A Fully Parallelizable Space-Time Multilevel Monte Carlo Method for Stochastic Differential Equations with Additive Noise

Abstract

In this work a combination of parallelizable space-time multigrid methods for deterministic parabolic partial differential equations with multilevel Monte Carlo methods for stochastic differential equations with additive noise is developed. Instead of applying the backward Euler--Maruyama scheme sequentially for every time step, the basic idea for the considered space-time method is to solve a large linear system at once, for which a parallelizable multigrid algorithm is constructed that inherits the space-time hierarchy of the multilevel Monte Carlo method. Overall, this results in a fully parallelizable algorithm with respect to space, time, and probability. As model problems for the numerical testing of the proposed method serve in finite dimensions the Ornstein--Uhlenbeck process and in infinite dimensions the stochastic heat equation in 1, 2, and 3 space dimensions.