Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
Abdellah Chkifa, Albert Cohen, Ronald DeVore, Christoph Schwab
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.
ESAIM: Mathematical Modelling and Numerical Analysis
https://doi.org/10.1051/m2an/2012027
Published - Nov 29 2012
Acknowledged KAUST grant number(s): KUS-C1-016-04
Acknowledgements: 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.
This publication acknowledges KAUST support, but has no KAUST affiliated authors.
10.1051/m2an/2012027
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Parametric Mathematics 100%
Number Mathematics 100%
Polynomial Mathematics 66%
PDE Mathematics 66%
Partial Differential Equation Physics 66%
Diffusivity Physics 66%
Independent Variables Physics 66%
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(1), 253-280. https://doi.org/10.1051/m2an/2012027
Chkifa, Abdellah ; Cohen, Albert ; DeVore, Ronald et al. / Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. In: ESAIM: Mathematical Modelling and Numerical Analysis. 2012 ; Vol. 47, No. 1. pp. 253-280.
@article{71bbcdfb095d424083f5c1dcc2b13bbb,
title = "Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs",
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. {\textcopyright} 2012 EDP Sciences, SMAI.",
author = "Abdellah Chkifa and Albert Cohen and Ronald DeVore and Christoph Schwab",
note = "KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.",
doi = "10.1051/m2an/2012027",
journal = "ESAIM: Mathematical Modelling and Numerical Analysis",
publisher = "EDP Sciences",
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, vol. 47, no. 1, pp. 253-280. https://doi.org/10.1051/m2an/2012027
Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. / Chkifa, Abdellah; Cohen, Albert; DeVore, Ronald et al.
In: ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 47, No. 1, 29.11.2012, p. 253-280.
T1 - Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
AU - Chkifa, Abdellah
AU - Cohen, Albert
AU - DeVore, Ronald
AU - Schwab, Christoph
N1 - KAUST Repository Item: Exported on 2020-10-01 Acknowledged KAUST grant number(s): KUS-C1-016-04 Acknowledgements: 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. This publication acknowledges KAUST support, but has no KAUST affiliated authors.
N2 - 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.
AB - 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.
UR - http://www.esaim-m2an.org/10.1051/m2an/2012027
U2 - 10.1051/m2an/2012027
DO - 10.1051/m2an/2012027
JO - ESAIM: Mathematical Modelling and Numerical Analysis
JF - ESAIM: Mathematical Modelling and Numerical Analysis
Chkifa A, Cohen A, DeVore R, Schwab C. Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs. ESAIM: Mathematical Modelling and Numerical Analysis. 2012 Nov 29;47(1):253-280. doi: 10.1051/m2an/2012027