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On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method

Published online by Cambridge University Press:  03 June 2015

Jun Li ,
Philippe Vignal ,
Shuyu Sun  and
Victor M. Calo
Jun Li*
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
Philippe Vignal*
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Material Science and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
Shuyu Sun*
Affiliation:
Applied Mathematics and Computational Science, Earth Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
Victor M. Calo*
Affiliation:
Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia Applied Mathematics and Computational Science, Earth Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia
*
Corresponding author.Email:lijun04@gmail.com
Email:philippe.vignal@kaust.edu.sa
Email:shuyu.sun@kaust.edu.sa
Email:victor.calo@kaust.edu.sa
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Abstract

In Markov Chain Monte Carlo (MCMC) simulations, thermal equilibria quantities are estimated by ensemble average over a sample set containing a large number of correlated samples. These samples are selected in accordance with the probability distribution function, known from the partition function of equilibrium state. As the stochastic error of the simulation results is significant, it is desirable to understand the variance of the estimation by ensemble average, which depends on the sample size (i.e., the total number of samples in the set) and the sampling interval (i.e., cycle number between two consecutive samples). Although large sample sizes reduce the variance, they increase the computational cost of the simulation. For a given CPU time, the sample size can be reduced greatly by increasing the sampling interval, while having the corresponding increase in variance be negligible if the original sampling interval is very small. In this work, we report a few general rules that relate the variance with the sample size and the sampling interval. These results are observed and confirmed numerically. These variance rules are derived for the MCMC method but are also valid for the correlated samples obtained using other Monte Carlo methods. The main contribution of this work includes the theoretical proof of these numerical observations and the set of assumptions that lead to them.

Keywords

76T9982B0582B8062M0565C40Phase coexistenceGibbs ensemblemolecular simulationMarkov Chain Monte Carlo methodvariance estimationblocking method.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 2 , August 2014 , pp. 467 - 490
Copyright
Copyright © Global Science Press Limited 2014

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