# Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice

## Abstract

A well-known challenge in uncertainty quantification (UQ) is the "curse of dimensionality". However, many high-dimensional UQ problems are essentially low-dimensional, because the randomness of the quantity of interest (QoI) is caused only by uncertain parameters varying within a low-dimensional subspace, known as the sufficient dimension reduction (SDR) subspace. Motivated by this observation, we propose and demonstrate in this paper an inverse regression-based UQ approach (IRUQ) for high-dimensional problems. Specifically, we use an inverse regression procedure to estimate the SDR subspace and then convert the original problem to a low-dimensional one, which can be efficiently solved by building a response surface model such as a polynomial chaos expansion. The novelty and advantages of the proposed approach is seen in its computational efficiency and practicality. Comparing with Monte Carlo, the traditionally preferred approach for high-dimensional UQ, IRUQ with a comparable cost generally gives much more accurate solutions even for high-dimensional problems, and even when the dimension reduction is not exactly sufficient. Theoretically, IRUQ is proved to converge twice as fast as the approach it uses seeking the SDR subspace. For example, while a sliced inverse regression method converges to the SDR subspace at the rate of $$O(n^{-1/2})$$, the corresponding IRUQ converges at $$O(n^{-1})$$. IRUQ also provides several desired conveniences in practice. It is non-intrusive, requiring only a simulator to generate realizations of the QoI, and there is no need to compute the high-dimensional gradient of the QoI. Finally, error bars can be derived for the estimation results reported by IRUQ.

- Authors:

- Publication Date:

- Research Org.:
- Pacific Northwest National Lab. (PNNL), Richland, WA (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1290388

- Report Number(s):
- PNNL-SA-113365

Journal ID: ISSN 0021-9991; KJ0401000

- DOE Contract Number:
- AC05-76RL01830

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Computational Physics; Journal Volume: 321

- Country of Publication:
- United States

- Language:
- English

### Citation Formats

```
Li, Weixuan, Lin, Guang, and Li, Bing.
```*Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice*. United States: N. p., 2016.
Web. doi:10.1016/j.jcp.2016.05.040.

```
Li, Weixuan, Lin, Guang, & Li, Bing.
```*Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice*. United States. doi:10.1016/j.jcp.2016.05.040.

```
Li, Weixuan, Lin, Guang, and Li, Bing. Thu .
"Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice". United States.
doi:10.1016/j.jcp.2016.05.040.
```

```
@article{osti_1290388,
```

title = {Inverse regression-based uncertainty quantification algorithms for high-dimensional models: Theory and practice},

author = {Li, Weixuan and Lin, Guang and Li, Bing},

abstractNote = {A well-known challenge in uncertainty quantification (UQ) is the "curse of dimensionality". However, many high-dimensional UQ problems are essentially low-dimensional, because the randomness of the quantity of interest (QoI) is caused only by uncertain parameters varying within a low-dimensional subspace, known as the sufficient dimension reduction (SDR) subspace. Motivated by this observation, we propose and demonstrate in this paper an inverse regression-based UQ approach (IRUQ) for high-dimensional problems. Specifically, we use an inverse regression procedure to estimate the SDR subspace and then convert the original problem to a low-dimensional one, which can be efficiently solved by building a response surface model such as a polynomial chaos expansion. The novelty and advantages of the proposed approach is seen in its computational efficiency and practicality. Comparing with Monte Carlo, the traditionally preferred approach for high-dimensional UQ, IRUQ with a comparable cost generally gives much more accurate solutions even for high-dimensional problems, and even when the dimension reduction is not exactly sufficient. Theoretically, IRUQ is proved to converge twice as fast as the approach it uses seeking the SDR subspace. For example, while a sliced inverse regression method converges to the SDR subspace at the rate of $O(n^{-1/2})$, the corresponding IRUQ converges at $O(n^{-1})$. IRUQ also provides several desired conveniences in practice. It is non-intrusive, requiring only a simulator to generate realizations of the QoI, and there is no need to compute the high-dimensional gradient of the QoI. Finally, error bars can be derived for the estimation results reported by IRUQ.},

doi = {10.1016/j.jcp.2016.05.040},

journal = {Journal of Computational Physics},

number = ,

volume = 321,

place = {United States},

year = {Thu Sep 01 00:00:00 EDT 2016},

month = {Thu Sep 01 00:00:00 EDT 2016}

}