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Manifold learning for the emulation of spatial fields from computational models
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Xing, W. W., Triantafyllidis, Vasileios, Shah, Akeel A., Nair, P. B. and Zabaras , Nicholas (2016) Manifold learning for the emulation of spatial fields from computational models. Journal of Computational Physics, 326 . 666 - 690. doi:10.1016/j.jcp.2016.07.040 ISSN 0021-9991.
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Official URL: http://dx.doi.org/10.1016/j.jcp.2016.07.040
Abstract
Repeated evaluations of expensive computer models in applications such as design optimization and uncertainty quantification can be computationally infeasible. For partial differential equation (PDE) models, the outputs of interest are often spatial fields leading to high-dimensional output spaces. Although emulators can be used to find faithful and computationally inexpensive approximations of computer models, there are few methods for handling high-dimensional output spaces. For Gaussian process (GP) emulation, approximations of the correlation structure and/or dimensionality reduction are necessary. Linear dimensionality reduction will fail when the output space is not well approximated by a linear subspace of the ambient space in which it lies. Manifold learning can overcome the limitations of linear methods if an accurate inverse map is available. In this paper, we use kernel PCA and diffusion maps to construct GP emulators for very high-dimensional output spaces arising from PDE model simulations. For diffusion maps we develop a new inverse map approximation. Several examples are presented to demonstrate the accuracy of our approach.
Item Type: | Journal Article | ||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Engineering > Engineering | ||||||||||
Library of Congress Subject Headings (LCSH): | Uncertainty -- Mathematical models, Principal components analysis , Differential equations, Partial | ||||||||||
Journal or Publication Title: | Journal of Computational Physics | ||||||||||
Publisher: | Academic Press Inc. Elsevier Science | ||||||||||
ISSN: | 0021-9991 | ||||||||||
Official Date: | 1 December 2016 | ||||||||||
Dates: |
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Volume: | 326 | ||||||||||
Page Range: | 666 - 690 | ||||||||||
DOI: | 10.1016/j.jcp.2016.07.040 | ||||||||||
Status: | Peer Reviewed | ||||||||||
Publication Status: | Published | ||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||
Date of first compliant deposit: | 7 February 2017 | ||||||||||
Date of first compliant Open Access: | 3 September 2017 | ||||||||||
Funder: | Engineering and Physical Sciences Research Council (EPSRC), China Scholarship Council (CSC), Seventh Framework Programme (European Commission) (FP7), University of Notre Dame. College of Engineering, Oak Ridge national laboratory (ORNL), Royal Society (Great Britain). Wolfson Research Merit Award (RSWRMA), Technische Universität München | ||||||||||
Grant number: | EP/L027682/1 (EPSRC), Grant No. 314159 (FP7), DARPA EQUiPS (ORNL) | ||||||||||
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