Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs

Handle URI:
http://hdl.handle.net/10754/599679
Title:
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
Authors:
Chkifa, Abdellah; Cohen, Albert; DeVore, Ronald; Schwab, Christoph
Abstract:
The numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated in the Hilbert space V = H0 1(D) by multivariate sparse polynomials in the parameter vector y with a controlled number N of terms. The convergence rate in terms of N does not depend on the number of parameters in V, which may be arbitrarily large or countably infinite, thereby breaking the curse of dimensionality. However, these approximation results do not describe the concrete construction of these polynomial expansions, and should therefore rather be viewed as benchmark for the convergence analysis of numerical methods. The present paper presents an adaptive numerical algorithm for constructing a sequence of sparse polynomials that is proved to converge toward the solution with the optimal benchmark rate. Numerical experiments are presented in large parameter dimension, which confirm the effectiveness of the adaptive approach. © 2012 EDP Sciences, SMAI.
Citation:
Chkifa A, Cohen A, DeVore R, Schwab C (2012) Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis 47: 253–280. Available: http://dx.doi.org/10.1051/m2an/2012027.
Publisher:
EDP Sciences
Journal:
ESAIM: Mathematical Modelling and Numerical Analysis
KAUST Grant Number:
KUS-C1-016-04
Issue Date:
29-Nov-2012
DOI:
10.1051/m2an/2012027
Type:
Article
ISSN:
0764-583X; 1290-3841
Sponsors:
This research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107, the AFOSR Contract FA95500910500, the ARO/DoD Contracts W911NF-05-1-0227 and W911NF-07-1-0185, the National Science Foundation Grant DMS 0915231; the excellency chair of the Foundation "Science Mathematiques de Paris" awarded to Ronald DeVore in 2009. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). This research is also supported by the Swiss National Science Foundation under Grant SNF 200021-120290/1 and by the European Research Council under grant ERC AdG247277. CS acknowledges hospitality by the Hausdorff Institute for Mathematics, Bonn, Germany.
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Full metadata record

DC FieldValue Language
dc.contributor.authorChkifa, Abdellahen
dc.contributor.authorCohen, Alberten
dc.contributor.authorDeVore, Ronalden
dc.contributor.authorSchwab, Christophen
dc.date.accessioned2016-02-28T06:07:21Zen
dc.date.available2016-02-28T06:07:21Zen
dc.date.issued2012-11-29en
dc.identifier.citationChkifa A, Cohen A, DeVore R, Schwab C (2012) Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis 47: 253–280. Available: http://dx.doi.org/10.1051/m2an/2012027.en
dc.identifier.issn0764-583Xen
dc.identifier.issn1290-3841en
dc.identifier.doi10.1051/m2an/2012027en
dc.identifier.urihttp://hdl.handle.net/10754/599679en
dc.description.abstractThe numerical approximation of parametric partial differential equations is a computational challenge, in particular when the number of involved parameter is large. This paper considers a model class of second order, linear, parametric, elliptic PDEs on a bounded domain D with diffusion coefficients depending on the parameters in an affine manner. For such models, it was shown in [9, 10] that under very weak assumptions on the diffusion coefficients, the entire family of solutions to such equations can be simultaneously approximated in the Hilbert space V = H0 1(D) by multivariate sparse polynomials in the parameter vector y with a controlled number N of terms. The convergence rate in terms of N does not depend on the number of parameters in V, which may be arbitrarily large or countably infinite, thereby breaking the curse of dimensionality. However, these approximation results do not describe the concrete construction of these polynomial expansions, and should therefore rather be viewed as benchmark for the convergence analysis of numerical methods. The present paper presents an adaptive numerical algorithm for constructing a sequence of sparse polynomials that is proved to converge toward the solution with the optimal benchmark rate. Numerical experiments are presented in large parameter dimension, which confirm the effectiveness of the adaptive approach. © 2012 EDP Sciences, SMAI.en
dc.description.sponsorshipThis research was supported by the Office of Naval Research Contracts ONR-N00014-08-1-1113, ONR N00014-09-1-0107, the AFOSR Contract FA95500910500, the ARO/DoD Contracts W911NF-05-1-0227 and W911NF-07-1-0185, the National Science Foundation Grant DMS 0915231; the excellency chair of the Foundation "Science Mathematiques de Paris" awarded to Ronald DeVore in 2009. This publication is based on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST). This research is also supported by the Swiss National Science Foundation under Grant SNF 200021-120290/1 and by the European Research Council under grant ERC AdG247277. CS acknowledges hospitality by the Hausdorff Institute for Mathematics, Bonn, Germany.en
dc.publisherEDP Sciencesen
dc.subjectAdaptive algorithmsen
dc.subjectHigh dimensional problemsen
dc.subjectParametric and stochastic PDE'sen
dc.subjectSparse polynomial approximationen
dc.titleSparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEsen
dc.typeArticleen
dc.identifier.journalESAIM: Mathematical Modelling and Numerical Analysisen
dc.contributor.institutionUniversite Pierre et Marie Curie, Paris, Franceen
dc.contributor.institutionCNRS Centre National de la Recherche Scientifique, Paris, Franceen
dc.contributor.institutionTexas A and M University, College Station, United Statesen
dc.contributor.institutionEidgenossische Technische Hochschule Zurich, Zurich, Switzerlanden
kaust.grant.numberKUS-C1-016-04en
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