# Diagnostics for Undersampling and Clustering in Monte Carlo Criticality Calculations

## Abstract

For the past 40 years, it has been known from the theoretical work of Gelbard [1] and Brissenden [2] that the results from Monte Carlo (MC) criticality calculations are biased if the number of neutrons per generation is not sufficiently large. The computed k-effective eigenvalue will be lower than the actual eigenvalue, and the computed eigenfunction will be too low in high-importance regions and too high in low-importance regions. This difficulty is commonly called the undersampling problem for criticality calculations. The biases for undersampling were confirmed for practical, realistic problems with modern MC codes by Brown [3]. The solution to undersampling is simple - use a larger number of neutrons per generation to reduce the biases to levels that are negligible. Over the past few years, the OECD-NEA-WPNCS Expert Group on Advanced Monte Carlo Techniques (EGAMCT) [4,5,6] investigated an extreme form of undersampling called clustering. When the number of neutrons per generation is small, neutrons in successive generations tend to group together (cluster) due to correlations, and the clusters tend to migrate through the problem space. While the problems of undersampling and clustering are well known and well understood theoretically, there were no existing diagnostics for MC codes to determinemore »

- Authors:

- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Publication Date:

- Research Org.:
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1467193

- Report Number(s):
- LA-UR-18-27656

- DOE Contract Number:
- AC52-06NA25396

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; MCNP; neutron transport

### Citation Formats

```
Brown, Forrest B., and Josey, Colin James.
```*Diagnostics for Undersampling and Clustering in Monte Carlo Criticality Calculations*. United States: N. p., 2018.
Web. doi:10.2172/1467193.

```
Brown, Forrest B., & Josey, Colin James.
```*Diagnostics for Undersampling and Clustering in Monte Carlo Criticality Calculations*. United States. doi:10.2172/1467193.

```
Brown, Forrest B., and Josey, Colin James. Fri .
"Diagnostics for Undersampling and Clustering in Monte Carlo Criticality Calculations". United States. doi:10.2172/1467193. https://www.osti.gov/servlets/purl/1467193.
```

```
@article{osti_1467193,
```

title = {Diagnostics for Undersampling and Clustering in Monte Carlo Criticality Calculations},

author = {Brown, Forrest B. and Josey, Colin James},

abstractNote = {For the past 40 years, it has been known from the theoretical work of Gelbard [1] and Brissenden [2] that the results from Monte Carlo (MC) criticality calculations are biased if the number of neutrons per generation is not sufficiently large. The computed k-effective eigenvalue will be lower than the actual eigenvalue, and the computed eigenfunction will be too low in high-importance regions and too high in low-importance regions. This difficulty is commonly called the undersampling problem for criticality calculations. The biases for undersampling were confirmed for practical, realistic problems with modern MC codes by Brown [3]. The solution to undersampling is simple - use a larger number of neutrons per generation to reduce the biases to levels that are negligible. Over the past few years, the OECD-NEA-WPNCS Expert Group on Advanced Monte Carlo Techniques (EGAMCT) [4,5,6] investigated an extreme form of undersampling called clustering. When the number of neutrons per generation is small, neutrons in successive generations tend to group together (cluster) due to correlations, and the clusters tend to migrate through the problem space. While the problems of undersampling and clustering are well known and well understood theoretically, there were no existing diagnostics for MC codes to determine whether a given calculation is using sufficient neutrons per generation to eliminate these problems. That is, no tools were available to diagnose biases due to undersampling or clustering. The tools described herein resolve this longstanding problem by providing 2 robust methods for detecting undersampling and clustering in MC criticality calculations. The diagnosis of undersampling and clustering is part of a larger effort to thoroughly overhaul the calculational methods for criticality problems, providing for automated sampling of the initial fission distribution, acceleration of the iteration convergence process, automated detection of convergence in the iterations and starting of the active tally cycles, and the diagnosis of undersampling. All of these new methods have been prototyped in a local modified version of mcnp6.2 [7] and tested on a variety of critical systems.},

doi = {10.2172/1467193},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2018},

month = {8}

}