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.
@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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {48},
year = {2014},
pages = {1351-1379},
doi = {10.1051/m2an/2013142},
mrnumber = {3264357},
language = {en},
url = {http://dml.mathdoc.fr/item/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 - Modélisation Mathématique et Analyse Numérique, Tome 48 (2014) pp. 1351-1379. doi : 10.1051/m2an/2013142. http://gdmltest.u-ga.fr/item/M2AN_2014__48_5_1351_0/
[1] , , , and , Hierarchical fractional-step approximations and parallel kinetic Monte Carlo algorithms. J. Comput. Phys. 231 (2012) 7795-7841. | MR 2972869 | Zbl 1259.82110
[2] and , 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 1401964 | Zbl 0866.60049
[3] and , 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 1367666 | Zbl 0838.60051
[4] and , Pathwise accuracy and ergodicity of Metropolized integrators for SDEs. Commut. Pure Appl. Math. LXIII (2010) 0655-0696. | MR 2583309 | Zbl 1214.60031
[5] , and , Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett. 105 (1984) 495-500.
[6] , , and , Temporal integrators for fluctuating hydrodynamics. Phys. Rev. E 87 (2013) 11.
[7] and , Dynamical ensembles in nonequilibrium statistical mechanics. Phys. Rev. Lett. 74 (1995) 2694-2697.
[8] , Handbook of Stochastic Methods: for Physics, Chemistry and the Natural Sciences. Springer Series in Synergetics (1985). | MR 858704 | Zbl 0515.60002
[9] , : An Introduction for Physical Scientists. Academic Press, New York (1992). | MR 1133392 | Zbl 0743.60001
[10] , and , 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 2221614 | Zbl 1094.65125
[11] , and , Entropic fluctuations in statistical mechanics: I. classical dynamical systems. Nonlinearity 2 (2011) 699-763. | MR 2765481 | Zbl 1234.37012
[12] , Stochastic Stability of Differential Equations, 2nd edition. Springer (2010). | MR 2894052 | Zbl 1241.60002
[13] and , Numerical Solution Stochastic Differential Equations, 3rd edition. Springer-Verlag (1999). | MR 1214374 | Zbl 0752.60043
[14] and , A Gallavotti-Cohen type symmetry in the large deviation functional for stochastic dynamics. J. Stat. Phys. 95 (1999) 333-365. | MR 1705590 | Zbl 0934.60090
[15] , and , Free Energy Computations: A Math. Perspective. Imperial College Press (2010). | MR 2681239 | Zbl 1227.82002
[16] and , Minimum entropy production principle from a dynamical fluctuation law. J. Math. Phys. 48 (2007) 053306. | MR 2329862 | Zbl 1144.81384
[17] , and , Steady state statistics of driven diffusions. Phys. A 387 (2008) 2675-2689. | MR 2587167
[18] , and , On the definition of entropy production, via examples. J. Math. Phys. 41 (2000) 1528-1553. | MR 1757968 | Zbl 0977.82025
[19] , and , Convergence of numerical time-averaging and stationary measures via Poisson equations. SIAM J. Numer. Anal. 48 (2010) 552-577. | MR 2669996 | Zbl 1217.65014
[20] , and , Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise. Stoch. Process. Appl. 101 (2002) 185-232. | MR 1931266 | Zbl 1075.60072
[21] and , Markov Chains and Stochastic Stability. Springer-Verlag (1993). | MR 1287609 | Zbl 0925.60001
[22] and , Stochastic Numerics for Mathematical Physics for Springer (2004). | MR 2069903 | Zbl 1085.60004
[23] and , Self-Organization in Nonequilibrium Systems. Wiley, New York (1977). | MR 522141 | Zbl 0363.93005
[24] and , Exponential convergence to non-equilibrium stationary states in classical statistical mechanics. Comm. Math. Phys. 225 (2002) 305-329. | MR 1889227 | Zbl 0989.82023
[25] , Ergodic properties of Markov processes. In Open quantum systems. II, vol. 1881. Lect. Notes Math. Springer, Berlin (2006) 1-39. | MR 2248986 | Zbl 1126.60057
[26] and , Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 (1996) 95-110. | MR 1399158 | Zbl 0888.60064
[27] , Molecular Modeling and Simulation. Springer (2002). | MR 1921061 | Zbl 1011.92019
[28] , Network theory of microscopic and macroscopic behavior of master equation systems. Rev. Modern Phys. 48 (1976) 571-585. | MR 443796
[29] and , Semirigorous synchronous relaxation algorithm for parallel kinetic Monte Carlo simulations of thin film growth. Phys. Rev. B 71 (2005) 125-432. | Zbl 1087.82015
[30] , Second order discretization schemes of stochastic differential systems for the computation of the invariant law. Stochastics Stochastics Rep. 29 (1990) 13-36. | Zbl 0697.60066
[31] , 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 1924934 | Zbl 1011.60039
[32] and , Expansion of the global error for numerical schemes solving stochastic differential equations. Stoch. Anal. Appl. 8 (1990) 483-509. | MR 1091544 | Zbl 0718.60058
[33] , Stochastic Processes in Physics and Chemistry. North Holland (2006). | Zbl 0974.60020