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

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.

16 pages, 3 figures, updated to version published in the Proceedings of MCQMC14

References in corpus (2)

• Duality in refined Sobolev-Malliavin spaces and weak approximations of SPDE
• Weak convergence rates for Euler-type approximations of semilinear stochastic evolution equations with nonlinear diffusion coefficients

Cited by in corpus (4)

• Drift-preserving numerical integrators for stochastic Hamiltonian systems
• Numerical analysis of lognormal diffusions on the sphere
• Monte Carlo versus multilevel Monte Carlo in weak error simulations of SPDE approximations
• Hilbert--Schmidt regularity of symmetric integral operators on bounded domains with applications to SPDE approximations