The task of repeatedly solving parametrized partial differential equations (pPDEs) in, e.g. optimization or interactive applications, makes it imperative to design highly efficient and equally accurate surrogate models. The reduced basis method (RBM) presents as such an option. Enabled by a mathematically rigorous error estimator, RBM constructs a low-dimensional subspace of the parameter-induced high fidelity solution manifold from 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 through the empirical interpolation method (EIM) when the PDE is nonlinear or its parameter dependence nonaffine, is either challenging to implement, or severely degrades 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 the efficiency degradation inherent to a traditional application of EIM. Two critical ingredients of the scheme are collocation at about twice as many locations as the dimension of the reduced solution space, and an efficient L1-norm-based error indicator for the strategic selection of the parameter values to build the reduced solution space. Together, these two ingredients render the proposed L1-ROC scheme 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.

中文翻译：

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稳态和时间相关非线性方程的基于 L1 的减少过度搭配和超减少

在例如优化或交互式应用程序中重复求解参数化偏微分方程 (pPDE) 的任务使得设计高效且同样准确的替代模型势在必行。简化基法 (RBM) 就是这样一种选择。在数学上严格的误差估计器的支持下，RBM ​​构建了参数引起的高保真解流形的低维子空间，从中计算出近似解。它可以利用离线-在线分解过程将效率提高几个数量级。然而，当 PDE 为非线性或其参数依赖非仿射时，这种分解通常通过经验插值方法 (EIM) 实现，要么难以实现，要么严重降低在线效率。在本文中，我们增强和扩展了 EIM 方法作为直接求解器，而不是助手，用于在降低的水平上求解非线性 pPDE。由此产生的方法称为减少过度搭配方法 (ROC)，它稳定且能够避免 EIM 传统应用固有的效率下降。该方案的两个关键要素是在大约两倍于缩减解空间维度的位置的搭配，以及一个有效的基于 L1 范数的误差指标，用于策略选择参数值以构建缩减解空间。这两个因素一起使所提出的 L1-ROC 方案既离线又高效。一个显着的特点是避免了在利用 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，无论是线下阶段，还是线上阶段。对不同族的瞬态和稳态非线性问题的数值测试证明了 L1-ROC 的高效率和准确性及其卓越的稳定性性能。