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Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems

Published online by Cambridge University Press:  02 August 2007

Markos A. Katsoulakis ,
Petr Plecháč ,
Luc Rey-Bellet  and
Dimitrios K. Tsagkarogiannis
Markos A. Katsoulakis
Affiliation:
Department of Mathematics, University of Massachusetts, USA. markos@math.umass.edu; lr7q@math.umass.edu
Petr Plecháč
Affiliation:
Department of Mathematics, University of Tennessee, USA. plechac@math.utk.edu
Luc Rey-Bellet
Affiliation:
Department of Mathematics, University of Massachusetts, USA. markos@math.umass.edu; lr7q@math.umass.edu
Dimitrios K. Tsagkarogiannis
Affiliation:
Max Planck Institute for Mathematics in the Sciences, Germany. tsagkaro@mis.mpg.de
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Abstract

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.

Keywords

Coarse-graininga posteriori error estimaterelative entropylattice spin systemsMonte Carlo methodGibbs measurecluster expansionrenormalization group map.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 41 , Issue 3 , May 2007 , pp. 627 - 660
Copyright
© EDP Sciences, SMAI, 2007

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