Volume 17, Issue 1
Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems

Yalchin Efendiev, Bangti Jin, Michael Presho & Xiaosi Tan

Commun. Comput. Phys., 17 (2015), pp. 259-286.

Published online: 2018-04

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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.

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@Article{CiCP-17-259, author = {Yalchin Efendiev, Bangti Jin, Michael Presho and Xiaosi Tan}, title = {Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems}, journal = {Communications in Computational Physics}, year = {2018}, volume = {17}, number = {1}, pages = {259--286}, 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.

}, issn = {1991-7120}, doi = {https://doi.org/10.4208/cicp.021013.260614a}, url = {http://global-sci.org/intro/article_detail/cicp/10958.html} }
TY - JOUR T1 - Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems AU - Yalchin Efendiev, Bangti Jin, Michael Presho & Xiaosi Tan JO - Communications in Computational Physics VL - 1 SP - 259 EP - 286 PY - 2018 DA - 2018/04 SN - 17 DO - http://dor.org/10.4208/cicp.021013.260614a UR - https://global-sci.org/intro/cicp/10958.html KW - AB -

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.

Yalchin Efendiev, Bangti Jin, Michael Presho & Xiaosi Tan. (1970). 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
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