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View all- Xu TKalantzis VLi RXi YDillon GSaad Y(2022) parGeMSLRParallel Computing10.1016/j.parco.2022.102956113:COnline publication date: 1-Oct-2022
A Hierarchical Low Rank Schur Complement Preconditioner for Indefinite Linear Systems
Pages A2234 - A2252
Abstract
Nonsymmetric and highly indefinite linear systems can be quite difficult to solve by iterative methods. This paper combines ideas from the multilevel Schur low rank preconditioner developed by Y. Xi, R. Li, and Y. Saad [SIAM J. Matrix Anal., 37 (2016), pp. 235--259] with classic block preconditioning strategies in order to handle this case. The method to be described generates a tree structure $\mathcal{T}$ that represents a hierarchical decomposition of the original matrix. This decomposition gives rise to a block structured matrix at each level of $\mathcal{T}$. An approximate inverse of the original matrix based on its block $LU$ factorization is computed at each level via a low rank property that characterizes the difference between the inverses of the Schur complement and another block of the reordered matrix. The low rank correction matrix is computed by several steps of the Arnoldi process. Numerical results illustrate the robustness of the proposed preconditioner with respect to indefiniteness for a few discretized partial differential equations and publicly available test problems.
References
[1]
W. E. Arnoldi, The principle of minimized iterations in the solution of the matrix eigenvalue problem, Quart. Appl. Math., 9 (1951), pp. 17--29.
[2]
O. Axelsson and B. Polman, Block preconditioning and domain decomposition methods II, J. Comput. Appl. Math., 24 (1988), pp. 55--72.
[3]
Z. Bai and J. W. Demmel, On swapping diagonal blocks in real Schur form, Linear Algebra Appl., 186 (1993), pp. 75--95.
[4]
M. Bebendorf, Why finite element discretizations can be factored by triangular hierarchical matrices, SIAM J. Numer. Anal., 45 (2007), pp. 1472--1494, https://doi.org/10.1137/060669747.
[5]
M. Bebendorf and T. Fischer, On the purely algebraic data-sparse approximation of the inverse and the triangular factors of sparse matrices, Numerical Linear Algebra Appl., 18 (2011), pp. 105--122, http://doi.org/10.1002/nla.714.
[6]
M. Benzi, Preconditioning techniques for large linear systems: A survey, J. Comput. Phys., 182 (2002), pp. 418--477.
[7]
M. Benzi, G. H. Golub, and J. Liesen, Numerical solution of saddle point problems, Acta Numer., 14 (2005), pp. 1--137.
[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. T\ruma, A sparse approximate inverse preconditioner for nonsymmetric linear systems, SIAM J. Sci. Comput., 19 (1998), pp. 968--994.
[10]
D. Cai, E. Chow, Y. Saad, and Y. Xi, SMASH: Structured matrix approximation by separation and hierarchy, Preprint ys-2016-10, University of Minnesota, Minneapolis, MN, 2016.
[11]
E. Chow and Y. Saad, Approximate inverse techniques for block-partitioned matrices, SIAM J. Sci. Comput., 18 (1997), pp. 1657--1675.
[12]
E. Chow and Y. Saad, Experimental study of ILU preconditioners for indefinite matrices, J. Comput. Appl. Math., 86 (1997), pp. 387--414.
[13]
E. Chow and Y. Saad, Approximate inverse preconditioners via sparse-sparse iterations, SIAM J. Sci. Comput., 19 (1998), pp. 995--1023.
[14]
E. C. Cyr, J. N. Shadid, R. S. Tuminaro, R. P. Pawlowski, and L. Chacón, A new approximate block factorization preconditioner for two-dimensional incompressible (reduced) resistive MHD, SIAM J. Sci. Comput., 35 (2013), pp. B701--B730.
[15]
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.
[16]
P. R. Amestoy, T. A. Davis, and I. S. Duff, Algorithm 837: An approximate minimum degree ordering algorithm, ACM Trans. Math. Software, 30 (2004), pp. 381--388.
[17]
T. A. Davis and Y. Hu, The University of Florida Sparse Matrix Collection, ACM Trans. Math. Software, 38 (2011), 1.
[18]
H. Elman, V. Howle, J. Shadid, R. Shuttleworth, and R. Tuminaro, Block preconditioners based on approximate commutators, SIAM J. Sci. Comput., 27 (2006), pp. 1651--1668.
[19]
H. C. Elman, D. J. Silvester, and A. J. Wathen, Finite Elements and Fast Iterative Solvers, Oxford University Press, Oxford, 2005.
[20]
Y. A. Erlangga, C. W. Oosterlee, and C. Vuik, A novel multigrid based preconditioner for heterogeneous Helmholtz problems, SIAM J. Sci. Comput., 27 (2006), pp. 1471--1492.
[21]
A. George, Nested dissection of a regular finite element mesh, SIAM J. Numer. Anal., 10 (1973), pp. 345--363.
[22]
M. Grote and T. Huckle, Parallel preconditioning with sparse approximate inverses, SIAM J. Sci. Comput., 18 (1997), pp. 838--853.
[23]
W. Hackbusch, A sparse matrix arithmetic based on $\mathcal{H}$-matrices. Part I: Introduction to ${H}$-matrices, Computing, 62 (1999), pp. 89--108.
[24]
W. Hackbusch and B. N. Khoromskij, A sparse ${H}$-matrix arithmetic. Part II: Application to multi-dimensional problems, Computing, 64 (2000), pp. 21--47.
[25]
P. Hénon and Y. Saad, A parallel multistage ILU factorization based on a hierarchical graph decomposition, SIAM J. Sci. Comput., 28 (2006), pp. 2266--2293.
[26]
V. E. Howle and R. C. Kirby, Block preconditioners for finite element discretization of incompressible flow with thermal convection, Numerical Linear Algebra Appl., 19 (2012), pp. 427--440.
[27]
V. E. Howle, R. C. Kirby, and G. Dillon, Block preconditioners for coupled fluids problems, SIAM J. Sci. Comput., 35 (2013), pp. S368--S385.
[28]
I. C. F. Ipsen, A note on preconditioning nonsymmetric matrices, SIAM J. Sci. Comput., 23 (2001), pp. 1050--1051.
[29]
V. Kalantzis, Y. Xi, and Y. Saad, Beyond AMLS: Domain Decomposition with Rational Filtering, preprint, https://arxiv.org/abs/1711.09487, 2017.
[30]
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.
[31]
E.-J. Lee and J. Zhang, Hybrid reordering strategies for ILU preconditioning of indefinite sparse matrices, J. Appl. Math. Comput., 22 (2006), pp. 307--316.
[32]
R. Li and Y. Saad, Divide and conquer low-rank preconditioners for symmetric matrices, SIAM J. Sci. Comput., 35 (2013), pp. A2069--A2095.
[33]
R. Li and Y. Saad, GPU-accelerated preconditioned iterative linear solvers, J. Supercomput., 63 (2013), pp. 443--466.
[34]
R. Li, Y. Xi, and Y. Saad, Schur complement-based domain decomposition preconditioners with low-rank corrections, Numerical Linear Algebra Appl., 23 (2016), pp. 706--729.
[35]
M. Magolu Monga Made, R. Beauwens, and G. Warzee, Preconditioning of discrete Helmholtz operators perturbed by a diagonal complex matrix, Internat. J. Numer. Meth. Biomed. Eng., 16 (2000), pp. 801--817.
[36]
K.-A. Mardal and R. Winther, Preconditioning discretizations of systems of partial differential equations, Numerical Linear Algebra Appl., 18 (2011), pp. 1--40.
[37]
M. F. Murphy, G. H. Golub, and A. J. Wathen, A note on preconditioning for indefinite linear systems, SIAM J. Sci. Comput., 21 (2000), pp. 1969--1972.
[38]
D. Osei-Kuffour, R. Li, and Y. Saad, Matrix reordering using multilevel graph coarsening for ILU preconditioning, SIAM J. Sci. Comput., 37 (2015), pp. A391--419.
[39]
D. Osei-Kuffuor and Y. Saad, Preconditioning Helmholtz linear systems, Appl. Numer. Math., 60 (2010), pp. 420--431, http://doi.org/10.1016/j.apnum.2009.09.003.
[40]
Y. Saad, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14 (1993), pp. 461--469.
[41]
Y. Saad, ILUM: A multi-elimination ILU preconditioner for general sparse matrices, SIAM J. Sci. Comput., 17 (1996), pp. 830--847.
[42]
Y. Saad, Iterative Methods for Sparse Linear Systems, 2nd ed., SIAM, Philadelphia, 2003.
[43]
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.
[44]
G. Stewart, Algorithm 506: HQR3 and EXCHNG: Fortran subroutines for calculating and ordering the eigenvalues of a real upper Hessenberg matrix, ACM Trans. Math. Software, 2 (1976), pp. 275--280.
[45]
M. B. van Gijzen, Y. A. Erlangga, and C. Vuik, Spectral analysis of the discrete Helmholtz operator preconditioned with a shifted Laplacian, SIAM J. Sci. Comput., 29 (2007), pp. 1942--1958.
[46]
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.
[47]
Y. Xi and Y. Saad, Computing partial spectra with least-squares rational filters, SIAM J. Sci. Comput., 38 (2016), pp. A3020--A3045, https://doi.org/10.1137/16M1061965.
[48]
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.
[49]
Y. Xi and J. Xia, On the stability of some hierarchical rank structured matrix algorithms, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 1279--1303, https://doi.org/10.1137/15M1026195.
[50]
Y. Xi, J. Xia, S. Cauley, and V. Balakrishnan, Superfast and stable structured solvers for Toeplitz least squares via randomized sampling, SIAM J. Matrix Anal. Appl., 35 (2014), pp. 44--72.
[51]
J. Xia, S. Chandrasekaran, M. Gu, and X. Li, Fast algorithms for hierarchically semiseparable matrices, Numer. Linear Algebra Appl., 17 (2010), pp. 953--976.
[52]
J. Xia, Y. Xi, S. Cauley, and V. Balakrishnan, Fast sparse selected inversion, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 1283--1314, https://doi.org/10.1137/14095755X.
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DOI:10.1137/sjoce3.40.4
Issue’s Table of Contents© 2018, Society for Industrial and Applied Mathematics.
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Society for Industrial and Applied Mathematics
United States
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Published: 01 January 2018
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View all- Xu TKalantzis VLi RXi YDillon GSaad Y(2022) parGeMSLRParallel Computing10.1016/j.parco.2022.102956113:COnline publication date: 1-Oct-2022
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