Hostname: page-component-745bb68f8f-d8cs5 Total loading time: 0 Render date: 2025-02-05T04:00:36.680Z Has data issue: false hasContentIssue false
  • English
  • Français

Geometric integrators and the Hamiltonian Monte Carlo method

Published online by Cambridge University Press:  04 May 2018

Nawaf Bou-Rabee  and
J. M. Sanz-Serna
Nawaf Bou-Rabee
Affiliation:
Department of Mathematical Sciences, Rutgers University Camden, 311 N. Fifth Street, CamdenNJ 08102, USA E-mail: [email protected]
J. M. Sanz-Serna
Affiliation:
Departamento de Matemáticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, E–28911 Leganés (Madrid), Spain E-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Type
Research Article
Information
Acta Numerica , Volume 27 , 01 May 2018 , pp. 113 - 206
Copyright
© Cambridge University Press, 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES 14

Akhmatskaya, E. and Reich, S. (2008), ‘GSHMC: An efficient method for molecular simulation’, J. Comput. Phys. 227, 4937–4954.Google Scholar
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.Google Scholar
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, 3239–3257.Google Scholar
Allen, M. P. and Tildesley, D. J. (1987), Computer Simulation of Liquids, Clarendon Press.Google Scholar
Andrieu, C., de Freitas, N., Doucet, A. and Jordan, M. I. (2003), ‘An introduction to MCMC for machine learning’, Machine Learning 50, 5–43.Google Scholar
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.Google Scholar
Asmussen, S. and Glynn, P. W. (2007), Stochastic Simulation: Algorithms and Analysis, Vol. 57 of Stochastic Modelling and Applied Probability, Springer.Google Scholar
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, 1501–1534.Google Scholar
Beskos, A., Pinski, F. J., Sanz-Serna, J. M. and Stuart, A. M. (2011), ‘Hybrid Monte-Carlo on Hilbert spaces’, Stoch. Proc. Appl. 121, 2201–2230.Google Scholar
Beskos, A., Roberts, G., Stuart, A. and Voss, J. (2008), ‘MCMC methods for diffusion bridges’, Stoch. Dynam. 8, 319–350.Google Scholar
Bishop, C. M. (2006), Pattern Recognition and Machine Learning, Springer.Google Scholar
Blanes, S. and Casas, F. (2016), A Concise Introduction to Geometric Numerical Integration, Monographs and Research Notes in Mathematics, CRC Press.Google Scholar
Blanes, S., Casas, F. and Sanz-Serna, J. M. (2014), ‘Numerical integrators for the hybrid Monte Carlo method’, SIAM J. Sci. Comput. 36, A1556–A1580.Google Scholar
Bou-Rabee, N. (2014), ‘Time integrators for molecular dynamics’, Entropy 16, 138–162.Google Scholar
Bou-Rabee, N. (2017) Cayley splitting for second-order Langevin stochastic partial differential equations. arXiv:1707.05603 Google Scholar
N. Bou-Rabee and A. Eberle (2018), Coupling and convergence for exact randomized Hamiltonian Monte-Carlo. In preparation.Google Scholar
Bou-Rabee, N. and Hairer, M. (2013), ‘Non-asymptotic mixing of the MALA algorithm’, IMA J. Numer. Anal. 33, 80–110.Google Scholar
Bou-Rabee, N. and Sanz-Serna, J. M. (2017), ‘Randomized Hamiltonian Monte Carlo’, Ann. Appl. Probab. 27, 2159–2194.Google Scholar
Bou-Rabee, N. and Vanden-Eijnden, E. (2010), ‘Pathwise accuracy and ergodicity of Metropolized integrators for SDEs’, Comm. Pure Appl. Math. 63, 655–696.Google Scholar
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, 2565–2580.Google Scholar
Bou-Rabee, N., Donev, A. and Vanden-Eijnden, E. (2014), ‘Metropolis integration schemes for self-adjoint diffusions’, Multiscale Model. Simul. 12, 781–831.Google Scholar
Butcher, J. C. (2016), Numerical Methods for Ordinary Differential Equations, third edition, Wiley.Google Scholar
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, 936–952.Google Scholar
Calvo, M. P. and Sanz-Serna, J. M. (2009), ‘Instabilities and inaccuracies in the integration of highly oscillatory problems’, SIAM J. Sci. Comput. 31, 1653–1677.Google Scholar
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. 63–74.Google Scholar
Campos, C. M. and Sanz-Serna, J. M. (2015), ‘Extra chance generalized hybrid Monte Carlo’, J. Comput. Phys. 281, 365–374.Google Scholar
Campos, C. M. and Sanz-Serna, J. M. (2017), ‘Palindromic 3-stage splitting integrators: A roadmap’, J. Comput. Phys. 346, 340–355.Google Scholar
Cancès, E., Legoll, F. and Stoltz, G. (2007), ‘Theoretical and numerical comparison of some sampling methods for molecular dynamics’, Math. Model. Numer. Anal. 41, 351–389.Google Scholar
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, 1391–1417.Google Scholar
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, 57–75.Google Scholar
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, 1–37.Google Scholar
Chen, Z. (2003), ‘Bayesian filtering: From Kalman filters to particle filters, and beyond’, Statistics 182, 1–69.Google Scholar
Cipra, B. A. (2000), ‘The best of the 20th century: Editors name top 10 algorithms’, SIAM News 33(4).Google Scholar
Da Prato, G. and Zabczyk, J. (2014), Stochastic Equations in Infinite Dimensions, Cambridge University Press.Google Scholar
Diaconis, P. (2009), ‘The Markov chain Monte Carlo revolution’, Bull. Am. Math. Soc. 46, 179–205.Google Scholar
Diaconis, P., Holmes, S. and Neal, R. M. (2000), ‘Analysis of a nonreversible Markov chain sampler’, Ann. Appl. Probab. 10, 726–752.Google Scholar
Duane, S., Kennedy, A. D., Pendleton, B. J. and Roweth, D. (1987), ‘Hybrid Monte-Carlo’, Phys. Lett. B 195, 216–222.Google Scholar
Eberle, A. (2016), ‘Reflection couplings and contraction rates for diffusions’, Probab. Theory Rel. Fields 166, 851–886.Google Scholar
Eberle, A. (2018), A coupling approach to the kinetic Langevin equation on the torus. In preparation.Google Scholar
Eberle, A., Guillin, A. and Zimmer, R. (2016) Couplings and quantitative contraction rates for Langevin dynamics. arXiv:1703.01617 Google Scholar
Evensen, G. (2009), Data Assimilation: The Ensemble Kalman Filter, Springer Science & Business Media.Google Scholar
Fang, Y., Sanz-Serna, J. M. and Skeel, R. D. (2014), ‘Compressible generalized hybrid Monte Carlo’, J. Chem. Phys. 140(17), 174108.Google Scholar
Fathi, M. (2014) Theoretical and numerical study of a few stochastic models of statistical physics. PhD thesis, Université Pierre et Marie Curie – Paris VI.Google Scholar
Fathi, M., Homman, A.-A. and Stoltz, G. (2015), ‘Error analysis of the transport properties of Metropolized schemes’, ESAIM Proc. Surv. 48, 341–363.Google Scholar
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.Google Scholar
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, 434–449.Google Scholar
Frantz, D. D., Freeman, D. L. and Doll, J. D. (1990), ‘Reducing quasi-ergodic behavior in Monte Carlo simulations by $J$ -walking: Applications to atomic clusters’, J. Chem. Phys. 93, 2769–2784.Google Scholar
Frenkel, D. and Smit, B. (2002), Understanding Molecular Simulation: From algorithms to Applications, second edition, Academic Press.Google Scholar
Gelfand, A. E. and Smith, A. F. M. (1990), ‘Sampling-based approaches to calculating marginal densities’, J. Amer. Statist. Assoc. 85(410), 398–409.Google Scholar
Geman, S. and Geman, D. (1984), ‘Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images’, IEEE Trans. Pattern Anal. Machine Intel. PAMI‐6, 721–741.Google Scholar
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. 156–163.Google Scholar
Geyer, C. J. (1992), ‘Practical Markov chain Monte Carlo’, Statist. Sci. 7, 473–483.Google Scholar
Geyer, C. J. and Thompson, E. A. (1995), ‘Annealing Markov chain Monte Carlo with applications to ancestral inference’, J. Amer. Statist. Assoc. 90(431), 909–920.Google Scholar
Ghahramani, Z. (2015), ‘‘Probabilistic machine learning and artificial intelligence’’, Nature 521(7553), 452.Google Scholar
Girolami, M. and Calderhead, B. (2011), ‘Riemann manifold Langevin and Hamiltonian Monte Carlo methods’, J. Royal Statist. Soc. B 73, 123–214.Google Scholar
Green, P. J. and Mira, A. (2001), ‘Delayed rejection in reversible jump Metropolis–Hastings’, Biometrika 88, 1035–1053.Google Scholar
Griffiths, D. F. and Higham, D. J. (2010), Numerical Methods for Ordinary Differential Equations: Initial Value Problems, Springer Undergraduate Mathematics Series, Springer.Google Scholar
Griffiths, D. F. and Sanz-Serna, J. M. (1986), ‘On the scope of the method of modified equations’, SIAM J. Sci. Statist. Comput. 7, 994–1008.Google Scholar
Gustafson, P. (1998), ‘A guided walk Metropolis algorithm’, Statist. Comput. 8, 357–364.Google Scholar
Hadfield, J. D. (2010), ‘MCMC methods for multi-response generalized linear mixed models: The MCMCglmm R package’, J. Statist. Softw. 33, 1–22.Google Scholar
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.Google Scholar
Hairer, E., Lubich, C. and Wanner, G. (2010), Geometric Numerical Integration, Springer.Google Scholar
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.Google Scholar
Hairer, M., Stuart, A. M. and Voss, J. (2009), ‘Sampling conditioned diffusions’, Trends Stoch. Anal. 353, 159–186.Google Scholar
Hansmann, U. H. E. (1997), ‘Parallel tempering algorithm for conformational studies of biological molecules’, Chem. Phys. Lett. 281, 140–150.Google Scholar
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, 1333–1338.Google Scholar
Hastings, W. K. (1970), ‘Monte-Carlo methods using Markov chains and their applications’, Biometrika 57, 97–109.Google Scholar
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, 435–447.Google Scholar
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, 1593–1623.Google Scholar
Horowitz, A. M. (1991), ‘A generalized guided Monte-Carlo algorithm’, Phys. Lett. B 268, 247–252.Google Scholar
Iserles, A. and Quispel, G. R. W. (2017) Why geometric integration? arXiv:1602.07755 Google Scholar
Izaguirre, J. A. and Hampton, S. S. (2004), ‘Shadow hybrid Monte Carlo: An efficient propagator in phase space of macromolecules’, J. Comput. Phys. 200, 581–604.Google Scholar
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, 188–204.Google Scholar
Ji, H. and Wong, W. H. (2006), ‘Computational biology: Toward deciphering gene regulatory information in mammalian genomes’, Biometrics 62, 645–663.Google Scholar
Kennedy, A. D. and Pendleton, B. (2001), ‘Cost of the generalized hybrid Monte Carlo algorithm for free field theory’, Nucl. Phys. B 607, 456–510.Google Scholar
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, 335–338.Google Scholar
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, 1581–1619.Google Scholar
Krauth, W. (2006), Statistical Mechanics: Algorithms and Computations, Oxford University Press.Google Scholar
Lamb, J. S. W. and Roberts, J. A. G. (1998), ‘Time-reversal symmetry in dynamical systems: A survey’, Nonlinear Phenomena 112, 1–39.Google Scholar
Landau, D. P. and Binder, K. (2014), A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press.Google Scholar
Leimkuhler, B. and Reich, S. (2004), Simulating Hamiltonian Dynamics, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.Google Scholar
Lelièvre, T., Rousset, M. and Stoltz, G. (2010), Free Energy Computations: A Mathematical Perspective, Imperial College Press.Google Scholar
Liang, F. and Wong, W. H. (2001), ‘Real-parameter evolutionary Monte Carlo with applications to Bayesian mixture models’, J. Amer. Statist. Assoc. 96(454), 653–666.Google Scholar
Link, W. A. and Barker, R. J. (2009), Bayesian Inference: With Ecological Applications, Academic Press.Google Scholar
Liu, J. S. (2008), Monte Carlo Strategies in Scientific Computing, second edition, Springer.Google Scholar
Lunn, D. J., Thomas, A., Best, N. and Spiegelhalter, D. (2000), ‘WinBUGS – a Bayesian modelling framework: Concepts, structure, and extensibility’, Statist. Comput. 10, 325–337.Google Scholar
Lunn, D., Jackson, C., Best, N., Thomas, A. and Spiegelhalter, D. (2012), The BUGS Book: A Practical Introduction to Bayesian Analysis, CRC press.Google Scholar
Lunn, D., Spiegelhalter, D., Thomas, A. and Best, N. (2009), ‘The BUGS project: Evolution, critique and future directions’, Statist. Medicine 28(25), 3049–3067.Google Scholar
Mackenzie, P. B. (1989), ‘An improved hybrid Monte Carlo method’, Phys. Lett. B 226, 369–371.Google Scholar
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, 500–509.CrossRefGoogle Scholar
Marinari, E. and Parisi, G. (1992), ‘Simulated tempering: A new Monte Carlo scheme’, Europhys. Lett. 19, 451.Google Scholar
Marsden, J. and Ratiu, T. (1999), Introduction to Mechanics and Symmetry, Springer Texts in Applied Mathematics, Springer.Google Scholar
Martin, A. D., Quinn, K. M. and Park, J. H. (2011), ‘MCMCpack: Markov chain Monte Carlo in R’, J. Statist. Softw. 42, 1–21.Google Scholar
Mclachlan, R. I. (1995), ‘On the numerical integration of ordinary differential equations by symmetric composition methods’, SIAM J. Sci. Comput. 16, 151–168.Google Scholar
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, 1087–1092.Google Scholar
Mira, A. (2001), ‘On Metropolis–Hastings algorithms with delayed rejection’, Metron 59, 231–241.Google Scholar
Murua, A. and Sanz-Serna, J. M. (1999), ‘Order conditions for numerical integrators obtained by composing simpler integrators’, Math. Phys. Eng. Sci. 357(1754), 1079–1100.Google Scholar
Murua, A. and Sanz-Serna, J. M. (2017), ‘Word series for dynamical systems and their numerical integrators’, Found. Comput. Math. 17, 675–712.Google Scholar
Neal, R. M. (1994), ‘An improved acceptance procedure for the hybrid Monte Carlo algorithm’, J. Comput. Phys. 111, 194–203.Google Scholar
Neal, R. M. (1996), ‘Sampling from multimodal distributions using tempered transitions’, Statist. Comput. 6, 353–366.Google Scholar
Neal, R. M. (2003), ‘Slice sampling’, Ann. Statist. 31, 705–741.Google Scholar
Neal, R. M. (2011), MCMC using Hamiltonian dynamics. In Handbook of Markov Chain Monte Carlo (Brooks, S. et al. , eds), Chapman & Hall/CRC, pp. 113–162.Google Scholar
Patil, A., Huard, D. and Fonnesbeck, C. J. (2010), ‘PyMC: Bayesian stochastic modelling in Python’, J. Statist. Softw. 35, 1.Google Scholar
Reznikoff, M. G. and Vanden-Eijnden, E. (2005), ‘Invariant measures of stochastic partial differential equations and conditioned diffusions’, Comptes Rendus Math. 340, 305–308.Google Scholar
Roberts, G. O. and Rosenthal, J. S. (1998), ‘Optimal scaling of discrete approximations to Langevin diffusions’, J. Roy. Statist. Soc. Ser. B 60, 255–268.Google Scholar
Roberts, G. O. and Tweedie, R. L. (1996a), ‘Exponential convergence of Langevin distributions and their discrete approximations’, Bernoulli 2, 341–363.Google Scholar
Roberts, G. O. and Tweedie, R. L. (1996b), ‘Geometric convergence and central limit theorems for multidimensional Hastings and Metropolis algorithms’, Biometrika 1, 95–110.Google Scholar
Roberts, G. O., Gelman, A. and Gilks, W. R. (1997), ‘Weak convergence and optimal scaling of random walk Metropolis algorithms’, Ann. Appl. Probab. 7, 110–120.Google Scholar
Sanz-Serna, J. M. (1991), Two topics in nonlinear stability. In Advances in Numerical Analysis, Vol. I (Light, W., ed.), Clarendon Press, pp. 147–174.Google Scholar
Sanz-Serna, J. M. (1992), Symplectic integrators for Hamiltonian problems: An overview. In Acta Numerica, Vol. 1, Cambridge University Press, pp. 243–286.Google Scholar
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. 193–205.Google Scholar
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. 121–143.Google Scholar
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. 39–88.Google Scholar
Sanz-Serna, J. M. (2016), ‘Symplectic Runge–Kutta schemes for adjoint equations, automatic differentiation, optimal control, and more’, SIAM Rev. 58, 3–33.Google Scholar
Sanz-Serna, J. M. and Calvo, M. P. (1994), Chapman & Hall.Google Scholar
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. 311–331.Google Scholar
Schlick, T. (2002), Molecular Modeling and Simulation: An Interdisciplinary Guide, Vol. 21 of Interdisciplinary Applied Mathematics, Springer.Google Scholar
Schütte, C. (1999) Conformational dynamics: Modeling, theory, algorithm, and application to biomolecules. Habilitation, Freie Universität Berlin.Google Scholar
Skeel, R. D. and Hardy, D. J. (2001), ‘Practical construction of modified Hamiltonians’, SIAM J. Sci. Comput. 23, 1172–1188.Google Scholar
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. 131–192.Google Scholar
Stoltz, G. (2007) Some mathematical methods for molecular and multiscale simulation. PhD thesis, École Nationale des Ponts et Chaussées.Google Scholar
Strang, G. (1963), ‘Accurate partial difference methods I: Linear Cauchy problems’, Arch. Rational Mech. Anal. 12, 392–402.Google Scholar
Stuart, A. M. (2010), Inverse problems: A Bayesian perspective. In Acta Numerica, Vol. 19, Cambridge University Press, pp. 451–559.Google Scholar
Sugita, Y. and Okamoto, Y. (1999), ‘Replica-exchange molecular dynamics method for protein folding’, Chem. Phys. Lett. 314, 141–151.Google Scholar
Sullivan, T. J. (2015), Introduction to Uncertainty Quantification, Vol. 63 of Texts in Applied Mathematics, Springer.Google Scholar
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.Google Scholar
Thrun, S., Burgard, W. and Fox, D. (2005), Probabilistic Robotics, MIT press.Google Scholar
Tierney, L. (1994), ‘Markov chains for exploring posterior distributions’, Ann. Statist. 22, 1701–1728.Google Scholar
Tuckerman, M. (2010), Statistical Mechanics: Theory and Molecular Simulation, Oxford University Press.Google Scholar
Wales, D. (2003), Energy Landscapes: Applications to Clusters, Biomolecules and Glasses, Cambridge University Press.Google Scholar
Webb, A. R. (2003), Statistical Pattern Recognition, Wiley.Google Scholar
Yoshida, H. (1990), ‘Construction of higher order symplectic integrators’, Phys. Lett. A 150, 262–268.Google Scholar