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    Multilevel markov chain monte carlo method for high-contrast single-phase flow problems

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    Type
    Article
    Authors
    Efendiev, Yalchin R. cc
    Jin, Bangti
    Michael, Presho
    Tan, Xiaosi
    KAUST Department
    Numerical Porous Media SRI Center (NumPor)
    Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
    Date
    2014-12-19
    Online Publication Date
    2014-12-19
    Print Publication Date
    2015-01
    Permanent link to this record
    http://hdl.handle.net/10754/563926
    
    Metadata
    Show full item record
    Abstract
    In this paper we propose a general framework for the uncertainty quantification of quantities of interest for high-contrast single-phase flow problems. It is based on the generalized multiscale finite element method (GMsFEM) and multilevel Monte Carlo (MLMC) methods. The former provides a hierarchy of approximations of different resolution, whereas the latter gives an efficient way to estimate quantities of interest using samples on different levels. The number of basis functions in the online GMsFEM stage can be varied to determine the solution resolution and the computational cost, and to efficiently generate samples at different levels. In particular, it is cheap to generate samples on coarse grids but with low resolution, and it is expensive to generate samples on fine grids with high accuracy. By suitably choosing the number of samples at different levels, one can leverage the expensive computation in larger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracy of the final Monte Carlo estimate. Further, we describe a multilevel Markov chain Monte Carlo method, which sequentially screens the proposal with different levels of approximations and reduces the number of evaluations required on fine grids, while combining the samples at different levels to arrive at an accurate estimate. The framework seamlessly integrates the multiscale features of the GMsFEM with the multilevel feature of the MLMC methods following the work in [26], and our numerical experiments illustrate its efficiency and accuracy in comparison with standard Monte Carlo estimates. © Global Science Press Limited 2015.
    Citation
    Efendiev, Y., Jin, B., Michael, P., & Tan, X. (2014). Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems. Communications in Computational Physics, 17(1), 259–286. doi:10.4208/cicp.021013.260614a
    Sponsors
    Y. Efendiev's work is partially supported by the U.S. Department of Energy Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program under Award Number DE-FG02-13ER26165 and the DoD Army ARO Project. The research of B. Jin is partly supported by NSF Grant DMS-1319052.
    Publisher
    Global Science Press
    Journal
    Communications in Computational Physics
    DOI
    10.4208/cicp.021013.260614a
    arXiv
    1402.5068
    ae974a485f413a2113503eed53cd6c53
    10.4208/cicp.021013.260614a
    Scopus Count
    Collections
    Articles; Computer, Electrical and Mathematical Science and Engineering (CEMSE) Division

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