A Full Multigrid-Multilevel Quasi-Monte Carlo Approach for Elliptic PDE with Random Coefficients

Abstract

The subsurface flow is usually subject to uncertain porous media structures. However, in most cases we only have partial knowledge about the porous media properties. A common approach is to model the uncertain parameters as random fields, then the expectation of Quantity of Interest(QoI) can be evaluated by the Monte Carlo method. In this study, we develop a full multigrid-multilevel Monte Carlo (FMG-MLMC) method to speed up the evaluation of random parameters effects on single-phase porous flows. In general, MLMC method applies a series of discretization with increasing resolution and computes the QoI on each of them, the success of which lies in the effective variance reduction. We exploit the similar hierarchies of MLMC and multigrid methods, and obtain the solution on coarse mesh Qcl as a byproduct of the multigrid solution on fine mesh Qfl on each level l. In the cases considered in this thesis, the computational saving is 20% theoretically. In addition, a comparison of Monte Carlo and Quasi-Monte Carlo (QMC) methods reveals a smaller estimator variance and faster convergence rate of the latter method in this study.