Spectral Methods for Multiscale Stochastic Differential Equations - Abdulle,A; Pavliotis,GA; Vaes,U (SIAM/ASA Journal on Uncertainty Quantification, 2017)
This paper presents a new method for the solution of multiscale stochastic differential equations atthe diffusive time scale. In contrast to averaging-based methods, e.g., the heterogeneous multiscalemethod (HMM) or the equation-free method, which rely on Monte Carlo simulations, in this paperwe introduce a new numerical methodology that is based on a spectral method. In particular, we usean expansion in Hermite functions to approximate the solution of an appropriate Poisson equation,which is used in order to calculate the coefficients of the homogenized equation. Spectral convergenceis proved under suitable assumptions. Numerical experiments corroborate the theory and illustratethe performance of the method. A comparison with the HMM and an application to singularlyperturbed stochastic PDEs are also presented.