  The need for multiple interactive, real-time simulations using different
parameter values has driven the design of fast numerical algorithms with
certifiable accuracies. The reduced basis method (RBM) presents itself as such
an option. RBM features a mathematically rigorous error estimator which drives
the construction of a low-dimensional subspace. A surrogate solution is then
sought in this low-dimensional space approximating the parameter-induced high
fidelity solution manifold. However when the system is nonlinear or its
parameter dependence nonaffine, this efficiency gain degrades tremendously, an
inherent drawback of the application of the empirical interpolation method
(EIM).
  In this paper, we augment and extend the EIM approach as a direct solver, as
opposed to an assistant, for solving nonlinear partial differential equations
on the reduced level. The resulting method, called Reduced Over-Collocation
method (ROC), is stable and capable of avoiding the efficiency degradation. Two
critical ingredients of the scheme are collocation at about twice as many
locations as the number of basis elements for the reduced approximation space,
and an efficient error indicator for the strategic building of the reduced
solution space. The latter, the main contribution of this paper, results from
an adaptive hyper reduction of the residuals for the reduced solution.
Together, these two ingredients render the proposed R2-ROC scheme both offline-
and online-efficient. A distinctive feature is that the efficiency degradation
appearing in traditional RBM approaches that utilize EIM for nonlinear and
nonaffine problems is circumvented, both in the offline and online stages.
Numerical tests on different families of time-dependent and steady-state
nonlinear problems demonstrate the high efficiency and accuracy of our R2-ROC
and its superior stability performance.
