## A spectral analysis of the domain decomposed Monte Carlo method for linear systems

## Abstract

The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear oper- ator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approxi- mation and the mean chord approximation are applied to estimate the leakage frac- tion of random walks from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem in numerical experiments to test the models for symmetric operators with spectral qualities similar to light water reactor problems. We find, in general, the derived approximations show good agreement with random walk lengths and leakage fractions computed by the numerical experiments.

- Authors:

- Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
- Univ. of Wisconsin, Madison, WI (United States)

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Science (SC)

- OSTI Identifier:
- 1265476

- Alternate Identifier(s):
- OSTI ID: 1247411

- Grant/Contract Number:
- AC05-00OR22725

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Nuclear Engineering and Design

- Additional Journal Information:
- Journal Volume: 295; Journal Issue: C; Journal ID: ISSN 0029-5493

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Slattery, Stuart R., Evans, Thomas M., and Wilson, Paul P. H. A spectral analysis of the domain decomposed Monte Carlo method for linear systems. United States: N. p., 2015.
Web. doi:10.1016/j.nucengdes.2015.07.054.
```

```
Slattery, Stuart R., Evans, Thomas M., & Wilson, Paul P. H. A spectral analysis of the domain decomposed Monte Carlo method for linear systems. United States. doi:10.1016/j.nucengdes.2015.07.054.
```

```
Slattery, Stuart R., Evans, Thomas M., and Wilson, Paul P. H. Tue .
"A spectral analysis of the domain decomposed Monte Carlo method for linear systems". United States. doi:10.1016/j.nucengdes.2015.07.054. https://www.osti.gov/servlets/purl/1265476.
```

```
@article{osti_1265476,
```

title = {A spectral analysis of the domain decomposed Monte Carlo method for linear systems},

author = {Slattery, Stuart R. and Evans, Thomas M. and Wilson, Paul P. H.},

abstractNote = {The domain decomposed behavior of the adjoint Neumann-Ulam Monte Carlo method for solving linear systems is analyzed using the spectral properties of the linear oper- ator. Relationships for the average length of the adjoint random walks, a measure of convergence speed and serial performance, are made with respect to the eigenvalues of the linear operator. In addition, relationships for the effective optical thickness of a domain in the decomposition are presented based on the spectral analysis and diffusion theory. Using the effective optical thickness, the Wigner rational approxi- mation and the mean chord approximation are applied to estimate the leakage frac- tion of random walks from a domain in the decomposition as a measure of parallel performance and potential communication costs. The one-speed, two-dimensional neutron diffusion equation is used as a model problem in numerical experiments to test the models for symmetric operators with spectral qualities similar to light water reactor problems. We find, in general, the derived approximations show good agreement with random walk lengths and leakage fractions computed by the numerical experiments.},

doi = {10.1016/j.nucengdes.2015.07.054},

journal = {Nuclear Engineering and Design},

number = C,

volume = 295,

place = {United States},

year = {2015},

month = {9}

}