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The task of repeatedly solving parametrized partial differential equations(pPDEs) in, e.g. optimization or interactive applications, makes it imperativeto design highly efficient and equally accurate surrogate models. The reducedbasis method (RBM) presents as such an option. Enabled by a mathematicallyrigorous error estimator, RBM constructs a low-dimensional subspace of theparameter-induced high fidelity solution manifold from which an approximatesolution is computed. It can improve efficiency by several orders of magnitudesleveraging an offline-online decomposition procedure. However, thisdecomposition, usually through the empirical interpolation method (EIM) whenthe PDE is nonlinear or its parameter dependence nonaffine, is eitherchallenging to implement, or severely degrades online efficiency.  In this paper, we augment and extend the EIM approach as a direct solver, asopposed to an assistant, for solving nonlinear pPDEs on the reduced level. Theresulting method, called Reduced Over-Collocation method (ROC), is stable andcapable of avoiding the efficiency degradation inherent to a traditionalapplication of EIM. Two critical ingredients of the scheme are collocation atabout twice as many locations as the dimension of the reduced solution space,and an efficient L1-norm-based error indicator for the strategic selection ofthe parameter values to build the reduced solution space. Together, these twoingredients render the proposed L1-ROC scheme both offline- andonline-efficient. A distinctive feature is that the efficiency degradationappearing in alternative RBM approaches that utilize EIM for nonlinear andnonaffine problems is circumvented, both in the offline and online stages.Numerical tests on different families of time-dependent and steady-statenonlinear problems demonstrate the high efficiency and accuracy of L1-ROC andits superior stability performance.

L1-based reduced over collocation and hyper reduction for steady state and time-dependent nonlinear equations