Convergence of adaptive biasing potential methods for diffusionsReport as inadecuate

Convergence of adaptive biasing potential methods for diffusions - Download this document for free, or read online. Document in PDF available to download.

1 UNINE - Institut de Mathématiques 2 ICJ - Institut Camille Jordan Villeurbanne

Abstract : We prove the consistency of an adaptive importance sampling strategy based on biasing the potential energy function V of a diffusion process dX 0 t = −∇V X 0 tdt + dW t ; for the sake of simplicity, periodic boundary conditions are assumed, so that X 0 t lives on the flat d-dimensional torus. The goal is to sample its invariant distribution µ = Z −1 exp −V x dx. The bias V t −V , where V t is the new random and time-dependent potential function, acts only on some coordinates of the system, and is designed to flatten the corresponding empirical occupation measure of the diffusion X in the large time regime. The diffusion process writes dX t = −∇V t X tdt + dW t , where the bias V t − V is function of the key quantity µ t : a probability occupation measure which depends on the past of the process, i.e. on X s s∈0,t. We are thus dealing with a self-interacting diffusion. In this note, we prove that when t goes to infinity, µ t almost surely converges to µ. Moreover, the approach is justified by the convergence of the bias to a limit which has an intepretation in terms of a free energy. The main argument is a change of variables, which formally validates the consistency of the approach. The convergence is then rigorously proven adapting the ODE method from stochastic approximation .

Author: Michel Benaïm - Charles-Edouard Bréhier -



Related documents