The gaptooth scheme, patch dynamics and equation-free controller design for distributed complex/multiscale processes

We present an equation-free multiscale computational framework for the design of "coarse" controllers for spatially distributed processes described by microscopic/mesoscopic evolution rules. In particular, we exploit the smoothness in space of the process observables to estimate the unknown coarse system dynamics. This is accomplished through appropriately initialized and linked ensembles of microscopic simulations realizing only a small portion of the macroscopic spatial domain (the so-called gaptooth and patch-dynamics schemes, (I.G. Kevrekidis, et al., 2003). We illustrate this framework by designing discrete-time, coarse linear controllers for a Lattice-Boltzmann (LB) scheme modelling a reaction-diffusion process (a kinetic-theory based realization of the FitzHugh-Nagumo equation in one spatial dimension).