Abstract
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.
Highlights
Let μ‹ be a probability distribution on the d-dimensional flat torus Td, of the type: eβV pxq dμ‹pxq “ dx, Z pβ q żZpβq “ eβV pxqdx, Td (1.1)Analysis of an ABF method based on self-interacting dynamics where dx is the normalized Lebesgue measure on Td
For applications in physics and chemistry, μ‹ is referred to as the Boltzmann-Gibbs distribution associated with the potential energy function V and the inverse temperature parameter β ą 0
To explain the construction of the method and to justify its efficiency, we assume that the reaction coordinate is representative of the metastable behavior of the system: roughly, this means that only the exploration in the z variable is affected by the metastability, whereas the exploration in the y variable is much faster
Summary
Let μ‹ be a probability distribution on the d-dimensional flat torus Td (with T “ R{Z), of the type: eβV pxq dμ‹pxq “. Analysis of an ABF method based on self-interacting dynamics where dx is the normalized Lebesgue measure on Td. For applications in physics and chemistry (e.g. in molecular dynamics), μ‹ is referred to as the Boltzmann-Gibbs distribution associated with the potential energy function V and the inverse temperature parameter β ą 0. For applications in statistics (e.g. in Bayesian statistics), ́βV is referred to as the log-likelihood. The function V : Td Ñ R is assumed to be smooth. In order to estimate integrals of the type ş φdμ‹, with φ : Td Ñ R, probabilistic methods are used, especially when d is large.
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