Multilevel Markov Chain Monte Carlo Method for High-Contrast Single-Phase Flow Problems (Communications in Computational Physics, 2018)
In this paper we propose a general framework for the uncertainty quantificationof quantities of interest for high-contrast single-phase flow problems. It isbased on the generalized multiscale finite element method (GMsFEM) and multilevelMonte Carlo (MLMC) methods. The former provides a hierarchy of approximations ofdifferent resolution, whereas the latter gives an efficient way to estimate quantities ofinterest using samples on different levels. The number of basis functions in the onlineGMsFEM stage can be varied to determine the solution resolution and the computationalcost, and to efficiently generate samples at different levels. In particular, it ischeap to generate samples on coarse grids but with low resolution, and it is expensiveto generate samples on fine grids with high accuracy. By suitably choosing thenumber of samples at different levels, one can leverage the expensive computation inlarger fine-grid spaces toward smaller coarse-grid spaces, while retaining the accuracyof the final Monte Carlo estimate. Further, we describe a multilevel Markov chainMonte Carlo method, which sequentially screens the proposal with different levels ofapproximations and reduces the number of evaluations required on fine grids, whilecombining the samples at different levels to arrive at an accurate estimate. The frameworkseamlessly integrates the multiscale features of the GMsFEM with the multilevelfeature of the MLMC methods following the work in [26], and our numerical experimentsillustrate its efficiency and accuracy in comparison with standard Monte Carloestimates.