  Reduced order models, in particular the reduced basis method, rely on
empirically built and problem dependent basis functions that are constructed
during an off-line stage. In the on-line stage, the precomputed
problem-dependent solution space, that is spanned by the basis functions, can
then be used in order to reduce the size of the computational problem. For
complex problems, the number of basis functions required to guarantee a certain
error tolerance can become too large in order to benefit computationally from
the model reduction. To overcome this, the present work introduces a framework
where local approximation spaces (in parameter space) are used to define the
reduced order approximation in order to have explicit control over the on-line
cost. This approach also adapts the local approximation spaces to local
anisotropic behavior in the parameter space. We present the algorithm and
numerous numerical tests.
