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2020 Analysis of an Adaptive Biasing Force method based on self-interacting dynamics
Michel Benaïm, Charles-Edouard Bréhier, Pierre Monmarché
Electron. J. Probab. 25: 1-28 (2020). DOI: 10.1214/20-EJP490


This article fills a gap in the mathematical analysis of Adaptive Biasing algorithms, which are extensively used in molecular dynamics computations. Given a reaction coordinate, ideally, the biasing force in the overdamped Langevin dynamics would be given by the gradient of the associated free energy function, which is unknown. We consider an adaptive biased version of the overdamped dynamics, where the biasing force depends on the past of the trajectory and is designed to approximate the free energy.

The main result of this article is the consistency and efficiency of this approach. More precisely we prove the almost sure convergence of the biasing force as time goes to infinity, and that the limit is close to the ideal biasing force, as an auxiliary parameter of the algorithm goes to $0$.

The proof is based on interpreting the process as a self-interacting dynamics, and on the study of a non-trivial fixed point problem for the limiting flow obtained using the ODE method.


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Michel Benaïm. Charles-Edouard Bréhier. Pierre Monmarché. "Analysis of an Adaptive Biasing Force method based on self-interacting dynamics." Electron. J. Probab. 25 1 - 28, 2020.


Received: 26 October 2019; Accepted: 27 June 2020; Published: 2020
First available in Project Euclid: 28 July 2020

zbMATH: 07252720
MathSciNet: MR4136468
Digital Object Identifier: 10.1214/20-EJP490

Primary: 60J60, 65C50


Vol.25 • 2020
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