Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method

Fabien Casenave; Alexandre Ern; Tony Lelièvre

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2014)

  • Volume: 48, Issue: 1, page 207-229
  • ISSN: 0764-583X

Abstract

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The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539–542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the empirical interpolation method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.

How to cite

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Casenave, Fabien, Ern, Alexandre, and Lelièvre, Tony. "Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 48.1 (2014): 207-229. <http://eudml.org/doc/273118>.

@article{Casenave2014,
abstract = {The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539–542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the empirical interpolation method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.},
author = {Casenave, Fabien, Ern, Alexandre, Lelièvre, Tony},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {reduced basis method; a posteriori error bound; round-off errors; boundary element method; empirical interpolation method; acoustics; Helmholtz equation; numerical examples; diffusion problem; acoustic scattering},
language = {eng},
number = {1},
pages = {207-229},
publisher = {EDP-Sciences},
title = {Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method},
url = {http://eudml.org/doc/273118},
volume = {48},
year = {2014},
}

TY - JOUR
AU - Casenave, Fabien
AU - Ern, Alexandre
AU - Lelièvre, Tony
TI - Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2014
PB - EDP-Sciences
VL - 48
IS - 1
SP - 207
EP - 229
AB - The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive to round-off errors. We propose herein an explanation of this fact. A first remedy has been proposed in [F. Casenave, Accurate a posteriori error evaluation in the reduced basis method. C. R. Math. Acad. Sci. Paris 350 (2012) 539–542.]. Herein, we improve this remedy by proposing a new approximation of the error bound using the empirical interpolation method (EIM). This method achieves higher levels of accuracy and requires potentially less precomputations than the usual formula. A version of the EIM stabilized with respect to round-off errors is also derived. The method is illustrated on a simple one-dimensional diffusion problem and a three-dimensional acoustic scattering problem solved by a boundary element method.
LA - eng
KW - reduced basis method; a posteriori error bound; round-off errors; boundary element method; empirical interpolation method; acoustics; Helmholtz equation; numerical examples; diffusion problem; acoustic scattering
UR - http://eudml.org/doc/273118
ER -

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