Advertisement

Statistics and Computing

, Volume 27, Issue 1, pp 147–168 | Cite as

Self-healing umbrella sampling: convergence and efficiency

  • Gersende FortEmail author
  • Benjamin Jourdain
  • Tony Lelièvre
  • Gabriel Stoltz
Article

Abstract

The Self-Healing Umbrella Sampling (SHUS) algorithm is an adaptive biasing algorithm which has been proposed in Marsili et al. (J Phys Chem B 110(29):14011–14013, 2006) in order to efficiently sample a multimodal probability measure. We show that this method can be seen as a variant of the well-known Wang–Landau algorithm Wang and Landau (Phys Rev E 64:056101, 2001a; Phys Rev Lett 86(10):2050–2053, 2001b). Adapting results on the convergence of the Wang-Landau algorithm obtained in Fort et al. (Math Comput 84(295):2297–2327, 2014a), we prove the convergence of the SHUS algorithm. We also compare the two methods in terms of efficiency. We finally propose a modification of the SHUS algorithm in order to increase its efficiency, and exhibit some similarities of SHUS with the well-tempered metadynamics method Barducci et al. (Phys Rev Lett 100:020,603, 2008).

Keywords

Wang–Landau algorithm Stochastic approximation algorithm Free energy biasing techniques 

Notes

Acknowledgments

This work is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement number 614492 and by the French National Research Agency under the grants ANR-12-BS01-0019 (STAB) and ANR-14-CE23-0012 (COSMOS). We also benefited from the scientific environment of the Laboratoire International Associé between the Centre National de la Recherche Scientifique and the University of Illinois at Urbana-Champaign.

References

  1. Andrieu, C., Moulines, E., Priouret, P.: Stability of stochastic approximation under verifiable conditions. SIAM J. Control Optim. 44, 283–312 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. Barducci, A., Bussi, G., Parrinello, M.: Well-tempered metadynamics: a smoothly converging and tunable free-energy method. Phys. Rev. Lett. 100, 020,603 (2008)CrossRefGoogle Scholar
  3. Bussi, G., Laio, A., Parrinello, M.: Equilibrium free energies from nonequilibrium metadynamics. Phys. Rev. Lett. 96, 090,601 (2006)CrossRefGoogle Scholar
  4. Chopin, N., Lelièvre, T., Stoltz, G.: Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors. Stat. Comput. 22(4), 897–916 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dama, J., Parrinello, M., Voth, G.: Well-tempered metadynamics converges asymptotically. Phys. Rev. Lett. 112, 240,602(1-6) (2014)CrossRefGoogle Scholar
  6. Dickson, B., Legoll, F., Lelièvre, T., Stoltz, G., Fleurat-Lessard, P.: Free energy calculations: an efficient adaptive biasing potential method. J. Phys. Chem. B 114, 5823–5830 (2010)CrossRefGoogle Scholar
  7. Fort, G.: Central limit theorems for stochastic approximation with controlled Markov chain dynamics. ESAIM: Probab. Stat. 19, 60–80 (2015)Google Scholar
  8. Fort, G., Jourdain, B., Kuhn, E., Lelièvre, T., Stoltz, G.: Convergence of the Wang-Landau. Math. Comput. 84(295), 2297–2327 (2014a)Google Scholar
  9. Fort, G., Jourdain, B., Kuhn, E., Lelièvre, T., Stoltz, G.: Efficiency of the Wang-Landau algorithm: a simple test case. Appl. Math. Res. Express 2014(2), 275–311 (2014b)Google Scholar
  10. Fort, G., Moulines, E., Priouret, P.: Convergence of adaptive and interacting Markov chain Monte Carlo algorithms. Ann. Stat. 39(6), 3262–3289 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hall, P., Heyde, P.: Martingale Limit Theory and its Application. Academic Press, New York (1980)zbMATHGoogle Scholar
  12. Hastings, W.K.: Monte Carlo sampling methods using Markov chains and their applications. Biometrika 57, 97–109 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Jacob, P.E., Ryder, R.J.: The Wang-Landau algorithm reaches the flat histogram criterion in finite time. Ann. Appl. Probab. 24(1), 34–53 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Laio, A., Parrinello, M.: Escaping free-energy minima. Proc. Natl. Acad. Sci. USA 99, 12562–12566 (2002)CrossRefGoogle Scholar
  15. Lelièvre, T.: Two mathematical tools to analyze metastable stochastic processes. In: Cangiani, A., Davidchack, R.L., Georgoulis, E., Gorban, A.N., Levesley, J., Tretyakov, M.V. (eds.) Numerical Mathematics and Advanced Applications 2011, pp. 791–810. Springer, Berlin (2013)CrossRefGoogle Scholar
  16. Lelièvre, T., Rousset, M., Stoltz, G.: Free Energy Computations: A Mathematical Perspective. Imperial College Press, London (2010)CrossRefzbMATHGoogle Scholar
  17. Marinari, E., Parisi, G.: Simulated tempering—a new Monte-Carlo scheme. Europhys. Lett. 19(6), 451–458 (1992)CrossRefGoogle Scholar
  18. Marsili, S., Barducci, A., Chelli, R., Procacci, P., Schettino, V.: Self-healing umbrella sampling: a non-equilibrium approach for quantitative free energy calculations. J. Phys. Chem. B 110(29), 14011–14013 (2006)Google Scholar
  19. Mengersen, K., Robert, C.: IID sampling with self-avoiding particle filters: the pinball sampler. In: Bernardo, J., David, A., Berger, J., West, M. (eds.) Bayesian Statistics 7 (2003)Google Scholar
  20. Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., Teller, A.H., Teller, E.: Equations of state calculations by fast computing machines. J. Chem. Phys. 21(6), 1087–1091 (1953)CrossRefGoogle Scholar
  21. Park, S., Sener, M.K., Lu, D., Schulten, K.: Reaction paths based on mean first-passage times. J. Chem. Phys. 119(3), 1313–1319 (2003)CrossRefGoogle Scholar
  22. Polyak, B.T., Juditsky, A.B.: Acceleration of stochastic approximation by averaging. SIAM J. Control Optim. 30(4), 838–855 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  23. Roberts, G.O., Tweedie, R.L.: Exponential convergence of Langevin distributions and their discrete approximations. Bernoulli 2(4), 341–363 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. Wang, F., Landau, D.: Determining the density of states for classical statistical models: a random walk algorithm to produce a flat histogram. Phys. Rev. E 64, 056101 (2001a)Google Scholar
  25. Wang, F.G., Landau, D.P.: Efficient, multiple-range random walk algorithm to calculate the density of states. Phys. Rev. Lett. 86(10), 2050–2053 (2001b)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Gersende Fort
    • 1
    Email author
  • Benjamin Jourdain
    • 2
  • Tony Lelièvre
    • 2
  • Gabriel Stoltz
    • 2
  1. 1.LTCI, CNRS & Telecom Paris TechParis Cedex 13France
  2. 2.Université Paris-Est, CERMICS (ENPC), INRIAMarne-la-ValléeFrance

Personalised recommendations