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).
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Notes
With the notation of the metadynamics works, what we call here a is denoted \(\varDelta T/(T + \varDelta T)\) where T is the temperature and \(\varDelta T > 0\) is a parameter. The limiting regime \(a = 1\) is recovered in the limit \(\varDelta T \rightarrow +\infty \), which corresponds to the standard metadynamics (Laio and Parrinello 2002; Bussi et al. 2006.
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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.
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Fort, G., Jourdain, B., Lelièvre, T. et al. Self-healing umbrella sampling: convergence and efficiency. Stat Comput 27, 147–168 (2017). https://doi.org/10.1007/s11222-015-9613-2
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DOI: https://doi.org/10.1007/s11222-015-9613-2