L1- and R2-based Reduced Over-Collocation methods for parametrized nonlinear partial differential equations
摘要：Repeatedly resolving certain parametrized partial differential equations (pPDEs) in, e.g. the optimization context, makes it imperative to design vastly more efficient numerical solvers without sacrificing any accuracy. The reduced basis method (RBM) presents itself as such an option. RBM seeks a surrogate solution in a carefully-built subspace of the parameter-induced high fidelity solution manifold. 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 degrading to the online efficiency. In this talk, we introduce the RBM and extend the EIM approach in the context of solving pPDEs in two different ways, resulting in the Reduced Over-Collocation methods (ROC), more exactly, the L1-ROC and R2-ROC. These are stable and capable of avoiding the efficiency degradation inherent to a direct application of EIM. Numerical tests on three different families of nonlinear problems demonstrate the high efficiency and accuracy of these new algorithms and its superior stability performance.