  An adaptive scheme to generate reduced-order models for parametric nonlinear
dynamical systems is proposed. It aims to automatize the POD-Greedy algorithm
combined with empirical interpolation. At each iteration, it is able to
adaptively determine the number of the reduced basis vectors and the number of
the interpolation basis vectors for basis construction. The proposed technique
is able to derive a suitable match between the reduced basis and the
interpolation basis vectors, making the generation of a stable, compact and
reliable reduced-order model possible. This is achieved by adaptively adding
new basis vectors or removing unnecessary ones, at each iteration of the greedy
algorithm. An efficient output error indicator plays a key role in the adaptive
scheme. We also propose an improved output error indicator based on previous
work. Upon convergence of the POD-Greedy algorithm, the new error indicator is
shown to be sharper than the existing ones, implicating that a more reliable
reduced-order model can be constructed. The proposed method is tested on
several nonlinear dynamical systems, namely, the viscous Burgers' equation and
two other models from chemical engineering.
