  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.
