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The onerous task of repeatedly resolving certain parametrized partialdifferential equations (pPDEs) in, e.g. the optimization context, makes itimperative to design vastly more efficient numerical solvers withoutsacrificing any accuracy. The reduced basis method (RBM) presents itself assuch an option. With a mathematically rigorous error estimator, RBM seeks asurrogate solution in a carefully-built subspace of the parameter-induced highfidelity solution manifold. It can improve efficiency by several orders ofmagnitudes leveraging 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 degrading to the online efficiency.  In this paper, we augment and extend the EIM approach in the context ofsolving pPDEs in two different ways, resulting in the Reduced Over-Collocationmethods (ROC). These are stable and capable of avoiding the efficiencydegradation inherent to a direct application of EIM. There are two ingredientsof these methods. First is a strategy to collocate at about twice as manylocations as the number of bases for the surrogate space. The second is anefficient approach for the strategic selection of the parameter values to buildthe reduced solution space for which we study two choices, a recent empiricalL1 approach and a new indicator based on the reduced residual. Together, thesetwo ingredients render the schemes, L1-ROC and R2-ROC, online efficient andimmune from the efficiency degradation of EIM for nonlinear and nonaffineproblems offline and online. Numerical tests on three different families ofnonlinear problems demonstrate the high efficiency and accuracy of these newalgorithms and their superior stability performance.

L1-ROC and R2-ROC: L1- and R2-based Reduced Over-Collocation methods for parametrized nonlinear partial differential equations