On stochastic error and computational efficiency of the Markov Chain Monte Carlo method

Handle URI:
http://hdl.handle.net/10754/563327
Title:
On stochastic error and computational efficiency of the Markov Chain Monte Carlo method
Authors:
Li, Jun; Vignal, Philippe ( 0000-0001-5300-6930 ) ; Sun, Shuyu ( 0000-0002-3078-864X ) ; Calo, Victor M. ( 0000-0002-1805-4045 )
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 theMCMCmethod 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. © 2014 Global-Science Press.
KAUST Department:
Numerical Porous Media SRI Center (NumPor); Materials Science and Engineering Program; Applied Mathematics and Computational Science Program; Earth Science and Engineering Program; Physical Sciences and Engineering (PSE) Division; Environmental Science and Engineering Program; Computational Transport Phenomena Lab
Publisher:
Global Science Press
Journal:
Communications in Computational Physics
Issue Date:
1-Jan-2014
DOI:
10.4208/cicp.110613.280214a
Type:
Article
ISSN:
18152406
Sponsors:
This work was supported in part by the King Abdullah University of Science and Technology (KAUST) Center for Numerical Porous Media. In addition, S. Sun would also like to acknowledge the support of this study by a research award from King Abdulaziz City for Science and Technology (KACST) through a project entitled "Study of Sulfur Solubility using Thermodynamics Model and Quantum Chemistry".
Appears in Collections:
Articles; Environmental Science and Engineering Program; Applied Mathematics and Computational Science Program; Physical Sciences and Engineering (PSE) Division; Earth Science and Engineering Program; Materials Science and Engineering Program; Computational Transport Phenomena Lab

Full metadata record

DC FieldValue Language
dc.contributor.authorLi, Junen
dc.contributor.authorVignal, Philippeen
dc.contributor.authorSun, Shuyuen
dc.contributor.authorCalo, Victor M.en
dc.date.accessioned2015-08-03T11:45:50Zen
dc.date.available2015-08-03T11:45:50Zen
dc.date.issued2014-01-01en
dc.identifier.issn18152406en
dc.identifier.doi10.4208/cicp.110613.280214aen
dc.identifier.urihttp://hdl.handle.net/10754/563327en
dc.description.abstractIn 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 theMCMCmethod 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. © 2014 Global-Science Press.en
dc.description.sponsorshipThis work was supported in part by the King Abdullah University of Science and Technology (KAUST) Center for Numerical Porous Media. In addition, S. Sun would also like to acknowledge the support of this study by a research award from King Abdulaziz City for Science and Technology (KACST) through a project entitled "Study of Sulfur Solubility using Thermodynamics Model and Quantum Chemistry".en
dc.publisherGlobal Science Pressen
dc.subjectBlocking methoden
dc.subjectGibbs ensembleen
dc.subjectMarkov Chain Monte Carlo methoden
dc.subjectMolecular simulationen
dc.subjectPhase coexistenceen
dc.subjectVariance estimationen
dc.titleOn stochastic error and computational efficiency of the Markov Chain Monte Carlo methoden
dc.typeArticleen
dc.contributor.departmentNumerical Porous Media SRI Center (NumPor)en
dc.contributor.departmentMaterials Science and Engineering Programen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentEarth Science and Engineering Programen
dc.contributor.departmentPhysical Sciences and Engineering (PSE) Divisionen
dc.contributor.departmentEnvironmental Science and Engineering Programen
dc.contributor.departmentComputational Transport Phenomena Laben
dc.identifier.journalCommunications in Computational Physicsen
kaust.authorVignal, Philippeen
kaust.authorSun, Shuyuen
kaust.authorCalo, Victor M.en
kaust.authorLi, Junen
All Items in KAUST are protected by copyright, with all rights reserved, unless otherwise indicated.