# Improving variance estimation in Monte Carlo eigenvalue simulations

## Abstract

We report that Monte Carlo (MC) methods have been widely used to solve eigenvalue problems in complex nuclear systems. Once a stationary fission source is obtained in MC simulations, the sample mean of many stationary cycles is calculated. Variance or standard deviation of the sample mean is needed to indicate the level of statistical uncertainty of the simulation and to understand the convergence of the sample mean. Current MC codes typically use sample variance to estimate the statistical uncertainty of the simulation and assume that the MC stationary cycles are independent. However, there is a correlation between these cycles, and estimators of the variance that ignore these correlations will systematically underestimate the variance. This paper discusses some statistical properties of the sample mean and the asymptotic variance and introduces two novel estimators based on (a) covariance-adjusted methods and (b) bootstrap methods to reduce the variance underestimation. For three test problems, it has been observed that both new methods can improve the estimation of the standard deviation of the sample mean by more than an order of magnitude. In addition, some interesting patterns were revealed for these estimates over the spatial regions, providing additional insights into MC simulations for nuclear systems.more »

- Authors:

- Texas A&M Univ., College Station, TX (United States)
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Publication Date:

- Research Org.:
- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1394244

- DOE Contract Number:
- AC05-00OR22725

- Resource Type:
- Journal Article

- Journal Name:
- Annals of Nuclear Energy (Oxford)

- Additional Journal Information:
- Journal Volume: 110; Journal Issue: C; Journal ID: ISSN 0306-4549

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Monte Carlo; Variance estimation; Bootstrap

### Citation Formats

```
Jin, Lei, Banerjee, Kaushik, Hamilton, Steven P., and Davidson, Gregory G.
```*Improving variance estimation in Monte Carlo eigenvalue simulations*. United States: N. p., 2017.
Web. doi:10.1016/j.anucene.2017.07.016.

```
Jin, Lei, Banerjee, Kaushik, Hamilton, Steven P., & Davidson, Gregory G.
```*Improving variance estimation in Monte Carlo eigenvalue simulations*. United States. doi:10.1016/j.anucene.2017.07.016.

```
Jin, Lei, Banerjee, Kaushik, Hamilton, Steven P., and Davidson, Gregory G. Thu .
"Improving variance estimation in Monte Carlo eigenvalue simulations". United States. doi:10.1016/j.anucene.2017.07.016.
```

```
@article{osti_1394244,
```

title = {Improving variance estimation in Monte Carlo eigenvalue simulations},

author = {Jin, Lei and Banerjee, Kaushik and Hamilton, Steven P. and Davidson, Gregory G.},

abstractNote = {We report that Monte Carlo (MC) methods have been widely used to solve eigenvalue problems in complex nuclear systems. Once a stationary fission source is obtained in MC simulations, the sample mean of many stationary cycles is calculated. Variance or standard deviation of the sample mean is needed to indicate the level of statistical uncertainty of the simulation and to understand the convergence of the sample mean. Current MC codes typically use sample variance to estimate the statistical uncertainty of the simulation and assume that the MC stationary cycles are independent. However, there is a correlation between these cycles, and estimators of the variance that ignore these correlations will systematically underestimate the variance. This paper discusses some statistical properties of the sample mean and the asymptotic variance and introduces two novel estimators based on (a) covariance-adjusted methods and (b) bootstrap methods to reduce the variance underestimation. For three test problems, it has been observed that both new methods can improve the estimation of the standard deviation of the sample mean by more than an order of magnitude. In addition, some interesting patterns were revealed for these estimates over the spatial regions, providing additional insights into MC simulations for nuclear systems. These new methodologies are based on post-processing the tally results and are therefore easy to implement and code agnostic.},

doi = {10.1016/j.anucene.2017.07.016},

journal = {Annals of Nuclear Energy (Oxford)},

issn = {0306-4549},

number = C,

volume = 110,

place = {United States},

year = {2017},

month = {7}

}