TY - JOUR
T1 - A multigrid multilevel monte carlo method using high-order finite-volume scheme for lognormal diffusion problems
AU - Kumar, Prashant
AU - Oosterlee, Cornelis W.
AU - Dwight, Richard P.
PY - 2017
Y1 - 2017
N2 - 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.
AB - 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.
KW - Fourth-order discretization
KW - Full multigrid
KW - Groundwater flow
KW - Multilevel Monte Carlo
KW - Random fields
KW - Stochastic partial differential equations
UR - http://www.scopus.com/inward/record.url?scp=85015961540&partnerID=8YFLogxK
U2 - 10.1615/Int.J.UncertaintyQuantification.2016018677
DO - 10.1615/Int.J.UncertaintyQuantification.2016018677
M3 - Article
AN - SCOPUS:85015961540
SN - 2152-5080
VL - 7
SP - 57
EP - 81
JO - International Journal of Uncertainty Quantification
JF - International Journal of Uncertainty Quantification
IS - 1
ER -