A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system

P. Kumar; P. Luo; F.J. Gaspar; C.W. Oosterlee

doi:10.1016/j.jcp.2018.05.046

A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system

P. Kumar, P. Luo, F.J. Gaspar, C.W. Oosterlee

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Research output: Contribution to journal › Article › Academic › peer-review

Abstract

A multilevel Monte Carlo (MLMC) method for Uncertainty Quantification (UQ) of advection-dominated contaminant transport in a coupled Darcy–Stokes flow system is described. In particular, we focus on high-dimensional epistemic uncertainty due to an unknown permeability field in the Darcy domain that is modelled as a lognormal random field. This paper explores different numerical strategies for the subproblems and suggests an optimal combination for the MLMC estimator. We propose a specific monolithic multigrid algorithm to efficiently solve the steady-state Darcy–Stokes flow with a highly heterogeneous diffusion coefficient. Furthermore, we describe an Alternating Direction Implicit (ADI) based time-stepping for the flux-limited quadratic upwinding discretization for the transport problem. Numerical experiments illustrating the multigrid convergence and cost of the MLMC estimator with respect to the smoothness of permeability field are presented.

Original language	English
Journal	Journal of Computational Physics
Volume	371
DOIs	https://doi.org/10.1016/j.jcp.2018.05.046
Publication status	Published - 2018
Externally published	Yes

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10.1016/j.jcp.2018.05.046

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title = "A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system",

abstract = "A multilevel Monte Carlo (MLMC) method for Uncertainty Quantification (UQ) of advection-dominated contaminant transport in a coupled Darcy–Stokes flow system is described. In particular, we focus on high-dimensional epistemic uncertainty due to an unknown permeability field in the Darcy domain that is modelled as a lognormal random field. This paper explores different numerical strategies for the subproblems and suggests an optimal combination for the MLMC estimator. We propose a specific monolithic multigrid algorithm to efficiently solve the steady-state Darcy–Stokes flow with a highly heterogeneous diffusion coefficient. Furthermore, we describe an Alternating Direction Implicit (ADI) based time-stepping for the flux-limited quadratic upwinding discretization for the transport problem. Numerical experiments illustrating the multigrid convergence and cost of the MLMC estimator with respect to the smoothness of permeability field are presented.",

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AU - Kumar, P.

AU - Luo, P.

AU - Gaspar, F.J.

AU - Oosterlee, C.W.

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AB - A multilevel Monte Carlo (MLMC) method for Uncertainty Quantification (UQ) of advection-dominated contaminant transport in a coupled Darcy–Stokes flow system is described. In particular, we focus on high-dimensional epistemic uncertainty due to an unknown permeability field in the Darcy domain that is modelled as a lognormal random field. This paper explores different numerical strategies for the subproblems and suggests an optimal combination for the MLMC estimator. We propose a specific monolithic multigrid algorithm to efficiently solve the steady-state Darcy–Stokes flow with a highly heterogeneous diffusion coefficient. Furthermore, we describe an Alternating Direction Implicit (ADI) based time-stepping for the flux-limited quadratic upwinding discretization for the transport problem. Numerical experiments illustrating the multigrid convergence and cost of the MLMC estimator with respect to the smoothness of permeability field are presented.

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JO - Journal of Computational Physics

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A multigrid multilevel Monte Carlo method for transport in the Darcy–Stokes system

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