Published 6 issues per year
ISSN Print: 2152-5080
ISSN Online: 2152-5099
Indexed in
A MULTIGRID MULTILEVEL MONTE CARLO METHOD USING HIGH-ORDER FINITE-VOLUME SCHEME FOR LOGNORMAL DIFFUSION PROBLEMS
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.
-
Luo Peiyao, Rodrigo Carmen, Gaspar Francisco J., Oosterlee Cornelis W., Uzawa Smoother in Multigrid for the Coupled Porous Medium and Stokes Flow System, SIAM Journal on Scientific Computing, 39, 5, 2017. Crossref
-
Kumar Prashant, Luo Peiyao, Gaspar Francisco J., Oosterlee Cornelis W., A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system, Journal of Computational Physics, 371, 2018. Crossref
-
Robbe Pieterjan, Nuyens Dirk, Vandewalle Stefan, Recycling Samples in the Multigrid Multilevel (Quasi-)Monte Carlo Method, SIAM Journal on Scientific Computing, 41, 5, 2019. Crossref
-
Herrmann Lukas, Strong convergence analysis of iterative solvers for random operator equations, Calcolo, 56, 4, 2019. Crossref
-
Kumar Prashant, Schmelzer Martin, Dwight Richard P., Stochastic turbulence modeling in RANS simulations via multilevel Monte Carlo, Computers & Fluids, 201, 2020. Crossref
-
Robbe Pieterjan, Nuyens Dirk, Vandewalle Stefan, Enhanced multi‐index Monte Carlo by means of multiple semicoarsened multigrid for anisotropic diffusion problems, Numerical Linear Algebra with Applications, 28, 3, 2021. Crossref