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-time behavior of a nonlinear stochastic differential equation. This nonlinearity comes from a conditional expectation computed with respect to one coordinate of the solution. The long-time convergence of the solutions to this equation has been proved in [Lelièvre et al., Nonlinearity 21 (2008) 1155-1181], under some existence and regularity assumptions. In this paper, we prove existence and uniqueness under suitable conditions for the nonlinear equation, and we study a particle approximation technique based on a Nadaraya-Watson estimator of the conditional expectation. The particle system converges to the solution of the nonlinear equation if the number of particles goes to infinity and then the kernel used in the Nadaraya-Watson approximation tends to a Dirac mass. We derive a rate for this convergence, and illustrate it by numerical examples on a toy model.
@article{M2AN_2010__44_5_831_0,
author = {Jourdain, Benjamin and Leli\`evre, Tony and Roux, Rapha\"el},
title = {Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {44},
year = {2010},
pages = {831-865},
doi = {10.1051/m2an/2010044},
mrnumber = {2731395},
zbl = {1201.65011},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2010__44_5_831_0}
}
Jourdain, Benjamin; Lelièvre, Tony; Roux, Raphaël. Existence, uniqueness and convergence of a particle approximation for the adaptive biasing force process. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 44 (2010) pp. 831-865. doi : 10.1051/m2an/2010044. http://gdmltest.u-ga.fr/item/M2AN_2010__44_5_831_0/
[1] , Sobolev spaces. Academic Press (1978). | Zbl 1098.46001
[2] , and , On conditional McKean Lagrangian stochastic models. Prob. Theor. Relat. Fields (to appear).
[3] , Analyse fonctionnelle. Théorie et applications. Collection Mathématiques appliquées pour la maîtrise, Masson, Paris (1983). | Zbl 0511.46001
[4] and Eds., Free Energy Calculations, Springer Series in Chemical Physics 86. Springer (2007).
[5] and , Calculating free energy using average forces. J. Chem. Phys. 115 (2001) 9169-9183.
[6] and , Mathematical Analysis and Numerical Methods for Science and Technology. Springer Verlag (1999).
[7] , Propagation and conditional propagation of chaos for pressureless gas equations. Prob. Theor. Relat. Fields 126 (2003) 459-479. | Zbl 1029.60048
[8] and , Overcoming free energy barriers using unconstrained molecular dynamics simulations. J. Chem. Phys. 121 (2004) 2904-2914.
[9] and , Strong solutions of stochastic equations with singular time dependent drift. Prob. Theor. Relat. Fields 131 (2005) 154-196. | Zbl 1072.60050
[10] , and , Computation of free energy profiles with parallel adaptive dynamics. J. Chem. Phys. 126 (2007) 134111.
[11] , and , Long-time convergence of an adaptive biasing force method. Nonlinearity 21 (2008) 1155-1181. | Zbl 1146.35320
[12] , Quelques méthodes de résolution des problèmes aux limites non-linéaires. Dunod (1969). | Zbl 0189.40603
[13] and , Problèmes aux limites non homogènes et applications. Dunod, Paris (1968-1970). | Zbl 0165.10801
[14] , and , Illustration of transition path theory on a collection of simple examples. J. Chem. Phys. 125 (2006) 084110.
[15] , Topics in propagation of chaos, Lecture notes in mathematics 1464. Springer-Verlag (1989). | Zbl 0732.60114
[16] and , A stochastic particle method with random weights for the computation of statistical solutions of McKean-Vlasov equations. Ann. Appl. Prob. 13 (2003) 140-180. | Zbl 1026.60110
[17] , Navier-Stokes equations and nonlinear functionnal analysis. North Holland, Amsterdam (1979). | Zbl 0522.35002
[18] , A wavelet particle approximation for McKean-Vlasov and 2D-Navier-Stokes statistical solutions. Stoch. Proc. Appl. 118 (2008) 284-318. | Zbl 1148.62024
[19] , Introduction à l'estimation non-paramétrique. Springer (2004). | Zbl 1029.62034