Skip to main content

Geometric integrators and the Hamiltonian Monte Carlo method

  • Nawaf Bou-Rabee (a1) and J. M. Sanz-Serna (a2)

This paper surveys in detail the relations between numerical integration and the Hamiltonian (or hybrid) Monte Carlo method (HMC). Since the computational cost of HMC mainly lies in the numerical integrations, these should be performed as efficiently as possible. However, HMC requires methods that have the geometric properties of being volume-preserving and reversible, and this limits the number of integrators that may be used. On the other hand, these geometric properties have important quantitative implications for the integration error, which in turn have an impact on the acceptance rate of the proposal. While at present the velocity Verlet algorithm is the method of choice for good reasons, we argue that Verlet can be improved upon. We also discuss in detail the behaviour of HMC as the dimensionality of the target distribution increases.

Hide All
Akhmatskaya, E. and Reich, S. (2008), ‘GSHMC: An efficient method for molecular simulation’, J. Comput. Phys. 227, 49374954.
Akhmatskaya, E., Fernández-Pendás, M., Radivojević, T. and Sanz-Serna, J. M. (2017) ‘Adaptive splitting integrators for enhancing sampling efficiency of modified Hamiltonian Monte Carlo methods in molecular simulation’, in Tribute to Keith Gubbins, Pioneer in the Theory of Liquids, special issue of Langmuir, 33, 11530–11542.
Alamo, A. and Sanz-Serna, J. M. (2016), ‘A technique for studying strong and weak local errors of splitting stochastic integrators’, SIAM J. Numer. Anal. 54, 32393257.
Allen, M. P. and Tildesley, D. J. (1987), Computer Simulation of Liquids, Clarendon Press.
Andrieu, C., de Freitas, N., Doucet, A. and Jordan, M. I. (2003), ‘An introduction to MCMC for machine learning’, Machine Learning 50, 543.
Arnol’d, V. I. (1989), Mathematical Methods of Classical Mechanics (translated from the Russian by K. Vogtmann and A. Weinstein), Vol. 60 of Graduate Texts in Mathematics, Springer.
Asmussen, S. and Glynn, P. W. (2007), Stochastic Simulation: Algorithms and Analysis, Vol. 57 of Stochastic Modelling and Applied Probability, Springer.
Beskos, A., Pillai, N. S., Roberts, G. O., Sanz-Serna, J. M. and Stuart, A. M. (2013), ‘Optimal tuning of hybrid Monte-Carlo algorithm’, Bernoulli 19, 15011534.
Beskos, A., Pinski, F. J., Sanz-Serna, J. M. and Stuart, A. M. (2011), ‘Hybrid Monte-Carlo on Hilbert spaces’, Stoch. Proc. Appl. 121, 22012230.
Beskos, A., Roberts, G., Stuart, A. and Voss, J. (2008), ‘MCMC methods for diffusion bridges’, Stoch. Dynam. 8, 319350.
Bishop, C. M. (2006), Pattern Recognition and Machine Learning, Springer.
Blanes, S. and Casas, F. (2016), A Concise Introduction to Geometric Numerical Integration, Monographs and Research Notes in Mathematics, CRC Press.
Blanes, S., Casas, F. and Sanz-Serna, J. M. (2014), ‘Numerical integrators for the hybrid Monte Carlo method’, SIAM J. Sci. Comput. 36, A1556A1580.
Bou-Rabee, N. (2014), ‘Time integrators for molecular dynamics’, Entropy 16, 138162.
Bou-Rabee, N. (2017) Cayley splitting for second-order Langevin stochastic partial differential equations. arXiv:1707.05603
N. Bou-Rabee and A. Eberle (2018), Coupling and convergence for exact randomized Hamiltonian Monte-Carlo. In preparation.
Bou-Rabee, N. and Hairer, M. (2013), ‘Non-asymptotic mixing of the MALA algorithm’, IMA J. Numer. Anal. 33, 80110.
Bou-Rabee, N. and Sanz-Serna, J. M. (2017), ‘Randomized Hamiltonian Monte Carlo’, Ann. Appl. Probab. 27, 21592194.
Bou-Rabee, N. and Vanden-Eijnden, E. (2010), ‘Pathwise accuracy and ergodicity of Metropolized integrators for SDEs’, Comm. Pure Appl. Math. 63, 655696.
Bou-Rabee, N. and Vanden-Eijnden, E. (2012), ‘A patch that imparts unconditional stability to explicit integrators for Langevin-like equations’, J. Comput. Phys. 231, 25652580.
Bou-Rabee, N., Donev, A. and Vanden-Eijnden, E. (2014), ‘Metropolis integration schemes for self-adjoint diffusions’, Multiscale Model. Simul. 12, 781831.
Butcher, J. C. (2016), Numerical Methods for Ordinary Differential Equations, third edition, Wiley.
Calvo, M. P. and Sanz-Serna, J. M. (1993), ‘The development of variable-step symplectic integrators, with application to the two-body problem’, SIAM J. Sci. Comput. 14, 936952.
Calvo, M. P. and Sanz-Serna, J. M. (2009), ‘Instabilities and inaccuracies in the integration of highly oscillatory problems’, SIAM J. Sci. Comput. 31, 16531677.
Calvo, M. P., Murua, A. and Sanz-Serna, J. M. (1994), Modified equations for ODEs. In Chaotic Numerics (Kloeden, P. E. and Palmer, K. J., eds), Vol. 172 of Contemporary Mathematics, AMS, pp. 6374.
Campos, C. M. and Sanz-Serna, J. M. (2015), ‘Extra chance generalized hybrid Monte Carlo’, J. Comput. Phys. 281, 365374.
Campos, C. M. and Sanz-Serna, J. M. (2017), ‘Palindromic 3-stage splitting integrators: A roadmap’, J. Comput. Phys. 346, 340355.
Cancès, E., Legoll, F. and Stoltz, G. (2007), ‘Theoretical and numerical comparison of some sampling methods for molecular dynamics’, Math. Model. Numer. Anal. 41, 351389.
Cano, B. and Sanz-Serna, J. M. (1997), ‘Error growth in the numerical integration of periodic orbits, with application to Hamiltonian and reversible systems’, SIAM J. Numer. Anal. 34, 13911417.
Cano, B. and Sanz-Serna, J. M. (1998), ‘Error growth in the numerical integration of periodic orbits by multistep methods, with application to reversible systems’, IMA J. Numer. Anal. 18, 5775.
Carpenter, B., Gelman, A., Hoffman, M., Lee, D., Goodrich, B., Betancourt, M., Brubaker, M. A., Guo, J., Li, P. and Riddell, A. (2016), ‘Stan: A probabilistic programming language’, J. Statist. Softw. 20, 137.
Chen, Z. (2003), ‘Bayesian filtering: From Kalman filters to particle filters, and beyond’, Statistics 182, 169.
Cipra, B. A. (2000), ‘The best of the 20th century: Editors name top 10 algorithms’, SIAM News 33(4).
Da Prato, G. and Zabczyk, J. (2014), Stochastic Equations in Infinite Dimensions, Cambridge University Press.
Diaconis, P. (2009), ‘The Markov chain Monte Carlo revolution’, Bull. Am. Math. Soc. 46, 179205.
Diaconis, P., Holmes, S. and Neal, R. M. (2000), ‘Analysis of a nonreversible Markov chain sampler’, Ann. Appl. Probab. 10, 726752.
Duane, S., Kennedy, A. D., Pendleton, B. J. and Roweth, D. (1987), ‘Hybrid Monte-Carlo’, Phys. Lett. B 195, 216222.
Eberle, A. (2016), ‘Reflection couplings and contraction rates for diffusions’, Probab. Theory Rel. Fields 166, 851886.
Eberle, A. (2018), A coupling approach to the kinetic Langevin equation on the torus. In preparation.
Eberle, A., Guillin, A. and Zimmer, R. (2016) Couplings and quantitative contraction rates for Langevin dynamics. arXiv:1703.01617
Evensen, G. (2009), Data Assimilation: The Ensemble Kalman Filter, Springer Science & Business Media.
Fang, Y., Sanz-Serna, J. M. and Skeel, R. D. (2014), ‘Compressible generalized hybrid Monte Carlo’, J. Chem. Phys. 140(17), 174108.
Fathi, M. (2014) Theoretical and numerical study of a few stochastic models of statistical physics. PhD thesis, Université Pierre et Marie Curie – Paris VI.
Fathi, M., Homman, A.-A. and Stoltz, G. (2015), ‘Error analysis of the transport properties of Metropolized schemes’, ESAIM Proc. Surv. 48, 341363.
Feng, K. and Qin, M. (2010), Symplectic Geometric Algorithms for Hamiltonian Systems (translated and revised from the Chinese original), Zhejiang Science and Technology, Hangzhou, and Springer.
Fernández-Pendás, M., Akhmatskaya, E. and Sanz-Serna, J. M. (2016), ‘Adaptive multi-stage integrators for optimal energy conservation in molecular simulations’, J. Comput. Phys. 327, 434449.
Frantz, D. D., Freeman, D. L. and Doll, J. D. (1990), ‘Reducing quasi-ergodic behavior in Monte Carlo simulations by -walking: Applications to atomic clusters’, J. Chem. Phys. 93, 27692784.
Frenkel, D. and Smit, B. (2002), Understanding Molecular Simulation: From algorithms to Applications, second edition, Academic Press.
Gelfand, A. E. and Smith, A. F. M. (1990), ‘Sampling-based approaches to calculating marginal densities’, J. Amer. Statist. Assoc. 85(410), 398409.
Geman, S. and Geman, D. (1984), ‘Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images’, IEEE Trans. Pattern Anal. Machine Intel. PAMI‐6, 721741.
Geyer, C. J. (1991), Markov chain Monte Carlo maximum likelihood. In Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface (Keramides, E. M., ed.), Interface Foundation of North America, pp. 156163.
Geyer, C. J. (1992), ‘Practical Markov chain Monte Carlo’, Statist. Sci. 7, 473483.
Geyer, C. J. and Thompson, E. A. (1995), ‘Annealing Markov chain Monte Carlo with applications to ancestral inference’, J. Amer. Statist. Assoc. 90(431), 909920.
Ghahramani, Z. (2015), ‘‘Probabilistic machine learning and artificial intelligence’’, Nature 521(7553), 452.
Girolami, M. and Calderhead, B. (2011), ‘Riemann manifold Langevin and Hamiltonian Monte Carlo methods’, J. Royal Statist. Soc. B 73, 123214.
Green, P. J. and Mira, A. (2001), ‘Delayed rejection in reversible jump Metropolis–Hastings’, Biometrika 88, 10351053.
Griffiths, D. F. and Higham, D. J. (2010), Numerical Methods for Ordinary Differential Equations: Initial Value Problems, Springer Undergraduate Mathematics Series, Springer.
Griffiths, D. F. and Sanz-Serna, J. M. (1986), ‘On the scope of the method of modified equations’, SIAM J. Sci. Statist. Comput. 7, 9941008.
Gustafson, P. (1998), ‘A guided walk Metropolis algorithm’, Statist. Comput. 8, 357364.
Hadfield, J. D. (2010), ‘MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package’, J. Statist. Softw. 33, 122.
Hairer, E. and Wanner, G. (2010), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Vol. 14 of Springer Series in Computational Mathematics, second revised edition, Springer.
Hairer, E., Lubich, C. and Wanner, G. (2010), Geometric Numerical Integration, Springer.
Hairer, E., Nørsett, S. P. and Wanner, G. (1993), Solving Ordinary Differential Equations I: Nonstiff Problems, Vol. 8 of Springer Series in Computational Mathematics, Springer.
Hairer, M., Stuart, A. M. and Voss, J. (2009), ‘Sampling conditioned diffusions’, Trends Stoch. Anal. 353, 159186.
Hansmann, U. H. E. (1997), ‘Parallel tempering algorithm for conformational studies of biological molecules’, Chem. Phys. Lett. 281, 140150.
Hansmann, U. H. E. and Okamoto, Y. (1993), ‘Prediction of peptide conformation by multicanonical algorithm: New approach to the multiple-minima problem’, J. Comput. Chem. 14, 13331338.
Hastings, W. K. (1970), ‘Monte-Carlo methods using Markov chains and their applications’, Biometrika 57, 97109.
Hess, B., Kutzner, C., van der Spoel, D. and Lindahl, D. (2008), ‘GROMACS 4: Algorithms for highly efficient, load-balanced, and scalable molecular simulation’, J. Chem. Theory Comp. 4, 435447.
Homan, M. D. and Gelman, A. (2014), ‘The No-U-Turn Sampler: Adaptively setting path lengths in Hamiltonian Monte Carlo’, J. Mach. Learning Res. 15, 15931623.
Horowitz, A. M. (1991), ‘A generalized guided Monte-Carlo algorithm’, Phys. Lett. B 268, 247252.
Iserles, A. and Quispel, G. R. W. (2017) Why geometric integration? arXiv:1602.07755
Izaguirre, J. A. and Hampton, S. S. (2004), ‘Shadow hybrid Monte Carlo: An efficient propagator in phase space of macromolecules’, J. Comput. Phys. 200, 581604.
Jensen, S. T., Liu, X. S., Zhou, Q. and Liu, J. S. (2004), ‘Computational discovery of gene regulatory binding motifs: A Bayesian perspective’, Statist. Sci. 19, 188204.
Ji, H. and Wong, W. H. (2006), ‘Computational biology: Toward deciphering gene regulatory information in mammalian genomes’, Biometrics 62, 645663.
Kennedy, A. D. and Pendleton, B. (2001), ‘Cost of the generalized hybrid Monte Carlo algorithm for free field theory’, Nucl. Phys. B 607, 456510.
Kikuchi, K., Yoshida, M., Maekawa, T. and Watanabe, H. (1991), ‘Metropolis Monte Carlo method as a numerical technique to solve the Fokker–Planck equation’, Chem. Phys. Lett. 185, 335338.
Kou, S. C., Zhou, Q. and Wong, W. H. (2006), ‘Discussion paper: Equi-energy sampler with applications in statistical inference and statistical mechanics’, Ann. Statist. 34, 15811619.
Krauth, W. (2006), Statistical Mechanics: Algorithms and Computations, Oxford University Press.
Lamb, J. S. W. and Roberts, J. A. G. (1998), ‘Time-reversal symmetry in dynamical systems: A survey’, Nonlinear Phenomena 112, 139.
Landau, D. P. and Binder, K. (2014), A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press.
Leimkuhler, B. and Reich, S. (2004), Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.
Lelièvre, T., Rousset, M. and Stoltz, G. (2010), Free Energy Computations: A Mathematical Perspective, Imperial College Press.
Liang, F. and Wong, W. H. (2001), ‘Real-parameter evolutionary Monte Carlo with applications to Bayesian mixture models’, J. Amer. Statist. Assoc. 96(454), 653666.
Link, W. A. and Barker, R. J. (2009), Bayesian Inference: With Ecological Applications, Academic Press.
Liu, J. S. (2008), Monte Carlo Strategies in Scientific Computing, second edition, Springer.
Lunn, D. J., Thomas, A., Best, N. and Spiegelhalter, D. (2000), ‘WinBUGS – a Bayesian modelling framework: Concepts, structure, and extensibility’, Statist. Comput. 10, 325337.
Lunn, D., Jackson, C., Best, N., Thomas, A. and Spiegelhalter, D. (2012), The BUGS Book: A Practical Introduction to Bayesian Analysis, CRC press.
Lunn, D., Spiegelhalter, D., Thomas, A. and Best, N. (2009), ‘The BUGS project: Evolution, critique and future directions’, Statist. Medicine 28(25), 30493067.
Mackenzie, P. B. (1989), ‘An improved hybrid Monte Carlo method’, Phys. Lett. B 226, 369371.
Mannseth, J., Kleppe, T. S. and Skaug, H. J. (2018), ‘On the application of improved symplectic integrators in Hamiltonian Monte Carlo’, Commun. Statist. Simul. Comput. 47, 500509.
Marinari, E. and Parisi, G. (1992), ‘Simulated tempering: A new Monte Carlo scheme’, Europhys. Lett. 19, 451.
Marsden, J. and Ratiu, T. (1999), Introduction to Mechanics and Symmetry, Springer Texts in Applied Mathematics, Springer.
Martin, A. D., Quinn, K. M. and Park, J. H. (2011), ‘MCMCpack: Markov chain Monte Carlo in R’, J. Statist. Softw. 42, 121.
Mclachlan, R. I. (1995), ‘On the numerical integration of ordinary differential equations by symmetric composition methods’, SIAM J. Sci. Comput. 16, 151168.
Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E. (1953), ‘Equations of state calculations by fast computing machines’, J. Chem. Phys. 21, 10871092.
Mira, A. (2001), ‘On Metropolis–Hastings algorithms with delayed rejection’, Metron 59, 231241.
Murua, A. and Sanz-Serna, J. M. (1999), ‘Order conditions for numerical integrators obtained by composing simpler integrators’, Math. Phys. Eng. Sci. 357(1754), 10791100.
Murua, A. and Sanz-Serna, J. M. (2017), ‘Word series for dynamical systems and their numerical integrators’, Found. Comput. Math. 17, 675712.
Neal, R. M. (1994), ‘An improved acceptance procedure for the hybrid Monte Carlo algorithm’, J. Comput. Phys. 111, 194203.
Neal, R. M. (1996), ‘Sampling from multimodal distributions using tempered transitions’, Statist. Comput. 6, 353366.
Neal, R. M. (2003), ‘Slice sampling’, Ann. Statist. 31, 705741.
Neal, R. M. (2011), MCMC using Hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo (Brooks, S. et al. , eds), Chapman & Hall/CRC, pp. 113162.
Patil, A., Huard, D. and Fonnesbeck, C. J. (2010), ‘PyMC: Bayesian stochastic modelling in Python’, J. Statist. Softw. 35, 1.
Reznikoff, M. G. and Vanden-Eijnden, E. (2005), ‘Invariant measures of stochastic partial differential equations and conditioned diffusions’, Comptes Rendus Math. 340, 305308.
Roberts, G. O. and Rosenthal, J. S. (1998), ‘Optimal scaling of discrete approximations to Langevin diffusions’, J. Roy. Statist. Soc. Ser. B 60, 255268.
Roberts, G. O. and Tweedie, R. L. (1996a), ‘Exponential convergence of Langevin distributions and their discrete approximations’, Bernoulli 2, 341363.
Roberts, G. O. and Tweedie, R. L. (1996b), ‘Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms’, Biometrika 1, 95110.
Roberts, G. O., Gelman, A. and Gilks, W. R. (1997), ‘Weak convergence and optimal scaling of random walk Metropolis algorithms’, Ann. Appl. Probab. 7, 110120.
Sanz-Serna, J. M. (1991), Two topics in nonlinear stability. In Advances in Numerical Analysis, Vol. I (Light, W., ed.), Clarendon Press, pp. 147174.
Sanz-Serna, J. M. (1992), Symplectic integrators for Hamiltonian problems: An overview. In Acta Numerica, Vol. 1, Cambridge University Press, pp. 243286.
Sanz-Serna, J. M. (1996), Backward error analysis of symplectic integrators. In Integration Algorithms and Classical Mechanics (Marsden, J. E. et al. , eds), Vol. 10 of Fields Institute Communications, AMS, pp. 193205.
Sanz-Serna, J. M. (1997), Geometric integration. In The State of the Art in Numerical Analysis (Duff, I. S. and Watson, G. A., eds), Vol. 63 of Institute of Mathematics and its Applications Conference Series, Oxford University Press, pp. 121143.
Sanz-Serna, J. M. (2014), Markov chain Monte Carlo and numerical differential equations. In Current Challenges in Stability Issues for Numerical Differential Equations (Dieci, L. and Guglielmi, N., eds), Vol. 2082 of Lecture Notes in Mathematics, Springer, pp. 3988.
Sanz-Serna, J. M. (2016), ‘Symplectic Runge–Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more’, SIAM Rev. 58, 333.
Sanz-Serna, J. M. and Calvo, M. P. (1994), Chapman & Hall.
Sanz-Serna, J. M. and Murua, A. (2015), Formal series and numerical integrators: Some history and some new techniques. In 8th International Congress on Industrial and Applied Mathematics, Higher Education Press, pp. 311331.
Schlick, T. (2002), Molecular Modeling and Simulation: An Interdisciplinary Guide, Vol. 21 of Interdisciplinary Applied Mathematics, Springer.
Schütte, C. (1999) Conformational dynamics: Modeling, theory, algorithm, and application to biomolecules. Habilitation, Freie Universität Berlin.
Skeel, R. D. and Hardy, D. J. (2001), ‘Practical construction of modified Hamiltonians’, SIAM J. Sci. Comput. 23, 11721188.
Sokal, A. (1997), Monte Carlo methods in statistical mechanics: Foundations and new algorithms. In Functional Integration: Basics and Applications (Dewitt-Morette, C. and Folacci, A., eds), Springer, pp. 131192.
Stoltz, G. (2007) Some mathematical methods for molecular and multiscale simulation. PhD thesis, École Nationale des Ponts et Chaussées.
Strang, G. (1963), ‘Accurate partial difference methods I: Linear Cauchy problems’, Arch. Rational Mech. Anal. 12, 392402.
Stuart, A. M. (2010), Inverse problems: A Bayesian perspective. In Acta Numerica, Vol. 19, Cambridge University Press, pp. 451559.
Sugita, Y. and Okamoto, Y. (1999), ‘Replica-exchange molecular dynamics method for protein folding’, Chem. Phys. Lett. 314, 141151.
Sullivan, T. J. (2015), Introduction to Uncertainty Quantification, Vol. 63 of Texts in Applied Mathematics, Springer.
Sweet, C. R., Hampton, S. S., Skeel, R. D. and Izaguirre, J. A. (2009), ‘A separable shadow Hamiltonian hybrid Monte Carlo method’, J. Chem. Phys. 131(17), 174106.
Thrun, S., Burgard, W. and Fox, D. (2005), Probabilistic Robotics, MIT press.
Tierney, L. (1994), ‘Markov chains for exploring posterior distributions’, Ann. Statist. 22, 17011728.
Tuckerman, M. (2010), Statistical Mechanics: Theory and Molecular Simulation, Oxford University Press.
Wales, D. (2003), Energy Landscapes: Applications to Clusters, Biomolecules and Glasses, Cambridge University Press.
Webb, A. R. (2003), Statistical Pattern Recognition, Wiley.
Yoshida, H. (1990), ‘Construction of higher order symplectic integrators’, Phys. Lett. A 150, 262268.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Acta Numerica
  • ISSN: 0962-4929
  • EISSN: 1474-0508
  • URL: /core/journals/acta-numerica
Please enter your name
Please enter a valid email address
Who would you like to send this to? *


Full text views

Total number of HTML views: 3
Total number of PDF views: 93 *
Loading metrics...

Abstract views

Total abstract views: 458 *
Loading metrics...

* Views captured on Cambridge Core between 4th May 2018 - 14th August 2018. This data will be updated every 24 hours.