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 \cite{ketelson2013}, and our numerical
experiments illustrate its efficiency and accuracy in comparison with standard
Monte Carlo estimates.Comment: 29 pages, 6 figure