arXiv subjects

Tony Lelievre

Publications and source records attributed to Tony Lelievre.

25 records · Page 2Linked to original sources

arXiv2009

A Variance Reduction Method for Parametrized Stochastic Differential Equations using the Reduced Basis Paradigm

In this work, we develop a reduced-basis approach for the efficient computation of parametrized expected values, for a large number of parameter values, using the control variate method to reduce the variance. Two algorithms are proposed to compute online, through a cheap reduced-basis approximation, the control variates for the computation of a large number of expectations of a functional of a parametrized Ito stochastic process (solution to a parametrized stochastic differential equation). For each algorithm, a reduced basis of control variates is pre-computed offline, following a so-called greedy procedure, which minimizes the variance among a trial sample of the output parametrized expectations. Numerical results in situations relevant to practical applications (calibration of volatility in option pricing, and parameter-driven evolution of a vector field following a Langevin equation from kinetic theory) illustrate the efficiency of the method.

math.NA↗

arXiv2009

Effective dynamics using conditional expectations

The question of coarse-graining is ubiquitous in molecular dynamics. In this article, we are interested in deriving effective properties for the dynamics of a coarse-grained variable $ξ(x)$, where $x$ describes the configuration of the system in a high-dimensional space $\R^n$, and $ξ$ is a smooth function with value in $\R$ (typically a reaction coordinate). It is well known that, given a Boltzmann-Gibbs distribution on $x \in \R^n$, the equilibrium properties on $ξ(x)$ are completely determined by the free energy. On the other hand, the question of the effective dynamics on $ξ(x)$ is much more difficult to address. Starting from an overdamped Langevin equation on $x \in \R^n$, we propose an effective dynamics for $ξ(x) \in \R$ using conditional expectations. Using entropy methods, we give sufficient conditions for the time marginals of the effective dynamics to be close to the original ones. We check numerically on some toy examples that these sufficient conditions yield an effective dynamics which accurately reproduces the residence times in the potential energy wells. We also discuss the accuracy of the effective dynamics in a pathwise sense, and the relevance of the free energy to build a coarse-grained dynamics.

math.AP↗

arXiv2007

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.

math.AP↗

arXiv2007

Diffusion Monte Carlo method: numerical analysis in a simple case

The Diffusion Monte Carlo method is devoted to the computation of electronic ground-state energies of molecules. In this paper, we focus on implementations of this method which consist in exploring the configuration space with a {\bf fixed} number of random walkers evolving according to a Stochastic Differential Equation discretized in time. We allow stochastic reconfigurations of the walkers to reduce the discrepancy between the weights that they carry. On a simple one-dimensional example, we prove the convergence of the method for a fixed number of reconfigurations when the number of walkers tends to $+\infty$ while the timestep tends to 0. We confirm our theoretical rates of convergence by numerical experiments. Various resampling algorithms are investigated, both theoretically and numerically

math.NA↗

arXiv2007

New entropy estimates for the Oldroyd-B model, and related models

This short note presents the derivation of a new {\it a priori} estimate for the Oldroyd-B model. Such an estimate may provide useful information when investigating the long-time behaviour of macro-macro models, and the stability of numerical schemes. We show how this estimate can be used as a guideline to derive new estimates for other macroscopic models, like the FENE-P model.

math.NA↗

arXiv2007

Computation of free energy differences through nonequilibrium stochastic dynamics: the reaction coordinate case

The computation of free energy differences through an exponential weighting of out of equilibrium paths (known as the Jarzynski equality) is often used for transitions between states described by an external parameter $λ$ in the Hamiltonian. We present here an extension to transitions between states defined by different values of some reaction coordinate, using a projected Brownian dynamics. In contrast with other approaches, we use a projection rather than a constraining potential to let the constraints associated with the reaction coordinate evolve. We show how to use the Lagrange multipliers associated with these constraints to compute the work associated with a given trajectory. Appropriate discretizations are proposed. Some numerical results demonstrate the applicability of the method for the computation of free energy difference profiles.

cond-mat.stat-mech↗

arXiv2007

Computation of free energy profiles with parallel adaptive dynamics

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.

cond-mat.stat-mech↗