  We present a subsampling strategy for the offline stage of the Reduced Basis
Method. The approach is aimed at bringing down the considerable offline costs
associated with using a finely-sampled training set. The proposed algorithm
exploits the potential of the pivoted QR decomposition and the discrete
empirical interpolation method to identify important parameter samples. It
consists of two stages. In the first stage, we construct a low-fidelity
approximation to the solution manifold over a fine training set. Then, for the
available low-fidelity snapshots of the output variable, we apply the pivoted
QR decomposition or the discrete empirical interpolation method to identify a
set of sparse sampling locations in the parameter domain. These points reveal
the structure of the parametric dependence of the output variable. The second
stage proceeds with a subsampled training set containing a by far smaller
number of parameters than the initial training set. Different subsampling
strategies inspired from recent variants of the empirical interpolation method
are also considered. Tests on benchmark examples justify the new approach and
show its potential to substantially speed up the offline stage of the Reduced
Basis Method, while generating reliable reduced-order models.
