The aim of this paper is to show that a high-order discretization can be used to improve the convergence of a multilevel Monte Carlo method for elliptic partial differential equations with lognormal random coefficients in combination with the multigrid solution method. To demonstrate this, we consider a fourth-order accurate finite-volume discretization. With the help of the Matérn family of covariance functions, we simulate the coefficient field with different degrees of smoothness. The idea behind using a fourth-order scheme is to capture the additional regularity in the solution introduced due to higher smoothness of the random field. Second-order schemes previously utilized for these types of problems are not able to fully exploit this additional regularity. We also propose a practical way of combining a full multigrid solver with the multilevel Monte Carlo estimator constructed on the same mesh hierarchy. Through this integration, one full multigrid solve at any level provides a valid sample for all the preceding Monte Carlo levels. The numerical results show that the fourth-order multilevel estimator consistently outperforms the second-order variant. In addition, we observe an asymptotic gain for the standard Monte Carlo estimator.

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International Journal for Uncertainty Quantification
Uncertainty Quantication in Hydraulic Fracturing using Multi-Level Monte Carlo and Multigrid
Scientific Computing

Kumar, P., Oosterlee, K., & Dwight, R. (2017). A multigrid multilevel Monte Carlo method using high-order finite-volume scheme for lognormal diffusion problems. International Journal for Uncertainty Quantification, 7(1), 57–81. doi:10.1615/Int.J.UncertaintyQuantification.2016018677