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The need for multiple interactive, real-time simulations using differentparameter values has driven the design of fast numerical algorithms withcertifiable accuracies. The reduced basis method (RBM) presents itself as suchan option. RBM features a mathematically rigorous error estimator which drivesthe construction of a low-dimensional subspace. A surrogate solution is thensought in this low-dimensional space approximating the parameter-induced highfidelity solution manifold. However when the system is nonlinear or itsparameter dependence nonaffine, this efficiency gain degrades tremendously, aninherent drawback of the application of the empirical interpolation method(EIM).  In this paper, we augment and extend the EIM approach as a direct solver, asopposed to an assistant, for solving nonlinear partial differential equationson the reduced level. The resulting method, called Reduced Over-Collocationmethod (ROC), is stable and capable of avoiding the efficiency degradation. Twocritical ingredients of the scheme are collocation at about twice as manylocations as the number of basis elements for the reduced approximation space,and an efficient error indicator for the strategic building of the reducedsolution space. The latter, the main contribution of this paper, results froman 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 degradationappearing in traditional 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 our R2-ROCand its superior stability performance.

An EIM-degradation free reduced basis method via over collocation and residual hyper reduction-based error estimation