TY - JOUR

T1 - Inverse regression-based uncertainty quantification algorithms for high-dimensional models

T2 - Theory and practice

AU - Li, Weixuan

AU - Lin, Guang

AU - Li, Bing

N1 - Funding Information:
This work is supported by the U.S. Department of Energy, Office of Science, Office of Advanced Scientific Computing Research, Applied Mathematics program as part of the Multifaceted Mathematics for Complex Energy Systems (M 2 ACS) project and part of the Collaboratory on Mathematics for Mesoscopic Modeling of Materials project, and National Science Foundation Grants DMS-1555072 and DMS-1407537 . Pacific Northwest National Laboratory is operated by Battelle for the DOE under Contract DE-AC05-76RL01830 .
Publisher Copyright:
© 2016 Elsevier Inc.

PY - 2016/9/15

Y1 - 2016/9/15

N2 - Many uncertainty quantification (UQ) approaches suffer from the curse of dimensionality, that is, their computational costs become intractable for problems involving a large number of uncertainty parameters. In these situations, the classic Monte Carlo often remains the preferred method of choice because its convergence rate O(n-1/2), where n is the required number of model simulations, does not depend on the dimension of the problem. However, many high-dimensional UQ problems are intrinsically low-dimensional, because the variation of the quantity of interest (QoI) is often caused by only a few latent parameters varying within a low-dimensional subspace, known as the sufficient dimension reduction (SDR) subspace in the statistics literature. Motivated by this observation, we propose two inverse regression-based UQ algorithms (IRUQ) for high-dimensional problems. Both algorithms use inverse regression to convert the original high-dimensional problem to a low-dimensional one, which is then efficiently solved by building a response surface for the reduced model, for example via the polynomial chaos expansion. The first algorithm, which is for the situations where an exact SDR subspace exists, is proved to converge at rate O(n-1), hence much faster than MC. The second algorithm, which doesn't require an exact SDR, employs the reduced model as a control variate to reduce the error of the MC estimate. The accuracy gain could still be significant, depending on how well the reduced model approximates the original high-dimensional one. IRUQ also provides several additional practical advantages: it is non-intrusive; it does not require computing the high-dimensional gradient of the QoI; and it reports an error bar so the user knows how reliable the result is.

AB - Many uncertainty quantification (UQ) approaches suffer from the curse of dimensionality, that is, their computational costs become intractable for problems involving a large number of uncertainty parameters. In these situations, the classic Monte Carlo often remains the preferred method of choice because its convergence rate O(n-1/2), where n is the required number of model simulations, does not depend on the dimension of the problem. However, many high-dimensional UQ problems are intrinsically low-dimensional, because the variation of the quantity of interest (QoI) is often caused by only a few latent parameters varying within a low-dimensional subspace, known as the sufficient dimension reduction (SDR) subspace in the statistics literature. Motivated by this observation, we propose two inverse regression-based UQ algorithms (IRUQ) for high-dimensional problems. Both algorithms use inverse regression to convert the original high-dimensional problem to a low-dimensional one, which is then efficiently solved by building a response surface for the reduced model, for example via the polynomial chaos expansion. The first algorithm, which is for the situations where an exact SDR subspace exists, is proved to converge at rate O(n-1), hence much faster than MC. The second algorithm, which doesn't require an exact SDR, employs the reduced model as a control variate to reduce the error of the MC estimate. The accuracy gain could still be significant, depending on how well the reduced model approximates the original high-dimensional one. IRUQ also provides several additional practical advantages: it is non-intrusive; it does not require computing the high-dimensional gradient of the QoI; and it reports an error bar so the user knows how reliable the result is.

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U2 - 10.1016/j.jcp.2016.05.040

DO - 10.1016/j.jcp.2016.05.040

M3 - Article

AN - SCOPUS:84971637794

SN - 0021-9991

VL - 321

SP - 259

EP - 278

JO - Journal of Computational Physics

JF - Journal of Computational Physics

ER -