  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.
