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

A mathematical formalization of the parallel replica dynamics

The purpose of this article is to lay the mathematical foundations of a well known numerical approach in computational statistical physics and molecular dynamics, namely the parallel replica dynamics introduced by A.F. Voter. The aim of the approach is to efficiently generate a coarse-grained evolution (in terms of state-to-state dynamics) of a given stochastic process. The approach formally consists in concurrently considering several realizations of the stochastic process, and tracking among the realizations that which, the soonest, undergoes an important transition. Using specific properties of the dynamics generated, a computational speed-up is obtained. In the best cases, this speed-up approaches the number of realizations considered. By drawing connections with the theory of Markov processes and, in particular, exploiting the notion of quasi-stationary distribution, we provide a mathematical setting appropriate for assessing theoretically the performance of the approach, and possibly improving it

Efficiency of the Wang-Landau algorithm: a simple test case

We analyze the efficiency of the Wang-Landau algorithm to sample a multimodal distribution on a prototypical simple test case. We show that the exit time from a metastable state is much smaller for the Wang Landau dynamics than for the original standard Metropolis-Hastings algorithm, in some asymptotic regime. Our results are confirmed by numerical experiments on a more realistic test case

Micro-macro models for viscoelastic fluids: modelling, mathematics and numerics

This paper is an introduction to the modelling of viscoelastic fluids, with an emphasis on micro-macro (or multiscale) models. Some elements of mathematical and numerical analysis are provided. These notes closely follow the lectures delivered by the second author at the Chinese Academy of Science during the Workshop "Stress Tensor Effects on Fluid Mechanics", in January 2010

Virtual gardening: Identifying problems and potential directions for 'ecological awareness' through soil management and plant recognition gaming

Games are increasingly proven to be effective learning tools through a multitude of methodologies and approaches and this is no different for issues relating to the environment and the place of humans within it. We collaborated with the Eden Project to create a mobile game addressing some concerns on the ecological awareness of visitors that they raised with us: a mobile garden management game with a plant recognition technology. Such a project proved a valuable opportunity to understand how a game for smart devices might promote short-term ecological awareness for a general audience. Using a research creation methodology, we analyse, document and run a limited empirical study through user experience testing on players to investigate if the game had an effect on their ecological awareness

Measurement of the neutron electric dipole moment via spin rotation in a non-centrosymmetric crystal

We have measured the neutron electric dipole moment using spin rotation in a non-centrosymmetric crystal. Our result is d_n = (2.5 +- 6.5(stat) +- 5.5(syst)) 10^{-24} e cm. The dominating contribution to the systematic uncertainty is statistical in nature and will reduce with improved statistics. The statistical sensitivity can be increased to 2 10^{-26} e cm in 100 days data taking with an improved setup. We state technical requirements for a systematic uncertainty at the same level.Comment: submitted to Phys. Lett.

Optimal importance sampling for overdamped Langevin dynamics

Calculating averages with respect to multimodal probability distributions is often necessary in applications. Markov chain Monte Carlo (MCMC) methods to this end, which are based on time averages along a realization of a Markov process ergodic with respect to the target probability distribution, are usually plagued by a large variance due to the metastability of the process. In this work, we mathematically analyze an importance sampling approach for MCMC methods that rely on the overdamped Langevin dynamics. Specifically, we study an estimator based on an ergodic average along a realization of an overdamped Langevin process for a modified potential. The estimator we consider incorporates a reweighting term in order to rectify the bias that would otherwise be introduced by this modification of the potential. We obtain an explicit expression in dimension 1 for the biasing potential that minimizes the asymptotic variance of the estimator for a given observable, and propose a general numerical approach for approximating the optimal potential in the multi-dimensional setting. We also investigate an alternative approach where, instead of the asymptotic variance for a given observable, a weighted average of the asymptotic variances corresponding to a class of observables is minimized. Finally, we demonstrate the capabilities of the proposed method by means of numerical experiments

Optimal friction matrix for underdamped Langevin sampling

A systematic procedure for optimising the friction coefficient in underdamped Langevin dynamics as a sampling tool is given by taking the gradient of the associated asymptotic variance with respect to friction. We give an expression for this gradient in terms of the solution to an appropriate Poisson equation and show that it can be approximated by short simulations of the associated first variation/tangent process under concavity assumptions on the log density. Our algorithm is applied to the estimation of posterior means in Bayesian inference problems and reduced variance is demonstrated when compared to the original underdamped and overdamped Langevin dynamics in both full and stochastic gradient cases

Approaching a parameter-free metadynamics

We present a unique derivation of metadynamics. The starting point for the derivation is an on-the-fly reweighting scheme but through an approximation we recover the standard metadynamics and the well-tempered metadynamics in a general form while never appealing to the extended Lagrangian framework. This work leads to a more robust understanding of the error in the computed free energy than what has been obtained previously. Moreover, a formula for the exact free energy is introduced. The formula can be used to post-process any existing well-tempered metadynamics data allowing one, in principle, to obtain an exact free energy regardless the metadynamics parameters.Comment: 4 pages, 1 figur