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Measuring the Irreversibility of Numerical Schemes forReversible Stochastic Differential Equations∗

Published online by Cambridge University Press:  13 August 2014

Markos Katsoulakis ,
Yannis Pantazis  and
Luc Rey-Bellet
Markos Katsoulakis
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA.. luc@math.umass.edu
Yannis Pantazis
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA.. luc@math.umass.edu
Luc Rey-Bellet
Affiliation:
Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, USA.. luc@math.umass.edu
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Abstract

For a stationary Markov process the detailed balance condition is equivalent to thetime-reversibility of the process. For stochastic differential equations (SDE’s), the timediscretization of numerical schemes usually destroys the time-reversibility property.Despite an extensive literature on the numerical analysis for SDE’s, their stabilityproperties, strong and/or weak error estimates, large deviations and infinite-timeestimates, no quantitative results are known on the lack of reversibility of discrete-timeapproximation processes. In this paper we provide such quantitative estimates by using theconcept of entropy production rate, inspired by ideas from non-equilibrium statisticalmechanics. The entropy production rate for a stochastic process is defined as the relativeentropy (per unit time) of the path measure of the process with respect to the pathmeasure of the time-reversed process. By construction the entropy production rate isnonnegative and it vanishes if and only if the process is reversible. Crucially, from anumerical point of view, the entropy production rate is an a posterioriquantity, hence it can be computed in the course of a simulation as the ergodicaverage of a certain functional of the process (the so-called Gallavotti−Cohen (GC) action functional). We computethe entropy production for various numerical schemes such as explicit Euler−Maruyama and explicit Milstein’s forreversible SDEs with additive or multiplicative noise. In addition we analyze the entropyproduction for the BBK integrator for the Langevin equation. The order (in thetime-discretization step Δt) of the entropy production rate provides a tool toclassify numerical schemes in terms of their (discretization-induced) irreversibility. Ourresults show that the type of the noise critically affects the behavior of the entropyproduction rate. As a striking example of our results we show that the Euler scheme formultiplicative noise is not an adequate scheme from a reversibilitypoint of view since its entropy production rate does not decrease withΔt.

Keywords

Stochastic differential equationsdetailed balancereversibilityrelative entropyentropy productionnumerical integration(overdamped) Langevin process
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 5 , September 2014 , pp. 1351 - 1379
Copyright
© EDP Sciences, SMAI 2014

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