Diagnosing Undersampling in Monte Carlo Eigenvalue and Flux Tally Estimates
Abstract
This study explored the impact of undersampling on the accuracy of tally estimates in Monte Carlo (MC) calculations. Steadystate MC simulations were performed for models of several critical systems with varying degrees of spatial and isotopic complexity, and the impact of undersampling on eigenvalue and fuel pin flux/fission estimates was examined. This study observed biases in MC eigenvalue estimates as large as several percent and biases in fuel pin flux/fission tally estimates that exceeded tens, and in some cases hundreds, of percent. This study also investigated five statistical metrics for predicting the occurrence of undersampling biases in MC simulations. Three of the metrics (the HeidelbergerWelch RHW, the Geweke ZScore, and the GelmanRubin diagnostics) are commonly used for diagnosing the convergence of Markov chains, and two of the methods (the Contributing Particles per Generation and Tally Entropy) are new convergence metrics developed in the course of this study. These metrics were implemented in the KENO MC code within the SCALE code system and were evaluated for their reliability at predicting the onset and magnitude of undersampling biases in MC eigenvalue and flux tally estimates in two of the critical models. Of the five methods investigated, the HeidelbergerWelch RHW, the GelmanRubin diagnostics,more »
 Authors:
 ORNL
 Publication Date:
 Research Org.:
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1221732
 DOE Contract Number:
 DEAC0500OR22725
 Resource Type:
 Conference
 Resource Relation:
 Conference: ICNC 2015, Charlotte, NC, USA, 20150913, 20150917, San Jose, California United States, 813 June 2014
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; 22 GENERAL STUDIES OF NUCLEAR REACTORS; 42 ENGINEERING
Citation Formats
Perfetti, Christopher M, and Rearden, Bradley T. Diagnosing Undersampling in Monte Carlo Eigenvalue and Flux Tally Estimates. United States: N. p., 2015.
Web.
Perfetti, Christopher M, & Rearden, Bradley T. Diagnosing Undersampling in Monte Carlo Eigenvalue and Flux Tally Estimates. United States.
Perfetti, Christopher M, and Rearden, Bradley T. 2015.
"Diagnosing Undersampling in Monte Carlo Eigenvalue and Flux Tally Estimates". United States.
doi:. https://www.osti.gov/servlets/purl/1221732.
@article{osti_1221732,
title = {Diagnosing Undersampling in Monte Carlo Eigenvalue and Flux Tally Estimates},
author = {Perfetti, Christopher M and Rearden, Bradley T},
abstractNote = {This study explored the impact of undersampling on the accuracy of tally estimates in Monte Carlo (MC) calculations. Steadystate MC simulations were performed for models of several critical systems with varying degrees of spatial and isotopic complexity, and the impact of undersampling on eigenvalue and fuel pin flux/fission estimates was examined. This study observed biases in MC eigenvalue estimates as large as several percent and biases in fuel pin flux/fission tally estimates that exceeded tens, and in some cases hundreds, of percent. This study also investigated five statistical metrics for predicting the occurrence of undersampling biases in MC simulations. Three of the metrics (the HeidelbergerWelch RHW, the Geweke ZScore, and the GelmanRubin diagnostics) are commonly used for diagnosing the convergence of Markov chains, and two of the methods (the Contributing Particles per Generation and Tally Entropy) are new convergence metrics developed in the course of this study. These metrics were implemented in the KENO MC code within the SCALE code system and were evaluated for their reliability at predicting the onset and magnitude of undersampling biases in MC eigenvalue and flux tally estimates in two of the critical models. Of the five methods investigated, the HeidelbergerWelch RHW, the GelmanRubin diagnostics, and Tally Entropy produced test metrics that correlated strongly to the size of the observed undersampling biases, indicating their potential to effectively predict the size and prevalence of undersampling biases in MC simulations.},
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journal = {},
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place = {United States},
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Here, this study focuses on understanding the phenomena in Monte Carlo simulations known as undersampling, in which Monte Carlo tally estimates may not encounter a sufficient number of particles during each generation to obtain unbiased tally estimates. Steadystate Monte Carlo simulations were performed using the KENO Monte Carlo tools within the SCALE code system for models of several burnup credit applications with varying degrees of spatial and isotopic complexities, and the incidence and impact of undersampling on eigenvalue and flux estimates were examined. Using an inadequate number of particle histories in each generation was found to produce a maximum bias of ~100 pcm in eigenvalue estimates and biases that exceeded 10% in fuel pin flux tally estimates. Having quantified the potential magnitude of undersampling biases in eigenvalue and flux tally estimates in these systems, this study then investigated whether Markov Chain Monte Carlo convergence metrics could be integrated into Monte Carlo simulations to predict the onset and magnitude of undersampling biases. Five potential metrics for identifying undersampling biases were implemented in the SCALE code system and evaluated for their ability to predict undersampling biases by comparing the test metric scores with the observed undersampling biases. Finally, of the five convergence metrics that were investigated, three (the HeidelbergerWelch relative halfwidth, the GelmanRubin ^{more » $$\hat{R}_c$$} diagnostic, and tally entropy) showed the potential to accurately predict the behavior of undersampling biases in the responses examined.

Metrics for Diagnosing Undersampling in Monte Carlo Tally Estimates
This study explored the potential of using Markov chain convergence diagnostics to predict the prevalence and magnitude of biases due to undersampling in Monte Carlo eigenvalue and flux tally estimates. Five metrics were applied to two models of pressurized water reactor fuel assemblies and their potential for identifying undersampling biases was evaluated by comparing the calculated test metrics with known biases in the tallies. Three of the five undersampling metrics showed the potential to accurately predict the behavior of undersampling biases in the responses examined in this study. 
UnderPrediction of Localized Tally Uncertainties in Monte Carlo Eigenvalue Calculations
Modeling and simulation using Monte Carlo methods is widely used in nuclear reactor criticality benchmarking applications. However, obtaining good statistics not only takes a large amount of computational time, but it has been shown that localized tally uncertainties may be underpredicted by a factor of five or more in select cases. The primary components of this underprediction include poor sampling due to improper source convergence and cycletocycle correlations in the fission source. Additional components relate to the flux shape and the size of the tally cells. These issues must be understood and dealt with in order to support the practicalmore » 
Quantifying the Effect of Undersampling in Monte Carlo Simulations Using SCALE
This study explores the effect of undersampling in Monte Carlo calculations on tally estimates and tally variance estimates for burnup credit applications. Steadystate Monte Carlo simulations were performed for models of several critical systems with varying degrees of spatial and isotopic complexity and the impact of undersampling on eigenvalue and flux estimates was examined. Using an inadequate number of particle histories in each generation was found to produce an approximately 100 pcm bias in the eigenvalue estimates, and biases that exceeded 10% in fuel pin flux estimates.