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://doi.org/10.4208/cicp.021013.260614a
UR - https://global-sci.org/intro/article_detail/cicp/10958.html
KW - 
AB - <p style="text-align: justify;">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.</p>