Spatial two-level interacting particle simulations and information theory-based error quantification

We propose a hierarchy of two-level kinetic Monte Carlo methods for sampling highdimensional, stochastic lattice particle dynamics with complex interactions. The method is based on the efficient coupling of different spatial resolution levels, taking advantage of the low sampling cost in a coarse space and developing local reconstruction strategies from coarse-grained dynamics. Furthermore, a natural extension to a multilevel kinetic coarse-grained Monte Carlo is presented. Microscopic reconstruction corrects possibly significant errors introduced through coarse-graining, leading to the controlled-error approximation of the sampled stochastic process. In this manner, the proposed algorithm overcomes known shortcomings of coarse-graining of particle systems with complex interactions such as combined long- and short-range particle interactions and/or complex lattice geometries. Specifically, we provide error analysis for the approximation of long-time stationary dynamics in terms of relative entropy, measuring the information loss of the path measures per unit time. We show that this observable either can be estimated a priori, or it can be tracked computationally a posteriori in the course of a simulation. The stationary regime is of critical importance to molecular simulations as it is relevant to long-time sampling, obtaining phase diagrams, and in studying metastability properties of high-dimensional complex systems. Finally, the multilevel nature of the method provides flexibility in combining rejection-free and null-event implementations, generating a hierarchy of algorithms with an adjustable number of rejections that includes well-known rejection-free and null-event algorithms.