  The task of repeatedly solving parametrized partial differential equations
(pPDEs) in, e.g. optimization or interactive applications, makes it imperative
to design highly efficient and equally accurate surrogate models. The reduced
basis method (RBM) presents as such an option. Enabled by a mathematically
rigorous error estimator, RBM constructs a low-dimensional subspace of the
parameter-induced high fidelity solution manifold from which an approximate
solution is computed. It can improve efficiency by several orders of magnitudes
leveraging an offline-online decomposition procedure. However, this
decomposition, usually through the empirical interpolation method (EIM) when
the PDE is nonlinear or its parameter dependence nonaffine, is either
challenging to implement, or severely degrades online efficiency.
  In this paper, we augment and extend the EIM approach as a direct solver, as
opposed to an assistant, for solving nonlinear pPDEs on the reduced level. The
resulting method, called Reduced Over-Collocation method (ROC), is stable and
capable of avoiding the efficiency degradation inherent to a traditional
application of EIM. Two critical ingredients of the scheme are collocation at
about twice as many locations as the dimension of the reduced solution space,
and an efficient L1-norm-based error indicator for the strategic selection of
the parameter values to build the reduced solution space. Together, these two
ingredients render the proposed L1-ROC scheme both offline- and
online-efficient. A distinctive feature is that the efficiency degradation
appearing in alternative 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 L1-ROC and
its superior stability performance.
