Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
160
result(s) for
"Lelièvre, Tony"
Sort by:
Partial differential equations and stochastic methods in molecular dynamics
The objective of molecular dynamics computations is to infer macroscopic properties
of matter from atomistic models via averages with respect to probability measures
dictated by the principles of statistical physics. Obtaining accurate results
requires efficient sampling of atomistic configurations, which are typically
generated using very long trajectories of stochastic differential equations in high
dimensions, such as Langevin dynamics and its overdamped limit. Depending on the
quantities of interest at the macroscopic level, one may also be interested in
dynamical properties computed from averages over paths of these dynamics. This review describes how techniques from the analysis of partial differential
equations can be used to devise good algorithms and to quantify their efficiency and
accuracy. In particular, a crucial role is played by the study of the long-time
behaviour of the solution to the Fokker–Planck equation associated with
the stochastic dynamics.
Journal Article
Free energy computations
This monograph provides a general introduction to advanced computational methods for free energy calculations, from the systematic and rigorous point of view of applied mathematics. Free energy calculations in molecular dynamics have become an outstanding and increasingly broad computational field in physics, chemistry and molecular biology within the past few years, by making possible the analysis of complex molecular systems. This work proposes a new, general and rigorous presentation, intended both for practitioners interested in a mathematical treatment, and for applied mathematicians interested in molecular dynamics.
eBook
Mathematical methods for the magnetohydrodynamics of liquid metals
This text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis on the magnetohydrodynamics of liquid metals, on two-fluid flows, and on a prototypical industrial application. The approach is a highly mathematical one, based on the rigorous analysis of the equations at hand, and a solid numerical analysis of the discretization methods. Up-to-date techniques, both on the theoretical side and the numerical side, are introduced to deal with the nonlinearities of the multifluid magnetohydrodynamics equations. At each stage of the exposition, examples of numerical simulations are provided, first on academic test cases to illustrate the approach, next on benchmarks well documented in the professional literature, and finally on real industrial cases. The simulation of aluminium electrolysis cells is used as a guideline throughout the book to motivate the study of a particular setting of the magnetohydrodynamics equations.
eBook
Langevin dynamics with constraints and computation of free energy differences
In this paper, we consider Langevin processes with mechanical constraints. The latter are a fundamental tool in molecular dynamics simulation for sampling purposes and for the computation of free energy differences. The results of this paper can be divided into three parts. (i) We propose a simple discretization of the constrained Langevin process based on a splitting strategy. We show how to correct the scheme so that it samples exactly the canonical measure restricted on a submanifold, using a Metropolis-Hastings correction in the spirit of the Generalized Hybrid Monte Carlo (GHMC) algorithm. Moreover, we obtain, in some limiting regime, a consistent discretization of the overdamped Langevin (Brownian) dynamics on a submanifold, also sampling exactly the correct canonical measure with constraints. (ii) For free energy computation using thermodynamic integration, we rigorously prove that the longtime average of the Lagrange multipliers of the constrained Langevin dynamics yields the gradient of a rigid version of the free energy associated with the constraints. A second order time discretization using the Lagrange multipliers is proposed. (iii) The Jarzynski-Crooks fluctuation relation is proved for Langevin processes with mechanical constraints evolving in time. An original numerical discretization without time discretization error is proposed, and its overdamped limit is studied. Numerical illustrations are provided for (ii) and (iii).
Journal Article
OPTIMAL SCALING FOR THE TRANSIENT PHASE OF THE RANDOM WALK METROPOLIS ALGORITHM: THE MEAN-FIELD LIMIT
We consider the random walk Metropolis algorithm on ℝn with Gaussian proposals, and when the target probability measure is the n-fold product of a one-dimensional law. In the limit n → ∞, it is well known (see [Ann. Appl. Probab. 7 (1997) 110–120]) that, when the variance of the proposal scales inversely proportional to the dimension n whereas time is accelerated by the factor n, a diffusive limit is obtained for each component of the Markov chain if this chain starts at equilibrium. This paper extends this result when the initial distribution is not the target probability measure. Remarking that the interaction between the components of the chain due to the common acceptance/rejection of the proposed moves is of mean-field type, we obtain a propagation of chaos result under the same scaling as in the stationary case. This proves that, in terms of the dimension n, the same scaling holds for the transient phase of the Metropolis–Hastings algorithm as near stationarity. The diffusive and mean-field limit of each component is a diffusion process nonlinear in the sense of McKean. This opens the route to new investigations of the optimal choice for the variance of the proposal distribution in order to accelerate convergence to equilibrium (see [Optimal scaling for the transient phase of Metropolis–Hastings algorithms: The longtime behavior Bernoulli (2014) To appear]).
Journal Article
CONVERGENCE OF METADYNAMICS
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 hold.
Journal Article
UNBIASEDNESS OF SOME GENERALIZED ADAPTIVE MULTILEVEL SPLITTING ALGORITHMS
We introduce a generalization of the Adaptive Multilevel Splitting algorithm in the discrete time dynamic setting, namely when it is applied to sample rare events associated with paths of Markov chains. We build an estimator of the rare event probability (and of any nonnormalized quantity associated with this event) which is unbiased, whatever the choice of the importance function and the number of replicas. This has practical consequences on the use of this algorithm, which are illustrated through various numerical experiments.
Journal Article
Convergence of the Wang-Landau algorithm
We analyze the convergence properties of the Wang-Landau algorithm. This sampling method belongs to the general class of adaptive importance sampling strategies which use the free energy along a chosen reaction coordinate as a bias. Such algorithms are very helpful to enhance the sampling properties of Markov Chain Monte Carlo algorithms, when the dynamics is metastable. We prove the convergence of the Wang-Landau algorithm and an associated central limit theorem.
Journal Article
Sharp Asymptotics of the First Exit Point Density
We consider the exit event from a metastable state for the overdamped Langevin dynamics
d
X
t
=
-
∇
f
(
X
t
)
d
t
+
h
d
B
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
→
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.
Journal Article