Measuring the Irreversibility of Numerical Schemes for Reversible Stochastic Differential Equations
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1351-1379.

For a stationary Markov process the detailed balance condition is equivalent to the time-reversibility of the process. For stochastic differential equations (SDE's), the time discretization of numerical schemes usually destroys the time-reversibility property. Despite an extensive literature on the numerical analysis for SDE's, their stability properties, strong and/or weak error estimates, large deviations and infinite-time estimates, no quantitative results are known on the lack of reversibility of discrete-time approximation processes. In this paper we provide such quantitative estimates by using the concept of entropy production rate, inspired by ideas from non-equilibrium statistical mechanics. The entropy production rate for a stochastic process is defined as the relative entropy (per unit time) of the path measure of the process with respect to the path measure of the time-reversed process. By construction the entropy production rate is nonnegative and it vanishes if and only if the process is reversible. Crucially, from a numerical point of view, the entropy production rate is an a posteriori quantity, hence it can be computed in the course of a simulation as the ergodic average of a certain functional of the process (the so-called Gallavotti-Cohen (GC) action functional). We compute the entropy production for various numerical schemes such as explicit Euler-Maruyama and explicit Milstein's for reversible SDEs with additive or multiplicative noise. In addition we analyze the entropy production for the BBK integrator for the Langevin equation. The order (in the time-discretization step Δt) of the entropy production rate provides a tool to classify numerical schemes in terms of their (discretization-induced) irreversibility. Our results show that the type of the noise critically affects the behavior of the entropy production rate. As a striking example of our results we show that the Euler scheme for multiplicative noise is not an adequate scheme from a reversibility point of view since its entropy production rate does not decrease with Δt.

DOI : 10.1051/m2an/2013142
Classification : 65C30, 82C3, 60H10
Mots-clés : stochastic differential equations, detailed balance, reversibility, relative entropy, entropy production, numerical integration, (overdamped) Langevin process
@article{M2AN_2014__48_5_1351_0,
     author = {Katsoulakis, Markos and Pantazis, Yannis and Rey-Bellet, Luc},
     title = {Measuring the {Irreversibility} of {Numerical} {Schemes} for {Reversible} {Stochastic} {Differential} {Equations}},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1351--1379},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {5},
     year = {2014},
     doi = {10.1051/m2an/2013142},
     mrnumber = {3264357},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2013142/}
}
TY  - JOUR
AU  - Katsoulakis, Markos
AU  - Pantazis, Yannis
AU  - Rey-Bellet, Luc
TI  - Measuring the Irreversibility of Numerical Schemes for Reversible Stochastic Differential Equations
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2014
SP  - 1351
EP  - 1379
VL  - 48
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2013142/
DO  - 10.1051/m2an/2013142
LA  - en
ID  - M2AN_2014__48_5_1351_0
ER  - 
%0 Journal Article
%A Katsoulakis, Markos
%A Pantazis, Yannis
%A Rey-Bellet, Luc
%T Measuring the Irreversibility of Numerical Schemes for Reversible Stochastic Differential Equations
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2014
%P 1351-1379
%V 48
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2013142/
%R 10.1051/m2an/2013142
%G en
%F M2AN_2014__48_5_1351_0
Katsoulakis, Markos; Pantazis, Yannis; Rey-Bellet, Luc. Measuring the Irreversibility of Numerical Schemes for Reversible Stochastic Differential Equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 5, pp. 1351-1379. doi : 10.1051/m2an/2013142. http://archive.numdam.org/articles/10.1051/m2an/2013142/

[1] G. Arampatzis, M.A. Katsoulakis, P. Plechac, M. Taufer and L. Xu, Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms. J. Comput. Phys. 231 (2012) 7795-7841. | MR | Zbl

[2] V. Bally and D. Talay, The law of the Euler scheme for stochastic differential equations. I. Convergence rate of the density. Monte Carlo Methods Appl. 2 (1996) 93-128. | MR | Zbl

[3] V. Bally and D. Talay, 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. | MR | Zbl

[4] N. Bou-Rabee and E. Vanden-Eijnden, Pathwise accuracy and ergodicity of Metropolized integrators for SDEs. Commut. Pure Appl. Math. LXIII (2010) 0655-0696. | MR | Zbl

[5] A. Brunger, C.B. Brooks and M. Karplus, Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett. 105 (1984) 495-500.

[6] S. Delong, B.E. Griffith, E. Vanden-Eijnden and A. Donev, Temporal integrators for fluctuating hydrodynamics. Phys. Rev. E 87 (2013) 11.

[7] G. Gallavotti and E.G.D. Cohen, Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74 (1995) 2694-2697.

[8] C. Gardiner, Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Springer Series in Synergetics (1985). | MR | Zbl

[9] D.T. Gillespie, Markov Processes: An Introduction for Physical Scientists. Academic Press, New York (1992). | MR | Zbl

[10] 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). | MR | Zbl

[11] V. Jakšić, C.-A. Pillet and L. Rey-Bellet, Entropic fluctuations in statistical mechanics: I. classical dynamical systems. Nonlinearity 2 (2011) 699-763. | MR | Zbl

[12] R. Khasminskii, Stochastic Stability of Differential Equations, 2nd edition. Springer (2010). | MR | Zbl

[13] P.E. Kloeden and E. Platen, Numerical Solution Stochastic Differential Equations, 3rd edition. Springer-Verlag (1999). | MR | Zbl

[14] J.L. Lebowitz and H. Spohn, A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95 (1999) 333-365. | MR | Zbl

[15] T. Lelievre, M. Rousset and G. Stoltz, Free Energy Computations: A Math. Perspective. Imperial College Press (2010). | MR | Zbl

[16] C. Maes and K. Netočný, Minimum entropy production principle from a dynamical fluctuation law. J. Math. Phys. 48 (2007) 053306. | MR | Zbl

[17] C. Maes, K. Netočný and B. Wynants, Steady state statistics of driven diffusions. Phys. A 387 (2008) 2675-2689. | MR

[18] C. Maes, F. Redig and A. Van Moffaert, On the definition of entropy production, via examples. J. Math. Phys. 41 (2000) 1528-1553. | MR | Zbl

[19] J.C. Mattingly, A.M. Stuart and M.V. Tretyakov, Convergence of numerical time-averaging and stationary measures via Poisson equations. SIAM J. Numer. Anal. 48 (2010) 552-577. | MR | Zbl

[20] J.C. Mattingly, A.M. Stuart and D.J. Higham, Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Process. Appl. 101 (2002) 185-232. | MR | Zbl

[21] S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability. Springer-Verlag (1993). | MR | Zbl

[22] G. Milstein and M. Tretyakov, Stochastic Numerics for Mathematical Physics for Springer (2004). | MR | Zbl

[23] G. Nicolis and I. Prigogine, Self-Organization in Nonequilibrium Systems. Wiley, New York (1977). | MR | Zbl

[24] L. Rey-Bellet and L.E. Thomas, Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Comm. Math. Phys. 225 (2002) 305-329. | MR | Zbl

[25] L. Rey-Bellet, Ergodic properties of Markov processes. In Open quantum systems. II, vol. 1881. Lect. Notes Math. Springer, Berlin (2006) 1-39. | MR | Zbl

[26] G.O. Roberts and R.L. Tweedie, Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 (1996) 95-110. | MR | Zbl

[27] T. Schlick, Molecular Modeling and Simulation. Springer (2002). | MR | Zbl

[28] J. Schnakenberg, Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Modern Phys. 48 (1976) 571-585. | MR

[29] Yunsic Shim and J.G. Amar, Semirigorous synchronous relaxation algorithm for parallel kinetic Monte Carlo simulations of thin film growth. Phys. Rev. B 71 (2005) 125-432. | Zbl

[30] D. Talay, Second order discretization schemes of stochastic differential systems for the computation of the invariant law. Stochastics Stochastics Rep. 29 (1990) 13-36. | Zbl

[31] D. Talay, 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. | MR | Zbl

[32] D. Talay and L. Tubaro, Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 (1990) 483-509. | MR | Zbl

[33] N.G. Van Kampen, Stochastic Processes in Physics and Chemistry. North Holland (2006). | Zbl

Cité par Sources :