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    Multilevel Monte Carlo methods using ensemble level mixed MsFEM for two-phase flow and transport simulations

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    Type
    Article
    Authors
    Efendiev, Yalchin R. cc
    Iliev, Oleg cc
    Kronsbein, C.
    KAUST Department
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Numerical Porous Media SRI Center (NumPor)
    Technology Transfer
    Date
    2013-08-21
    Online Publication Date
    2013-08-21
    Print Publication Date
    2013-10
    Permanent link to this record
    http://hdl.handle.net/10754/562914
    
    Metadata
    Show full item record
    Abstract
    In this paper, we propose multilevel Monte Carlo (MLMC) methods that use ensemble level mixed multiscale methods in the simulations of multiphase flow and transport. The contribution of this paper is twofold: (1) a design of ensemble level mixed multiscale finite element methods and (2) a novel use of mixed multiscale finite element methods within multilevel Monte Carlo techniques to speed up the computations. The main idea of ensemble level multiscale methods is to construct local multiscale basis functions that can be used for any member of the ensemble. In this paper, we consider two ensemble level mixed multiscale finite element methods: (1) the no-local-solve-online ensemble level method (NLSO); and (2) the local-solve-online ensemble level method (LSO). The first approach was proposed in Aarnes and Efendiev (SIAM J. Sci. Comput. 30(5):2319-2339, 2008) while the second approach is new. Both mixed multiscale methods use a number of snapshots of the permeability media in generating multiscale basis functions. As a result, in the off-line stage, we construct multiple basis functions for each coarse region where basis functions correspond to different realizations. In the no-local-solve-online ensemble level method, one uses the whole set of precomputed basis functions to approximate the solution for an arbitrary realization. In the local-solve-online ensemble level method, one uses the precomputed functions to construct a multiscale basis for a particular realization. With this basis, the solution corresponding to this particular realization is approximated in LSO mixed multiscale finite element method (MsFEM). In both approaches, the accuracy of the method is related to the number of snapshots computed based on different realizations that one uses to precompute a multiscale basis. In this paper, ensemble level multiscale methods are used in multilevel Monte Carlo methods (Giles 2008a, Oper.Res. 56(3):607-617, b). In multilevel Monte Carlo methods, more accurate (and expensive) forward simulations are run with fewer samples, while less accurate (and inexpensive) forward simulations are run with a larger number of samples. Selecting the number of expensive and inexpensive simulations based on the number of coarse degrees of freedom, one can show that MLMC methods can provide better accuracy at the same cost as Monte Carlo (MC) methods. The main objective of the paper is twofold. First, we would like to compare NLSO and LSO mixed MsFEMs. Further, we use both approaches in the context of MLMC to speedup MC calculations. © 2013 Springer Science+Business Media Dordrecht.
    Citation
    Efendiev, Y., Iliev, O., & Kronsbein, C. (2013). Multilevel Monte Carlo methods using ensemble level mixed MsFEM for two-phase flow and transport simulations. Computational Geosciences, 17(5), 833–850. doi:10.1007/s10596-013-9358-y
    Sponsors
    The research of O. Iliev and C. Kronsbein was supported by the DFG Project IL 55/1 - 2: "Multiscale analysis of two-phase flow in porous media with complex heterogeneities." The implementation of the mixed MsFEM is based on the code of Aarnes. For further details, we refer to [4, 18].
    Publisher
    Springer Nature
    Journal
    Computational Geosciences
    DOI
    10.1007/s10596-013-9358-y
    ae974a485f413a2113503eed53cd6c53
    10.1007/s10596-013-9358-y
    Scopus Count
    Collections
    Articles; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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