Improving MLMC for SDEs with application to the Langevin equation
For researchers using MLMC for SDEs, especially Langevin-type equations, this work offers incremental improvements by combining known techniques, with notable gains for small-noise problems.
This paper improves the efficiency of the Multilevel Monte Carlo method for SDEs, particularly the Langevin equation, by applying techniques like modified equations analysis, operator splitting, extrapolation, and discrete random variables. For small-noise problems, discrete random variables yield nearly two orders of magnitude efficiency gain at practical accuracy levels.
This paper applies several well-known tricks from the numerical treatment of deterministic differential equations to improve the efficiency of the Multilevel Monte Carlo (MLMC) method for stochastic differential equations (SDEs) and especially the Langevin equation. We use modified equations analysis to circumvent the need for a strong-approximation theory for the integrator, and we apply this to introduce MLMC for Langevin-type equations with integrators based on operator splitting. We combine this with extrapolation and investigate the use of discrete random variables in place of the Gaussian increments, which is a well-known technique for the weak approximation of SDEs. We show that, for small-noise problems, discrete random variables can lead to an increase in efficiency of almost two orders of magnitude for practical levels of accuracy.