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

Published online by Cambridge University Press:  10 January 2014

Fabien Casenave ,
Alexandre Ern  and
Tony Lelièvre
Fabien Casenave
Affiliation:
UniversitéParis-Est, CERMICS (ENPC), 6-8 Avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France
Alexandre Ern
Affiliation:
UniversitéParis-Est, CERMICS (ENPC), 6-8 Avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France
Tony Lelièvre
Affiliation:
UniversitéParis-Est, CERMICS (ENPC), 6-8 Avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France INRIA Rocquencourt, MICMAC Team-Project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France
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Abstract

The reduced basis method is a model reduction technique yielding substantial savings ofcomputational time when a solution to a parametrized equation has to be computed for manyvalues of the parameter. Certification of the approximation is possible by means of ana posteriori error bound. Under appropriate assumptions, this errorbound is computed with an algorithm of complexity independent of the size of the fullproblem. In practice, the evaluation of the error bound can become very sensitive toround-off errors. We propose herein an explanation of this fact. A first remedy has beenproposed in [F. Casenave, Accurate a posteriori error evaluation in thereduced basis method. C. R. Math. Acad. Sci. Paris 350(2012) 539–542.]. Herein, we improve this remedy by proposing a new approximationof the error bound using the empirical interpolation method (EIM). This method achieveshigher levels of accuracy and requires potentially less precomputations than the usualformula. 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 athree-dimensional acoustic scattering problem solved by a boundary element method.

Keywords

Reduced basis methoda posteriori error boundround-off errorsboundary element methodempirical interpolation methodacoustics
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 1 , January 2014 , pp. 207 - 229
Copyright
© EDP Sciences, SMAI, 2014

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