Improving multilevel Monte Carlo for stochastic differential equations with application to the Langevin equation

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 analysi...

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Bibliographic Details
Published in:Proceedings of the Royal Society. A, Mathematical, physical, and engineering sciences Mathematical, physical, and engineering sciences, 2015-04, Vol.471 (2176), p.20140679-20140679
Main Authors: Müller, Eike H., Scheichl, Rob, Shardlow, Tony
Format: Article
Language:English
Subjects:
Extrapolation
Geometrical Integrators
Modified Equations
Numerical Solution Of Stochastic Differential Equations
Weak Approximation
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Summary: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 as an alternative to 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.
ISSN:1364-5021
1471-2946
DOI:10.1098/rspa.2014.0679