# Monte Carlo methods for flux expansion solutions of transport problems

## Abstract

Adaptive Monte Carlo methods, based on the use of either correlated sampling or importance sampling, to obtain global solutions to certain transport problems have recently been described. The resulting learning algorithms are capable of achieving geometric convergence when applied to the estimation of a finite number of coefficients in a flux expansion representation of the global solution. However, because of the nonphysical nature of the random walk simulations needed to perform importance sampling, conventional transport estimators and source sampling techniques require modification to be used successfully in conjunction with such flux expansion methods. It is shown how these problems can be overcome. First, the traditional path length estimators in wide use in particle transport simulations are generalized to include rather general detector functions (which, in this application, are the individual basis functions chosen for the flus expansion). Second, it is shown how to sample from the signed probabilities that arise as source density functions in these applications, without destroying the zero variance property needed to ensure geometric convergence to zero error.

- Authors:

- Claremont Graduate Univ., CA (United States). Dept. of Mathematics

- Publication Date:

- OSTI Identifier:
- 687543

- Resource Type:
- Journal Article

- Journal Name:
- Nuclear Science and Engineering

- Additional Journal Information:
- Journal Volume: 133; Journal Issue: 1; Other Information: PBD: Sep 1999

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 66 PHYSICS; MONTE CARLO METHOD; SAMPLING; TRANSPORT THEORY; ALGORITHMS; CONVERGENCE; RADIATION FLUX; PROBABILISTIC ESTIMATION

### Citation Formats

```
Spanier, J.
```*Monte Carlo methods for flux expansion solutions of transport problems*. United States: N. p., 1999.
Web.

```
Spanier, J.
```*Monte Carlo methods for flux expansion solutions of transport problems*. United States.

```
Spanier, J. Wed .
"Monte Carlo methods for flux expansion solutions of transport problems". United States.
```

```
@article{osti_687543,
```

title = {Monte Carlo methods for flux expansion solutions of transport problems},

author = {Spanier, J.},

abstractNote = {Adaptive Monte Carlo methods, based on the use of either correlated sampling or importance sampling, to obtain global solutions to certain transport problems have recently been described. The resulting learning algorithms are capable of achieving geometric convergence when applied to the estimation of a finite number of coefficients in a flux expansion representation of the global solution. However, because of the nonphysical nature of the random walk simulations needed to perform importance sampling, conventional transport estimators and source sampling techniques require modification to be used successfully in conjunction with such flux expansion methods. It is shown how these problems can be overcome. First, the traditional path length estimators in wide use in particle transport simulations are generalized to include rather general detector functions (which, in this application, are the individual basis functions chosen for the flus expansion). Second, it is shown how to sample from the signed probabilities that arise as source density functions in these applications, without destroying the zero variance property needed to ensure geometric convergence to zero error.},

doi = {},

journal = {Nuclear Science and Engineering},

number = 1,

volume = 133,

place = {United States},

year = {1999},

month = {9}

}