arXiv subjects

Tony Lelièvre

Publications and source records attributed to Tony Lelièvre.

At least 37 records · Page 2Linked to original sources

arXiv2023

Estimation of statistics of transitions and Hill relation for Langevin dynamics

In molecular dynamics, statistics of transitions, such as the mean transition time, are macroscopic observables which provide important dynamical information on the underlying microscopic stochastic process. A direct estimation using simulations of microscopic trajectories over long time scales is typically computationally intractable in metastable situations. To overcome this issue, several numerical methods rely on a potential-theoretic identity, sometimes attributed to Hill in the computational statistical physics litterature, which expresses statistics of transitions in terms of the invariant measure of the sequence of configurations by which the underlying process enters metastable sets. The use of this identity then allows to replace the long time simulation problem with a rare event sampling problem, for which efficient algorithms are available. In this article, we rigorously analyse such a method for molecular systems modelled by the Langevin dynamics. Our main contributions are twofold. First, we prove the Hill relation in the fairly general context of positive Harris recurrent chains, and show that this formula applies to the Langevin dynamics. Second, we provide an explicit expression of the invariant measure involved in the Hill relation, and describe an elementary exact simulation procedure. Overall, this yields a simple and complete numerical method to estimate statistics of transitions.

math.PR↗

arXiv2023

Computing Surface Reaction Rates by Adaptive Multilevel Splitting Combined with Machine Learning and Ab Initio Molecular Dynamics

Computing accurate rate constants for catalytic events occurring at the surface of a given material represents a challenging task with multiple potential applications in chemistry. To address this question, we propose an approach based on a combination of the rare event sampling method called Adaptive Multilevel Splitting (AMS) and ab initio molecular dynamics (AIMD). The AMS method requires a one dimensional reaction coordinate to index the progress of the transition. Identifying a good reaction coordinate is difficult, especially for high dimensional problems such a those encountered in catalysis. We probe various approaches to build reaction coordinates such as Support Vector Machine and path collective variables. The AMS is implemented so as to communicate with a DFT-plane wave code. A relevant case study in catalysis: the change of conformation and the dissociation of a water molecule chemisorbed on the (100) $γ$-alumina surface is used to evaluate our approach. The calculated rate constants and transition mechanisms are discussed and compared to those obtained by a conventional static approach based on the Eyring-Polanyi equation with harmonic approximation. It is revealed that the AMS method may provide rate constants which are smaller than the static approach by up to two orders of magnitude due to entropic effects involved in the chemisorbed water.

physics.chem-ph↗

arXiv2023

Generative methods for sampling transition paths in molecular dynamics

Molecular systems often remain trapped for long times around some local minimum of the potential energy function, before switching to another one -- a behavior known as metastability. Simulating transition paths linking one metastable state to another one is difficult by direct numerical methods. In view of the promises of machine learning techniques, we explore in this work two approaches to more efficiently generate transition paths: sampling methods based on generative models such as variational autoencoders, and importance sampling methods based on reinforcement learning.

stat.ML↗

arXiv2022

On the Hill relation and the mean reaction time for metastable processes

We illustrate how the Hill relation and the notion of quasi-stationary distribution can be used to analyse the biasing error introduced by many numerical procedures that have been proposed in the literature, in particular in molecular dynamics, to compute mean reaction times between metastable states for Markov processes. The theoretical findings are illustrated on various examples demonstrating the sharpness of the biasing error analysis as well as the applicability of our study to elliptic diffusions.

math.PR↗

arXiv2022

Enhanced sampling methods for molecular dynamics simulations

Enhanced sampling algorithms have emerged as powerful methods to extend the utility of molecular dynamics simulations and allow the sampling of larger portions of the configuration space of complex systems in a given amount of simulation time. This review aims to present the unifying principles and differences of many of the computational methods currenly used for enhanced sampling in molecular simulations of biomolecules, soft matter and molecular crystals. Indeed, despite the apparent abundance and divergence of such methods, the principles at their core can be boiled down to a relatively limited number of statistical and physical principles. To enable comparisons, the various methods are introduced using similar terminology and notation. We then illustrate in which ways many different methods combine principles from a smaller class of enhanced sampling concepts. This review is intended for scientists with an understanding of the basics of molecular dynamics simulations and statistical physics who want a deeper understanding of the ideas that underlie various enhanced sampling methods and the relationships between them. This living review is intended to be updated to continue to reflect the wealth of sampling methods as they continue to emerge in the literature.

cond-mat.stat-mech↗

arXiv2022

Eyring-Kramers exit rates for the overdamped Langevin dynamics: the case with saddle points on the boundary

Let $(X_t)_{t\ge 0}$ be the stochastic process solution to the overdamped Langevin dynamics $$dX_t=-\nabla f(X_t) \, dt +\sqrt h \, dB_t$$ and let $Ω\subset \mathbb R^d $ be the basin of attraction of a local minimum of $f: \mathbb R^d \to \mathbb R$. Up to a small perturbation of $Ω$ to make it smooth, we prove that the exit rates of $(X_t)_{t\ge 0}$ from $Ω$ through each of the saddle points of $f$ on $\partial Ω$ can be parametrized by the celebrated Eyring-Kramers laws, in the limit $h \to 0$. This result provides firm mathematical grounds to jump Markov models which are used to model the evolution of molecular systems, as well as to some numerical methods which use these underlying jump Markov models to efficiently sample metastable trajectories of the overdamped Langevin dynamics.

math.PR↗

arXiv2022

A probabilistic study of the kinetic Fokker-Planck equation in cylindrical domains

We consider classical solutions to the kinetic Fokker-Planck equation on a bounded domain $\mathcal O \subset~\mathbb{R}^d$ in position, and we obtain a probabilistic representation of the solutions using the Langevin diffusion process with absorbing boundary conditions on the boundary of the phase-space cylindrical domain $D = \mathcal O \times \mathbb{R}^d$. Furthermore, a Harnack inequality, as well as a maximum principle, is provided on $D$ for solutions to this kinetic Fokker-Planck equation, together with the existence of a smooth transition density for the associated absorbed Langevin process. This transition density is shown to satisfy an explicit Gaussian upper-bound. Finally, the continuity and positivity of this transition density at the boundary of $D$ is also studied. All these results are in particular crucial to study the behavior of the Langevin diffusion process when it is trapped in a metastable state defined in terms of positions.

math.PR↗

arXiv2021

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.

stat.CO↗

arXiv2021

Chasing Collective Variables using Autoencoders and biased trajectories

Free energy biasing methods have proven to be powerful tools to accelerate the simulation of important conformational changes of molecules by modifying the sampling measure. However, most of these methods rely on the prior knowledge of low-dimensional slow degrees of freedom, i.e. Collective Variables (CV). Alternatively, such CVs can be identified using machine learning (ML) and dimensionality reduction algorithms. In this context, approaches where the CVs are learned in an iterative way using adaptive biasing have been proposed: at each iteration, the learned CV is used to perform free energy adaptive biasing to generate new data and learn a new CV. In this paper, we introduce a new iterative method involving CV learning with autoencoders: Free Energy Biasing and Iterative Learning with AutoEncoders (FEBILAE). Our method includes a reweighting scheme to ensure that the learning model optimizes the same loss at each iteration, and achieves CV convergence. Using the alanine dipeptide system and the solvated chignolin mini-protein system as examples, we present results of our algorithm using the extended adaptive biasing force as the free energy adaptive biasing method.

physics.bio-ph↗

arXiv2021

Multiple projection MCMC algorithms on submanifolds

We propose new Markov Chain Monte Carlo algorithms to sample probability distributions on submanifolds, which generalize previous methods by allowing the use of set-valued maps in the proposal step of the MCMC algorithms. The motivation for this generalization is that the numerical solvers used to project proposed moves to the submanifold of interest may find several solutions. We show that the new algorithms indeed sample the target probability measure correctly, thanks to some carefully enforced reversibility property. We demonstrate the interest of the new MCMC algorithms on illustrative numerical examples.

math.NA↗

arXiv2021

Quasi-stationary distribution for the Langevin process in cylindrical domains, part I: existence, uniqueness and long-time convergence

Consider the Langevin process, described by a vector (position,momentum) in $\mathbb{R}^{d}\times\mathbb{R}^d$. Let $\mathcal O$ be a $\mathcal{C}^2$ open bounded and connected set of $\mathbb{R}^d$. We prove the compactness of the semigroup of the Langevin process absorbed at the boundary of the domain $D:=\mathcal{O}\times\mathbb{R}^d$. We then obtain the existence of a unique quasi-stationary distribution (QSD) for the Langevin process on $D$. We also provide a spectral interpretation of this QSD and obtain an exponential convergence of the Langevin process conditioned on non-absorption towards the QSD.

math.PR↗

arXiv2021

Influence of sampling on the convergence rates of greedy algorithms for parameter-dependent random variables

The main focus of this article is to provide a mathematical study of the algorithm proposed in \cite{boyaval2010variance} where the authors proposed a variance reduction technique for the computation of parameter-dependent expectations using a reduced basis paradigm. We study the effect of Monte-Carlo sampling on the theoretical properties of greedy algorithms. In particular, using concentration inequalities for the empirical measure in Wasserstein distance proved in \cite{fournier2015rate}, we provide sufficient conditions on the number of samples used for the computation of empirical variances at each iteration of the greedy procedure to guarantee that the resulting method algorithm is a weak greedy algorithm with high probability. These theoretical results are not fully practical and we therefore propose a heuristic procedure to choose the number of Monte-Carlo samples at each iteration, inspired from this theoretical study, which provides satisfactory results on several numerical test cases.

math.NA↗

arXiv2021

The Adaptive Biasing Force algorithm with non-conservative forces and related topics

We propose a study of the Adaptive Biasing Force method's robustness under generic (possibly non-conservative) forces. We first ensure the flat histogram property is satisfied in all cases. We then introduce a fixed point problem yielding the existence of a stationary state for both the Adaptive Biasing Force and Projected Adapted Biasing Force algorithms, relying on generic bounds on the invariant probability measures of homogeneous diffusions. Using classical entropy techniques, we prove the exponential convergence of both biasing force and law as time goes to infinity, for both the Adaptive Biasing Force and the Projected Adaptive Biasing Force methods.

math.AP↗

arXiv2020

The exit from a metastable state: concentration of the exit point distribution on the low energy saddle points, part 2

We consider the first exit point distribution from a bounded domain $Ω$ of the stochastic process $(X_t)_{t\ge 0}$ solution to the overdamped Langevin dynamics $$d X_t = -\nabla f(X_t) d t + \sqrt{h} \ d B_t$$ starting from deterministic initial conditions in $Ω$, under rather general assumptions on $f$ (for instance, $f$ may have several critical points in $Ω$). This work is a continuation of the previous paper \cite{DLLN-saddle1} where the exit point distribution from $Ω$ is studied when $X_0$ is initially distributed according to the quasi-stationary distribution of $(X_t)_{t\ge 0}$ in $Ω$. The proofs are based on analytical results on the dependency of the exit point distribution on the initial condition, large deviation techniques and results on the genericity of Morse functions.

math.AP↗

arXiv2020

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

A modification of the Adaptive Biasing Force method is introduced, in which the free energy is approximated by a sum of tensor products of one-dimensional functions. This enables to handle a larger number of reaction coordinates than the classical algorithm. We prove the algorithm is well-defined and prove the long-time convergence toward a regularized version of the free energy for an idealized version of the algorithm. Numerical experiments demonstrate that the method is able to capture correlations between reaction coordinates.

math.PR↗

arXiv2019

Parareal computation of stochastic differential equations with time-scale separation: a numerical study

The parareal algorithm is known to allow for a significant reduction in wall clock time for accurate numerical solutions by parallelising across the time dimension. We present and test a micro-macro version of parareal, in which the fine propagator is based on a (high-dimensional, slow-fast) stochastic microscopic model, and the coarse propagator is based on a low-dimensional approximate effective dynamics at slow time scales. At the microscopic level, we use an ensemble of Monte Carlo particles, whereas the approximate coarse propagator uses the (deterministic) Fokker-Planck equation for the slow degrees of freedom. The required coupling between microscopic and macroscopic representations of the system introduces several design options, specifically on how to generate a microscopic probability distribution consistent with a required macroscopic probability distribution and how to perform the coarse-level updating of the macroscopic probability distribution in a meaningful manner. We numerically study how these design options affect the efficiency of the algorithm in a number of situations. The choice of the coarse-level updating operator strongly impacts the result, with a superior performance if addition and subtraction of the quantile function (inverse cumulative distribution) is used. How microscopic states are generated has a less pronounced impact, provided a suitable prior microscopic state is used.

math.NA↗

arXiv2019

Local and Global Perspectives on Diffusion Maps in the Analysis of Molecular Systems

Diffusion maps approximate the generator of Langevin dynamics from simulation data. They afford a means of identifying the slowly-evolving principal modes of high-dimensional molecular systems. When combined with a biasing mechanism, diffusion maps can accelerate the sampling of the stationary Boltzmann-Gibbs distribution. In this work, we contrast the local and global perspectives on diffusion maps, based on whether or not the data distribution has been fully explored. In the global setting, we use diffusion maps to identify metastable sets and to approximate the corresponding committor functions of transitions between them. We also discuss the use of diffusion maps within the metastable sets, formalising the locality via the concept of the quasi-stationary distribution and justifying the convergence of diffusion maps within a local equilibrium. This perspective allows us to propose an enhanced sampling algorithm. We demonstrate the practical relevance of these approaches both for simple models and for molecular dynamics problems (alanine dipeptide and deca-alanine).

physics.data-an↗

arXiv2019

Statistical Estimation of the Poincar{é} constant and Application to Sampling Multimodal Distributions

Poincar{é} inequalities are ubiquitous in probability and analysis and have various applications in statistics (concentration of measure, rate of convergence of Markov chains). The Poincar{é} constant, for which the inequality is tight, is related to the typical convergence rate of diffusions to their equilibrium measure. In this paper, we show both theoretically and experimentally that, given sufficiently many samples of a measure, we can estimate its Poincar{é} constant. As a by-product of the estimation of the Poincar{é} constant, we derive an algorithm that captures a low dimensional representation of the data by finding directions which are difficult to sample. These directions are of crucial importance for sampling or in fields like molecular dynamics, where they are called reaction coordinates. Their knowledge can leverage, with a simple conditioning step, computational bottlenecks by using importance sampling techniques.

math.PR↗