ESAIM. Mathematical Modelling and Numerical Analysis vol:45 issue:3 pages:541-561
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 investigate a numerical procedure to compute the solution of this Fokker–Planck equation directly on the coarse level, based on the estimation of the unknown pa-
rameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale, individual-based system. As these parameters might be solution-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 and present an analysis of the propagation of the estimation error in the numerical solution of the coarse Fokker–Planck equation. The results show that for decreasing spatial discretization error, the total error grows rapidly due to the use of estimated coefficients. This effect can be avoided by increasing the quality of the estimates when the spatial discretization decreases. Although the procedure is illustrated for a specific class of multiscale stochastic systems, it is devised so that it can easily be generalized to other stochastic or particle models.