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Coupled Coarse Graining and Markov Chain Monte Carlo for Lattice Systems

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Numerical Analysis of Multiscale Computations

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 82))

Abstract

We propose an efficient Markov Chain Monte Carlo method for sampling equilibrium distributions for stochastic lattice models. The method is capable of handling correctly and efficiently long and short-range particle interactions. The proposed method is a Metropolis-type algorithm with the proposal probability transition matrix based on the coarse-grained approximating measures introduced in (Katsoulakis et al. Proc. Natl. Acad. Sci. 100(3):782–787, 2003; Katsoulakis et al. ESAIM-Math. Model. Numer. Anal. 41(3):627–660, 2007). The proposed algorithm reduces the computational cost due to energy differences and has comparable mixing properties with the classical microscopic Metropolis algorithm, controlled by the level of coarsening and reconstruction procedure. The properties and effectiveness of the algorithm are demonstrated with an exactly solvable example of a one dimensional Ising-type model, comparing efficiency of the single spin-flip Metropolis dynamics and the proposed coupled Metropolis algorithm.

* The research of E.K. was supported by the National Science Foundation under the grant NSFCMMI- 0835582.

† The research of M.A. K. was supported by the National Science Foundation through the grants NSF-DMS-0715125 and the CDI – Type II award NSF-CMMI-0835673.

‡ The research of P.P. was partially supported by the National Science Foundation under the 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.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Tennessee, 1900 Laurel Ave, Knoxville, TN, 37916, USA

    Petr Plecháč

  2. Joint Institute for Computational Sciences, University of Tennessee and Oak Ridge National Laboratory, 2008, Oak Ridge, TN, 37831, USA

    Evangelia Kalligiannaki

  3. Department of Applied Mathematics, University of Massachusetts, University of Crete and Foundation of Research and Technology-Hellas, Nikolaou Plastira 100, Vassilika Vouton, GR – 700 13, Heraklion, Crete, Greece

    Markos A. Katsoulakis

Authors
  1. Evangelia Kalligiannaki
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  2. Markos A. Katsoulakis
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  3. Petr Plecháč
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Corresponding author

Correspondence to Petr Plecháč .

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Editors and Affiliations

  1. , Department of Mathematics, University of Texas, University Station C 1200 1, Austin, 78712-0257, Texas, USA

    Björn Engquist

  2. Dept. Numerical Analysis &, Computer Science (NADA), Royal Institute of Technology (KTH), Lindstedtsvägen 3, Stockholm, 100 44, Sweden

    Olof Runborg

  3. , Department of Mathematics, University of Texas at Austin, University Station C1200 1, Austin, 78712-0257, Texas, USA

    Yen-Hsi R. Tsai

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Kalligiannaki, E., Katsoulakis, M.A., Plecháč, P. (2012). Coupled Coarse Graining and Markov Chain Monte Carlo for Lattice Systems. In: Engquist, B., Runborg, O., Tsai, YH. (eds) Numerical Analysis of Multiscale Computations. Lecture Notes in Computational Science and Engineering, vol 82. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21943-6_11

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