Multi-level Monte Carlo Finite Element method for parabolic stochastic partial differential equations

Abstract

We analyze the convergence and complexity of multi-level Monte Carlo (MLMC) discretizations of a class of abstract stochastic, parabolic equations driven by square integrable martingales. We show, under regularity assumptions on the solution that are minimal under certain criteria, that the judicious combination of piecewise linear, continuous multi-level Finite Element discretizations in space and Euler--Maruyama discretizations in time yields mean square convergence of order one in space and of order $1/2$ in time to the expected value of the mild solution. The complexity of the multi-level estimator is shown to scale log-linearly with respect to the corresponding work to generate a single solution path on the finest mesh, resp. of the corresponding deterministic parabolic problem on the finest mesh. Examples are provided for Lévy driven SPDEs as well as equations for randomly forced surface diffusions.