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

Markos A. Katsoulakis; Petr Plecháč; Luc Rey-Bellet; Dimitrios K. Tsagkarogiannis

ESAIM: Mathematical Modelling and Numerical Analysis (2007)

  • Volume: 41, Issue: 3, page 627-660
  • ISSN: 0764-583X

Abstract

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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.

How to cite

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Katsoulakis, Markos A., et al. "Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems." ESAIM: Mathematical Modelling and Numerical Analysis 41.3 (2007): 627-660. <http://eudml.org/doc/250071>.

@article{Katsoulakis2007,
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. },
author = {Katsoulakis, Markos A., Plecháč, Petr, Rey-Bellet, Luc, Tsagkarogiannis, Dimitrios K.},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Coarse-graining; a posteriori error estimate; relative entropy; lattice spin systems; Monte Carlo method; Gibbs measure; cluster expansion; renormalization group map.; coarse-graining; a posteriori error estimate; lattice spin systems; renormalization group map},
language = {eng},
month = {8},
number = {3},
pages = {627-660},
publisher = {EDP Sciences},
title = {Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems},
url = {http://eudml.org/doc/250071},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Katsoulakis, Markos A.
AU - Plecháč, Petr
AU - Rey-Bellet, Luc
AU - Tsagkarogiannis, Dimitrios K.
TI - Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2007/8//
PB - EDP Sciences
VL - 41
IS - 3
SP - 627
EP - 660
AB - 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.
LA - eng
KW - Coarse-graining; a posteriori error estimate; relative entropy; lattice spin systems; Monte Carlo method; Gibbs measure; cluster expansion; renormalization group map.; coarse-graining; a posteriori error estimate; lattice spin systems; renormalization group map
UR - http://eudml.org/doc/250071
ER -

References

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