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

Benjamin Jourdain; Tony Lelièvre; Raphaël Roux

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 44, Issue: 5, page 831-865
  • ISSN: 0764-583X

Abstract

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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., 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, 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.

How to cite

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Jourdain, Benjamin, Lelièvre, Tony, and Roux, Raphaël. "Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process." ESAIM: Mathematical Modelling and Numerical Analysis 44.5 (2010): 831-865. <http://eudml.org/doc/250783>.

@article{Jourdain2010,
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-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., 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, 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. },
author = {Jourdain, Benjamin, Lelièvre, Tony, Roux, Raphaël},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Conditional McKean nonlinearity; interacting particle systems; Adaptive Biasing Force method; conditional McKean nonlinearity, interacting particle systems; adaptive biasing force method; nonlinear stochastic differential equation; Nadaraya-Watson estimator; convergence; numerical examples},
language = {eng},
month = {8},
number = {5},
pages = {831-865},
publisher = {EDP Sciences},
title = {Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process},
url = {http://eudml.org/doc/250783},
volume = {44},
year = {2010},
}

TY - JOUR
AU - Jourdain, Benjamin
AU - Lelièvre, Tony
AU - Roux, Raphaël
TI - Existence, uniqueness and convergence of a particle approximation for the Adaptive Biasing Force process
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/8//
PB - EDP Sciences
VL - 44
IS - 5
SP - 831
EP - 865
AB - 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., 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, 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.
LA - eng
KW - Conditional McKean nonlinearity; interacting particle systems; Adaptive Biasing Force method; conditional McKean nonlinearity, interacting particle systems; adaptive biasing force method; nonlinear stochastic differential equation; Nadaraya-Watson estimator; convergence; numerical examples
UR - http://eudml.org/doc/250783
ER -

References

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