Tony Lelièvre

Free energy computations : a mathematical perspective

Sampling Methods Stochastic Differential Equations Meta-Stability Free Energy Perturbation Thermodynamic Integration Constrained Dynamics Non-Equilibrium Methods Fluctuation Identities Jarzynski Identity Adaptive Techniques Long Time Convergence Replica Selection Methods Selection Mechanisms Parallel Computation.

Mathematical methods for the magnetohydrodynamics of liquid metals

Preface 1. The magnetohydrodynamics equations 2. Mathematical analysis of one-fluid problems 3. Numerical approximations of one-fluid problems 4. Mathematical analysis of two-fluid problems 5. Numerical simulation of two-fluid problem 6. MHD models for one industrial application References Index

Existence of solution for a micro–macro model of polymeric fluid: the FENE model

Abstract We analyse a non-linear micro–macro model of polymeric fluids in the case of a shear flow. More precisely, we consider the FENE dumbbell model, which models polymers by nonlinear springs, accounting for the finite extensibility of the polymer chain. We prove the existence of a unique solution to the stochastic differential equation which rules the evolution of a representative polymer in the flow and next deduce a local-in-time existence and uniqueness result on the system coupling the stochastic differential equation and the momentum equation on the fluid.

The adaptive biasing force method: everything you always wanted to know but were afraid to ask.

In the host of numerical schemes devised to calculate free energy differences by way of geometric transformations, the adaptive biasing force algorithm has emerged as a promising route to map complex free-energy landscapes. It relies upon the simple concept that as a simulation progresses, a continuously updated biasing force is added to the equations of motion, such that in the long-time limit it yields a Hamiltonian devoid of an average force acting along the transition coordinate of interest. This means that sampling proceeds uniformly on a flat free-energy surface, thus providing reliable free-energy estimates. Much of the appeal of the algorithm to the practitioner is in its physically intuitive underlying ideas and the absence of any requirements for prior knowledge about free-energy landscapes. Since its inception in 2001, the adaptive biasing force scheme has been the subject of considerable attention, from in-depth mathematical analysis of convergence properties to novel developments and extensions. The method has also been successfully applied to many challenging problems in chemistry and biology. In this contribution, the method is presented in a comprehensive, self-contained fashion, discussing with a critical eye its properties, applicability, and inherent limitations, as well as introducing novel extensions. Through free-energy calculations of prototypical molecular systems, many methodological aspects are examined, from stratification strategies to overcoming the so-called hidden barriers in orthogonal space, relevant not only to the adaptive biasing force algorithm but also to other importance-sampling schemes. On the basis of the discussions in this paper, a number of good practices for improving the efficiency and reliability of the computed free-energy differences are proposed.

Computation of free energy profiles with parallel adaptive dynamics

We propose a formulation of an adaptive computation of free energy differences, in the adaptive biasing force or nonequilibrium metadynamics spirit, using conditional distributions of samples of configurations which evolve in time. This allows us to present a truly unifying framework for these methods, and to prove convergence results for certain classes of algorithms. From a numerical viewpoint, a parallel implementation of these methods is very natural, the replicas interacting through the reconstructed free energy. We demonstrate how to improve this parallel implementation by resorting to some selection mechanism on the replicas. This is illustrated by computations on a model system of conformational changes.

Long-time convergence of an adaptive biasing force method

We propose a proof of convergence of an adaptive method used in molecular dynamics to compute free energy profiles. Mathematically, it amounts to studying the long-time behavior of a stochastic process which satisfies a non-linear stochastic differential equation, where the drift depends on conditional expectations of some functionals of the process. We use entropy techniques to prove exponential convergence to the stationary state.

NUMERICAL ANALYSIS OF MICRO–MACRO SIMULATIONS OF POLYMERIC FLUID FLOWS: A SIMPLE CASE

We present in this paper the numerical analysis of a simple micro–macro simulation of a polymeric fluid flow, namely the shear flow for the Hookean dumbbells model. Although restricted to this academic case (which is however used in practice as a test problem for new numerical strategies to be applied to more sophisticated cases), our study can be considered as a first step towards that of more complicated models. Our main result states the convergence of the fully discretized scheme (finite element in space, finite difference in time, plus Monte Carlo realizations) towards the coupled solution of a partial differential equation/stochastic differential equation system.

A mathematical formalization of the parallel replica dynamics

Abstract. We propose a mathematical analysis of a well-known numerical approach used in molecular dynamics to efficiently sample a coarse-grained description of the original trajectory (in terms of state-to-state dynamics). This technique is called parallel replica dynamics and has been introduced by Arthur F. Voter. The principle is to introduce many replicas of the original dynamics, and to consider the first transition event observed among all the replicas. The effective physical time is obtained by summing up all the times elapsed for all replicas. Using a parallel implementation, a speed-up of the order of the number of replicas can thus be obtained, allowing longer time scales to be computed. 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.

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 standard splitting strategy. We show how to correct the scheme so that it samples {\em exactly} the canonical measure restricted on a submanifold, using a Metropolis rule 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. The corresponding numerical methods can be used to sample (without any bias) a probability measure supported by a submanifold. (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-step error is proposed. Numerical illustrations are provided for (ii) and (iii).

Efficient Pricing of Asian Options by the PDE Approach

We consider the partial differential equation (PDE) proposed by Rogers and Shi (1995) and explain the main difficulties encountered in applying standard numerical schemes on this PDE as such. We then propose a scheme that is able to produce accurate results (to at least five decimal places) very quickly (in less than one second on a PC equipped with a 1-GHz Intel Pentium III microprocessor). We compare our approach with the schemes proposed in the literature.