Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems
ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 3, pp. 627-660.

The primary objective of this work is to develop coarse-graining schemes for stochastic many-body microscopic models and quantify their effectiveness in terms of a priori and a posteriori error analysis. In this paper we focus on stochastic lattice systems of interacting particles at equilibrium. The proposed algorithms are derived from an initial coarse-grained approximation that is directly computable by Monte Carlo simulations, and the corresponding numerical error is calculated using the specific relative entropy between the exact and approximate coarse-grained equilibrium measures. Subsequently we carry out a cluster expansion around this first - and often inadequate - approximation and obtain more accurate coarse-graining schemes. The cluster expansions yield also sharp a posteriori error estimates for the coarse-grained approximations that can be used for the construction of adaptive coarse-graining methods. We present a number of numerical examples that demonstrate that the coarse-graining schemes developed here allow for accurate predictions of critical behavior and hysteresis in systems with intermediate and long-range interactions. We also present examples where they substantially improve predictions of earlier coarse-graining schemes for short-range interactions.

DOI : 10.1051/m2an:2007032
Classification : 65C05, 65C20, 82B20, 82B80, 82-08
Mots clés : Coarse-graining, a posteriori error estimate, relative entropy, lattice spin systems, Monte Carlo method, Gibbs measure, cluster expansion, renormalization group map
Katsoulakis, Markos A.  ; Plecháč, Petr  ; Rey-Bellet, Luc  ; Tsagkarogiannis, Dimitrios K. 1

1 Max Planck Institute for Mathematics in the Sciences, Germany
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Katsoulakis, Markos A.; Plecháč, Petr; Rey-Bellet, Luc; Tsagkarogiannis, Dimitrios K. Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems. ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 3, pp. 627-660. doi : 10.1051/m2an:2007032. http://www.numdam.org/articles/10.1051/m2an:2007032/

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