  In this paper, we study numerically the linear damped second-order hyperbolic
partial differential equation (PDE) with affine parameter dependence using a
goal-oriented approach by finite element (FE) and reduced basis (RB) methods.
The main contribution of this paper is the "goal-oriented" proper orthogonal
decomposition (POD)-Greedy sampling procedure within the RB approximation
context. First, we introduce the RB recipe: Galerkin projection onto a space
$Y_N$ spanned by solutions of the governing PDE at $N$ selected points in
parameter space. This set of $N$ parameter points is constructed by the
standard POD-Greedy sampling procedure already developed. Second, based on the
affine parameter dependence, we make use of the offline-online computational
procedures: in the offline stage, we generate the RB space; in the online
stage, given a new parameter value, we calculate rapidly and accurately the
space-time RB output of interest and its associated asymptotic error. The
proposed goal-oriented POD-Greedy sampling procedure can now be implemented and
will look for the parameter points such that it minimizes this (asymptotic)
output error rather than the solution error (or, error indicator which is the
dual norm of residual) as in the standard POD-Greedy procedure. Numerical
results show that the new goal-oriented POD-Greedy sampling procedure improves
significantly the accuracy of the space-time output computation in comparison
with the standard POD-Greedy one. The method is thus ideally suited for
repeated, rapid and reliable evaluation of input-output relationships within
the space-time setting.
