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

TL;DR

This paper introduces a full multigrid-multilevel Monte Carlo method to efficiently evaluate the impact of uncertain porous media properties on single-phase subsurface flow, achieving significant computational savings.

Contribution

It develops a novel FMG-MLMC approach that combines multigrid and multilevel Monte Carlo techniques for faster, more efficient simulations of subsurface flow with random coefficients.

Findings

Achieves 20% computational savings with coarse mesh sampling.

QMC method shows smaller variance and faster convergence than MC.

Demonstrates effectiveness of FMG-MLMC in subsurface flow simulations.

Abstract

The subsurface flow is usually subject to uncertain porous media structures. In most cases, however, we only have partial knowledge about the porous media properties. A common approach is to model the uncertain parameters of porous media as random fields, then the statistical moments (e.g. expectation) of the 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 effective variance reduction is the success of the method. We exploit the similar hierarchies of MLMC and multigrid methods and obtain the solution on coarse mesh $Q^c_l$ 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 computational saving due to the coarse mesh samples saving is $20\%$ asymptotically. Besides, a comparison of Monte Carlo and Quasi-Monte Carlo (QMC) methods reveals a smaller estimator variance and a faster convergence rate of the latter approach in this study.