# Stochastic richardson extrapolation based numerical error estimation for kinetic plasma simulations.

## Abstract

We present a numerical error estimation technique specifically formulated to deal with stochastic code output with multiple discretization parameters. This method is based on multiple fits to an error model with arbitrary convergence rates and cross-coupling terms, performed using nonlinear optimization. The fitting approach varies by the type of residual norm which influences the importance of outliers, and weights which influences the relative importance of data points in the coarse and refined regions of discretization parameter space. To account for the influence of stochastic noise, these fits are performed on multiple bootstrap values based on the underlying response data set. Using an automated discretization domain selection scheme, the fits are performed on a series of reduced sets of discretization levels in order to find an optimal fully-converged result estimate in the minimum variance sense; this automated approach enables straightforward analysis of multiple quantities of interest and/or time and spatially-dependent response data. The overall numerical error analysis method is useful for verification and validation problems for stochastic simulation methods and forms a key component in the overall uncertainty quantification process. The method was demonstrated for steady and unsteady electron diode problems simulated using a particle-in-cell kinetic plasma code, demonstrating excellent results.

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

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

- OSTI Identifier:
- 1504853

- Report Number(s):
- SAND2015-8620

607347

- DOE Contract Number:
- AC04-94AL85000

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Radtke, Gregg Arthur, Cartwright, Keith, and Musson, Lawrence.
```*Stochastic richardson extrapolation based numerical error estimation for kinetic plasma simulations.*. United States: N. p., 2015.
Web. doi:10.2172/1504853.

```
Radtke, Gregg Arthur, Cartwright, Keith, & Musson, Lawrence.
```*Stochastic richardson extrapolation based numerical error estimation for kinetic plasma simulations.*. United States. doi:10.2172/1504853.

```
Radtke, Gregg Arthur, Cartwright, Keith, and Musson, Lawrence. Thu .
"Stochastic richardson extrapolation based numerical error estimation for kinetic plasma simulations.". United States. doi:10.2172/1504853. https://www.osti.gov/servlets/purl/1504853.
```

```
@article{osti_1504853,
```

title = {Stochastic richardson extrapolation based numerical error estimation for kinetic plasma simulations.},

author = {Radtke, Gregg Arthur and Cartwright, Keith and Musson, Lawrence},

abstractNote = {We present a numerical error estimation technique specifically formulated to deal with stochastic code output with multiple discretization parameters. This method is based on multiple fits to an error model with arbitrary convergence rates and cross-coupling terms, performed using nonlinear optimization. The fitting approach varies by the type of residual norm which influences the importance of outliers, and weights which influences the relative importance of data points in the coarse and refined regions of discretization parameter space. To account for the influence of stochastic noise, these fits are performed on multiple bootstrap values based on the underlying response data set. Using an automated discretization domain selection scheme, the fits are performed on a series of reduced sets of discretization levels in order to find an optimal fully-converged result estimate in the minimum variance sense; this automated approach enables straightforward analysis of multiple quantities of interest and/or time and spatially-dependent response data. The overall numerical error analysis method is useful for verification and validation problems for stochastic simulation methods and forms a key component in the overall uncertainty quantification process. The method was demonstrated for steady and unsteady electron diode problems simulated using a particle-in-cell kinetic plasma code, demonstrating excellent results.},

doi = {10.2172/1504853},

journal = {},

number = ,

volume = ,

place = {United States},

year = {2015},

month = {10}

}