Some Remarks on Preconditioning Molecular Dynamics
The SMAI Journal of computational mathematics, Volume 4 (2018), pp. 57-80.

We consider a Preconditioned Overdamped Langevin algorithm that does not alter the invariant distribution (up to controllable discretisation errors) and ask whether preconditioning improves the standard model in terms of reducing the asymptotic variance and of accelerating convergence to equilibrium. We present a detailed study of the dependence of the asymptotic variance on preconditioning in some elementary toy models related to molecular simulation. Our theoretical results are supported by numerical simulations.

Published online:
DOI: 10.5802/smai-jcm.29
Keywords: Preconditioned Overdamped Langevin algorithm, Asymptotic Variance, Central Limit Theorem, Model Hamiltonians, Lattice Model
AlRachid, Houssam 1; Mones, Letif 2; Ortner, Christoph 2

1 IDP Laboratory, Orleans University, Orleans, France
2 Mathematics Institute, University of Warwick, Coventry, United Kingdom
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AlRachid, Houssam; Mones, Letif; Ortner, Christoph. Some Remarks on Preconditioning Molecular Dynamics. The SMAI Journal of computational mathematics, Volume 4 (2018), pp. 57-80. doi : 10.5802/smai-jcm.29. http://www.numdam.org/articles/10.5802/smai-jcm.29/

[1] Barducci, A.; Bussi, G.; Parrinello, M. Well-Tempered Metadynamics: A Smoothly Converging and Tunable Free-Energy Method, Phys. Rev. Lett., Volume 100(2) (2008), 020603 pages

[2] Beskos, A.; Roberts, G.; Stuart, A. Optimal scalings for local Metropolis–Hastings chains on nonproduct targets in high dimensions, The Annals of Applied Probability, Volume 19(3) (2009), p. 863-98 | DOI | MR | Zbl

[3] Bhattacharya, R. A central limit theorem for diffusions with periodic coefficients, The Annals of Probability, Volume 13(2) (1985), pp. 385-396 | DOI | MR | Zbl

[4] Brascamp, H.J.; Lieb, E.H. On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, Journal of functional analysis, Volume 22(4) (1976), pp. 366-389 | DOI | Zbl

[5] Brünger, A.; Brooks, C. L.; Karplus, M. Stochastic boundary conditions for molecular dynamics simulations of ST2 water, Chemical Physics Letters, Volume 105(5) (1984), pp. 495-500 | DOI

[6] Cattiaux, P.; Chafai, D.; Guillin, A. Central limit theorems for additive functionals of ergodic Markov diffusions processes, ALEA, Volume 9(2) (2012), pp. 337-382 | Zbl

[7] Duncan, A. B.; Lelièvre, T.; Pavliotis, G. A. Variance Reduction using Nonreversible Langevin Samplers, Journal of Statistical Physics, Volume 163(3) (2016), pp. 457-491 | DOI | MR | Zbl

[8] Girolami, M.; Calderhead, B. Riemann Manifold Langevin and Hamiltonian Monte Carlo methods, Journal of the Royal Statistical Society: Series B (Statistical Methodology), Volume 73(2) (2011), pp. 123-214 | DOI | MR | Zbl

[9] Glynn, P. W.; Meyn, S.P. A Liapounov bound for solutions of the Poisson equation, The Annals of Probability (1996), pp. 916-931 | DOI | MR | Zbl

[10] Hairer, E.; Lubich, C.; Wanner, G. Geometric numerical integration illustrated by the Stormer-Verlet method, Acta numerica, Volume 12(12) (2003), pp. 399-450 | DOI | MR | Zbl

[11] Helland, I. S. Central limit theorems for martingales with discrete or continuous time, Scandinavian Journal of Statistics, Volume 9(2) (1982), pp. 79-94 | Zbl

[12] Hitrik, M.; Pravda-Starov, K. Spectra and semigroup smoothing for non-elliptic quadratic operators, Math. Ann., Volume 344(4) (2009), pp. 801-846 | DOI | MR | Zbl

[13] Hornak, W.; Abel, R.; Okur, A.; Strockbine, B.; Roitberg, A.; Simmerling, C. Comparison of multiple Amber force fields and development of improved protein backbone parameters, Prot. Struct. Funct. Bioinf., Volume 65(3) (2006), pp. 712-725 | DOI

[14] Hou, F.; Goodman, J.; Hogg, D.W.; Weare, J.; Schwab, C. An affine-invariant sampler for exoplanet fitting and discovery in radial velocity data, The Astrophysical Journal, Volume 745(2) (2012)

[15] Iserles, A. A First Course in the Numerical Analysis of Differential Equations, Cambridge Texts in Applied Mathematics, Cambridge University Press (2008)

[16] J. C. Mattingly, A. M. Stuart; Higham, D. J. Ergodicity for SDEs and approximations: locally Lipschitz vector fields and degenerate noise, Stochastic processes and their applications, Volume 101(2) (2002), pp. 185-232 | MR | Zbl

[17] Kästner, J.; Thiel, W. Bridging the gap between thermodynamic integration and umbrella sampling provides a novel analysis method: Umbrella integration, J. Chem. Phys., Volume 123 (2005) | DOI

[18] Kipnis, C.; Varadhan, S.R.S. OCentral limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions, Communications in Mathematical Physics, Volume 104(1) (1986), pp. 1-19 | DOI | Zbl

[19] Laio, A.; Parrinello, M. Escaping free-energy minima, Proceedings of the National Academy of Sciences, Volume 99 (2002), pp. 12562-12566 | DOI

[20] Lelièvre, T. Personal communication (2017)

[21] Lelièvre, T.; Nier, F.; Pavliotis, G. Optimal non-reversible linear drift for the convergence to equilibrium of a diffusion, Journal of Statistical Physics, Volume 152(2) (2013), pp. 237-274 | DOI | MR | Zbl

[22] Martin, J.; Wilcox, L. C.; Burstedde, C.; Ghattas, O. A stochastic Newton MCMC method for large-scale statistical inverse problems with application to seismic inversion, SIAM Journal on Scientific Computing, Volume 34(3) (2012), p. A1460-A1487 | DOI | MR | Zbl

[23] Meyn, S. P.; Tweedie, R. L. Stability of Markovian processes III: Foster-Lyapunov criteria for continuous-time processes, Advances in Applied Probability, Volume 25(3) (1993), pp. 518-548 | DOI | MR | Zbl

[24] Meyn, S. P.; Tweedie, R. L. A survey of Foster-Lyapunov techniques for general state space Markov processes, In Proceedings of the Workshop on Stochastic Stability and Stochastic Stabilization, Metz, France. Citeseer (1993)

[25] Mones, L.; Bernstein, N.; Csanyi, G. Exploration, Sampling, And Reconstruction of Free Energy Surfaces with Gaussian Process Regression, Journal of Chemical Theory and Computation, Volume 12(10) (2016), pp. 5100-5110 | DOI

[26] Mones, L.; Csanyi, G.; Ortner, C. in preparation (2017)

[27] Olson, D.; Ortner, C. Regularity and locality of point defects in multilattices, to appear in AMRX, arXiv:1608.08930 (2016) | Zbl

[28] Ottobre, M.; Pavliotis, G. A.; Pravda-Starov, K. Exponential return to equilibrium for hypoelliptic quadratic systems, Journal of Functional Analysis, Volume 262(9) (2012), pp. 4000-4039 | DOI | MR | Zbl

[29] Packwood, D.; Kermode, J.; Mones, L.; Bernstein, N.; J. Woolley, N. I. M. Gould; Ortner, C.; Csanyi, G. universal preconditioner for simulating condensed phase materials, J. Chem. Phys., Volume 144(16) (2016)

[30] Pillai, N. S.; Stuart, A. M.; Thiéry, A. H. Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions, The Annals of Applied Probability, Volume 22(6) (2017), pp. 2320-2356

[31] Robert, C. P.; Casella, G. Monte Carlo Statistical Methods, 2nd ed. Springer, New York (2004) | Zbl

[32] Roberts, G. O.; Tweedie, R. L. Exponential convergence of Langevin distributions and their discrete approximations, Bernoulli, Volume 2(4) (1996), pp. 341-363 | DOI | MR | Zbl

[33] Villani, V. Hypocoercivity, Bull. Am. Math. Soc., Volume 46(2) (2009) | Zbl

[34] Xifara, T.; Sherlock, C.; Livingstone, S.; Byrne, S.; Girolami, M. Langevin diffusions and the Metropolis-adjusted Langevin algorithm, Statistics & Probability Letters, Volume 91 (2014), pp. 14-19 | DOI | MR | Zbl

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