

Preprint 39/2008
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.