# A kinetic theory for nonanalog Monte Carlo particle transport algorithms: Exponential transform with angular biasing in planar-geometry anisotropically scattering media

## Abstract

The authors show that Monte Carlo simulations of neutral particle transport in planargeometry anisotropically scattering media, using the exponential transform with angular biasing as a variance reduction device, are governed by a new Boltzman Monte Carlo (BMC) equation, which includes particle weight as an extra independent variable. The weight moments of the solution of the BMC equation determine the moments of the score and the mean number of collisions per history in the nonanalog Monte Carlo simulations. Therefore, the solution of the BMC equation predicts the variance of the score and the figure of merit in the simulation. Also, by (1) using an angular biasing function that is closely related to the ``asymptotic`` solution of the linear Boltzman equation and (2) requiring isotropic weight changes as collisions, they derive a new angular biasing scheme. Using the BMC equation, they propose a universal ``safe`` upper limit of the transform parameter, valid for any type of exponential transform. In numerical calculations, they demonstrate that the behavior of the Monte Carlo simulations and the performance predicted by deterministically solving the BMC equation agree well, and that the new angular biasing scheme is always advantageous.

- Authors:

- Univ. of Michigan, Ann Arbor, MI (United States). Dept. of Nuclear Engineering and Radiological Sciences

- Publication Date:

- OSTI Identifier:
- 653492

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Computational Physics

- Additional Journal Information:
- Journal Volume: 145; Journal Issue: 1; Other Information: PBD: 1 Sep 1998

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 66 PHYSICS; NEUTRAL-PARTICLE TRANSPORT; MONTE CARLO METHOD; ANISOTROPY; BOLTZMANN EQUATION; ALGORITHMS; WEIGHTING FUNCTIONS

### Citation Formats

```
Ueki, T., and Larsen, E.W.
```*A kinetic theory for nonanalog Monte Carlo particle transport algorithms: Exponential transform with angular biasing in planar-geometry anisotropically scattering media*. United States: N. p., 1998.
Web. doi:10.1006/jcph.1998.6039.

```
Ueki, T., & Larsen, E.W.
```*A kinetic theory for nonanalog Monte Carlo particle transport algorithms: Exponential transform with angular biasing in planar-geometry anisotropically scattering media*. United States. doi:10.1006/jcph.1998.6039.

```
Ueki, T., and Larsen, E.W. Tue .
"A kinetic theory for nonanalog Monte Carlo particle transport algorithms: Exponential transform with angular biasing in planar-geometry anisotropically scattering media". United States. doi:10.1006/jcph.1998.6039.
```

```
@article{osti_653492,
```

title = {A kinetic theory for nonanalog Monte Carlo particle transport algorithms: Exponential transform with angular biasing in planar-geometry anisotropically scattering media},

author = {Ueki, T. and Larsen, E.W.},

abstractNote = {The authors show that Monte Carlo simulations of neutral particle transport in planargeometry anisotropically scattering media, using the exponential transform with angular biasing as a variance reduction device, are governed by a new Boltzman Monte Carlo (BMC) equation, which includes particle weight as an extra independent variable. The weight moments of the solution of the BMC equation determine the moments of the score and the mean number of collisions per history in the nonanalog Monte Carlo simulations. Therefore, the solution of the BMC equation predicts the variance of the score and the figure of merit in the simulation. Also, by (1) using an angular biasing function that is closely related to the ``asymptotic`` solution of the linear Boltzman equation and (2) requiring isotropic weight changes as collisions, they derive a new angular biasing scheme. Using the BMC equation, they propose a universal ``safe`` upper limit of the transform parameter, valid for any type of exponential transform. In numerical calculations, they demonstrate that the behavior of the Monte Carlo simulations and the performance predicted by deterministically solving the BMC equation agree well, and that the new angular biasing scheme is always advantageous.},

doi = {10.1006/jcph.1998.6039},

journal = {Journal of Computational Physics},

number = 1,

volume = 145,

place = {United States},

year = {1998},

month = {9}

}