Polynomial metamodels with canonical lowrank approximations: Numerical insights and comparison to sparse polynomial chaos expansions
Abstract
The growing need for uncertainty analysis of complex computational models has led to an expanding use of metamodels across engineering and sciences. The efficiency of metamodeling techniques relies on their ability to provide statisticallyequivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing metamodels in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing metamodels with polynomial functions in highdimensional problems is offered by the newly emerged lowrank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRAmore »
 Authors:
 Publication Date:
 OSTI Identifier:
 22572358
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 321; Other Information: Copyright (c) 2016 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; CHAOS THEORY; EFFICIENCY; ERRORS; FINITE ELEMENT METHOD; MULTIVARIATE ANALYSIS; POLYNOMIALS; THERMAL CONDUCTION
Citation Formats
Konakli, Katerina, Email: konakli@ibk.baug.ethz.ch, and Sudret, Bruno. Polynomial metamodels with canonical lowrank approximations: Numerical insights and comparison to sparse polynomial chaos expansions. United States: N. p., 2016.
Web. doi:10.1016/J.JCP.2016.06.005.
Konakli, Katerina, Email: konakli@ibk.baug.ethz.ch, & Sudret, Bruno. Polynomial metamodels with canonical lowrank approximations: Numerical insights and comparison to sparse polynomial chaos expansions. United States. doi:10.1016/J.JCP.2016.06.005.
Konakli, Katerina, Email: konakli@ibk.baug.ethz.ch, and Sudret, Bruno. 2016.
"Polynomial metamodels with canonical lowrank approximations: Numerical insights and comparison to sparse polynomial chaos expansions". United States.
doi:10.1016/J.JCP.2016.06.005.
@article{osti_22572358,
title = {Polynomial metamodels with canonical lowrank approximations: Numerical insights and comparison to sparse polynomial chaos expansions},
author = {Konakli, Katerina, Email: konakli@ibk.baug.ethz.ch and Sudret, Bruno},
abstractNote = {The growing need for uncertainty analysis of complex computational models has led to an expanding use of metamodels across engineering and sciences. The efficiency of metamodeling techniques relies on their ability to provide statisticallyequivalent analytical representations based on relatively few evaluations of the original model. Polynomial chaos expansions (PCE) have proven a powerful tool for developing metamodels in a wide range of applications; the key idea thereof is to expand the model response onto a basis made of multivariate polynomials obtained as tensor products of appropriate univariate polynomials. The classical PCE approach nevertheless faces the “curse of dimensionality”, namely the exponential increase of the basis size with increasing input dimension. To address this limitation, the sparse PCE technique has been proposed, in which the expansion is carried out on only a few relevant basis terms that are automatically selected by a suitable algorithm. An alternative for developing metamodels with polynomial functions in highdimensional problems is offered by the newly emerged lowrank approximations (LRA) approach. By exploiting the tensor–product structure of the multivariate basis, LRA can provide polynomial representations in highly compressed formats. Through extensive numerical investigations, we herein first shed light on issues relating to the construction of canonical LRA with a particular greedy algorithm involving a sequential updating of the polynomial coefficients along separate dimensions. Specifically, we examine the selection of optimal rank, stopping criteria in the updating of the polynomial coefficients and error estimation. In the sequel, we confront canonical LRA to sparse PCE in structuralmechanics and heatconduction applications based on finiteelement solutions. Canonical LRA exhibit smaller errors than sparse PCE in cases when the number of available model evaluations is small with respect to the input dimension, a situation that is often encountered in reallife problems. By introducing the conditional generalization error, we further demonstrate that canonical LRA tend to outperform sparse PCE in the prediction of extreme model responses, which is critical in reliability analysis.},
doi = {10.1016/J.JCP.2016.06.005},
journal = {Journal of Computational Physics},
number = ,
volume = 321,
place = {United States},
year = 2016,
month = 9
}

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