Long-time convergence of an adaptive biasing force method: Variance reduction by Helmholtz projection

Alrachid, Houssam; Lelièvre, Tony

doi:10.5802/smai-jcm.4

Cited by 22 publications

(61 citation statements)

References 10 publications

(27 reference statements)

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“…Moreover, the interest of this projection is that the variance of the force is reduced, because the nonconservative part is set to zero. This aspect is analyzed in ref ( 100 ).…”

Section: Addressing Nonergodicity Scenariosmentioning

confidence: 99%

The Adaptive Biasing Force Method: Everything You Always Wanted To Know but Were Afraid To Ask

et al. 2014

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In the host of numerical schemes devised to calculate free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an average force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy estimates. Much of the appeal of the algorithm to the practitioner is in its physically intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth mathematical analysis of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chemistry and biology. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a critical eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calculations of prototypical molecular systems, many methodological aspects are examined, from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a number of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.

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“…Moreover, the interest of this projection is that the variance of the force is reduced, because the nonconservative part is set to zero. This aspect is analyzed in ref ( 100 ).…”

Section: Addressing Nonergodicity Scenariosmentioning

confidence: 99%

The Adaptive Biasing Force Method: Everything You Always Wanted To Know but Were Afraid To Ask

et al. 2014

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“…Finally, Alrachid and Lelièvre (2015) have proved the convergence of a variant of the ABF method, where the biasing force is projected onto a gradient field (which makes sense since the expected limit, the mean force, is indeed a gradient).…”

Section: Extensions and Related Workmentioning

confidence: 99%

Partial differential equations and stochastic methods in molecular dynamics

Lelièvre¹,

Stoltz²

2016

Acta Numerica

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The objective of molecular dynamics computations is to infer macroscopic properties of matter from atomistic models via averages with respect to probability measures dictated by the principles of statistical physics. Obtaining accurate results requires efficient sampling of atomistic configurations, which are typically generated using very long trajectories of stochastic differential equations in high dimensions, such as Langevin dynamics and its overdamped limit. Depending on the quantities of interest at the macroscopic level, one may also be interested in dynamical properties computed from averages over paths of these dynamics.This review describes how techniques from the analysis of partial differential equations can be used to devise good algorithms and to quantify their efficiency and accuracy. In particular, a crucial role is played by the study of the long-time behaviour of the solution to the Fokker–Planck equation associated with the stochastic dynamics.

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“…Since, numerically, we use Monte-Carlo methods to approximate F t and ∇A t , the variance is an important quantity to assess the quality of the result. The following second main result is a variance reduction result and proved in [1].…”

Section: Projected Adaptive Biasing Force Methods (Pabf)mentioning

confidence: 92%

“…We apply now ABF and PABF dynamics to the trimer problem described above. One can refer to [1] for more detailed descriptions of the model and the used ABF and PABF algorithms.…”

Section: Numerical Experimentsmentioning

confidence: 99%

Variance Reduction Result for a Projected Adaptive Biasing Force Method

Alrachid

Lelièvre

2017

Computational Mathematics, Numerical Analysis and Applications

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This paper is committed to investigate an extension of the classical adaptive biasing force method, which is used to compute the free energy related to the Boltzmann-Gibbs measure and a reaction coordinate function. The issue of this technique is that the approximated gradient of the free energy, called biasing force, is not a gradient. The commitment to this field is to project the estimated biasing force on a gradient using the Helmholtz decomposition. The variance of the biasing force is reduced using this technique, which makes the algorithm more efficient than the standard ABF method. We prove exponential convergence to equilibrium of the estimated free energy, with a precise rate of convergence in function of Logarithmic Sobolev inequality constants.

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Long-time convergence of an adaptive biasing force method: Variance reduction by Helmholtz projection

Cited by 22 publications

References 10 publications

The Adaptive Biasing Force Method: Everything You Always Wanted To Know but Were Afraid To Ask

The Adaptive Biasing Force Method: Everything You Always Wanted To Know but Were Afraid To Ask

Partial differential equations and stochastic methods in molecular dynamics

Variance Reduction Result for a Projected Adaptive Biasing Force Method

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