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

## TL;DR

This paper introduces a parametric acceleration technique for multilevel Monte Carlo methods applied to nonlinear variably saturated flow, significantly improving computational efficiency in uncertainty quantification of Richards' equation.

## Contribution

It develops a novel MLMC estimator with parametric continuation for nonlinear stochastic Richards' equation, enhancing efficiency over traditional methods.

## Key findings

- Achieves substantial computational savings with the new estimator.
- Demonstrates robustness of the solver for highly nonlinear systems.
- Validates the approach through numerical experiments.

## 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.