Commun. Comput. Phys., 16 (2014), pp. 467-490. |
On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method Jun Li ^{1*}, Philippe Vignal ^{2}, Shuyu Sun ^{3}, Victor M. Calo ^{4} 1 Center for Numerical Porous Media, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia.2 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. 3 Applied Mathematics and Computational Science, Earth Sciences and Engineering, King Abdullah University of Science and Technology, Thuwal, Saudi Arabia. 4 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. Received 11 June 2013; Accepted (in revised version) 28 February 2014 Available online 12 June 2014 doi:10.4208/cicp.110613.280214a 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. AMS subject classifications: 76T99, 82B05, 82B80, 62M05, 65C40Notice: Undefined variable: pac in /var/www/html/readabs.php on line 165 Key words: Phase coexistence, Gibbs ensemble, molecular simulation, Markov Chain Monte Carlo method, variance estimation, blocking method. *Corresponding author. Email: lijun04@gmail.com (J. Li), philippe.vignal@kaust.edu.sa (P. Vignal), shuyu.sun@kaust.edu.sa (S. Sun), victor.calo@kaust.edu.sa (V. M. Calo) |