Coarse-graining schemes and a posteriori error estimates for stochastic lattice systems
Markos Katsoulakis, Petr Plechac, Luc Rey-Bellet, and Dimitrios Tsagkarogiannis
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Submission date: 19. Aug. 2006
published in: ESAIM / Mathematical modelling and numerical analysis, 41 (2007) 3, p. 627-660
DOI number (of the published article): 10.1051/m2an:2007032
MSC-Numbers: 65C05, 65C20, 82B20, 82B80, 82-08
Keywords and phrases: coarse-graining, a posteriori error estimate, relative entropy, renormalization group map
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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 phase transitions 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.