## This content will become publicly available on January 17, 2020

## Gradient-Free Construction of Active Subspaces for Dimension Reduction in Complex Models with Applications to Neutronics

## Abstract

Recent developments in the field of reduced-order modeling---and, in particular, active subspace construction---have made it possible to efficiently approximate complex models by constructing low-order response surfaces based upon a small subspace of the original high-dimensional parameter space. These methods rely upon the fact that the response tends to vary more prominently in a few dominant directions defined by linear combinations of the original inputs, allowing for a rotation of the coordinate axis and a consequent transformation of the parameters. In this work, we discuss a gradient-free active subspace algorithm that is feasible for high-dimensional parameter spaces where finite-difference techniques are impractical. This analysis extends the gradient-free algorithm introduced in [A. Lewis, R. Smith, and B. Williams, Comput. Math. Appl., 72 (2016), pp. 1603--1615] in two significant ways: (i) we introduce an initialization algorithm to identify lower-dimensional subspaces of influential directions to seed the gradient-free algorithm for high-dimensional problems, and (ii) we analyze dimension selection criteria to verify the algorithms. Finally, we illustrate the initialized gradient-free active subspace algorithm for a neutronics example implemented with SCALE6.1 for input dimensions up to 7700.

- Authors:

- North Carolina State Univ., Raleigh, NC (United States)
- Lafayette College, Easton, PA (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Iowa State Univ., Ames, IA (United States)
- University of Sharjah (United Arab Emirates)

- Publication Date:

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

- Sponsoring Org.:
- USDOE Office of Nuclear Energy (NE)

- OSTI Identifier:
- 1530778

- Report Number(s):
- LA-UR-18-25392

Journal ID: ISSN 2166-2525

- Grant/Contract Number:
- 89233218CNA000001; AC05-00OR22725

- Resource Type:
- Accepted Manuscript

- Journal Name:
- SIAM/ASA Journal on Uncertainty Quantification

- Additional Journal Information:
- Journal Volume: 7; Journal Issue: 1; Journal ID: ISSN 2166-2525

- Publisher:
- SIAM

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; gradient-free active subspaces; neutronics applications

### Citation Formats

```
Coleman, Kayla D., Lewis, Allison, Smith, Ralph C., Williams, Brian J., Morris, Max D., and Khuwaileh, Bassam. Gradient-Free Construction of Active Subspaces for Dimension Reduction in Complex Models with Applications to Neutronics. United States: N. p., 2019.
Web. doi:10.1137/16M1075119.
```

```
Coleman, Kayla D., Lewis, Allison, Smith, Ralph C., Williams, Brian J., Morris, Max D., & Khuwaileh, Bassam. Gradient-Free Construction of Active Subspaces for Dimension Reduction in Complex Models with Applications to Neutronics. United States. doi:10.1137/16M1075119.
```

```
Coleman, Kayla D., Lewis, Allison, Smith, Ralph C., Williams, Brian J., Morris, Max D., and Khuwaileh, Bassam. Thu .
"Gradient-Free Construction of Active Subspaces for Dimension Reduction in Complex Models with Applications to Neutronics". United States. doi:10.1137/16M1075119.
```

```
@article{osti_1530778,
```

title = {Gradient-Free Construction of Active Subspaces for Dimension Reduction in Complex Models with Applications to Neutronics},

author = {Coleman, Kayla D. and Lewis, Allison and Smith, Ralph C. and Williams, Brian J. and Morris, Max D. and Khuwaileh, Bassam},

abstractNote = {Recent developments in the field of reduced-order modeling---and, in particular, active subspace construction---have made it possible to efficiently approximate complex models by constructing low-order response surfaces based upon a small subspace of the original high-dimensional parameter space. These methods rely upon the fact that the response tends to vary more prominently in a few dominant directions defined by linear combinations of the original inputs, allowing for a rotation of the coordinate axis and a consequent transformation of the parameters. In this work, we discuss a gradient-free active subspace algorithm that is feasible for high-dimensional parameter spaces where finite-difference techniques are impractical. This analysis extends the gradient-free algorithm introduced in [A. Lewis, R. Smith, and B. Williams, Comput. Math. Appl., 72 (2016), pp. 1603--1615] in two significant ways: (i) we introduce an initialization algorithm to identify lower-dimensional subspaces of influential directions to seed the gradient-free algorithm for high-dimensional problems, and (ii) we analyze dimension selection criteria to verify the algorithms. Finally, we illustrate the initialized gradient-free active subspace algorithm for a neutronics example implemented with SCALE6.1 for input dimensions up to 7700.},

doi = {10.1137/16M1075119},

journal = {SIAM/ASA Journal on Uncertainty Quantification},

number = 1,

volume = 7,

place = {United States},

year = {2019},

month = {1}

}