A Note on the Importance of Weak Convergence Rates for SPDE Approximations in Multilevel Monte Carlo Schemes

TL;DR

This paper explores how weak convergence rates of SPDE approximations impact the efficiency of multilevel Monte Carlo schemes, showing that better weak convergence results can significantly reduce computational costs.

Contribution

It demonstrates the influence of weak convergence rates on sample complexity in multilevel Monte Carlo methods for SPDEs, providing insights for more efficient simulations.

Findings

Weak convergence rates directly affect the number of samples needed.

Improved weak convergence results lead to reduced computational complexity.

The study clarifies the role of weak convergence in SPDE approximation efficiency.

Abstract

It is a well-known rule of thumb that approximations of stochastic partial differential equations have essentially twice the order of weak convergence compared to the corresponding order of strong convergence. This is already known for many approximations of stochastic (ordinary) differential equations while it is recent research for stochastic partial differential equations. In this note it is shown how the availability of weak convergence results influences the number of samples in multilevel Monte Carlo schemes and therefore reduces the computational complexity of these schemes for a given accuracy of the approximations.