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The subsurface flow is usually subject to uncertain porous media structures.In most cases, however, we only have partial knowledge about the porous mediaproperties. A common approach is to model the uncertain parameters of porousmedia as random fields, then the statistical moments (e.g. expectation) of theQuantity of Interest(QoI) can be evaluated by the Monte Carlo method. In thisstudy, we develop a full multigrid-multilevel Monte Carlo (FMG-MLMC) method tospeed up the evaluation of random parameters effects on single-phase porousflows. In general, MLMC method applies a series of discretization withincreasing resolution and computes the QoI on each of them. The effectivevariance reduction is the success of the method. We exploit the similarhierarchies of MLMC and multigrid methods and obtain the solution on coarsemesh $Q^c_l$ as a byproduct of the full multigrid solution on fine mesh $Q^f_l$on each level $l$. In the cases considered in this work, the computationalsaving due to the coarse mesh samples saving is $20\%$ asymptotically. Besides,a comparison of Monte Carlo and Quasi-Monte Carlo (QMC) methods reveals asmaller estimator variance and a faster convergence rate of the latter approachin this study.

A full multigrid multilevel Monte Carlo method for the single phase subsurface flow with random coefficients