  The reduced basis method (RBM) empowers repeated and rapid evaluation of
parametrized partial differential equations through an offline-online
decomposition, a.k.a. a learning-execution process. A key feature of the method
is a greedy algorithm repeatedly scanning the training set, a fine
discretization of the parameter domain, to identify the next dimension of the
parameter-induced solution manifold along which we expand the surrogate
solution space. Although successfully applied to problems with fairly high
parametric dimensions, the challenge is that this scanning cost dominates the
offline cost due to it being proportional to the cardinality of the training
set which is exponential with respect to the parameter dimension. In this work,
we review three recent attempts in effectively delaying this curse of
dimensionality, and propose two new hybrid strategies through successive
refinement and multilevel maximization of the error estimate over the training
set. All five offline-enhanced methods and the original greedy algorithm are
tested and compared on {two types of problems: the thermal block problem and
the geometrically parameterized Helmholtz problem.
