L1-Based Reduced Over Collocation and Hyper Reduction for Steady State and Time-Dependent Nonlinear Equations

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. reduced basis method (RBM) presents itself as such an option. Accompanied by a mathematically rigorous error estimator, RBM carefully constructs low-dimensional subspace the parameter-induced high fidelity solution manifold on which approximate is computed. It can improve efficiency several orders magnitudes leveraging offline-online decomposition procedure. However this decomposition, usually implemented with aid from empirical interpolation (EIM) for nonlinear and/or parametric-nonaffine PDEs, be challenging implement, results severely degraded online efficiency. In paper, we augment extend EIM approach direct solver, opposed assistant, pPDEs level. resulting method, called Reduced Over-Collocation (ROC), stable capable avoiding degradation exhibited traditional EIM. Two critical ingredients scheme are collocation at about twice many locations dimension approximation space, L1-norm-based indicator strategic selection parameter values whose snapshots span space. Together, these two ensure that proposed L1-ROC both offline- online-efficient. A distinctive feature appearing alternative approaches utilize nonaffine problems circumvented, offline stages. Numerical tests different families time-dependent steady-state demonstrate accuracy its superior stability performance.