Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method. (10th January 2014)

• Record Type: Journal Article
• Authors: Casenave, Fabien; Ern, Alexandre; Lelièvre, Tony
• Is Part Of: Mathematical modelling and numerical analysis. Volume 48:Part 1(2013)
• Page Start: 207
• Page End: 229
• Publication Date: 2014-01-10
• Subjects: Numerical analysis -- Periodicals; Mathematical models -- Periodicals; 510
• DOI: 10.1051/m2an/2013097
• Languages: English
• ISSNs: 0764-583X

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>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 <italic>a posteriori </italic>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 <italic>a posteriori </italic>error evaluation in the reduced basis method. <italic>C. R. Math. Acad. Sci. Paris </italic><bold>350 </bold>(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.</p> </abstract>