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International Journal for Uncertainty Quantification

Impact factor: 1.000

ISSN Print: 2152-5080
ISSN Online: 2152-5099

Open Access

International Journal for Uncertainty Quantification

DOI: 10.1615/Int.J.UncertaintyQuantification.2016018677
pages 57-81

A MULTIGRID MULTILEVEL MONTE CARLO METHOD USING HIGH-ORDER FINITE-VOLUME SCHEME FOR LOGNORMAL DIFFUSION PROBLEMS

Prashant Kumar
CWI-Centrum Wiskunde & Informatica, Amsterdam, The Netherlands
Cornelis W. Oosterlee
CWI-Centrum Wiskunde & Informatica, Amsterdam, The Netherlands; Delft University of Technology, Delft Institute of Applied Mathematics, Delft, The Netherlands
Richard P. Dwight
Delft University of Technology, Faculty of Aerospace Engineering, Delft, The Netherlands

ABSTRACT

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.