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


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).


Wang–Landau algorithm Stochastic approximation algorithm Free energy biasing techniques 



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.


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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

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