Improved Multilevel Monte Carlo Methods for Finite Volume Discretisations of Darcy Flow in Randomly Layered Media

TL;DR

This paper advances multilevel Monte Carlo methods for simulating steady state Darcy flow in randomly layered porous media, improving accuracy and efficiency through new convergence results and innovative variance reduction techniques.

Contribution

It introduces new convergence analysis for finite volume discretisations and a novel variance reduction method called Coarse Grid Variates for Darcy flow simulations.

Findings

Proved new convergence results for spatial discretisation errors.

Developed an optimal implementation using algebraic multigrid methods.

Introduced Coarse Grid Variates for variance reduction.

Abstract

We consider the application of multilevel Monte Carlo methods to steady state Darcy flow in a random porous medium, described mathematically by elliptic partial differential equations with random coefficients. The levels in the multilevel estimator are defined by finite volume discretisations of the governing equations with different mesh parameters. To simulate different layers in the subsurface, the permeability is modelled as a piecewise constant or piecewise spatially correlated random field, including the possibility of piecewise log-normal random fields. The location of the layers is assumed unknown, and modelled by a random process. We prove new convergence results of the spatial discretisation error required to quantify the mean square error of the multilevel estimator, and provide an optimal implementation of the method based on algebraic multigrid methods and a novel variance reduction technique termed Coarse Grid Variates.