{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,25]],"date-time":"2026-04-25T03:24:23Z","timestamp":1777087463529,"version":"3.51.4"},"reference-count":33,"publisher":"American Mathematical Society (AMS)","issue":"295","license":[{"start":{"date-parts":[[2016,3,23]],"date-time":"2016-03-23T00:00:00Z","timestamp":1458691200000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We analyze the convergence properties of the Wang-Landau algorithm. This sampling method belongs to the general class of adaptive importance sampling strategies which use the free energy along a chosen reaction coordinate as a bias. Such algorithms are very helpful to enhance the sampling properties of Markov Chain Monte Carlo algorithms, when the dynamics is metastable. We prove the convergence of the Wang-Landau algorithm and an associated central limit theorem.<\/p>","DOI":"10.1090\/s0025-5718-2015-02952-4","type":"journal-article","created":{"date-parts":[[2015,3,23]],"date-time":"2015-03-23T11:57:22Z","timestamp":1427111842000},"page":"2297-2327","source":"Crossref","is-referenced-by-count":18,"title":["Convergence of the Wang-Landau algorithm"],"prefix":"10.1090","volume":"84","author":[{"given":"Gersende","family":"Fort","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Benjamin","family":"Jourdain","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Estelle","family":"Kuhn","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tony","family":"Leli\u00e8vre","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gabriel","family":"Stoltz","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2015,3,23]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"283","DOI":"10.1137\/S0363012902417267","article-title":"Stability of stochastic approximation under verifiable conditions","volume":"44","author":"Andrieu, Christophe","year":"2005","journal-title":"SIAM J. Control Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0363-0129","issn-type":"print"},{"issue":"1","key":"2","first-page":"209","article-title":"The Wang-Landau algorithm in general state spaces: applications and convergence analysis","volume":"20","author":"Atchad\u00e9, Yves F.","year":"2010","journal-title":"Statist. Sinica","ISSN":"https:\/\/id.crossref.org\/issn\/1017-0405","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"V. Babin, C. Roland, and C. Sagui, Adaptively biased molecular dynamics for free energy calculations, J. Chem. Phys. 128 (2008), 134101.","DOI":"10.1063\/1.2844595"},{"key":"4","unstructured":"A. Benveniste, M. M\u00e9tivier, and Priouret P., Adaptive Algorithms and Stochastic Approximations, Springer-Verlag, 1987."},{"issue":"3","key":"5","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1080\/10618600.2012.723569","article-title":"An adaptive interacting Wang-Landau algorithm for automatic density exploration","volume":"22","author":"Bornn, Luke","year":"2013","journal-title":"J. Comput. Graph. Statist.","ISSN":"https:\/\/id.crossref.org\/issn\/1061-8600","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1016\/0304-4149(87)90039-1","article-title":"Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds","volume":"27","author":"Chen, Han Fu","year":"1988","journal-title":"Stochastic Process. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0304-4149","issn-type":"print"},{"issue":"4","key":"7","doi-asserted-by":"publisher","first-page":"897","DOI":"10.1007\/s11222-011-9257-9","article-title":"Free energy methods for Bayesian inference: efficient exploration of univariate Gaussian mixture posteriors","volume":"22","author":"Chopin, Nicolas","year":"2012","journal-title":"Stat. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0960-3174","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"E. Darve and A. Pohorille, Calculating free energies using average force, J. Chem. Phys. 115 (2001), no. 20, 9169\u20139183.","DOI":"10.1063\/1.1410978"},{"key":"9","doi-asserted-by":"crossref","unstructured":"B. Dickson, F. Legoll, T. Leli\u00e8vre, G. Stoltz, and P. Fleurat-Lessard, Free energy calculations: An efficient adaptive biasing potential method, J. Phys. Chem. B 114 (2010), 5823\u20135830.","DOI":"10.1021\/jp100926h"},{"key":"10","doi-asserted-by":"crossref","unstructured":"G. Fort, Central Limit Theorems for Stochastic Approximation with Controlled Markov chain Dynamics, to appear in ESAIM:PS (2015).","DOI":"10.1051\/ps\/2014013"},{"key":"11","unstructured":"G. Fort, B. Jourdain, T. Leli\u00e8vre and G. Stoltz, Self-healing umbrella sampling: convergence and efficiency, http:\/\/arxiv.org\/abs\/1410.2109."},{"key":"12","doi-asserted-by":"crossref","unstructured":"G. Fort, B. Jourdain, E. Kuhn, T. Leli\u00e8vre and G. Stoltz, Efficiency of the Wang-Landau algorithm: a simple test case, Appl. Math. Res. Express AMRX 2014 (2014), no. 2, 275\u2013311.","DOI":"10.1093\/amrx\/abu003"},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"3262","DOI":"10.1214\/11-AOS938","article-title":"Convergence of adaptive and interacting Markov chain Monte Carlo algorithms","volume":"39","author":"Fort, G.","year":"2011","journal-title":"Ann. Statist.","ISSN":"https:\/\/id.crossref.org\/issn\/0090-5364","issn-type":"print"},{"key":"14","series-title":"Probability and Mathematical Statistics","isbn-type":"print","volume-title":"Martingale limit theory and its application","author":"Hall, P.","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0123193508"},{"key":"15","doi-asserted-by":"crossref","unstructured":"W. K. Hastings, Monte Carlo sampling methods using Markov chains and their applications, Biometrika 57 (1970), 97\u2013109.","DOI":"10.1093\/biomet\/57.1.97"},{"key":"16","doi-asserted-by":"crossref","unstructured":"J. H\u00e9nin and C. Chipot, Overcoming free energy barriers using unconstrained molecular dynamics simulations, J. Chem. Phys. 121 (2004), no. 7, 2904\u20132914.","DOI":"10.1063\/1.1773132"},{"issue":"1","key":"17","doi-asserted-by":"publisher","first-page":"34","DOI":"10.1214\/12-AAP913","article-title":"The Wang-Landau algorithm reaches the flat histogram criterion in finite time","volume":"24","author":"Jacob, Pierre E.","year":"2014","journal-title":"Ann. Appl. Probab.","ISSN":"https:\/\/id.crossref.org\/issn\/1050-5164","issn-type":"print"},{"issue":"5","key":"18","doi-asserted-by":"publisher","first-page":"831","DOI":"10.1051\/m2an\/2010044","article-title":"Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process","volume":"44","author":"Jourdain, Benjamin","year":"2010","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"19","series-title":"Applications of Mathematics (New York)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-2696-8","volume-title":"Stochastic approximation algorithms and applications","volume":"35","author":"Kushner, Harold J.","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/038794916X"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00205-011-0426-y","article-title":"Long-time convergence of an adaptive biasing force method: the bi-channel case","volume":"202","author":"Leli\u00e8vre, T.","year":"2011","journal-title":"Arch. Ration. Mech. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-9527","issn-type":"print"},{"key":"21","doi-asserted-by":"crossref","unstructured":"T. Leli\u00e8vre, M. Rousset, and G. Stoltz, Computation of free energy profiles with adaptive parallel dynamics, J. Chem. Phys. 126 (2007), 134111.","DOI":"10.1063\/1.2711185"},{"issue":"6","key":"22","doi-asserted-by":"publisher","first-page":"1155","DOI":"10.1088\/0951-7715\/21\/6\/001","article-title":"Long-time convergence of an adaptive biasing force method","volume":"21","author":"Leli\u00e8vre, Tony","year":"2008","journal-title":"Nonlinearity","ISSN":"https:\/\/id.crossref.org\/issn\/0951-7715","issn-type":"print"},{"key":"23","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/9781848162488","volume-title":"Free energy computations","author":"Leli\u00e8vre, Tony","year":"2010","ISBN":"https:\/\/id.crossref.org\/isbn\/9781848162471"},{"issue":"16","key":"24","doi-asserted-by":"publisher","first-page":"2632","DOI":"10.1016\/j.spl.2008.02.034","article-title":"Almost sure convergence of randomly truncated stochastic algorithms under verifiable conditions","volume":"78","author":"Lelong, J\u00e9r\u00f4me","year":"2008","journal-title":"Statist. Probab. Lett.","ISSN":"https:\/\/id.crossref.org\/issn\/0167-7152","issn-type":"print"},{"key":"25","doi-asserted-by":"crossref","unstructured":"F. Liang, A general Wang-Landau algorithm for Monte Carlo computation, J. Am. Stat. Assoc. 100 (2005), 1311\u20131327.","DOI":"10.1198\/016214505000000259"},{"issue":"477","key":"26","doi-asserted-by":"publisher","first-page":"305","DOI":"10.1198\/016214506000001202","article-title":"Stochastic approximation in Monte Carlo computation","volume":"102","author":"Liang, Faming","year":"2007","journal-title":"J. Amer. Statist. Assoc.","ISSN":"https:\/\/id.crossref.org\/issn\/0162-1459","issn-type":"print"},{"key":"27","doi-asserted-by":"crossref","unstructured":"S. Marsili, A. Barducci, R. Chelli, P. Procacci, and V. Schettino, Self-healing Umbrella Sampling: A non-equilibrium approach for quantitative free energy calculations, J. Phys. Chem. B 110 (2006), no. 29, 14011\u201314013.","DOI":"10.1021\/jp062755j"},{"key":"28","doi-asserted-by":"crossref","unstructured":"N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equations of state calculations by fast computing machines, J. Chem. Phys. 21 (1953), no. 6, 1087\u20131091.","DOI":"10.1063\/1.1699114"},{"key":"29","doi-asserted-by":"crossref","unstructured":"S. Meyn and R. L. Tweedie, Markov Chains and Stochastic Stability, Cambridge, 2009.","DOI":"10.1017\/CBO9780511626630"},{"key":"30","doi-asserted-by":"crossref","unstructured":"K. Minoukadeh, C. Chipot, and T. Leli\u00e8vre, Potential of mean force calculations: a multiple-walker adaptive biasing force approach, J. Chem. Th. Comput. 6 (2010), no. 4, 1008\u20131017.","DOI":"10.1021\/ct900524t"},{"key":"31","doi-asserted-by":"publisher","first-page":"400","DOI":"10.1214\/aoms\/1177729586","article-title":"A stochastic approximation method","volume":"22","author":"Robbins, Herbert","year":"1951","journal-title":"Ann. Math. Statistics","ISSN":"https:\/\/id.crossref.org\/issn\/0003-4851","issn-type":"print"},{"key":"32","doi-asserted-by":"crossref","unstructured":"F. G. Wang and D. P. Landau, Determining the density of states for classical statistical models: A random walk algorithm to produce a flat histogram, Phys. Rev. E 64 (2001), 056101.","DOI":"10.1103\/PhysRevE.64.056101"},{"key":"33","doi-asserted-by":"crossref","unstructured":"F. G. Wang and D. P. Landau, Efficient, multiple-range random walk algorithm to calculate the density of states, Phys. Rev. Lett. 86 (2001), no. 10, 2050\u20132053.","DOI":"10.1103\/PhysRevLett.86.2050"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2015-84-295\/S0025-5718-2015-02952-4\/S0025-5718-2015-02952-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-295\/S0025-5718-2015-02952-4\/S0025-5718-2015-02952-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:28:38Z","timestamp":1776796118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-295\/S0025-5718-2015-02952-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,3,23]]},"references-count":33,"journal-issue":{"issue":"295","published-print":{"date-parts":[[2015,9]]}},"alternative-id":["S0025-5718-2015-02952-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2015-02952-4","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2015,3,23]]}}}