Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs

Abdellah Chkifa; Albert Cohen; Ronald DeVore; Christoph Schwab

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 1, page 253-280
  • ISSN: 0764-583X

Abstract

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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 = H01(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.

How to cite

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Chkifa, Abdellah, et al. "Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.1 (2013): 253-280. <http://eudml.org/doc/273196>.

@article{Chkifa2013,
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 = H01(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.},
author = {Chkifa, Abdellah, Cohen, Albert, DeVore, Ronald, Schwab, Christoph},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {parametric and stochastic PDE’s; sparse polynomial approximation; high dimensional problems; adaptive algorithms; Taylor approximation; parametric and stochastic PDEs; high-dimensional problems; numerical experiments},
language = {eng},
number = {1},
pages = {253-280},
publisher = {EDP-Sciences},
title = {Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs},
url = {http://eudml.org/doc/273196},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Chkifa, Abdellah
AU - Cohen, Albert
AU - DeVore, Ronald
AU - Schwab, Christoph
TI - Sparse adaptive Taylor approximation algorithms for parametric and stochastic elliptic PDEs
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 1
SP - 253
EP - 280
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 = H01(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.
LA - eng
KW - parametric and stochastic PDE’s; sparse polynomial approximation; high dimensional problems; adaptive algorithms; Taylor approximation; parametric and stochastic PDEs; high-dimensional problems; numerical experiments
UR - http://eudml.org/doc/273196
ER -

References

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