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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Coarse-graining schemes for stochastic lattice systems with short and long-range interactions


Authors: Markos A. Katsoulakis, Petr Plecháč, Luc Rey-Bellet and Dimitrios K. Tsagkarogiannis
Journal: Math. Comp. 83 (2014), 1757-1793
MSC (2010): Primary 65C05, 65C20, 82B20, 82B80, 82-08
Published electronically: March 25, 2014
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Abstract: We develop coarse-graining schemes for stochastic many-particle microscopic models with competing short- and long-range interactions on a $ d$-dimensional lattice. We focus on the coarse-graining of equilibrium Gibbs states, and by using cluster expansions we analyze the corresponding renormalization group map. We quantify the approximation properties of the coarse-grained terms arising from different types of interactions and present a hierarchy of correction terms. We derive semi-analytical numerical coarse-graining schemes that are accompanied by a posteriori error estimates for lattice systems with short- and long-range interactions.


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Additional Information

Markos A. Katsoulakis
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003 and Department of Applied Mathematics, University of Crete and Foundation of Research – and — Technology-Hellas, Greece
Email: markos@math.umass.edu

Petr Plecháč
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark, Delaware 19716
Email: plechac@math.udel.edu

Luc Rey-Bellet
Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts 01003
Email: lr7q@math.umass.edu

Dimitrios K. Tsagkarogiannis
Affiliation: Hausdorff Center for Mathematics, University of Bonn, D-53115 Bonn, Germany
Email: dtsagkaro@gmail.com

DOI: http://dx.doi.org/10.1090/S0025-5718-2014-02806-8
PII: S 0025-5718(2014)02806-8
Keywords: Coarse-graining, lattice spin systems, Monte Carlo method, Gibbs measure, cluster expansion, renormalization group map, sub-grid scale modeling, multi-body interactions
Received by editor(s): March 8, 2010
Received by editor(s) in revised form: March 22, 2011, and November 9, 2011
Published electronically: March 25, 2014
Additional Notes: The research of the first author was supported by the National Science Foundation through grants NSF-DMS-0715125, the CDI -Type II award NSF-CMMI-0835673, and the European Commission FP7-REGPOT-2009-1 project “Archimedes Center for Modeling, Analysis and Computation”
The research of the second author was partially supported by the National Science Foundation under grant NSF-DMS-0813893 and by the Office of Advanced Scientific Computing Research, U.S. Department of Energy under DE-SC0001340; the work was partly done at the Oak Ridge National Laboratory, which is managed by UT-Battelle, LLC under Contract No. DE-AC05-00OR22725
The research of the third author was partially supported by grant NSF-DMS-06058
The research of the fourth author has been supported by a Marie Curie Intra European Fellowship within the 7th European Community Framework Program
Article copyright: © Copyright 2014 American Mathematical Society