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

Published online by Cambridge University Press:  30 November 2010

Yves Frederix ,
Giovanni Samaey  and
Dirk Roose
Yves Frederix
Affiliation:
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
Giovanni Samaey
Affiliation:
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
Dirk Roose
Affiliation:
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
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Abstract

We consider multiscale systems for which only a fine-scalemodel describing the evolution of individuals (atoms,molecules, bacteria, agents) is given, while we are interested in theevolution of the population density on coarse space and timescales. Typically, this evolution is described by a coarseFokker-Planck equation. In this paper, we consider a numerical procedure to compute the solution ofthis 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-basedsystem. As these parameters might be space- and time-dependent, theestimation is performed in every spatial discretization point and atevery time step. If the fine-scale model is stochastic, the estimationprocedure introduces noise on the coarse level.We investigate stability conditions for this procedure in thepresence of this noise and present ananalysis of the propagation of the estimation error in the numericalsolution of the coarse Fokker-Planck equation.

Keywords

Multiscale computingstochastic systemsFokker-Planckequationuncertainty propagation
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 3 , May 2011 , pp. 541 - 561
Copyright
© EDP Sciences, SMAI, 2010

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