Rapid covariance-based sampling of linear SPDE approximations in the multilevel Monte Carlo method

TL;DR

This paper introduces a fast covariance-based sampling scheme for linear SPDEs within a multilevel Monte Carlo framework, significantly reducing computational costs compared to traditional path-based methods.

Contribution

It presents a novel covariance computation scheme for non-nested Galerkin approximations in multilevel Monte Carlo for linear SPDEs, enabling more efficient sampling.

Findings

The new schemes are computationally more efficient than traditional methods.

Simulations confirm significant cost reductions under certain assumptions.

The approach does not require nested Galerkin subspaces at different levels.

Abstract

The efficient simulation of the mean value of a non-linear functional of the solution to a linear stochastic partial differential equation (SPDE) with additive Gaussian noise is considered. A Galerkin finite element method is employed along with an implicit Euler scheme to arrive at a fully discrete approximation of the mild solution to the equation. A scheme is presented to compute the covariance of this approximation, which allows for rapid sampling in a Monte Carlo method. This is then extended to a multilevel Monte Carlo method, for which a scheme to compute the cross-covariance between the approximations at different levels is presented. In contrast to traditional path-based methods it is not assumed that the Galerkin subspaces at these levels are nested. The computational complexities of the presented schemes are compared to traditional methods and simulations confirm that, under suitable assumptions, the costs of the new schemes are significantly lower.