Multiscale spatial Monte Carlo simulations: multigriding, computational singular perturbation, and hierarchical stochastic closures.

Monte Carlo (MC) simulation of most spatially distributed systems is plagued by several problems, namely, execution of one process at a time, large separation of time scales of various processes, and large length scales. Recently, a coarse-grained Monte Carlo (CGMC) method was introduced that can capture large length scales at reasonable computational times. An inherent assumption in this CGMC method revolves around a mean-field closure invoked in each coarse cell that is inaccurate for short-ranged interactions. Two new approaches are explored to improve upon this closure. The first employs the local quasichemical approximation, which is applicable to first nearest-neighbor interactions. The second, termed multiscale CGMC method, employs singular perturbation ideas on multiple grids to capture the entire cluster probability distribution function via short microscopic MC simulations on small, fine-grid lattices by taking advantage of the time scale separation of multiple processes. Computational strategies for coupling the fast process at small length scales (fine grid) with the slow processes at large length scales (coarse grid) are discussed. Finally, the binomial tau-leap method is combined with the multiscale CGMC method to execute multiple processes over the entire lattice and provide additional computational acceleration. Numerical simulations demonstrate that in the presence of fast diffusion and slow adsorption and desorption processes the two new approaches provide more accurate solutions in comparison to the previously introduced CGMC method.