A multilevel Monte Carlo (MLMC) method for quantifying model-form uncertainties associated with the Reynolds-Averaged Navier-Stokes (RANS) simulations is presented. Two, high-dimensional, stochastic extensions of the RANS equations are considered to demonstrate the applicability of the MLMC method. The first approach is based on global perturbation of the baseline eddy viscosity field using a lognormal random field. A more general second extension is considered based on the work of [Xiao et al. (2017)], where the entire Reynolds Stress Tensor (RST) is perturbed while maintaining realizability. For two fundamental flows, we show that the MLMC method based on a hierarchy of meshes is asymptotically faster than plain Monte Carlo. Additionally, we demonstrate that for some flows an optimal multilevel estimator can be obtained for which the cost scales with the same order as a single CFD solve on the finest grid level.

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We present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that is modeled using the Richards equation. We propose a stochastic extension for the empirical models that are typically employed to close the Richards equations. This is achieved by treating the soil parameters in these models as spatially correlated random fields with appropriately defined marginal distributions. As some of these parameters can only take values in a specific range, non-Gaussian models are utilized. The randomness in these parameters may result in path-wise highly nonlinear systems, so that a robust solver with respect to the random input is required. For this purpose, a solution method based on a combination of the modified Picard iteration and a cell-centered multigrid method for heterogeneous diffusion coefficients is utilized. Moreover, we propose a non-standard MLMC estimator to solve the resulting high-dimensional stochastic Richards equation. The improved efficiency of this multilevel estimator is achieved by parametric continuation that allows us to incorporate simpler nonlinear problems on coarser levels for variance reduction while the target strongly nonlinear problem is solved only on the finest level. Several numerical experiments are presented showing computational savings obtained by the new estimator compared with the original MC estimator.

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Uncertainty is ubiquitous in many areas of science and engineering. It may result from the inadequacy of mathematical models to represent the reality or from unknown physical parameters that are required as inputs for these models. Uncertainty may also arise due to the inherent randomness of the system being analyzed. For many problems of practical interest, uncertainty quantification (UQ) can involve computations that are intractable even for the modern supercomputers, if conventional mathematical techniques are utilized. The reason is typically a product of complexity factors associated with many samples needed to compute the statistics, and for each sample, complexity associated with the spatio-temporal scales characteristics to the system. The main objective of this research is to obtain multilevel solvers for stochastic fluid flow problems with high-dimensional uncertainties. In our approach, the complexity arising due to sampling is overcome by the multilevel Monte Carlo (MLMC) method and complexity due to spatio-temporal scales is eliminated via the multigrid solver. Historically, Monte Carlo (MC) type methods have been proven to be the methods of choice for problems with a large uncertainty dimension as they do not suffer from the curse of dimensionality. A well-known computational bottleneck associated with the plain MC method is the slow convergence of the sampling error. For problems involving a wide range of space and time scales, ensuring a low mean square error will require a large number of MC samples on a very fine computational mesh making the estimator very expensive. Inspired by the multigrid ideas, the MLMC method generalizes the standard MC to multiple grids, exhibiting an exceptional improvement. The efficiency of the MLMC method comes from solving the problem of interest on a coarse grid and subsequently adding corrections based on fewer mesh resolutions. On the coarsest grid, a large number of samples can be computed inexpensively. The corrections computed on fewer grids, have smaller variances and can be estimated accurately using only fewer samples. The estimates at different levels are then combined using a telescopic sum...@en

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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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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