Analysis of Discrete L2 Projection on Polynomial Spaces with Random Evaluations

Handle URI:
http://hdl.handle.net/10754/563433
Title:
Analysis of Discrete L2 Projection on Polynomial Spaces with Random Evaluations
Authors:
Migliorati, Giovanni; Nobile, Fabio; von Schwerin, Erik; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
We analyze the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is uncertainty quantification for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the univariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero and bounded from above, provided the number of samples scales quadratically with the dimension of the polynomial space. Optimality is meant in the sense that the weighted L2 norm of the error committed by the random discrete projection is bounded with high probability from above by the best L∞ error achievable in the given polynomial space, up to logarithmic factors. Several numerical tests are presented in both the univariate and multivariate cases, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function. © 2014 SFoCM.
KAUST Department:
Applied Mathematics and Computational Science Program; Center for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ); Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division; Stochastic Numerics Research Group
Publisher:
Springer Science + Business Media
Journal:
Foundations of Computational Mathematics
Issue Date:
5-Mar-2014
DOI:
10.1007/s10208-013-9186-4
Type:
Article
ISSN:
16153375
Sponsors:
The authors would like to recognize the support of the PECOS Center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar" and King Abdullah University of Science and Technology (KAUST) through the AEA projects "Predictability and Uncertainty Quantification for Models of Porous Media" and "Tracking Uncertainties in Computational Modeling of Reactive Systems" is also acknowledged. R. Tempone is a member of the KAUST SRI Center for Uncertainty Quantification. The first and second authors were supported by the Italian Grant FIRB-IDEAS (Project RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems." We are indebted to A. Cohen and R. DeVore for giving us valuable feedback on the convergence proof. We would also like to thank the anonymous referees for their useful comments that helped us to improve considerably the manuscript, and in particular for the suggestion in the proof of Theorem 2.
Appears in Collections:
Articles; Applied Mathematics and Computational Science Program; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorMigliorati, Giovannien
dc.contributor.authorNobile, Fabioen
dc.contributor.authorvon Schwerin, Eriken
dc.contributor.authorTempone, Raulen
dc.date.accessioned2015-08-03T11:51:21Zen
dc.date.available2015-08-03T11:51:21Zen
dc.date.issued2014-03-05en
dc.identifier.issn16153375en
dc.identifier.doi10.1007/s10208-013-9186-4en
dc.identifier.urihttp://hdl.handle.net/10754/563433en
dc.description.abstractWe analyze the problem of approximating a multivariate function by discrete least-squares projection on a polynomial space starting from random, noise-free observations. An area of possible application of such technique is uncertainty quantification for computational models. We prove an optimal convergence estimate, up to a logarithmic factor, in the univariate case, when the observation points are sampled in a bounded domain from a probability density function bounded away from zero and bounded from above, provided the number of samples scales quadratically with the dimension of the polynomial space. Optimality is meant in the sense that the weighted L2 norm of the error committed by the random discrete projection is bounded with high probability from above by the best L∞ error achievable in the given polynomial space, up to logarithmic factors. Several numerical tests are presented in both the univariate and multivariate cases, confirming our theoretical estimates. The numerical tests also clarify how the convergence rate depends on the number of sampling points, on the polynomial degree, and on the smoothness of the target function. © 2014 SFoCM.en
dc.description.sponsorshipThe authors would like to recognize the support of the PECOS Center at ICES, University of Texas at Austin (Project Number 024550, Center for Predictive Computational Science). Support from the VR project "Effektiva numeriska metoder for stokastiska differentialekvationer med tillampningar" and King Abdullah University of Science and Technology (KAUST) through the AEA projects "Predictability and Uncertainty Quantification for Models of Porous Media" and "Tracking Uncertainties in Computational Modeling of Reactive Systems" is also acknowledged. R. Tempone is a member of the KAUST SRI Center for Uncertainty Quantification. The first and second authors were supported by the Italian Grant FIRB-IDEAS (Project RBID08223Z) "Advanced numerical techniques for uncertainty quantification in engineering and life science problems." We are indebted to A. Cohen and R. DeVore for giving us valuable feedback on the convergence proof. We would also like to thank the anonymous referees for their useful comments that helped us to improve considerably the manuscript, and in particular for the suggestion in the proof of Theorem 2.en
dc.publisherSpringer Science + Business Mediaen
dc.subjectApproximation theoryen
dc.subjectError analysisen
dc.subjectGeneralized polynomial chaosen
dc.subjectMultivariate polynomial approximationen
dc.subjectNoise-free dataen
dc.subjectNonparametric regressionen
dc.subjectPoint collocationen
dc.titleAnalysis of Discrete L2 Projection on Polynomial Spaces with Random Evaluationsen
dc.typeArticleen
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.contributor.departmentCenter for Uncertainty Quantification in Computational Science and Engineering (SRI-UQ)en
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.contributor.departmentStochastic Numerics Research Groupen
dc.identifier.journalFoundations of Computational Mathematicsen
dc.contributor.institutionCSQI-MATHICSE, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerlanden
kaust.authorvon Schwerin, Eriken
kaust.authorTempone, Raulen
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