Theoretical and numerical comparison of some sampling methods for molecular dynamics
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 41 (2007) no. 2 , p. 351-389
doi : 10.1051/m2an:2007014
URL stable : http://www.numdam.org/item?id=M2AN_2007__41_2_351_0

Classification:  82B80,  37M25,  65C05,  65C40
The purpose of the present article is to compare different phase-space sampling methods, such as purely stochastic methods (Rejection method, Metropolized independence sampler, Importance Sampling), stochastically perturbed Molecular Dynamics methods (Hybrid Monte Carlo, Langevin Dynamics, Biased Random Walk), and purely deterministic methods (Nosé-Hoover chains, Nosé-Poincaré and Recursive Multiple Thermostats (RMT) methods). After recalling some theoretical convergence properties for the various methods, we provide some new convergence results for the Hybrid Monte Carlo scheme, requiring weaker (and easier to check) conditions than previously known conditions. We then turn to the numerical efficiency of the sampling schemes for a benchmark model of linear alkane molecules. In particular, the numerical distributions that are generated are compared in a systematic way, on the basis of some quantitative convergence indicators.

### Bibliographie

[1] E. Akhmatskaya and S. Reich, The targetted shadowing hybrid Monte Carlo (TSHMC) method, in New Algorithms for Macromolecular Simulation, Lecture Notes in Computational Science and Engineering 49, B. Leimkuhler, C. Chipot, R. Elber, A. Laaksonen, A. Mark, T. Schlick, C. Schuette and R. Skeel Eds., Springer Verlag, Berlin and New York (2006) 145-158. Zbl pre05049190

[2] M.P. Allen and D.J. Tildesley, Computer simulation of liquids. Oxford Science Publications (1987). Zbl 0703.68099

[3] H.C. Andersen, Molecular dynamics simulations at constant pressure and/or temperature J. Chem. Phys. 72 (1980) 2384-2393.

[4] E. Barth, B.J. Leimkuhler, and C.R. Sweet, Approach to thermal equilibrium in biomolecular simulation. Proceedings of AM3-2004 conference, available at the URL http://adrg.maths.ed.ac.uk/ADRG/FILES/Archive/BaLeSw2005.pdf Zbl 1094.92004

[5] S.D. Bond, B.J. Leimkuhler, and B.B. Laird, The Nosé-Poincaré method for constant temperature molecular dynamics. J. Comput. Phys. 151 (1999) 114-134. Zbl 0933.81058

[6] A. Brünger, C.B. Brooks, and M. Karplus, Stochastic boundary conditions for molecular dynamics simulations of ST2 water. Chem. Phys. Lett. 105 (1983) 495-500.

[7] E. Cancès, F. Castella, P. Chartier, E. Faou, C. Le Bris, F. Legoll and G. Turinici, High-order averaging schemes with error bounds for thermodynamical properties calculations by molecular dynamics simulations. J. Chem. Phys. 121 (2004) 10346-10355.

[8] E. Cancès, F. Castella, P. Chartier, E. Faou, C. Le Bris, F. Legoll and G. Turinici, Long-time averaging for integrable Hamiltonian dynamics. Numer. Math. 100 (2005) 211-232. Zbl 1084.65126

[9] E.A. Carter, G. Ciccotti, J.T. Hynes and R. Kapral, Constrained reaction coordinate dynamics for the simulation of rare events. Chem. Phys. Lett. 156 (1989) 472-477.

[10] Y. Chen, Another look at Rejection sampling through Importance sampling. Discussion papers 04-30, Institute of Statistics and Decision Science, Duke University (2004). Zbl 1084.62005 | MR 2153124

[11] G. Ciccotti, R. Kapral and E. Vanden-Eijnden, Blue Moon sampling, vectorial reaction coordinates, and unbiased constrained dynamics. Chem. Phys. Chem. 6 (2005) 1809-1814.

[12] G. Ciccotti, T. Lelièvre and E. Vanden-Eijnden, Projection of diffusions on submanifolds: Application to mean force computation. CERMICS preprint 309 (2006). Zbl 1185.82050 | MR 2376846 | Zbl pre05237828

[13] S. Duane, A.D. Kennedy, B. Pendleton and D. Roweth, Hybrid Monte Carlo. Phys. Letters B. 195 (1987) 216-222.

[14] M. Duflo, Random iterative models. Springer, Berlin, New York (1997). MR 1485774 | Zbl 0868.62069

[15] W. E, W. Ren and E. Vanden-Eijnden, Finite temperature string method for the study of rare events. J. Phys. Chem. B 109 (2005) 6688-6693.

[16] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in advanced mathematics. CRC Press, Chapman and Hall (1991). MR 1158660 | Zbl 0804.28001

[17] D. Frenkel and B. Smit, Understanding Molecular Simulation, From Algorithms to Applications, 2nd edn. Academic Press (2002). Zbl 0889.65132

[18] G. Grimett and D. Stirzaker, Probability and Random Processes. Oxford University Press (2001). MR 2059709 | Zbl 1015.60002

[19] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, Structure-Preserving Algorithms For Ordinary Differential Equations, Springer Series in Computational Mathematics 31, 2nd edn. Springer-Verlag, Berlin (2006). MR 2221614 | Zbl 1094.65125

[20] S. Hampton, P. Brenner, A. Wenger, S. Chatterjee and J.A. Izaguirre, Biomolecular Sampling: Algorithms, Test Molecules, and Metrics, in New Algorithms for Macromolecular Simulation, Lecture Notes in Computational Science and Engineering 49, B. Leimkuhler, C. Chipot, R. Elber, A. Laaksonen, A. Mark, T. Schlick, C. Schuette and R. Skeel Eds., Springer Verlag, Berlin and New York (2006) 103-123. Zbl 1094.92006

[21] R.Z. Has'Minskii, Stochastic Stability of Differential Equations. Sijthoff and Noordhoff (1980). Zbl 0441.60060

[22] W.K. Hastings, Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57 (1970) 97-109. Zbl 0219.65008

[23] F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential. Arch. Rational Mech. Anal. 171 (2004) 151-218. Zbl 1139.82323

[24] W.G. Hoover, Canonical dynamics: Equilibrium phase-space distributions. Phys. Rev. A 31 (1985) 1695-1697.

[25] F.C. Hoppensteadt, M. Rahman and B.D. Welfert, $\sqrt{n}$-Central limit theorems for Markov processes with applications to circular processes, preprint version (2003). Available at the URL http://math.asu.edu/$\sim$bdw/PAPERS/CLT.pdf

[26] A.M. Horowitz, A generalized guided Monte Carlo algorithms. Phys. Lett. B 268 (1991) 247-252.

[27] J.A. Izaguirre and S.S. Hampton, Shadow Hybrid Monte Carlo: an efficient propagator in phase space of macromolecules. J. Comput. Phys. 200 (2004) 581-604. Zbl 1115.65383

[28] A.D. Kennedy and B. Pendleton, Cost of the generalised hybrid Monte Carlo algorithm for free field theory. Nucl. Phys. B 607 (2001) 456-510. Zbl 0969.81639

[29] A. Laio and M. Parrinello, Escaping free energy minima. Proc. Natl. Acad. Sci. USA 99 (2002) 12562-12566.

[30] B. Lapeyre, E. Pardoux and R. Sentis, Méthodes de Monte Carlo pour les équations de transport et de diffusion, Mathématiques et applications 29, Springer (1998); B. Lapeyre, E. Pardoux and R. Sentis, translated by A. Craig and F. Craig, Introduction to Monte-Carlo methods for transport and diffusion equations. Oxford University Press (2003). MR 1621249 | Zbl 0886.65124

[31] F. Legoll, Molecular and Multiscale Methods for the Numerical Simulation of Materials. Ph.D. thesis, University of Paris VI, France (2004).

[32] F. Legoll, M. Luskin and R. Moeckel, Non-ergodicity of the Nosé-Hoover thermostatted harmonic oscillator. Arch. Rat. Mech. Anal. 184 (2007) 449-463. Zbl 1122.82002

[33] B.J. Leimkuhler and S. Reich, Simulating Hamiltonian dynamics, Cambridge monographs on applied and computational mathematics 14. Cambridge University Press (2005). MR 2132573 | Zbl 1069.65139

[34] B.J. Leimkuhler and C.R. Sweet, A Hamiltonian formulation for recursive multiple thermostats in a common timescale. SIAM J. Appl. Dyn. Syst. 4 (2005) 187-216. Zbl 1075.92057

[35] J.S. Liu, Monte Carlo strategies in Scientific Computing. Springer Series in Statistics (2001). MR 1842342 | Zbl 0991.65001

[36] P.B. Mackenze, An improved hybrid Monte Carlo. Phys. Lett. B. 226 (1989) 369-371.

[37] X. Mao, Stochastic differential equations and applications. Horwood, Chichester (1997). Zbl 1138.60005

[38] J.E. Marsden and M. West, Discrete mechanics and variational integrators. Acta Numer. 10 (2001) 357-514. Zbl 1123.37327

[39] M.G. Martin and J.I. Siepmann, Transferable potentials for phase equilibria 102 (1998) 2569-2577.

[40] G.J. Martyna, M.L. Klein and M.E. Tuckerman, Nosé-Hoover chains: The canonical ensemble via continuous dynamics. J. Chem. Phys. 97 (1992) 2635-2643.

[41] G.J. Martyna, M.E. Tuckerman, D.J. Tobias and M.L. Klein, Explicit reversible integrators for extended systems dynamics. Mol. Phys. 87 (1996) 1117-1157.

[42] J.C. Mattingly, A.M. Stuart and D.J. Higham, Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. Stoch. Proc. Appl. 101 (2002) 185-232. Zbl 1075.60072

[43] K.L. Mengersen and R.L. Tweedie, Rates of convergence in the Hastings-Metropolis algorithm. Ann. Statist. 24 (1996) 101-121. Zbl 0854.60065

[44] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller, Equations of state calculations by fast computing machines. J. Chem. Phys. 21 (1953) 1087-1091.

[45] S.P. Meyn and R.L. Tweedie, Stability of Markovian processes. II. Continuous-time processes and sampled chains. Adv. Appl. Probab. 24 (1993) 487-517. Zbl 0781.60052

[46] S.P. Meyn and R.L. Tweedie, Markov Chains and Stochastic Stability. Springer (1993). MR 1287609 | Zbl 0925.60001

[47] G.N. Milstein and M.V. Tretyakov, Quasi-symplectic methods for Langevin-type equations. IMA J. Numer. Anal. 23 (2003) 593-626. Zbl 1055.65141

[48] B. Mishra and T. Schlick, The notion of error in Langevin dynamics: I. Linear analysis. J. Chem. Phys. 105 (1996) 299-318.

[49] R.M. Neal, An improved acceptance procedure for the hybrid Monte-Carlo algorithm. J. Comput. Phys. 111 (1994) 194-203. Zbl 0797.65115

[50] N. Niederreiter, Random Number Generation and Quasi Monte-Carlo Methods. Society for Industrial and Applied Mathematics (1992). MR 1172997 | Zbl 0761.65002

[51] S. Nosé, A Molecular Dynamics method for simulations in the canonical ensemble, Mol. Phys. 52 (1984) 255-268.

[52] S. Nosé, A unified formulation of the constant temperature Molecular Dynamics method, J. Chem. Phys. 81 (1985) 511-519.

[53] G. Pagès, Sur quelques algorithmes récursifs pour les probabilités numériques. ESAIM: PS 5 (2001) 141-170. | Numdam | Zbl 0998.60073

[54] D.C. Rapaport, The Art of Molecular Dynamics Simulations. Cambridge University Press (1995). Zbl 1098.81009

[55] S. Reich, Backward error analysis for numerical integrators. SIAM J. Numer. Anal. 36 (1999) 1549-1570. Zbl 0935.65142

[56] G.O. Roberts and J.S. Rosenthal, Optimal scaling of discrete approximations to Langevin diffusions. J. Roy. Stat. Soc. B 60 (1998) 255-268. Zbl 0913.60060

[57] G.O. Roberts and R.L. Tweedie, Exponential convergence of Langevin diffusions and their discrete approximations. Bernoulli 2 (1996) 341-364. Zbl 0870.60027

[58] G.O. Roberts and R.L. Tweedie, Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms. Biometrika 83 (1996) 95-110. Zbl 0888.60064

[59] L.C.G. Rogers, Smooth transition densities for one-dimensional probabilities. Bull. London Math. Soc 17 (1985) 157-161. Zbl 0604.60077

[60] J.P. Ryckaert and A. Bellemans, Molecular dynamics of liquid alkanes. Faraday Discuss. 66 (1978) 95-106.

[61] A. Scemama, T. Lelièvre, G. Stoltz, E. Cancès and M. Caffarel, An efficient sampling algorithm for Variational Monte Carlo. J. Chem. Phys. 125 (2006) 114105.

[62] T. Schlick, Molecular Modeling and Simulation. Springer (2002). MR 1921061 | Zbl 1011.92019

[63] C. Schütte, Conformational dynamics: Modelling, Theory, Algorithm, and Application to Biomolecules. Habilitation Thesis, Free University Berlin (1999).

[64] C. Schütte and W. Huisinga, Biomolecular conformations can be identified as metastable sets of molecular dynamics, in Handbook of Numerical Analysis (Special volume on computational chemistry), Vol. X, P.G. Ciarlet and C. Le Bris Eds., Elsevier (2003) 699-744. Zbl 1066.81658

[65] C. Schütte, A. Fischer, W. Huisinga and P. Deuflhard, A direct approach to conformational dynamics based on Hybrid Monte-Carlo. J. Comp. Phys. 151 (1999) 146-168. Zbl 0933.65145

[66] T. Shardlow, Splitting for dissipative particle dynamics. SIAM J. Sci. Comput. 24 (2003) 1267-1282. Zbl 1043.60048

[67] R.D. Skeel, in The graduate student's guide to numerical analysis, Springer Series in Computational Mathematics, M. Ainsworth, J. Levesley and M. Marletta Eds., Springer-Verlag, Berlin (1999) 119-176. Zbl 0938.65150

[68] R.D. Skeel and J.A. Izaguirre, An impulse integrator for Langevin dynamics. Mol. Phys. 100 (2002) 3885-3891.

[69] M.R. Sorensen and A.F. Voter, Temperature accelerated dynamics for simulation of infrequent events. J. Chem. Phys. 112 (2000) 9599-9606.

[70] G. Stoltz, Quelques méthodes mathématiques pour la simulation moléculaire et multiéchelle. Ph.D. Thesis (in preparation).

[71] C.R. Sweet, Hamiltonian Thermostatting Techniques for Molecular Dynamics Simulation. Ph.D. Thesis, University of Leicester (2004).

[72] D. Talay, Second-order discretization schemes of stochastic differential systems for the computation of the invariant law. Stoch. Stoch. Rep. 29 (1990) 13-36. Zbl 0697.60066

[73] D. Talay, Approximation of invariant measures of nonlinear Hamiltonian and dissipative stochastic differential equations, in Progress in Stochastic Structural Dynamics, R. Bouc and C. Soize Eds., Publication du L.M.A.-C.N.R.S. 152 (1999) 139-169.

[74] D. Talay, Stochastic Hamiltonian dissipative systems: exponential convergence to the invariant measure, and discretization by the implicit Euler scheme. Markov Proc. Rel. Fields 8 (2002) 163-198. MR 1924934 | Zbl 1011.60039

[75] M.E. Tuckerman and G.J. Martyna, Understanding modern molecular dynamics: Techniques and applications. J. Phys. Chem. B 104 (2000) 159-178.

[76] L. Verlet, Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard-Jones molecules. Phys. Rev. 159 (1967) 98-103.

[77] A.F. Voter, A method for accelerating the molecular dynamics simulation of infrequent events. J. Chem. Phys. 106 (1997) 4665-4677.

[78] A.F. Voter, Parallel replica method for dynamics of infrequent events. Phys. Rev. B 57 (1998) 13985-13988.

[79] W. Wang and R.D. Skeel, Analysis of a few numerical integration methods for the Langevin equation. Mol. Phys. 101 (2003) 2149-2156.

[80] Z. Zhu, M.E. Tuckerman, S.O. Samuelson and G.J. Martyna, Using novel variable transformations to enhance conformational sampling in molecular dynamics. Phys. Rev. Lett. 88 (2002) 100201.