Claude Le Bris

High-order averaging schemes with error bounds for thermodynamical properties calculations by molecular dynamics simulations.

We introduce high-order formulas for the computation of statistical averages based on the long-time simulation of molecular dynamics trajectories. In some cases, this allows us to significantly improve the convergence rate of time averages toward ensemble averages. We provide some numerical examples that show the efficiency of our scheme. When trajectories are approximated using symplectic integration schemes (such as velocity Verlet), we give some error bounds that allow one to fix the parameters of the computation in order to reach a given desired accuracy in the most efficient manner.

Optimisation de géométrie dans le cadre des théories de type Thomas-Fermi pour les cristaux périodiques

Resume Nous presentons dans cette Note letude du probleme de loptimisation de geometrie pour un cristal periodique dans le cadre du modele TFW, cest-a-dire le probleme de minimisation de lenergie de ce modele en fonction du reseau qui definit le cristal. Nous montrons lexistence dun tel minimum, definissant au passage des modeles de polymeres lineiques et de films minces pour cette theorie.

Dimension-independent convergence rate for non-isotropic (1, λ) - ES

Based on the theory of non-negative super martingales, convergence results are proven for adaptive (1, λ) - ES (i.e. with Gaussian mutations), and geometrical convergence rates are derived. In the d-dimensional case (d > 1), the algorithm studied here uses a different step-size update in each direction. However, the critical value for the step-size, and the resulting convergence rate do not depend on the dimension. Those results are discussed with respect to previous works. Rigorous numerical investigations on some 1-dimensional functions validate the theoretical results. Trends for future research are indicated.

Examples of computational approaches for elliptic, possibly multiscale PDEs with random inputs

We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of multiscale problems, but the purpose is more general. On a set of prototypical situations, we investigate two critical issues present in many settings: variance reduction techniques to obtain sufficiently accurate results at a limited computational cost when solving PDEs with random coefficients, and finite element techniques that are sufficiently flexible to carry over to geometries with random fluctuations. Some elements of theoretical analysis and numerical analysis are briefly mentioned. Numerical experiments, although simple, provide convincing evidence of the efficiency of the approaches.

Some variance reduction methods for numerical stochastic homogenization

We give an overview of a series of recent studies devoted to variance reduction techniques for numerical stochastic homogenization. Numerical homogenization requires that a set of problems is solved at the microscale, the so-called corrector problems. In a random environment, these problems are stochastic and therefore need to be repeatedly solved, for several configurations of the medium considered. An empirical average over all configurations is then performed using the Monte Carlo approach, so as to approximate the effective coefficients necessary to determine the macroscopic behaviour. Variance severely affects the accuracy and the cost of such computations. Variance reduction approaches, borrowed from other contexts in the engineering sciences, can be useful. Some of these variance reduction techniques are presented, studied and tested here.

Domain Decomposition and Electronic Structure Computations: A Promising Approach

We describe a domain decomposition approach applied to the specific context of electronic structure calculations. The approach has been introduced in [BCHL07]. We survey here the computational context, and explain the peculiarities of the approach as compared to problems of seemingly the same type in other engineering sciences. Improvements of the original approach presented in [BCHL07], including algorithmic refinements and effective parallel implementation, are included here. Test cases supporting the interest of the method are also reported.