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

¹ UniversitéParis-Est, CERMICS (ENPC), 6-8 Avenue Blaise Pascal, Cité Descartes, 77455 Marne-la-Vallée, France
² INRIA Rocquencourt, MICMAC Team-Project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France

Received: 5 December 2012
Revised: 13 June 2013

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.