arXiv subjects

Tony Lelièvre

Publications and source records attributed to Tony Lelièvre.

At least 55 records · Page 3Linked to original sources

arXiv2019

Stochastic homogenization of a scalar viscoelastic model exhibiting stress-strain hysteresis

Motivated by rate-independent stress-strain hysteresis observed in filled rubber, this article considers a scalar viscoelastic model in which the constitutive law is random and varies on a lengthscale which is small relative to the overall size of the solid. Using stochastic two-scale convergence as introduced by Bourgeat, Mikelic and Wright, we obtain the homogenized limit of the evolution, and demonstrate that under certain hypotheses, the homogenized model exhibits hysteretic behaviour which persists under asymptotically slow loading. These results are illustrated by means of numerical simulations in a particular one-dimensional instance of the model.

math.AP↗

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

Convergence of metadynamics: discussion of the adiabatic hypothesis

By drawing a parallel between metadynamics and self interacting models for polymers, we study the longtime convergence of the original metadynamics algorithm in the adiabatic setting, namely when the dynamics along the collective variables decouples from the dynamics along the other degrees of freedom. We also discuss the bias which is introduced when the adiabatic assumption does not holds.

math.PR↗

arXiv2019

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

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 the quasi-stationary distribution in $Ω$. In the small temperature regime ($h\to 0$) and under rather general assumptions on $f$ (in particular, $f$ may have several critical points in $Ω$), it is proven that the support of the distribution of the first exit point concentrates on some points realizing the minimum of $f$ on $\partial Ω$. The proof relies on tools to study tunnelling effects in semi-classical analysis. Extensions of the results to more general initial distributions than the quasi-stationary distribution are also presented.

math.AP↗

arXiv2019

Sharp asymptotics of the first exit point density

We consider the exit event from a metastable state for the overdamped Langevin dynamics $dX_t = -\nabla f(X_t) dt + \sqrt{h} dB_t$. Using tools from semiclassical analysis, we prove that, starting from the quasi stationary distribution within the state, the exit event can be modeled using a jump Markov process parametrized with the Eyring-Kramers formula, in the small temperature regime $h \to 0$. We provide in particular sharp asymptotic estimates on the exit distribution which demonstrate the importance of the prefactors in the Eyring-Kramers formula. Numerical experiments indicate that the geometric assumptions we need to perform our analysis are likely to be necessary. These results also hold starting from deterministic initial conditions within the well which are sufficiently low in energy. From a modelling viewpoint, this gives a rigorous justification of the transition state theory and the Eyring-Kramers formula, which are used to relate the overdamped Langevin dynamics (a continuous state space Markov dynamics) to kinetic Monte Carlo or Markov state models (discrete state space Markov dynamics). From a theoretical viewpoint, our analysis paves a new route to study the exit event from a metastable state for a stochastic process.

math.AP↗

arXiv2018

gen.parRep: a first implementation of the Generalized Parallel Replica dynamics for the long time simulation of metastable biochemical systems

Metastability is one of the major encountered obstacle when performing long molecular dynamics simulations, and many methods were developed to address this challenge. The "Parallel Replica" (ParRep) dynamics is known for allowing to simulate very long trajectories of metastable Langevin dynamics in the materials science community, but it relies on assumptions that can hardly be transposed to the world of biochemical simulations. The later developed "Generalized ParRep" variant solves those issues, but it was not applied to significant systems of interest so far. In this article, we present the program gen.parRep, the first publicly available implementation of the Generalized Parallel Replica method (BSD 3-Clause license), targeting frequently encountered metastable biochemical systems, such as conformational equilibria or dissociation of protein-ligand complexes. It will be shown that the resulting C++ implementation exhibits a strong linear scalability, providing up to 70 % of the maximum possible speedup on several hundreds of CPUs.

physics.chem-ph↗

arXiv2018

Exit event from a metastable state and Eyring-Kramers law for the overdamped Langevin dynamics

In molecular dynamics, several algorithms have been designed over the past few years to accelerate the exit event from a metastable region of the configuration space. Some of them are based on the fact that the exit event from a metastable region is well approximated by a Markov jump process. In this work, we present recent results on the exit event from a metastable region for the overdamped Langevin dynamics obtained in [17, 18, 49]. These results aim in particular at justifying the use of a Markov jump process parametrized by the Eyring-Kramers law to model the exit event from a metastable region.

math.AP↗

arXiv2018

On a new class of score functions to estimate tail probabilities of some stochastic processes with Adaptive Multilevel Splitting

We investigate the application of the Adaptive Multilevel Splitting algorithm for the estimation of tail probabilities of solutions of Stochastic Differential Equations evaluated at a given time, and of associated temporal averages. We introduce a new, very general and effective family of score functions which is designed for these problems. We illustrate its behavior on a series of numerical experiments. In particular, we demonstrate how it can be used to estimate large deviation rate functionals for the longtime limit of temporal averages.

math.PR↗

arXiv2018

Effective dynamics for non-reversible stochastic differential equations: a quantitative study

Coarse-graining is central to reducing dimensionality in molecular dynamics, and is typically characterized by a mapping which projects the full state of the system to a smaller class of variables. While extensive literature has been devoted to coarse-graining starting from reversible systems, not much is known in the non-reversible setting. In this article, starting with a non-reversible dynamics, we introduce and study an effective dynamics which approximates the (non-closed) projected dynamics. Under fairly weak conditions on the system, we prove error bounds on the trajectorial error between the projected and the effective dynamics. In addition to extending existing results to the non-reversible setting, our error estimates also indicate that the notion of mean force motivated by this effective dynamics is a good one.

math.PR↗

arXiv2018

Analysis of the Adaptive Multilevel Splitting method on the isomerization of alanine dipeptide

We apply the Adaptive Multilevel Splitting method to the Ceq -> Cax transition of alanine dipeptide in vacuum. Some properties of the algorithm are numerically illustrated, such as the unbiasedness of the probability estimator and the robustness of the method with respect to the choice of the reaction coordinate. We also calculate the transition time obtained via the probability estimator, using an appropriate ensemble of initial conditions. Finally, we show how the Adaptive Multilevel Splitting method can be used to compute an approximation of the committor function.

physics.chem-ph↗

arXiv2018

Pathwise estimates for effective dynamics: the case of nonlinear vectorial reaction coordinates

Effective dynamics using conditional expectation was proposed in [F. Legoll and T. Lelièvre, Nonlinearity, 2010] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [F. Legoll, T. Lelièvre, and S. Olla, Stoch. Process. Appl, 2017], in this paper we obtain pathwise estimates for effective dynamics when the reaction coordinate function is either nonlinear or vector-valued.

math.PR↗

arXiv2018

Mathematical foundations of Accelerated Molecular Dynamics methods

The objective of this review article is to present recent results on the mathematical analysis of the Accelerated Dynamics algorithms introduced by A.F. Voter in collaboration with D. Perez and M. Sorensen. Using the notion of quasi-stationary distribution, one is able to rigorously justify the fact that the exit event from a metastable state for the Langevin or overdamped Langevin dynamics can be modeled by a kinetic Monte Carlo model. Moreover, under some geometric assumptions, one can prove that this kinetic Monte Carlo model can be parameterized using Eyring-Kramers formulas. These are the building blocks required to analyze the Accelerated Dynamics algorithms, to understand their efficiency and their accuracy, and to improve and generalize these techniques beyond their original scope.

math.NA↗

arXiv2018

Analysis of a micro-macro acceleration method with minimum relative entropy moment matching

We analyse convergence of a micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations with time-scale separation between the (fast) evolution of individual trajectories and the (slow) evolution of the macroscopic function of interest. We consider a class of methods, presented in [Debrabant, K., Samaey, G., Zieliński, P. A micro-macro acceleration method for the Monte Carlo simulation of stochastic differential equations. SINUM, 55 (2017) no. 6, 2745-2786], that performs short bursts of path simulations, combined with the extrapolation of a few macroscopic state variables forward in time. After extrapolation, a new microscopic state is then constructed, consistent with the extrapolated variable and minimising the perturbation caused by the extrapolation. In the present paper, we study a specific method in which this perturbation is minimised in a relative entropy sense. We discuss why relative entropy is a useful metric, both from a theoretical and practical point of view, and rigorously study local errors and numerical stability of the resulting method as a function of the extrapolation time step and the number of macroscopic state variables. Using these results, we discuss convergence to the full microscopic dynamics, in the limit when the extrapolation time step tends to zero and the number of macroscopic state variables tends to infinity.

math.NA↗

arXiv2017

Convergence and efficiency of adaptive importance sampling techniques with partial biasing

We consider a generalization of the discrete-time Self Healing Umbrella Sampling method, which is an adaptive importance technique useful to sample multimodal target distributions. The importance function is based on the weights (namely the relative probabilities) of disjoint sets which form a partition of the space. These weights are unknown but are learnt on the fly yielding an adaptive algorithm. In the context of computational statistical physics, the logarithm of these weights is, up to a multiplicative constant, the free energy, and the discrete valued function defining the partition is called the collective variable. The algorithm falls into the general class of Wang-Landau type methods, and is a generalization of the original Self Healing Umbrella Sampling method in two ways: (i) the updating strategy leads to a larger penalization strength of already visited sets in order to escape more quickly from metastable states, and (ii) the target distribution is biased using only a fraction of the free energy, in order to increase the effective sample size and reduce the variance of importance sampling estimators. The algorithm can also be seen as a generalization of well-tempered metadynamics. We prove the convergence of the algorithm and analyze numerically its efficiency on a toy example.

math.PR↗

arXiv2016

Jump Markov models and transition state theory: the Quasi-Stationary Distribution approach

We are interested in the connection between a metastable continuous state space Markov process (satisfying e.g. the Langevin or overdamped Langevin equation) and a jump Markov process in a discrete state space. More precisely, we use the notion of quasi-stationary distribution within a metastable state for the continuous state space Markov process to parametrize the exit event from the state. This approach is useful to analyze and justify methods which use the jump Markov process underlying a metastable dynamics as a support to efficiently sample the state-to-state dynamics (accelerated dynamics techniques). Moreover, it is possible by this approach to quantify the error on the exit event when the parametrization of the jump Markov model is based on the Eyring-Kramers formula. This therefore provides a mathematical framework to justify the use of transition state theory and the Eyring-Kramers formula to build kinetic Monte Carlo or Markov state models.

math.PR↗

arXiv2015

Variants of the Empirical Interpolation Method: symmetric formulation, choice of norms and rectangular extension

The Empirical Interpolation Method (EIM) is a greedy procedure that constructs approximate representations of two-variable functions in separated form. In its classical presentation, the two variables play a non-symmetric role. In this work, we give an equivalent definition of the EIM approximation, in which the two variables play symmetric roles. Then, we give a proof for the existence of this approximation, and extend it up to the convergence of the EIM, and for any norm chosen to compute the error in the greedy step. Finally, we introduce a way to compute a separated representation in the case where the number of selected values is different for each variable. In the case of a physical field measured by sensors, this is useful to discard a broken sensor while keeping the information provided by the associated selected field.

math.NA↗

arXiv2015

Local and global solution for a nonlocal Fokker-Planck equation related to the adaptive biaising force processes

We prove global existence, uniqueness and regularity of the mild, Lp and classical solution of a non-linear Fokker-Planck equation arising in an adaptive importance sampling method for molecular dynamics calculations. The non- linear term is related to a conditional expectation, and is thus non-local. The proof uses tools from the theory of semigroups of linear operators for the local existence result, and an a priori estimate based on a supersolution for the global existence result.

math.AP↗

arXiv2015

Computation of sensitivities for the invariant measure of a parameter dependent diffusion

We consider the solution to a stochastic differential equation with a drift function which depends smoothly on some real parameter $λ$, and admitting a unique invariant measure for any value of $λ$ around $λ$ = 0. Our aim is to compute the derivative with respect to $λ$ of averages with respect to the invariant measure, at $λ$ = 0. We analyze a numerical method which consists in simulating the process at $λ$ = 0 together with its derivative with respect to $λ$ on long time horizon. We give sufficient conditions implying uniform-in-time square integrability of this derivative. This allows in particular to compute efficiently the derivative with respect to $λ$ of the mean of an observable through Monte Carlo simulations.

math.PR↗