Abstract
Hierarchical (\(\mathcal {H}\)-) matrices provide a data-sparse way to approximate fully populated matrices. The two basic steps in the construction of an \(\mathcal {H}\)-matrix are (a) the hierarchical construction of a matrix block partition, and (b) the blockwise approximation of matrix data by low rank matrices. In the context of finite element discretisations of elliptic boundary value problems, \(\mathcal {H}\)-matrices can be used for the construction of preconditioners such as approximate \(\mathcal {H}\)-LU factors. In this paper, we develop a new black box approach to construct the necessary partition. This new approach is based on the matrix graph of the sparse stiffness matrix and no longer requires geometric data associated with the indices like the standard clustering algorithms. The black box clustering and a subsequent \(\mathcal {H}\)-LU factorisation have been implemented in parallel, and we provide numerical results in which the resulting black box \(\mathcal {H}\)-LU factorisation is used as a preconditioner in the iterative solution of the discrete (three-dimensional) convection-diffusion equation.
References
Bebendorf, M.: Why finite element discretizations can be factored by triangular hierarchical matrices. SIAM Num. Anal. 45, 1472–1494 (2007)
Börm, S., Grasedyck, L., Hackbusch, W.: Hierarchical matrices (2003). Lecture Notes No. 21, Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany, available online at http://www.mis.mpg.de/preprints/ln/, revised version June (2006)
Börm, S., Grasedyck, L., Hackbusch, W.: Introduction to hierarchical matrices with applications. Eng. Anal. Boundary Elements 27, 405–422 (2003)
Brainman, I., Toledo, S.: Nested-dissection orderings for sparse LU with partial pivoting. SIAM J. Mat. Anal. Appl. 23, 998–1012 (2002)
Brandt, A., McCormick, S., Ruge, J.: Algebraic multigrid (AMG) for sparse matrix equations. In: Sparsity and its Applications, pp. 257–284. Cambridge University Press, Cambridge (1984)
Bröker, O., Grote, M., Mayer, C., Reusken, A.: Robust parallel smoothing for multigrid via sparse approximate inverses. SIAM J. Sci. Comput. 23, 1396–1417 (2001)
Davis, T.A.: Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Soft. 30(2), 196–199 (2004)
George, A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10, 345–363 (1973)
Grama, A., Gupta, A., Karypis, G., Kumar, V.: Introduction to Parallel Computing. Addison Wesley, Reading (2003)
Grasedyck, L., Hackbusch, W.: Construction and arithmetics of \({{\mathcal{H}}}\)-matrices. Computing 70, 295–334 (2003)
Grasedyck, L., Kriemann, R., Borne, S.L.: Parallel Black Box Domain Decomposition Based \({\mathcal{H}}\)-LU Preconditioning. Tech. rep., Max-Planck-Institute MIS (2005)
Grasedyck, L., Kriemann, R., Le Borne, S.: Domain decomposition based \({{\mathcal{H}}}\)-LU preconditioning. Numer. Math. (2006) (submitted)
Grasedyck, L., Le Borne, S.: \({{\mathcal{H}}}\)-matrix preconditioners in convection-dominated problems. SIAM J. Mat. Anal. 27, 1172–1183 (2006)
Haase, G., Kuhn, M., Reitzinger, S.: Parallel AMG on distributed memory computers. SIAM J. Sci.Comp. 24(2), 410–427 (2002)
Hackbusch, W.: A sparse matrix arithmetic based on \({{\mathcal{H}}}\)-matrices. Part I: Introduction to \({{\mathcal{H}}}\)-matrices. Computing 62, 89–108 (1999)
Hackbusch, W.: Direct domain decomposition using the hierarchical matrix technique. In: Herrera, I., Keyes, D., Widlund, O., Yates, R. (eds.) Domain Decomposition Methods in Science and Engineering, pp. 39–50. UNAM (2003)
Hendrickson, B., Leland, R.: The Chaco User’s Guide: Version 2.0. Tech. Rep. SAND94–2692, Sandia National Laboratories (1994)
Hendrickson, B., Rothberg, E.: Improving the run time and quality of nested dissection ordering. SIAM J. Sci. Comp. 20, 468–489 (1998)
Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1999)
Kriemann, R.: Parallele Algorithmen für \({\mathcal {H}}\)-Matrizen. Ph.D. thesis, Universität Kiel (2005)
Le Borne, S., Oliveira, S., Yang, F.: \({{\mathcal{H}}}\)-matrix preconditioners for symmetric saddle-point systems from meshfree discretizations. Numer. Linear Algebra Appl. (2006) (in press)
Lintner, M.: The eigenvalue problem for the 2d Laplacian in \({{\mathcal{H}}}\)-matrix arithmetic and application to the heat and wave equation. Computing 72, 293–323 (2004)
Lipton, R.J., Rose, D.J., Tarjan, R.E.: Generalized nested dissection. SIAM J. Numer. Anal. 16, 346–358 (1979)
Oliveira, S., Yang, F.: An algebraic approach for \(\mathcal {H}\)-matrix preconditioners. Computing 80, 169–188 (2007)
Pellegrin, F.: SCOTCH 5.0 User’s guide. Tech. rep., LaBRI, Université Bordeaux I (2007)
Ruge, R.W., Stüben, K.: Efficient solution of finite difference and finite element equations by algebraic multigrid (AMG). In: Paddon, H.H.D.J. (ed.) Multigrid Methods for Integral and Differential Equations, pp. 169–212. Clarenden Press, Oxford (1985)
Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with PARDISO. J. Future Generat. Comput. Syst. 20, 475–487 (2004)
Schenk, O., Gärtner, K.: On fast factorization pivoting methods for symmetric indefinite systems. Elec. Trans. Numer. Anal. 23, 158–179 (2006)
Schenk, O., Gärtner, K., Fichtner, W.: Efficient sparse LU factorization with left-right looking strategy on shared memory multiprocessors. BIT 40, 158–176 (2000)
Open Access
This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Wittum.
Dedicated to Wolfgang Hackbusch on the occasion of his 60th birthday.
The work was supported in part by the US Department of Energy under Grant No. DE-FG02-04ER25649 and by the National Science Foundation under grant No. DMS-0408950.
Rights and permissions
Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Grasedyck, L., Kriemann, R. & Le Borne, S. Parallel black box \(\mathcal {H}\)-LU preconditioning for elliptic boundary value problems. Comput. Visual Sci. 11, 273–291 (2008). https://doi.org/10.1007/s00791-008-0098-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00791-008-0098-9
Keywords
- Hierarchical matrices
- Black box clustering
- Preconditioning
- LU
Mathematics Subject Classification (2000)
- 65F05
- 65F30
- 65F50
- 65N55