Spatially distributed stochastic systems: Equation-free and equation-assisted preconditioned computations

Spatially distributed problems are often approximately modeled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g., concentrations). The derivation of accurate such PDEs starting from finer scale, atomistic models, and using suitable averaging is often a challenging task; approximate PDEs are typically obtained through mathematical closure procedures (e.g., mean field approximations). In this paper, we show how such approximate macroscopic PDEs can be exploited in constructing preconditioners to accelerate stochastic computations for spatially distributed particle-based process models. We illustrate how such preconditioning can improve the convergence of equation-free coarse-grained methods based on coarse timesteppers. Our model problem is a stochastic reaction-diffusion model capable of exhibiting Turing instabilities.