Inverse regressionbased uncertainty quantification algorithms for highdimensional models: Theory and practice
Abstract
A wellknown challenge in uncertainty quantification (UQ) is the "curse of dimensionality". However, many highdimensional UQ problems are essentially lowdimensional, because the randomness of the quantity of interest (QoI) is caused only by uncertain parameters varying within a lowdimensional subspace, known as the sufficient dimension reduction (SDR) subspace. Motivated by this observation, we propose and demonstrate in this paper an inverse regressionbased UQ approach (IRUQ) for highdimensional problems. Specifically, we use an inverse regression procedure to estimate the SDR subspace and then convert the original problem to a lowdimensional 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 highdimensional UQ, IRUQ with a comparable cost generally gives much more accurate solutions even for highdimensional 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 nonintrusive, requiring only a simulator to generate realizations of the QoI, and there is no need to compute the highdimensional 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):
 PNNLSA113365
Journal ID: ISSN 00219991; KJ0401000
 DOE Contract Number:
 AC0576RL01830
 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 regressionbased uncertainty quantification algorithms for highdimensional 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 regressionbased uncertainty quantification algorithms for highdimensional models: Theory and practice. United States. doi:10.1016/j.jcp.2016.05.040.
Li, Weixuan, Lin, Guang, and Li, Bing. 2016.
"Inverse regressionbased uncertainty quantification algorithms for highdimensional models: Theory and practice". United States.
doi:10.1016/j.jcp.2016.05.040.
@article{osti_1290388,
title = {Inverse regressionbased uncertainty quantification algorithms for highdimensional models: Theory and practice},
author = {Li, Weixuan and Lin, Guang and Li, Bing},
abstractNote = {A wellknown challenge in uncertainty quantification (UQ) is the "curse of dimensionality". However, many highdimensional UQ problems are essentially lowdimensional, because the randomness of the quantity of interest (QoI) is caused only by uncertain parameters varying within a lowdimensional subspace, known as the sufficient dimension reduction (SDR) subspace. Motivated by this observation, we propose and demonstrate in this paper an inverse regressionbased UQ approach (IRUQ) for highdimensional problems. Specifically, we use an inverse regression procedure to estimate the SDR subspace and then convert the original problem to a lowdimensional 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 highdimensional UQ, IRUQ with a comparable cost generally gives much more accurate solutions even for highdimensional 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 nonintrusive, requiring only a simulator to generate realizations of the QoI, and there is no need to compute the highdimensional 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 = 2016,
month = 9
}

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