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.

Classification: 65C05, 65C20, 82B20, 82B80, 82-08

Keywords: Coarse-graining, a posteriori error estimate, relative entropy, lattice spin systems, Monte Carlo method, Gibbs measure, cluster expansion, renormalization group map

@article{M2AN_2007__41_3_627_0, author = {Katsoulakis, Markos A. and Plech\'a\v c, Petr and Rey-Bellet, Luc and Tsagkarogiannis, Dimitrios K.}, title = {Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {41}, number = {3}, year = {2007}, pages = {627-660}, doi = {10.1051/m2an:2007032}, zbl = {pre05289387}, mrnumber = {2355714}, language = {en}, url = {http://www.numdam.org/item/M2AN_2007__41_3_627_0} }

Katsoulakis, Markos A.; Plecháč, Petr; Rey-Bellet, Luc; Tsagkarogiannis, Dimitrios K. Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 3, pp. 627-660. doi : 10.1051/m2an:2007032. http://www.numdam.org/item/M2AN_2007__41_3_627_0/

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