  We use asymptotically optimal \emph{adaptive} numerical methods (here
specifically a wavelet scheme) for snapshot computations within the offline
phase of the Reduced Basis Method (RBM). The resulting discretizations for each
snapshot (i.e., parameter-dependent) do not permit the standard RB `truth
space', but allow for error estimation of the RB approximation with respect to
the exact solution of the considered parameterized partial differential
equation.
  The residual-based a posteriori error estimators are computed by an adaptive
dual wavelet expansion, which allows us to compute a surrogate of the dual norm
of the residual. The resulting adaptive RBM is analyzed. We show the
convergence of the resulting adaptive Greedy method. Numerical experiments for
stationary and instationary problems underline the potential of this approach.
