Analysis of Variational Integrators for Langevin Processes
Nawaf Bou-Rabee
Free University of Berlin

This talk analyzes generalizations of variational integrators from simple mechanical systems to Langevin processes that are relevant in, e.g., computations of molecular dynamics trajectories. An introduction to variational integrators and discrete mechanics is provided. Special attention is paid towards operating these integrators in multiscale systems at the verge of linear stability where autonomous backward error analysis often breaks down.

Against this backdrop stochastic variational integrators are presented based on Lie-Trotter and symmetric Strang splittings of the Focker-Planck operator. These discretizations of Langevin processes are quite natural, but seem to have only recently been proposed in the literature by Vanden-Eijnden & Ciccotti [2006] and Bussi & Parrinello [2007]. An analysis of these integrators reveals the necessity of rejection sampling for nonlinear stability. Geometric ergodicity of the resulting Metropolis-stabilized, discrete Markov chain is demonstrated. Numerics computing equilibrium statistics and autocorrelation functions in conformation changes of certain alkanes is presented.

This work is joint with Christof Schutte (FUB) and Eric Vanden-Eijnden (NYU).