Abstract

This paper presents a parallel preconditioning method for distributed sparse linear systems, based on an approximate inverse of the original matrix, that adopts a general framework of distributed sparse matrices and exploits domain decomposition (DD) and low-rank corrections. The DD approach decouples the matrix and, once inverted, a low-rank approximation is applied by exploiting the Sherman--Morrison--Woodbury formula, which yields two variants of the preconditioning methods. The low-rank expansion is computed by the Lanczos procedure with reorthogonalizations. Numerical experiments indicate that, when combined with Krylov subspace accelerators, this preconditioner can be efficient and robust for solving symmetric sparse linear systems. Comparisons with pARMS, a DD-based parallel incomplete LU (ILU) preconditioning method, are presented for solving Poisson's equation and linear elasticity problems.

Keywords

  1. Sherman--Morrison--Woodbury formula
  2. low-rank approximation
  3. distributed sparse linear systems
  4. parallel preconditioner
  5. incomplete LU factorization
  6. Krylov subspace method
  7. domain decomposition

MSC codes

  1. 65F08
  2. 65N22
  3. 65N55
  4. 65Y05
  5. 65Y20

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
GLVis: Accurate Finite Element Visualization, http://glvis.org.
2.
MFEM: Modular Finite Element Methods, http://mfem.org.
3.
S. Ambikasaran and E. Darve, An $\mathcal{O}(n\log n)$ fast direct solver for partial hierarchically semi-separable matrices, J. Sci. Comput., 57 (2013), pp. 477--501.
4.
P. R. Amestoy, T. A. Davis, and I. S. Duff, An approximate minimum degree ordering algorithm, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 886--905, https://doi.org/10.1137/S0895479894278952.
5.
P. R. Amestoy, T. A. Davis, and I. S. Duff, Algorithm 837: An approximate minimum degree ordering algorithm, ACM Trans. Math. Softw., 30 (2004), pp. 381--388.
6.
A. Aminfar, S. Ambikasaran, and E. Darve, A fast block low-rank dense solver with applications to finite-element matrices, J. Comput. Phys., 304 (2016), pp. 170--188.
7.
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, and H. Zhang, PETSc Users Manual, Tech. report ANL-95/11 - Revision 3.5, Argonne National Laboratory, Lemont, IL, 2014.
8.
S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, L. C. McInnes, K. Rupp, B. F. Smith, and H. Zhang, PETSc Web Page, http://www.mcs.anl.gov/petsc, 2014.
9.
S. Balay, W. D. Gropp, L. C. McInnes, and B. F. Smith, Efficient management of parallelism in object oriented numerical software libraries, in Modern Software Tools in Scientific Computing, E. Arge, A. M. Bruaset, and H. P. Langtangen, eds., Birkhäuser, Basel, 1997, pp. 163--202.
10.
D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd ed., Cambridge University Press, Cambridge, UK, 2007.
11.
X. Cai and Y. Saad, Overlapping domain decomposition algorithms for general sparse matrices, Numer. Linear Algebra Appl., 3 (1996), pp. 221--237.
12.
X.-C. Cai and M. Sarkis, A restricted additive Schwarz preconditioner for general sparse linear systems, SIAM J. Sci. Comput., 21 (1999), pp. 792--797, https://doi.org/10.1137/S106482759732678X.
13.
Ü. V. Çatalyurek and C. Aykanat, Hypergraph-partitioning-based decomposition for parallel sparse-matrix vector multiplication, IEEE Trans. Parallel Distributed Syst., 10 (1999), pp. 673--693.
14.
S. Chandrasekaran, P. Dewilde, M. Gu, W. Lyons, and T. Pals, A fast solver for HSS representations via sparse matrices, SIAM J. Matrix Anal. Appl., 29 (2006), pp. 67--81, https://doi.org/10.1137/050639028.
15.
S. Chandrasekaran, M. Gu, and T. Pals, A fast ULV decomposition solver for hierarchically semiseparable representations, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 603--622, https://doi.org/10.1137/S0895479803436652.
16.
E. Chow and Y. Saad, Approximate inverse preconditioners via sparse-sparse iterations, SIAM J. Sci. Comput., 19 (1998), pp. 995--1023, https://doi.org/10.1137/S1064827594270415.
17.
T. A. Davis, Direct Methods for Sparse Linear Systems, Fundam. Algorithms 2, SIAM, Philadelphia, 2006, https://doi.org/10.1137/1.9780898718881.
18.
M. Dryja and O. Widlund, An Additive Variant of the Schwarz Alternating Method for the Case of Many Subregions, Courant Institute of Mathematical Sciences, New York University, New York, 1987.
19.
M. Dryja and O. Widlund, Additive Schwarz methods for elliptic finite element problems in three dimensions, in Proceedings of the 5th International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, 1991, pp. 3--18.
20.
B. Engquist and L. Ying, Sweeping preconditioner for the Helmholtz equation: Hierarchical matrix representation, Comm. Pure Appl. Math., 64 (2011), pp. 697--735.
21.
P. Ghysels, X. S. Li, F.-H. Rouet, S. Williams, and A. Napov, An efficient multicore implementation of a novel HSS-structured multifrontal solver using randomized sampling, SIAM J. Sci. Comput., 38 (2016), pp. S358--S384, https://doi.org/10.1137/15M1010117.
22.
G. H. Golub and C. F. Van Loan, Matrix Computations, 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013.
23.
L. Grasedyck and W. Hackbusch, Construction and arithmetics of $\mathcal{H}$-matrices, Computing, 70 (2003), pp. 295--334.
24.
L. Grigori, F. Nataf, and S. Yousef, Robust Algebraic Schur Complement Preconditioners Based on Low Rank Corrections, Rapport de recherche RR-8557, INRIA, Paris, France, 2014.
25.
W. Hackbusch, A sparse matrix arithmetic based on $\mathcal{H}$-matrices. Part I: Introduction to $\mathcal{H}$-matrices, Computing, 62 (1999), pp. 89--108.
26.
W. Hackbusch and S. Börm, Data-sparse approximation by adaptive $\mathcal{H}^2$-matrices, Computing, 69 (2002), pp. 1--35.
27.
W. Hackbusch and S. Börm, $\mathcal{H}^2$-matrix approximation of integral operators by interpolation, Appl. Numer. Math., 43 (2002), pp. 129--143.
28.
W. Hackbusch and B. N. Khoromskij, A sparse $\mathcal{H}$-matrix arithmetic. Part II: Application to multi-dimensional problems, Computing, 64 (2000), pp. 21--47.
29.
N. Halko, P. G. Martinsson, and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), pp. 217--288, https://doi.org/10.1137/090771806.
30.
B. Hendrickson and R. Leland, The Chaco User's Guide Version 2, Sandia National Laboratories, Albuquerque, NM, 1994.
31.
A. S. Householder, Theory of Matrices in Numerical Analysis, Blaisdell, Johnson, CO, 1964.
32.
G. Karypis and V. Kumar, A fast and high quality multilevel scheme for partitioning irregular graphs, SIAM J. Sci. Comput., 20 (1998), pp. 359--392, https://doi.org/10.1137/S1064827595287997.
33.
G. Karypis and V. Kumar, A parallel algorithm for multilevel graph partitioning and sparse matrix ordering, J. Parallel Distributed Comput., 48 (1998), pp. 71--95.
34.
T. G. Kolda, Partitioning sparse rectangular matrices for parallel processing, in Solving Irregularly Structured Problems in Parallel (Berkeley, CA, 1998), Lecture Notes in Comput. Sci. 1457, Springer, Berlin, 1998, pp. 68--79.
35.
C. Lanczos, An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, J. Research Nat. Bur. Standards, 45 (1950), pp. 255--282.
36.
S. Le Borne, $\mathcal{H}$-matrices for convection-diffusion problems with constant convection, Computing, 70 (2003), pp. 261--274.
37.
S. Le Borne and L. Grasedyck, $\mathcal{H}$-matrix preconditioners in convection-dominated problems, SIAM J. Matrix Anal. Appl., 27 (2006), pp. 1172--1183, https://doi.org/10.1137/040615845.
38.
R. Li and Y. Saad, Divide and conquer low-rank preconditioners for symmetric matrices, SIAM J. Sci. Comput., 35 (2013), pp. A2069--A2095, https://doi.org/10.1137/120872735.
39.
R. Li and Y. Saad, GPU-accelerated preconditioned iterative linear solvers, J. Supercomputing, 63 (2013), pp. 443--466.
40.
R. Li, Y. Xi, and Y. Saad, Schur complement-based domain decomposition preconditioners with low-rank corrections, Numer. Linear Algebra Appl., 23 (2016), pp. 706--729.
41.
Z. Li, Y. Saad, and M. Sosonkina, pARMS: A parallel version of the algebraic recursive multilevel solver, Numer. Linear Algebra Appl., 10 (2003), pp. 485--509.
42.
B. N. Parlett, The Symmetric Eigenvalue Problem, Classics Appl. Math. 20, SIAM, Philadelphia, 1998, https://doi.org/10.1137/1.9781611971163.
43.
B. N. Parlett and D. S. Scott, The Lanczos algorithm with selective orthogonalization, Math. Comp., 33 (1979), pp. 217--238.
44.
F. Pellegrini, Scotch and libScotch 5.1 User's Guide, Université Bordeaux I, Talence, France, 2010; available online from https://gforge.inria.fr/docman/view.php/248/7104/scotch_user5.1.pdf.
45.
A. Pothen, H. D. Simon, and K.-P. Liou, Partitioning sparse matrices with eigenvectors of graphs, SIAM J. Matrix Anal. Appl., 11 (1990), pp. 430--452, https://doi.org/10.1137/0611030.
46.
F. Rouet, X. S. Li, P. Ghysels, and A. Napov, A distributed-memory package for dense hierarchically semi-separable matrix computations using randomization, ACM Trans. Math. Softw., 42 (2016), 27.
47.
Y. Saad, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14 (1993), pp. 461--469, https://doi.org/10.1137/0914028.
48.
Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003, https://doi.org/10.1137/1.9780898718003.
49.
Y. Saad and M. Sosonkina, pARMS: A package for solving general sparse linear systems on parallel computers, in Parallel Processing and Applied Mathematics, R. Wyrzykowski, J. Dongarra, M. Paprzycki, and J. Waśniewski, eds., Lecture Notes in Comput. Sci. 2328, Springer, Berlin, Heidelberg, 2002, pp. 446--457.
50.
Y. Saad and B. Suchomel, ARMS: An algebraic recursive multilevel solver for general sparse linear systems, Numer. Linear Algebra Appl., 9 (2002), pp. 359--378.
51.
H. D. Simon, The Lanczos algorithm with partial reorthogonalization, Math. Comp., 42 (1984), pp. 115--142.
52.
B. Smith, P. Bjø rstad, and W. Gropp, Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations, Cambridge University Press, New York, 1996.
53.
Y. Xi, R. Li, and Y. Saad, An algebraic multilevel preconditioner with low-rank corrections for sparse symmetric matrices, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 235--259, https://doi.org/10.1137/15M1021830.
54.
Y. Xi and Y. Saad, A rational function preconditioner for indefinite sparse linear systems, SIAM J. Sci. Comput., 39 (2017), pp. A1145--A1167, https://doi.org/10.1137/16M1078409.
55.
J. Xia, Efficient structured multifrontal factorization for general large sparse matrices, SIAM J. Sci. Comput., 35 (2013), pp. A832--A860, https://doi.org/10.1137/120867032.
56.
J. Xia, S. Chandrasekaran, M. Gu, and X. S. Li, Fast algorithms for hierarchically semiseparable matrices, Numer. Linear Algebra Appl., 17 (2010), pp. 953--976.
57.
J. Xia, S. Chandrasekaran, M. Gu, and X. S. Li, Superfast multifrontal method for large structured linear systems of equations, SIAM J. Matrix Anal. Appl., 31 (2009), pp. 1382--1411, https://doi.org/10.1137/09074543X.
58.
J. Xia and M. Gu, Robust approximate Cholesky factorization of rank-structured symmetric positive definite matrices, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2899--2920.
59.
Y. Zhou, Y. Saad, M. L. Tiago, and J. R. Chelikowsky, Parallel self-consistent-field calculations via Chebyshev-filtered subspace acceleration, Phys. Rev. E, 74 (2006), 066704.

Information & Authors

Information

Published In

cover image SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Matrix Analysis and Applications
Pages: 807 - 828
ISSN (online): 1095-7162

History

Submitted: 23 November 2016
Accepted: 19 April 2017
Published online: 1 August 2017

Keywords

  1. Sherman--Morrison--Woodbury formula
  2. low-rank approximation
  3. distributed sparse linear systems
  4. parallel preconditioner
  5. incomplete LU factorization
  6. Krylov subspace method
  7. domain decomposition

MSC codes

  1. 65F08
  2. 65N22
  3. 65N55
  4. 65Y05
  5. 65Y20

Authors

Affiliations

Funding Information

National Science Foundation https://doi.org/10.13039/100000001 : DMS-1216366, DMS-1521573
U.S. Department of Energy https://doi.org/10.13039/100000015 : DE-AC52-07NA27344 (LLNL-JRNL-727122)

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Figures

Tables

Media

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media