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Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems

Published online by Cambridge University Press:  19 December 2014

Yalchin Efendiev ,
Bangti Jin ,
Presho Michael  and
Xiaosi Tan
Yalchin Efendiev
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA SRI Center for Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia
Bangti Jin
Affiliation:
Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK
Presho Michael*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
Xiaosi Tan
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
*
*Email addresses: yalchinrefendiev@gmail.com (Y. Efendiev), bangti.jin@gmail.com (B. Jin), michaelpresho@gmail.com (M. Presho), tanxiaosi@gmail.com (X. Tan)
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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.

Keywords

65N3076M10Generalized multiscale finite element methodmultilevel Monte Carlo method, multilevel Markov chain Monte Carlouncertainty quantification
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 1 , January 2015 , pp. 259 - 286
Copyright
Copyright © Global Science Press Limited 2015 

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