Polynomial chaos representation of databases on manifolds
Abstract
Characterizing the polynomial chaos expansion (PCE) of a vectorvalued random variable with probability distribution concentrated on a manifold is a relevant problem in datadriven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by datadriven queries.
 Authors:
 Université ParisEst, Laboratoire Modélisation et Simulation MultiEchelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 MarneLaVallée, Cedex 2 (France)
 University of Southern California, 210 KAP Hall, Los Angeles, CA 90089 (United States)
 Publication Date:
 OSTI Identifier:
 22622278
 Resource Type:
 Journal Article
 Resource Relation:
 Journal Name: Journal of Computational Physics; Journal Volume: 335; Other Information: Copyright (c) 2017 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:
 97 MATHEMATICAL METHODS AND COMPUTING; CHAOS THEORY; CONCENTRATION RATIO; CONVERGENCE; DATASETS; DIFFUSION; DISTRIBUTION; EXPANSION; POLYNOMIALS; PROBABILITY; RANDOMNESS; SAMPLING; STATISTICS
Citation Formats
Soize, C., Email: christian.soize@univparisest.fr, and Ghanem, R., Email: ghanem@usc.edu. Polynomial chaos representation of databases on manifolds. United States: N. p., 2017.
Web. doi:10.1016/J.JCP.2017.01.031.
Soize, C., Email: christian.soize@univparisest.fr, & Ghanem, R., Email: ghanem@usc.edu. Polynomial chaos representation of databases on manifolds. United States. doi:10.1016/J.JCP.2017.01.031.
Soize, C., Email: christian.soize@univparisest.fr, and Ghanem, R., Email: ghanem@usc.edu. Sat .
"Polynomial chaos representation of databases on manifolds". United States.
doi:10.1016/J.JCP.2017.01.031.
@article{osti_22622278,
title = {Polynomial chaos representation of databases on manifolds},
author = {Soize, C., Email: christian.soize@univparisest.fr and Ghanem, R., Email: ghanem@usc.edu},
abstractNote = {Characterizing the polynomial chaos expansion (PCE) of a vectorvalued random variable with probability distribution concentrated on a manifold is a relevant problem in datadriven settings. The probability distribution of such random vectors is multimodal in general, leading to potentially very slow convergence of the PCE. In this paper, we build on a recent development for estimating and sampling from probabilities concentrated on a diffusion manifold. The proposed methodology constructs a PCE of the random vector together with an associated generator that samples from the target probability distribution which is estimated from data concentrated in the neighborhood of the manifold. The method is robust and remains efficient for high dimension and large datasets. The resulting polynomial chaos construction on manifolds permits the adaptation of many uncertainty quantification and statistical tools to emerging questions motivated by datadriven queries.},
doi = {10.1016/J.JCP.2017.01.031},
journal = {Journal of Computational Physics},
number = ,
volume = 335,
place = {United States},
year = {Sat Apr 15 00:00:00 EDT 2017},
month = {Sat Apr 15 00:00:00 EDT 2017}
}

Two numerical techniques are proposed to construct a polynomial chaos (PC) representation of an arbitrary secondorder random vector. In the first approach, a PC representation is constructed by matching a target joint probability density function (pdf) based on sequential conditioning (a sequence of conditional probability relations) in conjunction with the Rosenblatt transformation. In the second approach, the PC representation is obtained by having recourse to the Rosenblatt transformation and simultaneously matching a set of target marginal pdfs and target Spearman's rank correlation coefficient (SRCC) matrix. Both techniques are applied to model an experimental spatiotemporal data set, exhibiting strong nonstationary andmore »

Polynomial metamodels with canonical lowrank approximations: Numerical insights and comparison to sparse polynomial chaos expansions
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 themore »