# Variance Estimation in Monte Carlo Eigenvalue Simulations Using Spectral Analysis Method

## Abstract

Monte Carlo (MC) simulation is widely used to solve the eigenvalue form of the Boltzmann transport equation that mathematically represents the neutron transport process through complex multiplying (fissionable) systems. Monte Carlo eigenvalue simulation starts with an assumed fission source distribution and uses the fission sites from the previous iteration (cycle) as the starting source in the current iteration. Important system parameters (MC tallies) such as fuel pin-power distribution are estimated over several cycles after the convergence of the fission source distribution to a stationary distribution. However, the MC fission source iteration algorithm that uses fission source sites from the previous cycle introduces a cycle-to-cycle correlation. Monte Carlo simulations that do not account for the cycle-to-cycle correlation systematically underestimate the variance of the estimated system parameters (sample mean). This paper presents the relationship between the spectral density in the frequency domain at frequency zero and the variance of the sample mean. This paper introduces a novel method in the frequency domain for the MC variance estimation. For the three test problems used in this paper, researchers have observed that the new method results in an improvement of more than one order of magnitude to the standard deviation of the sample mean.more »

- Authors:

- Texas A&M Univ.–Corpus Christi, Corpus Christi, 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:
- 1459299

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Nuclear Science and Engineering

- Additional Journal Information:
- Journal Volume: 191; Journal Issue: 3; Journal ID: ISSN 0029-5639

- Publisher:
- American Nuclear Society - Taylor & Francis

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 73 NUCLEAR PHYSICS AND RADIATION PHYSICS; Monte Carlo; variance estimation; periodogram; spectral density

### Citation Formats

```
Jin, Lei, and Banerjee, Kaushik.
```*Variance Estimation in Monte Carlo Eigenvalue Simulations Using Spectral Analysis Method*. United States: N. p., 2018.
Web. doi:10.1080/00295639.2018.1471269.

```
Jin, Lei, & Banerjee, Kaushik.
```*Variance Estimation in Monte Carlo Eigenvalue Simulations Using Spectral Analysis Method*. United States. doi:10.1080/00295639.2018.1471269.

```
Jin, Lei, and Banerjee, Kaushik. Thu .
"Variance Estimation in Monte Carlo Eigenvalue Simulations Using Spectral Analysis Method". United States.
doi:10.1080/00295639.2018.1471269.
```

```
@article{osti_1459299,
```

title = {Variance Estimation in Monte Carlo Eigenvalue Simulations Using Spectral Analysis Method},

author = {Jin, Lei and Banerjee, Kaushik},

abstractNote = {Monte Carlo (MC) simulation is widely used to solve the eigenvalue form of the Boltzmann transport equation that mathematically represents the neutron transport process through complex multiplying (fissionable) systems. Monte Carlo eigenvalue simulation starts with an assumed fission source distribution and uses the fission sites from the previous iteration (cycle) as the starting source in the current iteration. Important system parameters (MC tallies) such as fuel pin-power distribution are estimated over several cycles after the convergence of the fission source distribution to a stationary distribution. However, the MC fission source iteration algorithm that uses fission source sites from the previous cycle introduces a cycle-to-cycle correlation. Monte Carlo simulations that do not account for the cycle-to-cycle correlation systematically underestimate the variance of the estimated system parameters (sample mean). This paper presents the relationship between the spectral density in the frequency domain at frequency zero and the variance of the sample mean. This paper introduces a novel method in the frequency domain for the MC variance estimation. For the three test problems used in this paper, researchers have observed that the new method results in an improvement of more than one order of magnitude to the standard deviation of the sample mean. The new method also compares favorably with the previously introduced batch, bootstrap, and covariance-adjusted methods when applied to the three test problems investigated in this paper. This new method does not require modification of the MC eigenvalue algorithm (power iteration), is code agnostic, and is therefore easy to use when implementing in any existing MC code. In conclusion, the new estimate can be calculated without saving tally results of all active/stationary cycles.},

doi = {10.1080/00295639.2018.1471269},

journal = {Nuclear Science and Engineering},

number = 3,

volume = 191,

place = {United States},

year = {Thu Jun 28 00:00:00 EDT 2018},

month = {Thu Jun 28 00:00:00 EDT 2018}

}