Variance Reduction Result for a Projected Adaptive Biasing Force Method

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

Two mathematical tools to analyze metastable stochastic processes

We present how entropy estimates and logarithmic Sobolev inequalities on the one hand, and the notion of quasi-stationary distribution on the other hand, are useful tools to analyze metastable overdamped Langevin dynamics, in particular to quantify the degree of metastability. We discuss the interest of these approaches to estimate the efficiency of some classical algorithms used to speed up the sampling, and to evaluate the error introduced by some coarse-graining procedures. This paper is a summary of a plenary talk given by the author at the ENUMATH 2011 conference

Accelerated dynamics: Mathematical foundations and algorithmic improvements

We present a review of recent works on the mathematical analysis of algorithms which have been proposed by A.F. Voter and co-workers in the late nineties in order to efficiently generate long trajectories of metastable processes. These techniques have been successfully applied in many contexts, in particular in the field of materials science. The mathematical analysis we propose relies on the notion of quasi stationary distribution

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

In this paper, we propose an improvement of the adaptive biasing force (ABF) method, by projecting the estimated mean force onto a gradient. The associated stochastic process satisfies a non linear stochastic differential equation. Using entropy techniques, we prove exponential convergence to the stationary state of this stochastic process. We finally show on some numerical examples that the variance of the approximated mean force is reduced using this technique, which makes the algorithm more efficient than the standard ABF method.Comment: 33 pages, 20 figure

Orbitwise countings in H(2) and quasimodular forms

We prove formulae for the countings by orbit of square-tiled surfaces of genus two with one singularity. These formulae were conjectured by Hubert & Leli\`{e}vre. We show that these countings admit quasimodular forms as generating functions.Comment: 22 pages, 6 figure

Prime arithmetic Teichmuller discs in H(2)

It is well-known that Teichmuller discs that pass through "integer points'' of the moduli space of abelian differentials are very special: they are closed complex geodesics. However, the structure of these special Teichmuller discs is mostly unexplored: their number, genus, area, cusps, etc. We prove that in genus two all translation surfaces in H(2) tiled by a prime number n > 3 of squares fall into exactly two Teichmuller discs, only one of them with elliptic points, and that the genus of these discs has a cubic growth rate in n.Comment: Accepted for publication in Israel Journal of Mathematics. A previous version circulated with the title "Square-tiled surfaces in H(2)''. Changes from v1: improved redaction, fixed typos, added reference

Enhanced sampling of multidimensional free-energy landscapes using adaptive biasing forces

We propose an adaptive biasing algorithm aimed at enhancing the sampling of multimodal measures by Langevin dynamics. The underlying idea consists in generalizing the standard adaptive biasing force method commonly used in conjunction with molecular dynamics to handle in a more effective fashion multidimensional reaction coordinates. The proposed approach is anticipated to be particularly useful for reaction coordinates, the components of which are weakly coupled, as illuminated in a mathematical analysis of the long-time convergence of the algorithm. The strength as well as the intrinsic limitation of the method are discussed and illustrated in two realistic test cases