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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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References

Arampatzis, G., Katsoulakis, M.A., Plechac, P., Taufer, M. and Xu, L., Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms. J. Comput. Phys. 231 (2012) 7795–7841. Google Scholar
Bally, V. and Talay, D., The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the density. Monte Carlo Methods Appl. 2 (1996) 93–128. Google Scholar
Bally, V. and Talay, D., The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the distribution function. Probab. Theory Related Fields 104 (1996) 43–60. Google Scholar
N. Bou-Rabee and E. Vanden-Eijnden, Pathwise accuracy and ergodicity of Metropolized integrators for SDEs. Commut. Pure Appl. Math. LXIII (2010) 0655–0696.
Brunger, A., Brooks, C.B. and Karplus, M., Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett. 105 (1984) 495–500. Google Scholar
Delong, S., Griffith, B.E., Vanden-Eijnden, E. and Donev, A., Temporal integrators for fluctuating hydrodynamics. Phys. Rev. E 87 (2013) 11. Google Scholar
Gallavotti, G. and Cohen, E.G.D., Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74 (1995) 2694–2697. Google ScholarPubMed
C. Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Springer Series in Synergetics (1985).
D.T. Gillespie, Markov Processes: An Introduction for Physical Scientists. Academic Press, New York (1992).
E. Hairer, Ch. Lubich and G. Wanner, Structure-preserving algorithms for ordinary differential equations, in Geometric Numerical Integration. vol. 31 of Springer Ser. Comput. Math., 2nd edition. Springer-Verlag, Berlin (2006).
Jakšić, V., Pillet, C.-A. and Rey-Bellet, L., Entropic fluctuations in statistical mechanics: I. classical dynamical systems. Nonlinearity 2 (2011) 699–763. Google Scholar
R. Khasminskii, Stochastic Stability of Differential Equations, 2nd edition. Springer (2010).
P.E. Kloeden and E. Platen, Numerical Solution Stochastic Differential Equations, 3rd edition. Springer-Verlag (1999).
Lebowitz, J.L. and Spohn, H., A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95 (1999) 333–365. Google Scholar
T. Lelievre, M. Rousset and G. Stoltz, Free Energy Computations: A Math. Perspective. Imperial College Press (2010).
Maes, C. and Netočný, K., Minimum entropy production principle from a dynamical fluctuation law. J. Math. Phys. 48 (2007) 053306. Google Scholar
Maes, C., Netočný, K. and Wynants, B., Steady state statistics of driven diffusions. Phys. A 387 (2008) 2675–2689. Google Scholar
Maes, C., Redig, F. and Van Moffaert, A., On the definition of entropy production, via examples. J. Math. Phys. 41 (2000) 1528–1553. Google Scholar
Mattingly, J.C., Stuart, A.M. and Tretyakov, M.V., Convergence of numerical time-averaging and stationary measures via Poisson equations. SIAM J. Numer. Anal. 48 (2010) 552–577. Google Scholar
Mattingly, J.C., Stuart, A.M. and Higham, D.J., Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Process. Appl. 101 (2002) 185–232. Google Scholar
S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability. Springer-Verlag (1993).
G. Milstein and M. Tretyakov, Stochastic Numerics for Mathematical Physics for Springer (2004).
G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems. Wiley, New York (1977).
Rey-Bellet, L. and Thomas, L.E., Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Comm. Math. Phys. 225 (2002) 305–329. Google Scholar
L. Rey-Bellet, Ergodic properties of Markov processes. In Open quantum systems. II, vol. 1881. Lect. Notes Math. Springer, Berlin (2006) 1–39.
Roberts, G.O. and Tweedie, R.L., Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 (1996) 95–110. Google Scholar
T. Schlick, Molecular Modeling and Simulation. Springer (2002).
Schnakenberg, J., Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Modern Phys. 48 (1976) 571–585. Google Scholar
Yunsic Shim and Amar, J.G., Semirigorous synchronous relaxation algorithm for parallel kinetic Monte Carlo simulations of thin film growth. Phys. Rev. B 71 (2005) 125–432. Google Scholar
Talay, D., Second order discretization schemes of stochastic differential systems for the computation of the invariant law. Stochastics Stochastics Rep. 29 (1990) 13–36. Google Scholar
Talay, D., Stochastic Hamiltonian systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Processes and Related Fields 8 (2002) 163–198. Google Scholar
Talay, D. and Tubaro, L., Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 (1990) 483–509. Google Scholar
N.G. van Kampen, Stochastic Processes in Physics and Chemistry. North Holland (2006).