On Dynamic Parallelization of Multilevel Monte Carlo Algorithm

Shegunov, Nikolay; Iliev, Oleg

doi:10.2478/cait-2020-0066

Abstract

MultiLevel Monte Carlo (MLMC) attracts great interest for numerical simulations of Stochastic Partial Differential Equations (SPDEs), due to its superiority over the standard Monte Carlo (MC) approach. MLMC combines in a proper manner many cheap fast simulations with few slow and expensive ones, the variance is reduced, and a significant speed up is achieved. Simulations with MC/MLMC consist of three main components: generating random fields, solving deterministic problem and reduction of the variance. Each part is subject to a different degree of parallelism. Compared to the classical MC, MLMC introduces “levels” on which the sampling is done. These levels have different computational cost, thus, efficiently utilizing the parallel resources becomes a non-trivial problem. The main focus of this paper is the parallelization of the MLMC Algorithm.

References

1. Giles, M. B. Multilevel Monte Carlo Methods. – Acta Numerica, Vol. 24, 2015, pp. 259-328. DOI: 10.1017/S096249291500001X.10.1017/S096249291500001X
Search in Google Scholar Back to article
2. Graham, G., F. Y. Kuo, D. Nuyens, R. Scheichl, I. H. Sloan. Quasi-Monte Carlo Methods for Elliptic PDEs with Random Coefficients and Applications. – Journal of Computational Physics, Vol. 230, 2011, No 10, pp. 3668-3694.10.1016/j.jcp.2011.01.023
Search in Google Scholar Back to article
3. Cliffe, K. A., M. B. Giles, R. Scheichl, A. L. Teckentrup. Multilevel Monte Carlo Methods and Applications to Elliptic PDEs with Random Coefficients. – Computing and Visualization in Science, Vol. 14, 2011, No 1. DOI: 10.1007/S00791-011-0160-X.10.1007/s00791-011-0160-x
Search in Google Scholar Back to article
4. Blaheta, R., M. Béreš, S. Domesová. A Study of Stochastic FEM Method for Porous Media Flow Problem. – In: R. Bris, P. Dao, Eds. Proc. of International Conference Applied Mathematics in Engineering and Reliability. CRC Press, 2016, pp. 281-289.10.1201/b21348-47
Search in Google Scholar Back to article
5. Mohring, J., R. Milk, A. Ngo, O. Klein, O. Iliev, M. Ohlberger, P. Bastian. Uncertainty Quantification for Porous Media Flow Using Multilevel Monte Carlo. – In: Proc. of Int. Conf. Large-Scale Scientific Computing, Springer International Publishing, 2015, pp. 145-152.10.1007/978-3-319-26520-9_15
Search in Google Scholar Back to article
6. Drzisga, D., B. Gmeiner, U. Rude, R. Scheichl. Scheduling Massively Parallel Multigrid for Multilevel Monte Carlo Methods. – SISC, Vol. 39, 2017, No 5, S873-97.10.1137/16M1083591
Search in Google Scholar Back to article
7. Iliev, O., J. Mohring, N. Shegunov. Renormalization Based MLMC Method for Scalar Elliptic SPDE. – In: Proc. of International Conference on Large-Scale Scientific Computing, Springer, 2017, pp. 145-152.10.1007/978-3-319-73441-5_31
Search in Google Scholar Back to article
8. Zakharov, P., O. Iliev, J. Mohring, N. Shegunov. Parallel Multilevel Monte Carlo Algorithms for Elliptic PDEs with Random Coefficients. – In: Lecture Notes in Computer Science. Vol. 11958. 2020, p. 463.
Search in Google Scholar Back to article
9. Bastian, P., M. Blatt, A. Dedner, C. Engwer, R. Klofkorn, M. Ohlberger, O. Sander. A Generic Grid Interface for Parallel and Adaptive Scientific Computing. Part I: Abstract Framework. – Computing, Vol. 82, 2008, No 2-3, pp. 103-119.10.1007/s00607-008-0003-x
Search in Google Scholar Back to article
10. Wen, X. H., J. J. Gomez-Hernández. Upscaling Hydraulic Conductivities in Heterogeneous Media: An Overview. – J. Hydrol., Vol. 183, 1996, No 1-2, pp. ix-xxxii.10.1016/S0022-1694(96)80030-8
Search in Google Scholar Back to article

On Dynamic Parallelization of Multilevel Monte Carlo Algorithm

Abstract

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