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

The efficient simulation of the mean value of a non-linear functional of the solution to a linear stochastic partial differential equations (SPDE) with additive Gaussian noise at a fixed time 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 (MLMC), 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 the costs of the latter are significantly greater than those of the former.