A parametric acceleration of multilevel Monte Carlo convergence for nonlinear variably saturated flow

Abstract

We present a multilevel Monte Carlo (MLMC) method for the uncertainty quantification of variably saturated porous media flow that are 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 to the original MC estimator.

Modified Picard with multigrid method for two‐phase flow problems in rigid porous media

“…The factor $$$ 1/2 $$$ in the previous expression is due to the lack of consistency of the operator $$$ {R}_h^{2h}{L}_h{P}_{2h}^h $$$ with the differential operator 55 . We have chosen a $$$ W\left(2,2\right)- $$$cycle since it was found in References 54 and 49 to be a very robust and efficient multigrid cycling strategy. The stopping criterion for the multigrid solver is set to reduce the initial residual by a factor $$$ TO{L}_{MG} $$$, i.e.…”

“…We will see that such a basic multigrid algorithm converges well even in the context of random heterogeneous fields. A similar multigrid algorithm was proposed in Reference 49 for solving Richard's equation.…”

Accelerated Multilevel Monte Carlo With Kernel‐Based Smoothing and Latinized Stratification

“…Most MLMC studies focus either on the estimation of statistical moments of a QoI (Kumar et al, 2019;Linde et al, 2017;Müller et al, 2013Müller et al, , 2016 or on the single-point evaluation of its CDF to estimate rare events, for example, probability of failure (Ullmann & Papaioannou, 2015). Much less work has been done on MLMC for estimation of the full CDF/PDF of a QoI.…”