Abstract
This paper focuses on minimizing further the communications in Monte Carlo methods for Linear Algebra and thus improving the overall performance. The focus is on producing set of small number of covering Markov chains which are much longer that the usually produced ones. This approach allows a very efficient communication pattern that enables to transmit the sampled portion of the matrix in parallel case. The approach is further applied to quasi-Monte Carlo. A comparison of the efficiency of the new approach in case of Sparse Approximate Matrix Inversion and hybrid Monte Carlo and quasi-Monte Carlo methods for solving Systems of Linear Algebraic Equations is carried out. Experimental results showing the efficiency of our approach on a set of test matrices are presented. The numerical experiments have been executed on the MareNostrum III supercomputer at the Barcelona Supercomputing Center (BSC) and on the Avitohol supercomputer at the Institute of Information and Communication Technologies (IICT).
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References
Golub, G., Loan, C.: Matrix Computations. Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore (1996)
Straßburg, J., Alexandrov, V.N.: Enhancing Monte Carlo preconditioning methods for matrix computations. In: Proceedings ICCS 2014, pp. 1580–1589 (2014)
Alexandrov, V.N., Esquivel-Flores, O.A.: Towards Monte Carlo preconditioning approach and hybrid Monte Carlo algorithms for matrix computations. CMA 70(11), 2709–2718 (2015)
Carpentieri, B., Duff, I., Giraud, L.: Some sparse pattern selection strategies for robust Frobenius norm minimization preconditioners in electromagnetism. Numer. Linear Algebra Appl. 7, 667–685 (2000)
Carpentieri, B., Duff, I., Giraud, L.: Experiments with sparse preconditioning of dense problems from electromagnetic applications, CERFACS, Toulouse, France. Technical report (2000)
Alléon, G., Benzi, M., Giraud, L.: Sparse approximate inverse preconditioning for dense linear systems arising in computational electromagnetics. Num. Algorithms 16(1), 1–15 (1997)
Evans, T., Hamilton, S., Joubert, W., Engelmann, C.: MCREX - Monte Carlo Resilient Exascale Project. http://www.csm.ornl.gov/newsite/documents
Benzi, M., Meyer, C., Tůma, M.: A sparse approximate inverse preconditioner for the conjugate gradient method. SIAM J. Sci. Comput. 17(5), 1135–1149 (1996)
Huckle, T., Kallischko, A., Roy, A., Sedlacek, M., Weinzierl, T.: An efficient parallel implementation of the MSPAI preconditioner. Parallel Comput. 36(5–6), 273–284 (2010)
Grote, M., Hagemann, M.: SPAI: SParse Approximate Inverse Preconditioner. Spaidoc. pdf paper in the SPAI, vol. 3, p. 1 (2006)
Huckle, T.: Factorized sparse approximate inverses for preconditioning. J. Supercomput. 25(2), 109–117 (2003)
Strassburg, J., Alexandrov, V.: On scalability behaviour of Monte Carlo sparse approximate inverse for matrix computations. In: Proceedings of the ScalA 2013 Workshop, Article no. 6. ACM (2013)
Vajargah, B.F.: A new algorithm with maximal rate convergence to obtain inverse matrix. Appl. Math. Comput. 191(1), 280–286 (2007)
Hoemmen, M., Vuduc, R., Nishtala, R.: BeBOP sparse matrix converter. University of California at Berkeley (2011)
Boisvert, R.F., Pozo, R., Remington, K., Barrett, R.F., Dongarra, J.J.: Matrix market: a web resource for test matrix collections. In: Boisvert, R.F. (ed.) QNS 1997. IFIPAICT, pp. 125–137. Springer, Boston (1997). https://doi.org/10.1007/978-1-5041-2940-4_9
Davis, T.A., Hu, Y.: The University of Florida sparse matrix collection. ACM Trans. Math. Softw. (TOMS) 38(1), 1 (2011)
Alexandrov, V., Esquivel-Flores, O., Ivanovska, S., Karaivanova, A.: On the preconditioned Quasi-Monte Carlo algorithm for matrix computations. In: Lirkov, I., Margenov, S.D., Waśniewski, J. (eds.) LSSC 2015. LNCS, vol. 9374, pp. 163–171. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-26520-9_17
Karaivanova, A.: Quasi-Monte Carlo methods for some linear algebra problems. Convergence and complexity. Serdica J. Comput. 4, 57–72 (2010)
Atanassov, E., Gurov, T., Karaivanova, A., Ivanovska, S., Durchova, M., Dimitrov, D.: On the parallelization approaches for Intel MIC architecture. In: AIP Conference Proceedings, vol. 1773, p. 070001 (2016). https://doi.org/10.1063/1.4964983
Acknowledgments
The work of the authors (V.A., D.D., and O.E-F.) is supported by Severo Ochoa program of excellence, Spain. The work of the authors (A.K. and T.G.) is supported by the NSF of Bulgaria under Grant DFNI-I02/8.
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Alexandrov, V., Davila, D., Esquivel-Flores, O., Karaivanova, A., Gurov, T., Atanassov, E. (2018). On Monte Carlo and Quasi-Monte Carlo for Matrix Computations. In: Lirkov, I., Margenov, S. (eds) Large-Scale Scientific Computing. LSSC 2017. Lecture Notes in Computer Science(), vol 10665. Springer, Cham. https://doi.org/10.1007/978-3-319-73441-5_26
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