The task of repeatedly solving parametrized partial differential equations (pPDEs) in optimization, control, or interactive applications makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents itself as such an option. Accompanied by a mathematically rigorous error estimator, RBM carefully constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold on 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 implemented with aid from the empirical interpolation method (EIM) for nonlinear and/or parametric-nonaffine PDEs, can be challenging to implement, or results in severely degraded 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 efficiency degradation exhibited in traditional applications of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced approximation space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values whose snapshots span the reduced approximation space. Together, these two ingredients ensure that the proposed L1-ROC scheme is 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.

在优化，控制或交互式应用程序中反复求解参数化偏微分方程（pPDE）的任务使得必须设计高效且同等准确的替代模型。减基法（RBM）本身就是这样一种选择。伴随着数学上严格的误差估计器，RBM精心构建了参数诱导的高保真度解决方案流形的低维子空间，在该子空间上可以计算出近似解。利用离线-在线分解过程，它可以将效率提高几个数量级。但是，通常在经验插值方法（EIM）的帮助下对非线性和/或参数非仿射PDE进行分解，可能难以实施，或者导致在线效率严重降低。在本文中，我们扩充和扩展了EIM方法，将其作为直接求解器（而不是助手）来解决简化水平上的非线性pPDE。所得方法称为减少过度配位法（ROC），该方法稳定且能够避免EIM传统应用中出现的效率下降。该方案的两个关键要素是在减少的近似空间维数的两倍位置上并置一个有效的基于L1范数的误差指示符，用于策略性选择参数值，其快照跨越减少的近似空间。这两个因素共同确保了所提出的L1-ROC方案既具有离线效率又具有在线效率。一个显着的特征是，在离线和在线阶段，都可以避免使用EIM解决非线性和非仿射问题的替代RBM方法中出现的效率下降。对不同类型的时变和稳态非线性问题进行数值测试，证明了L1-ROC的高效率和准确性以及出色的稳定性能。