Adaptive methods for efficient sampling. Applications in molecular dynamics

Thursday, May 14, 2009 - 11:15am - 12:15pm
Lind 409
Tony Lelievre (Ecole Nationale des Ponts et Chaussees)
One aim of molecular dynamics simulations is to sample Boltzmann-Gibbs measures associated to some potentials in high dimensional spaces, to compute macroscopic quantities (such as chemical reaction constants, or diffusions constants) by statistical means in the canonical (NVT) ensemble. Numerical methods typically rely on ergodic limits for processes solution to well-chosen stochastic differential equations (SDEs). The main difficulty comes from existence of metastable states in which the stochastic processes remain for long time: this may slow down dramatically the convergence of the ergodic limit. We present a class of adaptive importance sampling methods which enable fast exploration of the configurational space, by modifying the potential seen by the particles (the associated SDE becomes non-homogeneous and nonlinear). These methods accelerate the longtime convergence while they allow to obtain, in the longtime limit, the quantities of practical interest (the marginal law associated to the slow variables in the system). We propose a proof of convergence for some of these methods, based on entropy techniques.

- T. Lelievre, M. Rousset and G. Stoltz, Computation of free energy profiles with parallel adaptive dynamics, Journal of Chemical Physics 126, 134111 (2007)
- T. Lelievre, M. Rousset and G. Stoltz, Long-time convergence of an Adaptive Biasing Force method, Nonlinearity, 21, 1155-1181 (2008)
- T. Lelievre, A general two-scale criteria for logarithmic Sobolev inequalities, to appear in Journal of Functional Analysis