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Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems
ESAIM: Mathematical Modelling and Numerical Analysis (2007)
  • MA Katsoulakis, University of Massachusetts - Amherst
  • P Plechac
  • L Rey-Bellet, University of Massachusetts - Amherst
  • DK Tsagkarogiannis
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.
  • coarse-graining,
  • a posteriori error estimate,
  • relative entropy,
  • lattice spin systems,
  • Monte Carlo method,
  • Gibbs measure,
  • cluster expansion,
  • renormalization group map
Publication Date
May, 2007
Publisher Statement
DOI: 10.1051/m2an:2007032
Citation Information
MA Katsoulakis, P Plechac, L Rey-Bellet and DK Tsagkarogiannis. "Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems" ESAIM: Mathematical Modelling and Numerical Analysis Vol. 41 Iss. 3 (2007)
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