No abstract

We propose a formulation of adaptive computation of free energy differences, in the ABF or nonequilibrium metadynamics spirit, using conditional distributions of samples of configurations which evolve in time. This allows 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 show 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.

We propose a proof of convergence of an adaptive method used in molecular dynamics to compute free energy profiles (see Darve and Porohille 2001 J.

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)

We consider a general Schrödinger operator L + V on a domain E ⊂ R d and its associated positive ground state h solution to the maximal eigenvalue problem L(h) + V h = λh. In this work, an interacting particle model approximating the pair (h, λ) is studied. When V ≤ 0, a basic version of this particle system consists of N walkers evolving independently according to the Markov generator L, each walker dying at a rate given by the value of the potential |V | at the walker's current location; when a walker dies, any other one splits in two. The long time distribution of the particle system is then an estimator of h. Under some reasonable assumptions (with examples for E = R d ), we get a nonasymptotic control of the L p deviations (resp., the bias) of this estimator with the genuine rate of convergence in 1/ √ N (resp., 1/N ). We also compute explicitly the asymptotic standard deviation of the estimation of λ, which remains bounded in usual mild situations.

International audienceWe 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. By interpreting the algorithm as a sequential sampler in path space, we are able to build an estimator of the rare event probability (and of any non-normalized 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

Abstract. The Adaptive Multilevel Splitting algorithm [4] is a very powerful and versatile method to estimate rare events probabilities. It is an iterative procedure on an interacting particle system, where at each step, the k less well-adapted particles among n are killed while k new better adapted particles are resampled according to a conditional law. We analyze the algorithm in the idealized setting of an exact resampling and prove that the estimator of the rare event probability is unbiased whatever k. We also obtain a precise asymptotic expansion for the variance of the estimator and the cost of the algorithm in the large n limit, for a fixed k.

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 any 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. Mathematical setting and resultsWe first describe in Section 2.1 the target probability measures we are interested in sampling. The core of our analysis is presented in Section 2.2, where we show how to rigorously formalize the reversibility properties of discretizations of the Hamiltonian dynamics on a submanifold for possibly large timesteps. One crucial ingredient in this analysis is the Lagrange multiplier function, which associates to a state on the submanifold and another state close to the submanifold (obtained by an unconstrained step of the dynamics) a new state on the submanifold. Examples of such Lagrange multiplier functions are provided in Section 2.3. Once the discretization of the Hamiltonian part of the dynamics is clear, GHMC schemes follow in a straightforward way; see Section 2.4. Geometric settingLet us first introduce the measures we are interested in sampling, namely the target measure (2) below, as well as the probability measure (4) on an extended space, which admits (2) as a marginal. We only provide the most essential objects needed for our analysis. For more details and motivations for the definitions and results given here, we refer the reader to the standard reference textbooks [1,4], and also to [22, Chapter 3.3.2] for a self-contained presentation. Target measureWe denote by d the dimension of the ambient space, and consider a submanifold M ⊂ R d defined as the zero level set of a given smooth function ξ :For example, the function ξ encodes molecular constraints or reaction coordinates in molecular dynamics, or is a summary statistics in Approximate Bayesian Computations. Let M ∈ R d×d be a fixed symmetric positive definite matrix, which is interpreted as the mass tensor below. The ambient space R d is endowed with the scalar product q,q M = q T Mq. One could take for simplicity M = Id but it is sometimes useful to consider non identity mass matrices for numerical purposes [13]. The function ξ is assumed to be smooth on a neighborhood of M in R d and such thatis an invertible matrix for...