# 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

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topJourdain, 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 -

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