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.