arXiv subjects

Gabriel Stoltz

Publications and source records attributed to Gabriel Stoltz.

At least 55 records · Page 3Linked to original sources

arXiv2021

Computational statistical physics and hypocoercivity

This note provides an introduction to molecular dynamics, the computational implementation of the theory of statistical physics. The discussion is focused on the properties of Langevin dynamics, a degenerate stochastic differential equation which can be seen as a perturbation of Hamiltonian dynamics. From an analytical point of view, the generator of Langevin dynamics is a degenerate elliptic operator. The evolution of the law of the stochastic process is governed by the Fokker-Planck equation, and its longtime convergence can be obtained via hypocoercive techniques, some of which are reviewed here. One consequence of these analytical results in terms of error estimates for the computation of average properties of molecular systems is the estimation of the asymptotic variance of time averages in a central limit theorem.

math.AP↗

arXiv2021

Weighted $L^2$-contractivity of Langevin dynamics with singular potentials

Convergence to equilibrium of underdamped Langevin dynamics is studied under general assumptions on the potential $U$ allowing for singularities. By modifying the direct approach to convergence in $L^2$ pioneered by F. Hérau and developped by Dolbeault, Mouhot and Schmeiser, we show that the dynamics converges exponentially fast to equilibrium in the topologies $L^2(dμ)$ and $L^2(W^* dμ)$, where $μ$ denotes the invariant probability measure and $W^*$ is a suitable Lyapunov weight. In both norms, we make precise how the exponential convergence rate depends on the friction parameter $γ$ in Langevin dynamics, by providing a lower bound scaling as $\min(γ, γ^{-1})$. The results hold for usual polynomial-type potentials as well as potentials with singularities such as those arising from pairwise Lennard-Jones interactions between particles.

math.PR↗

arXiv2021

Martingale product estimators for sensitivity analysis in computational statistical physics

We introduce a new class of estimators for the linear response of steady states of stochastic dynamics. We generalize the likelihood ratio approach and formulate the linear response as a product of two martingales, hence the name "martingale product estimators". We present a systematic derivation of the martingale product estimator, and show how to construct such estimator so its bias is consistent with the weak order of the numerical scheme that approximates the underlying stochastic differential equation. Motivated by the estimation of transport properties in molecular systems, we present a rigorous numerical analysis of the bias and variance for these new estimators in the case of Langevin dynamics. We prove that the variance is uniformly bounded in time and derive a specific form of the estimator for second-order splitting schemes for Langevin dynamics. For comparison, we also study the bias and variance of a Green-Kubo estimator, motivated, in part, by its variance growing linearly in time. Presented analysis shows that the new martingale product estimators, having uniformly bounded variance in time, offer a competitive alternative to the traditional Green-Kubo estimator. We compare on illustrative numerical tests the new estimators with results obtained by the Green-Kubo method.

math.NA↗

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

Thermo-mechanical transport in rotor chains

We study the macroscopic profiles of temperature and angular momentum in the stationary state of chains of rotors under a thermo-mechanical forcing applied at the boundaries. These profiles are solutions of a system of diffusive partial differential equations with boundary conditions determined by the thermo-mechanical forcing. Instead of expensive Monte Carlo simulations of the underlying microscopic dynamics, we perform extensive numerical computations based on a finite difference method for the system of partial differential equations describing the macroscopic steady state. We first present a formal derivation of these stationary equations based on a linear response argument and local equilibrium assumptions. We then study various properties of the solutions to these equations. This allows to characterize the regime of parameters leading to uphill energy diffusion -- a situation in which the energy flows in the direction of the gradient of temperature -- and to identify regions of parameters corresponding to a negative energy conductivity (i.e. a positive linear response of the energy current to a gradient of temperature). The macroscopic equations we derive are consistent with some previous results obtained by numerical simulation of the microscopic physical system, which confirms their validity.

cond-mat.stat-mech↗

arXiv2020

Coulomb gases under constraint: some theoretical and numerical results

We consider Coulomb gas models for which the empirical measure typically concentrates, when the number of particles becomes large, on an equilibrium measure minimizing an electrostatic energy. We study the behavior when the gas is conditioned on a rare event. We first show that the special case of quadratic confinement and linear constraint is exactly solvable due to a remarkable factorization, and that the conditioning has then the simple effect of shifting the cloud of particles without deformation. To address more general cases, we perform a theoretical asymptotic analysis relying on a large deviations technique known as the Gibbs conditioning principle. The technical part amounts to establishing that the conditioning ensemble is an I-continuity set of the energy. This leads to characterizing the conditioned equilibrium measure as the solution of a modified variational problem. For simplicity, we focus on linear statistics and on quadratic statistics constraints. Finally, we numerically illustrate our predictions and explore cases in which no explicit solution is known. For this, we use a Generalized Hybrid Monte Carlo algorithm for sampling from the conditioned distribution for a finite but large system.

math.PR↗

arXiv2020

Large deviations of empirical measures of diffusions in weighted topologies

We consider large deviations of empirical measures of diffusion processes. In a first part, we present conditions to obtain a large deviations principle (LDP) for a precise class of unbounded functions. This provides an analogue to the standard Cramér condition in the context of diffusion processes, which turns out to be related to a spectral gap condition for a Witten-Schrödinger operator. Secondly, we study more precisely the properties of the Donsker-Varadhan rate functional associated with the LDP. We revisit and generalize some standard duality results as well as a more original decomposition of the rate functional with respect to the symmetric and antisymmetric parts of the dynamics. Finally, we apply our results to overdamped and underdamped Langevin dynamics, showing the applicability of our framework for degenerate diffusions in unbounded configuration spaces.

math.PR↗

arXiv2020

More on the long time stability of Feynman-Kac semigroups

Feynman-Kac semigroups appear in various areas of mathematics: non-linear filtering, large deviations theory, spectral analysis of Schrodinger operators among others. Their long time behavior provides important information, for example in terms of ground state energy of Schrodinger operators, or scaled cumulant generating function in large deviations theory. In this paper, we propose a simple and natural extension of the stability of Markov chains for these non-linear evolutions. As other classical ergodicity results, it relies on two assumptions: a Lyapunov condition that induces some compactness, and a minorization condition ensuring some mixing. Illustrative examples are provided, where the stability of the non-linear semigroup arises either from the underlying dynamics or from the Feynman-Kac weight function. We also use our technique to provide uniform in the time step convergence estimates for discretizations of stochastic differential equations

math.PR↗

arXiv2020

Machine learning force fields and coarse-grained variables in molecular dynamics: application to materials and biological systems

Machine learning encompasses a set of tools and algorithms which are now becoming popular in almost all scientific and technological fields. This is true for molecular dynamics as well, where machine learning offers promises of extracting valuable information from the enormous amounts of data generated by simulation of complex systems. We provide here a review of our current understanding of goals, benefits, and limitations of machine learning techniques for computational studies on atomistic systems, focusing on the construction of empirical force fields from ab-initio databases and the determination of reaction coordinates for free energy computation and enhanced sampling.

physics.comp-ph↗

arXiv2019

Convergence of the likelihood ratio method for linear response of non-equilibrium stationary states

We consider numerical schemes for computing the linear response of steady-state averages of stochastic dynamics with respect to a perturbation of the drift part of the stochastic differential equation. The schemes are based on Girsanov's change-of-measure theory to reweight trajectories with factors derived from a linearization of the Girsanov weights. We investigate both the discretization error and the finite time approximation error. The designed numerical schemes are shown to be of bounded variance with respect to the integration time, which is a desirable feature for long time simulation. We also show how the discretization error can be improved to second order accuracy in the time step by modifying the weight process in an appropriate way.

math.NA↗

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↗

arXiv2019

Hybrid Monte Carlo methods for sampling probability measures on submanifolds

Probability measures supported on submanifolds can be sampled by adding an extra momentum variable to the state of the system, and discretizing the associated Hamiltonian dynamics with some stochastic perturbation in the extra variable. In order to avoid biases in the invariant probability measures sampled by discretizations of these stochastically perturbed Hamiltonian dynamics, a Metropolis rejection procedure can be considered. The so-obtained scheme belongs to the class of generalized Hybrid Monte Carlo (GHMC) algorithms. We show here how to generalize to GHMC a procedure suggested by Goodman, Holmes-Cerfon and Zappa for Metropolis random walks on submanifolds, where a reverse projection check is performed to enforce the reversibility of the algorithm for large timesteps and hence avoid biases in the invariant measure. We also provide a full mathematical analysis of such procedures, as well as numerical experiments demonstrating the importance of the reverse projection check on simple toy examples.

math.NA↗

arXiv2019

A reduced Hartree-Fock model of slice-like defects in the Fermi sea

Studying the electronic structure of defects in materials is an important subject in condensed matter physics. From a mathematical point of view, nonlinear mean-field models of localized defects in insulators are well understood. We present here a mean-field model to study a particular instance of extended defects in metals. These extended defects typically correspond to taking out a slab of finite width in the three-dimensional homogeneous electron gas. We work in the framework of the reduced Hartree-Fock model with either Yukawa or Coulomb interactions. Using techniques developed in~[Frank2011, Frank2013] to study local perturbations of the free-electron gas, we show that our model admits minimizers, and that Yukawa ground state energies and density matrices converge to ground state Coulomb energies and density matrices as the Yukawa parameter tends to zero. We moreover present numerical simulations where we observe Friedel oscillations in the total electronic density.

math-ph↗

arXiv2019

Error estimates on ergodic properties of discretized Feynman-Kac semigroups

We consider the numerical analysis of the time discretization of Feynman-Kac semigroups associated with diffusion processes. These semigroups naturally appear in several fields, such as large deviation theory, Diffusion Monte Carlo or non-linear filtering. We present errors estimates a la Talay-Tubaro on their invariant measures when the underlying continuous stochastic differential equation is discretized; as well as on the leading eigenvalue of the generator of the dynamics, which corresponds to the rate of creation of probability. This provides criteria to construct efficient integration schemes of Feynman-Kac dynamics, as well as a mathematical justification of numerical results already observed in the Diffusion Monte Carlo community. Our analysis is illustrated by numerical simulations.

math.NA↗

arXiv2019

A mathematical justification of the finite time approximation of Becker-Döring equations by a Fokker-Planck dynamics

The Becker-Döring equations are an infinite dimensional system of ordinary differntial equations describing coagulation/fragmentation processes of species of integer sizes. Formal Taylor expansions motivate that its solution should be well described by a partial differential equation for large sizes, of advection-diffusion type, called Fokker-Planck equation. We rigorously prove the link between these two descriptions for evolutions on finite times rather than in some hydrodynamic limit, motivated by the results of numerical simulations and the construction of dedicated algorithms based on splitting strategies. In fact, the Becker-Döring equations and the Fokker-Planck equation are related through some pure diffusion with unbounded diffusion coefficient. The crucial point in the analysis is to obtain decay estimates for the solution of this pure diffusion and its derivates to control remainders in the Taylor expansions. The small parameter in this analysis is the inverse of the minimal size of the species.

math.CA↗

arXiv2019

A perturbative approach to control variates in molecular dynamics

We propose a general variance reduction strategy for diffusion processes. Our approach does not require the knowledge of the measure that is sampled, which may indeed be unknown as for nonequilibrium dynamics in statistical physics. We show by a perturbative argument that a control variate computed for a simplified version of the model can provide an efficient control variate for the actual problem at hand. We illustrate our method with numerical experiments and show how the control variate is built in three practical cases: the computation of the mobility of a particle in a periodic potential; the thermal flux in atom chains, relying on a harmonic approximation; and the mean length of a dimer in a solvent under shear, using a non-solvated dimer as the approximation.

math.NA↗

arXiv2018

Langevin dynamics with general kinetic energies

We study Langevin dynamics with a kinetic energy different from the standard, quadratic one in order to accelerate the sampling of Boltzmann-Gibbs distributions. In particular, this kinetic energy can be non-globally Lipschitz, which raises issues for the stability of discretizations of the associated Langevin dynamics. We first prove the exponential convergence of the law of the continuous process to the Boltzmann-Gibbs measure by a hypocoercive approach, and characterize the asymptotic variance of empirical averages over trajectories. We next develop numerical schemes which are stable and of weak order two, by considering splitting strategies where the discretizations of the fluctuation/dissipation are corrected by a Metropolis procedure. We use the newly developped schemes for two applications: optimizing the shape of the kinetic energy for the so-called adaptively restrained Langevin dynamics (which considers perturbations of standard quadratic kinetic energies vanishing around the origin); and reducing the metastability of some toy models using non-globally Lipschitz kinetic energies.

cond-mat.stat-mech↗