An analysis of noise propagation in the multiscale simulation of coarse Fokker-Planck equations

Department of Computer Science, K.U. Leuven, Celestijnenlaan 200A, 3001 Leuven, Belgium. Yves.Frederix@cs.kuleuven.be; Giovanni.Samaey@cs.kuleuven.be; Dirk.Roose@cs.kuleuven.be

Received: 24 October 2009
Revised: 3 May 2010
Revised: 26 July 2010

Abstract

We consider multiscale systems for which only a fine-scale model describing the evolution of individuals (atoms, molecules, bacteria, agents) is given, while we are interested in the evolution of the population density on coarse space and time scales. Typically, this evolution is described by a coarse Fokker-Planck equation. In this paper, we consider a numerical procedure to compute the solution of this Fokker-Planck equation directly on the coarse level, based on the estimation of the unknown parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale, individual-based system. As these parameters might be space- and time-dependent, the estimation is performed in every spatial discretization point and at every time step. If the fine-scale model is stochastic, the estimation procedure introduces noise on the coarse level. We investigate stability conditions for this procedure in the presence of this noise and present an analysis of the propagation of the estimation error in the numerical solution of the coarse Fokker-Planck equation.