Mathematical strategies in the coarse-graining of extensive systems: error quantification and adaptivity

Markos Katsoulakis, Petr Plechac, Luc Rey-Bellet, and Dimitrios Tsagkarogiannis

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Submission date: 25. Apr. 2008
Pages: 23
published in: Journal of non-Newtonian fluid mechanics, 152 (2008) 1/3, p. 101-112 
DOI number (of the published article): 10.1016/j.jnnfm.2007.05.005
Bibtex
MSC-Numbers: 65C05, 65C20, 82B20, 82B80, 82-08
Keywords and phrases: coarse-graining, a posteriori error estimate, adaptive coarse-graining, relative entropy, lattice spin systems, Coarse Grained Monte Carlo method, Gibbs measure, cluster expansion
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Abstract:
In this paper we continue our study of coarse-graining schemes for stochastic many-body microscopic models started in previous work, focusing on equilibrium stochastic lattice systems. Using cluster expansion techniques we expand the exact coarse-grained Hamiltonian around a first approximation and derive higher accuracy schemes by including more terms in the expansion. The accuracy of the coarse-graining schemes is measured in terms of information loss, i.e., relative entropy, between the exact and approximate coarse-grained Gibbs measures. We test the effectiveness of our schemes in systems with competing short and long range interactions, using an analytically solvable model as a computational benchmark. Furthermore, the cluster expansion yields sharp a posteriori error estimates for the coarse-grained approximations that can be computed on-the-fly during the simulation. Based on these estimates we develop a numerical strategy to assess the quality of the coarse-graining and suitably refine or coarsen the simulations. We demonstrate the use of this diagnostic tool in the numerical calculation of phase diagrams.