Multilevel approximations and coarse-graining of lattice jump processes

We discuss a hierarchy of approximation methods developed for accelerating sampling microscopic dynamics of stochastic lattice systems. The approach is based on efficient coupling of different resolution levels, taking advantage of the low sampling cost in a coarse space and the local reconstructions.

We provide error estimates for (a) long-time stationary dynamics in terms of relative entropy, and (b) finite-time weak error estimates that control mesoscale observables.

From the computational point of view the multilevel nature of the method allows for speeding up sampling algorithms such as kinetic Monte Carlo applied to systems with complex lattice geometries and particle interactions.

We also briefly discuss related mathematical, numerical and algorithmic issues arising in the parallelization of spatially distributed kinetic Monte Carlo simulations.