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Existence, uniqueness and convergence of a particle approximation for the AdaptiveBiasing Force process

Published online by Cambridge University Press:  26 August 2010

Benjamin Jourdain ,
Tony Lelièvre  and
Raphaël Roux
Benjamin Jourdain
Affiliation:
Université Paris-Est, CERMICS, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France.
Tony Lelièvre
Affiliation:
Université Paris-Est, CERMICS, Project-Team MICMAC ENPC-INRIA, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France. lelievre@cermics.enpc.fr
Raphaël Roux
Affiliation:
Université Paris-Est, CERMICS, Project-Team MICMAC ENPC-INRIA, 6 et 8 avenue Blaise Pascal, 77455 Marne-La-Vallée Cedex 2, France. lelievre@cermics.enpc.fr
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Abstract

We study a free energy computation procedure, introduced in[Darve and Pohorille, J. Chem. Phys.115 (2001) 9169–9183; Hénin and Chipot, J. Chem. Phys.121 (2004) 2904–2914], which relies on the long-timebehavior of a nonlinear stochasticdifferential equation. This nonlinearity comes from a conditionalexpectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions tothis equation has been provedin [Lelièvre et al., Nonlinearity21 (2008) 1155–1181], under some existence and regularity assumptions.In this paper, we prove existence and uniqueness under suitable conditions for the nonlinear equation, andwe study a particle approximation technique based on a Nadaraya-Watson estimator ofthe conditional expectation. The particle system converges to the solutionof the nonlinear equation if the number of particles goes to infinityand then the kernel used in the Nadaraya-Watson approximation tends to aDirac mass. We derive a rate for this convergence, and illustrate it by numericalexamples on a toy model.

Keywords

Conditional McKean nonlinearityinteracting particle systemsAdaptiveBiasing Force method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 44 , Issue 5: Special Issue on Probabilistic methods and their applications , September 2010 , pp. 831 - 865
Copyright
© EDP Sciences, SMAI, 2010

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