Multilevel weighted least squares polynomial approximation

Handle URI:
http://hdl.handle.net/10754/626523
Title:
Multilevel weighted least squares polynomial approximation
Authors:
Haji-Ali, Abdul-Lateef; Nobile, Fabio; Tempone, Raul ( 0000-0003-1967-4446 ) ; Wolfers, Sören
Abstract:
Weighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.
KAUST Department:
Applied Mathematics and Computational Science Program
Publisher:
arXiv
KAUST Grant Number:
2281; 2584
Issue Date:
30-Jun-2017
ARXIV:
arXiv:1707.00026
Type:
Preprint
Additional Links:
http://arxiv.org/abs/1707.00026v3; http://arxiv.org/pdf/1707.00026v3
Appears in Collections:
Other/General Submission; Applied Mathematics and Computational Science Program

Full metadata record

DC FieldValue Language
dc.contributor.authorHaji-Ali, Abdul-Lateefen
dc.contributor.authorNobile, Fabioen
dc.contributor.authorTempone, Raulen
dc.contributor.authorWolfers, Sörenen
dc.date.accessioned2017-12-28T07:32:14Z-
dc.date.available2017-12-28T07:32:14Z-
dc.date.issued2017-06-30en
dc.identifier.urihttp://hdl.handle.net/10754/626523-
dc.description.abstractWeighted least squares polynomial approximation uses random samples to determine projections of functions onto spaces of polynomials. It has been shown that, using an optimal distribution of sample locations, the number of samples required to achieve quasi-optimal approximation in a given polynomial subspace scales, up to a logarithmic factor, linearly in the dimension of this space. However, in many applications, the computation of samples includes a numerical discretization error. Thus, obtaining polynomial approximations with a single level method can become prohibitively expensive, as it requires a sufficiently large number of samples, each computed with a sufficiently small discretization error. As a solution to this problem, we propose a multilevel method that utilizes samples computed with different accuracies and is able to match the accuracy of single-level approximations with reduced computational cost. We derive complexity bounds under certain assumptions about polynomial approximability and sample work. Furthermore, we propose an adaptive algorithm for situations where such assumptions cannot be verified a priori. Finally, we provide an efficient algorithm for the sampling from optimal distributions and an analysis of computationally favorable alternative distributions. Numerical experiments underscore the practical applicability of our method.en
dc.publisherarXiven
dc.relation.urlhttp://arxiv.org/abs/1707.00026v3en
dc.relation.urlhttp://arxiv.org/pdf/1707.00026v3en
dc.rightsArchived with thanks to arXiven
dc.titleMultilevel weighted least squares polynomial approximationen
dc.typePreprinten
dc.contributor.departmentApplied Mathematics and Computational Science Programen
dc.eprint.versionPre-printen
dc.contributor.institutionOxford Universityen
dc.contributor.institutionÉcole Polytechnique Fédérale de Lausanne (EPFL), CSQI-MATHen
dc.identifier.arxividarXiv:1707.00026en
kaust.authorTempone, Raulen
kaust.authorWolfers, Sörenen
kaust.grant.number2281en
kaust.grant.number2584en
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