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.
A hyper-reduced MAC scheme for the parametric Stokes and Navier-Stokes equations
加速某些参数化系统的重复求解的需要激发了更有效的降阶方法的发展。经典的缩减基方法是流行的，由于离线-在线分解和数学上严格的{ \em a posterior}误差估计器，引导贪婪算法离线。对于非线性和非仿射问题，超约简技术已经被引入，使这种分解有效。然而，他们可能是棘手的实现，并经常降低在线计算效率。为了避免这种退化，减少残差减少过度配置（R2-ROC）被发明集成经验插值技术的解决方案快照和精心选择的残差，配置哲学，和简单的评估超减少精心选择的残差。在本文中，我们引入了一个自适应的丰富策略R2-ROC渲染它能够处理参数化流体流动问题。在此基础上，提出了一种新的超约化MAC算法，并在Stokes方程和Navier-斯托克斯方程上进行了测试，证明了该算法的高效性、稳定性和精确性.
Physics-informed deep learning and compressive collocation for high-dimensional diffusion-reaction equations: practical existence theory and numerics
Department of Mathematics and Statistics（数学与统计学系）；Concordia University（康科迪亚大学）；Department of Scientific Computing（科学计算系）；Florida State University（佛罗里达州立大学）；Department of Physics & Astronomy（物理与天文学系）；University of Victoria（维多利亚大学）
Hyperbolic model for Helmholtz equation with impedance boundary conditions
Ivane Javakhishvili Tbilisi State University（伊万·贾瓦赫什维利第比利斯国立大学）；Institute of Energy and Climate research 8: Troposphere, Research Center Jülich（于利希研究中心能源与气候研究所8：对流层）
Fractional Gray-Scott Model: Well-posedness, Discretization, and Simulations
Shandong University（山东大学）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）；Department of Mathematics, University of South Carolina（南卡罗来纳大学数学系）
Particle-without-Particle: a practical pseudospectral collocation method for linear partial differential equations with distributional sources
Institute of Space Sciences (ICE, CSIC)（空间科学研究所）；Institut d’Estudis Espacials de Catalunya (IEEC)（加泰罗尼亚空间研究学院）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Observatoire des Sciences de l’Univers en région Centre (OSUC), Université d’Orléans（奥尔良大学中部地区宇宙科学观测台）；Université d’Orléans（奥尔良大学）；Centro Brasileiro de Pesquisas Físicas (CBPF)（巴西物理研究中心）
