Adaptive force biasing algorithms: New convergence results and tensor approximations of the bias

Abstract

We analyze and propose variants of the adaptive biasing force method. First, we prove the convergence of a version of the algorithm where the biasing force is estimated using a weighted occupation measure, with an explicit asymptotic variance. Second, we propose a new flavour of the algorithm adapted to high-dimensional reaction coordinates, for which the standard approaches suffer from the curse of dimensionality. More precisely, the free energy is approximated by a sum of tensor products of one-dimensional functions. The consistency of the tensor approximation is established. Numerical experiments on five-dimensional reaction coordinates demonstrate that the method is indeed able to capture correlations between them.