Error Analysis of Coarse-Graining for Stochastic Lattice Dynamics

The coarse-grained Monte Carlo (CGMC) algorithm was originally proposed in the series of works [M. A. Katsoulakis, A. J. Majda, and D. G. Vlachos, J. Comput. Phys., 186 (2003), pp. 250-278; M. A. Katsoulakis, A. J. Majda, and D. G. Vlachos, Proc. Natl. Acad. Sci. USA, 100 (2003), pp. 782-787; M. A. Katsoulakis and D. G. Vlachos, J. Chem. Phys., 119 (2003), pp. 9412-9427]. In this paper we further investigate the approximation properties of the coarse-graining procedure and provide both analytical and numerical evidence that the hierarchy of the coarse models is built in a systematic way that allows for error control in both transient and long-time simulations. We demonstrate that the numerical accuracy of the CGMC algorithm as an approximation of stochastic lattice spin flip dynamics is of order two in terms of the coarse-graining ratio and that the natural small parameter is the coarse-graining ratio over the range of particle/particle interactions. The error estimate is shown to hold in the weak convergence sense. We employ the derived analytical results to guide CGMC algorithms and demonstrate a CPU speed-up in demanding computational regimes that involve nucleation, phase transitions, and metastability.

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