A Note on the Importance of Weak Convergence Rates for SPDE Approximations in Multilevel Monte Carlo Schemes
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.
KeywordsStochastic partial differential equations Multilevel Monte Carlo methods Finite element approximations Weak error analysis Stochastic heat equation
This research was supported in part by the Knut and Alice Wallenberg foundation as well as the Swedish Research Council under Reg. No. 621-2014-3995. The author thanks Lukas Herrmann, Andreas Petersson, and two anonymous referees for helpful comments.
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