Local diffusion models for stochastic reacting systems: estimation issues in equation-free numerics

Abstract

Extracting coarse-grained derivative information from fine scale, atomistic/stochastic simulations constitutes an important component of multiscale numerics for reacting systems. In this paper, we demonstrate the use of local parametric diffusion models; observing the output of short bursts of stochastic simulation, the drift and diffusion coefficients of such local models are obtained “on demand” by maximizing an approximation of the likelihood function. The parametric fit utilizes the likelihood expansion of Aït-Sahalia, giving closed-form expressions for the transition density of both scalar and multivariate processes. The fine scale simulations generating the data are the Gillespie stochastic simulation algorithm as well as stochastic differential equations (SDEs). The postulated local parametric diffusion models are SDEs with affine drift and diffusion functions; the information extracted from these (estimated) local parametric models is utilized in various “equation-free” computational tasks. We demonstrate how such local diffusion modeling can (1) contribute to linking stochastic simulation with traditional, continuum numerical methods such as fixed point and initial value problem solving algorithms as well as variational integration; (2) aid in extracting quantitative information about the invariant distribution associated with the underlying stochastic system; and (3) be used in computational model reduction motivated by singular perturbation theory.