Communications in Mathematical Sciences

Volume 17 (2019)

Number 1

Convergence analysis of adaptive biasing potential methods for diffusion processes

Pages: 81 – 130



Michel Benaïm (Université de Neuchâtel, Institut de Mathématiques, Neuchâtel, Switzerland)

Charles-Edouard Bréhier (Institut Camille Jordan, Université Claude Bernard Lyon, Villeurbanne, France)


This article is concerned with the mathematical analysis of a family of adaptive importance sampling algorithms applied to diffusion processes. These methods, referred to as Adaptive Biasing Potential methods, are designed to efficiently sample the invariant distribution of the diffusion process, thanks to the approximation of the associated free energy function (relative to a reaction coordinate). The bias which is introduced in the dynamics is computed adaptively; it depends on the past of the trajectory of the process through some time-averages.

We give a detailed and general construction of such methods. We prove the consistency of the approach (almost sure convergence of well-chosen weighted empirical probability distribution). We justify the efficiency thanks to several qualitative and quantitative additional arguments. To prove these results, we revisit and extend tools from stochastic approximation applied to self-interacting diffusions, in an original context.


adaptive biasing, self-interacting diffusions, free energy computation

2010 Mathematics Subject Classification

60Cxx, 60J60

Received 6 October 2017

Accepted 10 October 2018

Published 30 May 2019