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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*
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
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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.

Type
Research Article
Copyright
Copyright © Global Science Press Limited 2014

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References

[1]Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H. and Teller, E., Equation of state calculations by fast computing machines, J. Chem. Phys., 21 (1953), 10871092.CrossRefGoogle Scholar
[2]Bird, G. A., Molecular gas dynamics and the direct simulation of gas flows, Clarendon Press, Oxford, 1994.Google Scholar
[3]Barth, A., Lang, A., Multilevel Monte Carlo method with applications to stochastic partial differential equations, Int. J. Comput. Math., 89 (2012), 24792498.CrossRefGoogle Scholar
[4]Metropolis, N., The beginning of the Monte Carlo method, Los Alamos Science, 12 (1987), 125130.Google Scholar
[5]Frenkel, D., Smit, B., Understanding molecular simulation, from algorithms to applications, Academic press, 2002.Google Scholar
[6]Marshall, A. W., The use of multi-stage sampling schemes in Monte Carlo methods, Symposium on Monte Carlo methods, Wiley, New York, 1956, 123140.Google Scholar
[7]Liu, J. S., Monte Carlo strategies in scientific computing, Harvard university, 2001.Google Scholar
[8]Panagiotopoulos, A. Z., Direct determination of phase coexistence properties of fluids by Monte Carlo simulation in a new ensemble, Mol. Phys., 61 (1987), 813826.CrossRefGoogle Scholar
[9]Kofke, D. A., Gibbs-Duhem integration: A new method for direct evaluation of phase coexistence by molecular simulations, Mol. Phys., 78 (1993), 13311336.CrossRefGoogle Scholar
[10]Kofke, D. A., Direct evaluation of phase coexistence by molecular simulation via integration along the coexistence line, J. Chem. Phys., 98 (1993), 41494162.CrossRefGoogle Scholar
[11]Agrawal, R., Kofke, D. A., Solid-fluid coexistence for inverse-power potentials, Phys. Rev. Lett., 74 (1995), 122125.CrossRefGoogle ScholarPubMed
[12]Tilwani, P., Wu, D., Direct simulation of phase coexistence in solids using the Gibbs ensemble: Configuration annealing Monte Carlo, Master’s thesis, Department of Chemical Engineer-ing, Colorado School of Mines, Golden, Colorado, USA, 1999.Google Scholar
[13]Errington, J. R., Panagiotopoulos, A. Z., A fixed point charge model for water optimized to the vapor-liquid coexistence properties, J. Phys. Chem. B, 102 (1998), 74707475.CrossRefGoogle Scholar
[14]Smit, B., Karaborni, S., Siepmann, J. I., Computer simulation of vapor-liquid phase equilibria of n-alkanes, J. Chem. Phys., 102 (1995), 21262140.CrossRefGoogle Scholar
[15]Martin, M. G., Siepmann, J. I., Transferable models for phase equilibria 1. United-atom description of n-alkanes, J. Phys. Chem. B., 102 (1998), 25692577.CrossRefGoogle Scholar
[16]Nath, S. K., Escobedo, F. A., de Pablo, J. J., On the simulation of vapour-liquid equilibria for alkanes, J. Chem. Phys., 108 (1998), 99059911.CrossRefGoogle Scholar
[17]Errington, J. R., Panagiotopoulos, A. Z., A new intermolecular potential model for the n- alkane homologous series, J. Phys. Chem. B., 103 (1999), 63146322.CrossRefGoogle Scholar
[18]Potoff, J. J., Errington, J. R., Panagiotopoulos, A. Z., Molecular simulation of phase equilibria for mixtures of polar and non-polar components, Mol. Phys., 97 (1999), 10731083.CrossRefGoogle Scholar
[19]Ungerer, P., Lachet, V., Tavitian, B., Applications of molecular simulation in oil and gas production and processing, Oil & gas Science and technology - Rev. IFP, 61 (2006), 387403.CrossRefGoogle Scholar
[20]Hajipour, M., Aghamiri, S. F., Sabzyan, H., Seyedeyn-Azad, F., Extension of the exp-6 model to the simulation of vapor-liquid equilibria of primary alcohols and their mixtures, Fluid Phase Equilibria, 301 (2011), 7379.CrossRefGoogle Scholar
[21]Li, J., Sun, S., Calo, V. M., Monte Carlo molecular simulation of phase-coexistence for oil production and processing, SPE Reservoir Characterization and Simulation Conference and Exhibition, 2011, no. 148282.CrossRefGoogle Scholar
[22]Panagiotopoulos, A. Z., Quirke, N., Stapleton, M. R., Tildesley, D. J., Phase equilibria by simulations in the Gibbs ensemble: Alternative derivation, generalization and application to mixtures and membrane equilibria, Mol. Phys., 63 (1988), 527545.CrossRefGoogle Scholar
[23]Manousiouthakis, V. I., Deem, M. W., Strict detailed balance is unnecessary in Monte Carlo simulation, J. Chem. Phys., 110 (1999), 27532756.CrossRefGoogle Scholar
[24]Nicolas, J. J., Gubbins, K. E., Streett, W. B, Tildesley, D. J., Equation of state for the Lennard-Jones fluid, Mol. Phys., 37 (1979), 14291454.CrossRefGoogle Scholar
[25]Flyvbjerg, H., Petersen, H. G., Error estimates on averages of correlated data, J. Chem. Phys., 91(1) (1989), 461466.CrossRefGoogle Scholar
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On Stochastic Error and Computational Efficiency of the Markov Chain Monte Carlo Method
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