Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations

The simulation of the expectation of a stochastic quantity E[Y] by Monte Carlo methods is known to be computationally expensive especially if the stochastic quantity or its approximation Y-n is expensive to simulate, e.g., the solution of a stochastic partial differential equation. If the convergence of Y-n to Y in terms of the error |E[Y - Y-n]| is to be simulated, this will typically be done by a Monte Carlo method, i.e., |E[Y] - E-N [Y-n]| is computed. In this article upper and lower bounds for the additional error caused by this are determined and compared to those of |E-N [Y - Y-n]|, which are found to be smaller. Furthermore, the corresponding results for multilevel Monte Carlo estimators, for which the additional sampling error converges with the same rate as |E[Y - Y-n]|, are presented. Simulations of a stochastic heat equation driven by multiplicative Wiener noise and a geometric Brownian motion are performed which confirm the theoretical results and show the consequences of the presented theory for weak error simulations. (C) 2017 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.