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

TL;DR

This paper compares Monte Carlo and multilevel Monte Carlo methods for simulating weak errors in SPDE approximations, providing bounds, theoretical analysis, and simulations to evaluate their efficiency and accuracy.

Contribution

It introduces bounds for additional errors in weak error simulations and compares Monte Carlo and multilevel Monte Carlo methods in this context.

Findings

Multilevel Monte Carlo reduces sampling error more efficiently.

Bounds for additional errors are tighter for multilevel Monte Carlo.

Simulations confirm theoretical error estimates.

Abstract

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.