Casenave, Fabien; Ern, Alexandre; Lelièvre, Tony Accurate and online-efficient evaluation of the a posteriori error bound in the reduced basis method. (English) Zbl 1288.65157 ESAIM, Math. Model. Numer. Anal. 48, No. 1, 207-229 (2014). Summary: 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, C. R., Math., Acad. Sci. Paris 350, No. 9–10, 539–542 (2012; Zbl 1245.65105)]. 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. Cited in 11 Documents MSC: 65N15 Error bounds for boundary value problems involving PDEs 65N38 Boundary element methods for boundary value problems involving PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation 76Q05 Hydro- and aero-acoustics 76M15 Boundary element methods applied to problems in fluid mechanics Keywords:reduced basis method; a posteriori error bound; round-off errors; boundary element method; empirical interpolation method; Helmholtz equation; numerical examples; diffusion problem; acoustic scattering Citations:Zbl 1245.65105 Software:rbMIT PDFBibTeX XMLCite \textit{F. Casenave} et al., ESAIM, Math. Model. Numer. Anal. 48, No. 1, 207--229 (2014; Zbl 1288.65157) Full Text: DOI arXiv