIAM J. Numer. Anal. 8, 639--655.

[15]

Coleman, T. F., Edenbrandt, A., and Gilbert, J. R. 1986. Predicting fill for sparse orthogonal factorization. J. ACM 33, 517--532.

[16]

Davis, T. A. University of Florida sparse matrix collection. www.cise.ufl.edu/research/sparse.

[17]

Davis, T. A. 2004. Algorithm 832: UMFPACK V4.3, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30, 2.

[18]

Davis, T. A. and Duff, I. S. 1997. An unsymmetric-pattern multifrontal method for sparse LU factorization. SIAM J. Matrix Anal. Applic. 18, 1, 140--158.

[19]

Davis, T. A. and Duff, I. S. 1999. A combined unifrontal/multifrontal method for unsymmetric sparse matrices. ACM Trans. Math. Softw. 25, 1, 1--19.

[20]

Davis, T. A., Gilbert, J. R., Larimore, S. I., and Ng, E. G. 2004a. Algorithm: COLAMD, a column approximate minimum degree ordering algorithm. ACM Trans. Math. Softw., to appear.

[21]

Davis, T. A., Gilbert, J. R., Larimore, S. I., and Ng, E. G. 2004b. A column approximate minimum degree ordering algorithm. ACM Trans. Math. Softw., to appear.

[22]

Demmel, J. W., Eisenstat, S. C., Gilbert, J. R., Li, X. S., and Liu, J. W. H. 1999. A supernodal approach to sparse partial pivoting. SIAM J. Matrix Anal. Applic. 20, 3, 720--755.

[23]

Dongarra, J. J., Du Croz, J. J., Duff, I. S., and Hammarling, S. 1990. A set of Level 3 Basic Linear Algebra Subprograms. ACM Trans. Math. Softw. 16, 1--17.

[24]

Duff, I. S. 1977. On permutations to block triangular form. J. Inst. of Math. Appl. 19, 339--342.

[25]

Duff, I. S. 1981. On algorithms for obtaining a maximum transversal. ACM Trans. Math. Softw. 7, 315--330.

[26]

Duff, I. S. 1984. The solution of nearly symmetric sparse linear systems. In Computing methods in applied sciences and engineering, VI, R. Glowinski and J.-L. Lions, Eds. North Holland, Amsterdam New York and London, 57--74.

[27]

Duff, I. S. 2002. A new code for the solution of sparse symmetric definite and indefinite systems. Tech. Rep. TR-2002-024, Rutherford Appleton Laboratory.

[28]

Duff, I. S., Erisman, A. M., and Reid, J. K. 1986. Direct Methods for Sparse Matrices. London: Oxford Univ. Press.

[29]

Duff, I. S. and Koster, J. 2001. On algorithms for permuting large entries to the diagonal of a sparse matrix. SIAM J. Matrix Anal. Applic. 22, 4, 973--996.

[30]

Duff, I. S. and Reid, J. K. 1982. MA27 - a set of Fortran subroutines for solving sparse symmetric sets of linear equations. Tech. Rep. AERE-R-10533, AERE Harwell Laboratory, United Kingdom Atomic Energy Authority.

[31]

Duff, I. S. and Reid, J. K. 1983a. The multifrontal solution of indefinite sparse symmetric linear systems. ACM Trans. Math. Softw. 9, 302--325.

[32]

Duff, I. S. and Reid, J. K. 1983b. A note on the work involved in no-fill sparse matrix factorization. IMA J. Num. Anal. 3, 37--40.

[33]

Duff, I. S. and Reid, J. K. 1984. The multifrontal solution of unsymmetric sets of linear equations. SIAM J. Sci. Statist. Comput. 5, 3, 633--641.

[34]

Duff, I. S. and Reid, J. K. 1996. The design of MA48, a code for the direct solution of sparse unsymmetric linear systems of equations. ACM Trans. Math. Softw. 22, 2, 187--226.

[35]

Duff, I. S. and Scott, J. A. 1996. The design of a new frontal code for solving sparse unsymmetric systems. ACM Trans. Math. Softw. 22, 1, 30--45.

[36]

Eisenstat, S. C. and Liu, J. W. H. 1992. Exploiting structural symmetry in unsymmetric sparse symbolic factorization. SIAM J. Matrix Anal. Applic. 13, 1, 202--211.

[37]

George, A. and Liu, J. W. H. 1981. Computer Solution of Large Sparse Positive Definite Systems. Englewood Cliffs, New Jersey: Prentice-Hall.

[38]

George, A. and Liu, J. W. H. 1989. The evolution of the minimum degree ordering algorithm. SIAM Rev. 31, 1, 1--19.

[39]

George, A. and Ng, E. G. 1985. An implementation of Gaussian elimination with partial pivoting for sparse systems. SIAM J. Sci. Statist. Comput. 6, 2, 390--409.

[40]

George, A. and Ng, E. G. 1987. Symbolic factorization for sparse Gaussian elimination with partial pivoting. SIAM J. Sci. Statist. Comput. 8, 6 (Nov.), 877--898.

[41]

Gilbert, J. R., Li, X. S., Ng, E. G., and Peyton, B. W. 2001. Computing row and column counts for sparse QR and LU factorization. BIT 41, 4, 693--710.

[42]

Gilbert, J. R., Moler, C., and Schreiber, R. 1992. Sparse matrices in MATLAB: design and implementation. SIAM J. Matrix Anal. Applic. 13, 1, 333--356.

[43]

Gilbert, J. R. and Peierls, T. 1988. Sparse partial pivoting in time proportional to arithmetic operations. SIAM J. Sci. Statist. Comput. 9, 862--874.

[44]

Goto, K. and van de Geijn, R. 2002. On reducing TLB misses in matrix multiplication, FLAME working note 9. Tech. Rep. TR-2002-55, The University of Texas at Austin, Department of Computer Sciences.

[45]

Gupta, A. 2002. Improved symbolic and numerical factorization algorithms for unsymmetric sparse matrices. SIAM J. Matrix Anal. Applic. 24, 529--552.

[46]

Gupta, A., Gustavson, F., Joshi, M., Karypis, G., and Kumar, V. 1999. PSPASES: an efficient and scalable parallel sparse direct solver. In Kluwer Intl. Series in Engineering and Science, T. Yang, Ed. Vol. 515. Kluwer.

[47]

Hadfield, S. M. 1994. On the LU factorization of sequences of identically structured sparse matrices within a distributed memory environment. Ph.D. thesis, University of Florida, Gainesville, FL.

[48]

Hadfield, S. M. and Davis, T. A. 1995. The use of graph theory in a parallel multifrontal method for sequences of unsymmetric pattern sparse matrices. Cong. Numer. 108, 43--52.

[49]

Heath, M. T. and Raghavan, P. 1995. A Cartesian parallel nested dissection algorithm. SIAM J. Matrix Anal. Applic. 16, 1, 235--253.

[50]

Hendrickson, B. and Rothberg, E. 1999. Improving the runtime and quality of nested dissection ordering. SIAM J. Sci. Comput. 20, 468--489.

[51]

Karypis, G. and Kumar, V. 1998. A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20, 359--392.

[52]

Larimore, S. I. 1998. An approximate minimum degree column ordering algorithm. Tech. Rep. TR-98-016, Univ. of Florida, Gainesville, FL. www.cise.ufl.edu.

[53]

Liu, J. W. H. 1990. The role of elimination trees in sparse factorization. SIAM J. Matrix Anal. Applic. 11, 1, 134--172.

[54]

Liu, J. W. H. 1992. The multifrontal method for sparse matrix solution: Theory and practice. SIAM Rev. 34, 1, 82--109.

[55]

Matstoms, P. 1994. Sparse QR factorization in MATLAB. ACM Trans. Math. Softw. 20, 1, 136--159.

[56]

Ng, E. G. and Raghavan, P. 1999. Performance of greedy ordering heuristics for sparse Cholesky factorization. SIAM J. Matrix Anal. Applic. 20, 4, 902--914.

[57]

Rothberg, E. and Eisenstat, S. C. 1998. Node selection strategies for bottom-up sparse matrix orderings. SIAM J. Matrix Anal. Applic. 19, 3, 682--695.

[58]

Whaley, R. C., Petitet, A., and Dongarra, J. J. 2001. Automated emperical optimization of software and the ATLAS project. Parallel Computing 27, 1-2, 3--35.

[59]

Yannakakis, M. 1981. Computing the minimum fill-in is NP-Complete. SIAM J. Alg. Disc. Meth. 2, 77--79.

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Index Terms

A column pre-ordering strategy for the unsymmetric-pattern multifrontal method
1. Computing methodologies
  1. Symbolic and algebraic manipulation
    1. Symbolic and algebraic algorithms
      1. Linear algebra algorithms
2. Mathematics of computing
  1. Mathematical analysis
    1. Numerical analysis
      1. Computations on matrices
  2. Mathematical software

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Published In

cover image ACM Transactions on Mathematical Software

ACM Transactions on Mathematical Software Volume 30, Issue 2

June 2004

142 pages

ISSN:0098-3500

EISSN:1557-7295

DOI:10.1145/992200

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Copyright © 2004 ACM.

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 June 2004

Published in TOMS Volume 30, Issue 2

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Manimaran JBaskar AParthiban VShangerganesh L(2024)Analytical and numerical studies of a cancer invasion model with nonlocal diffusionJournal of Nonlinear, Complex and Data Science10.1515/jncds-2023-0021Online publication date: 8-Aug-2024
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