A Multilevel Monte Carlo Algorithm for Parabolic Advection-Diffusion Problems with Discontinuous Coefficients

TL;DR

This paper develops a multilevel Monte Carlo method tailored for parabolic advection-diffusion problems with discontinuous coefficients, enabling efficient computation of solution moments in complex heterogeneous media.

Contribution

It introduces a novel multilevel Monte Carlo algorithm that handles discontinuous random coefficients and derives optimal convergence rates for such stochastic PDEs.

Findings

Achieves optimal mean-square convergence rates.

Handles stochastic discontinuities in media coefficients.

Provides a stochastic grid discretization approach.

Abstract

The Richards' equation is a model for flow of water in unsaturated soils. The coefficients of this (nonlinear) partial differential equation describe the permeability of the medium. Insufficient or uncertain measurements are commonly modeled by random coefficients. For flows in heterogeneous\textbackslash fractured\textbackslash porous media, the coefficients are modeled as discontinuous random fields, where the interfaces along the stochastic discontinuities represent transitions in the media. More precisely, the random coefficient is given by the sum of a (continuous) Gaussian random field and a (discontinuous) jump part. In this work moments of the solution to the random partial differential equation are calculated using a path-wise numerical approximation combined with multilevel Monte Carlo sampling. The discontinuities dictate the spatial discretization, which leads to a stochastic grid. Hence, the refinement parameter and problem-dependent constants in the error analysis are random variables and we derive (optimal) a-priori convergence rates in a mean-square sense.