zbMATH — the first resource for mathematics

Scalable linear solvers based on enlarged Krylov subspaces with dynamic reduction of search directions. (English) Zbl 1425.65049
65F10 Iterative numerical methods for linear systems
65Y05 Parallel numerical computation
Full Text: DOI
[1] S. F. Ashby, T. A. Manteuffel, and P. E. Saylor, A taxonomy for conjugate gradient methods, SIAM J. Numer. Anal., 27 (1990), pp. 1542–1568, https://doi.org/10.1137/0727091. · Zbl 0723.65018
[2] S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune, K. Buschelman, L. Dalcin, V. Eijkhout, W. D. Gropp, D. Kaushik, M. G. Knepley, D. A. May, L. C. McInnes, K. Rupp, P. Sanan, B. F. Smith, S. Zampini, H. Zhang, and H. Zhang, PETSc Users Manual, Tech. Report ANL-95/11 - Revision 3.8, Argonne National Laboratory, Lemont, IL, 2017, http://www.mcs.anl.gov/petsc.
[3] E. Carson, N. Knight, and J. Demmel, Avoiding communication in nonsymmetric Lanczos-based Krylov subspace methods, SIAM J. Sci. Comput., 35 (2013), pp. S42–S61, https://doi.org/10.1137/120881191. · Zbl 1281.65057
[4] A. T. Chronopoulos, s-step iterative methods for (non)symmetric (in)definite linear systems, SIAM J. Numer. Anal., 28 (1991), pp. 1776–1789, https://doi.org/10.1137/0728088. · Zbl 0741.65026
[5] T. A. Davis and Y. Hu, The University of Florida sparse matrix collection, ACM Trans. Math. Software, 38 (2011), 1, https://doi.org/10.1145/2049662.2049663. · Zbl 1365.65123
[6] J. Demmel, L. Grigori, M. Hoemmen, and J. Langou, Communication-optimal parallel and sequential QR and LU factorizations, SIAM J. Sci. Comput., 34 (2012), pp. A206–A239, https://doi.org/10.1137/080731992. · Zbl 1241.65028
[7] J. Demmel, M. Hoemmen, M. Mohiyuddin, and K. Yelick, Minimizing communication in sparse matrix solvers, in Proceedings of the ACM/IEEE Supercomputing SC9 Conference, 2009.
[8] J. W. Demmel, L. Grigori, M. Gu, and H. Xiang, Communication avoiding rank revealing QR factorization with column pivoting, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 55–89, https://doi.org/10.1137/13092157X. · Zbl 1327.65078
[9] A. J. Dongarra, V. Eijkhout, and A. Kalhan, Reverse Communication Interface for Linear Algebra Templates for Iterative Methods, Tech. Report, 1995, http://www.netlib.org/lapack/lawnspdf/lawn99.pdf.
[10] Z. Dostál, Conjugate gradient method with preconditioning by projector, Int. J. Comput. Math., 23 (1988), pp. 315–323, https://doi.org/10.1080/00207168808803625. · Zbl 0668.65034
[11] A. Dubrulle, Retooling the method of block conjugate gradients, Electron. Trans. Numer. Anal., 12 (2001), pp. 216–233. · Zbl 0985.65021
[12] A. el Guennouni, K. Jbilou, and H. Sadok, A block version of BiCGSTAB for linear systems with multiple right-hand sides, Electron. Trans. Numer. Anal., 16 (2003), pp. 129–142. · Zbl 1065.65052
[13] P. Ghysels and W. Vanroose, Hiding global synchronization latency in the preconditioned conjugate gradient algorithm, Parallel Comput., 40 (2014), pp. 224–238, https://doi.org/10.1016/j.parco.2013.06.001.
[14] L. Grigori, S. Moufawad, and F. Nataf, Enlarged Krylov subspace conjugate gradient methods for reducing communication, SIAM J. Matrix Anal. Appl., 37 (2016), pp. 744–773, https://doi.org/10.1137/140989492. · Zbl 1382.65086
[15] L. Grigori, F. Nataf, and S. Yousef, Robust Algebraic Schur Complement Preconditioners Based on Low Rank Corrections, Research Report RR-8557, 2014, https://hal.inria.fr/hal-01017448.
[16] L. Grigori and O. Tissot, Reducing the Communication and Computational Costs of Enlarged Krylov Subspaces Conjugate Gradient, Research Report RR-9023, 2017, https://hal.inria.fr/hal-01451199.
[17] M. H. Gutknecht, Block Krylov space methods for linear systems with multiple right-hand sides: An introduction, in Modern Mathematical Models, Methods and Algorithms for Real World Systems, A. H. Siddiqi, I. S. Duff, and O. Christensen, eds., Anamaya, New Delhi, India, 2007, pp. 420–447.
[18] F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), pp. 251–265. · Zbl 1266.68090
[19] M. R. Hestenes and E. Stiefel, Methods of conjugate gradients for solving linear systems, J. Research Nat. Bur. Standards, 49 (1952), pp. 409–436.
[20] M. Hoemmen, Communication-Avoiding Krylov Subspace Methods, Ph.D. thesis, University of California, Berkeley, Berkeley, CA, 2010.
[21] H. Ji and Y. Li, A breakdown-free block conjugate gradient method, BIT, 57 (2017), pp. 379–403, https://doi.org/10.1007/s10543-016-0631-z. · Zbl 1392.65057
[22] P. Jiránek, M. Rozložník, and M. H. Gutknecht, How to make simpler GMRES and GCR more stable, SIAM J. Matrix Anal. Appl., 30 (2008), pp. 1483–1499, https://doi.org/10.1137/070707373. · Zbl 1176.65036
[23] P. Jolivet, Domain Decomposition Methods. Application to High-Performance Computing, theses, Université de Grenoble, Grenoble, France, 2014, https://tel.archives-ouvertes.fr/tel-01155718.
[24] P. Jolivet and P. Tournier, Block iterative methods and recycling for improved scalability of linear solvers, in Proceedings of the International Conference for High Performance Computing, Networking, Storage and Analysis, SC ’16, IEEE Press, Piscataway, NJ, 2016, 17, http://dl.acm.org/citation.cfm?id=3014904.3014927.
[25] 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. · Zbl 0915.68129
[26] M. Kreutzer, J. Thies, M. Röhrig-Zöllner, A. Pieper, F. Shahzad, M. Galgon, A. Basermann, H. Fehske, G. Hager, and G. Wellein, GHOST: Building blocks for high performance sparse linear algebra on heterogeneous systems, Internat. J. Parallel Programming, 45 (2017), pp. 1046–1072, https://doi.org/10.1007/s10766-016-0464-z.
[27] J. Langou, Iterative Methods for Solving Linear Systems with Multiple Right-Hand Sides, Ph.D. thesis, CERFACS, Toulouse, France, 2003.
[28] J. Málek and Z. Strakoš, Preconditioning and the Conjugate Gradient Method in the Context of Solving PDEs, SIAM Spotlights 1, SIAM, Philadelphia, 2015, https://doi.org/10.1137/1.9781611973846.
[29] D. P. O’Leary, The block conjugate gradient algorithm and related methods, Linear Algebra Appl., 29 (1980), pp. 293–322.
[30] M. Robbé and M. Sadkane, Exact and inexact breakdowns in the block GMRES method, Linear Algebra Appl., 419 (2006), pp. 265–285. · Zbl 1112.65028
[31] Y. Saad, On the rates of convergence of the Lanczos and the block-Lanczos methods, SIAM J. Numer. Anal., 17 (1980), pp. 687–706, https://doi.org/10.1137/0717059. · Zbl 0456.65016
[32] Y. Saad, Iterative Methods for Sparse Linear Systems, SIAM, Philadelphia, 2003, https://doi.org/10.1137/1.9780898718003.
[33] N. Spillane, An adaptive multipreconditioned conjugate gradient algorithm, SIAM J. Sci. Comput., 38 (2016), pp. A1896–A1918, https://doi.org/10.1137/15M1028534. · Zbl 1416.65087
[34] N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps, Numer. Math., 126 (2014), pp. 741–770, https://doi.org/10.1007/s00211-013-0576-y. · Zbl 1291.65109
[35] J. M. Tang, R. Nabben, C. Vuik, and Y. A. Erlangga, Comparison of two-level preconditioners derived from deflation, domain decomposition and multigrid methods, J. Sci. Comput., 39 (2009), pp. 340–370, https://doi.org/10.1007/s10915-009-9272-6. · Zbl 1203.65073
[36] R. Thakur, R. Rabenseifner, and W. Gropp, Optimization of collective communication operations in MPICH, Int. J. High Perform. Comput. Appl., 19 (2005), pp. 49–66, https://doi.org/10.1177/1094342005051521.
[37] E. Wang, Q. Zhang, B. Shen, G. Zhang, X. Lu, Q. Wu, and Y. Wang, Intel math kernel library, in High-Performance Computing on the Intel\textregistered Xeon Phi, Springer, Cham, 2014, pp. 167–188.
[38] I. Yamazaki, S. Tomov, and J. Dongarra, Mixed-precision Cholesky QR factorization and its case studies on multicore CPU with multiple GPUs, SIAM J. Sci. Comput., 37 (2015), pp. C307–C330, https://doi.org/10.1137/14M0973773. · Zbl 1320.65046
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.