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

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 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 parameters (drift and diffusion) using only appropriately chosen realizations of the fine-scale, individualbased 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. 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 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 parameters (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.

Semiparametric Drift and Diffusion Estimation for Multiscale Diffusions

“…The property that the variance of the error vanishes for N → ∞ reflects the fact that the estimator relies on an identity, i.e. on a direct (deterministic) computation (17) rather than on asymptotic time limits (i.e. on ergodicity).…”

“…We also mention that the effect of the multiscale structure on the evolution of the coarse-grained probability density using the Fokker-Planck equation (Kolmogorov's forward equation) was studied in [17]. In this study it was shown that when decreasing the spatial discretization in a finite difference approximation the error increases rapidly and that in order to avoid this, it is necessary to improve the accuracy of the estimators of the drift and diffusion coefficients.…”