We are interested in the numerical solution of coupled nonlinear partial differential equations (PDEs) in two and three dimensions. Under certain assumptions on the domain, we take advantage of the Kronecker structure arising in standard space discretizations of the differential operators and illustrate how the resulting system of ordinary differential equations (ODEs) can be treated directly in matrix or tensor form. Moreover, in the framework of the proper orthogonal decomposition (POD) and the discrete empirical interpolation method (DEIM) we derive a two- and three-sided model order reduction strategy that is applied directly to the ODE system in matrix and tensor form respectively. We discuss how to integrate the reduced order model and, in particular, how to solve the tensor-valued linear system arising at each timestep of a semi-implicit time discretization scheme. We illustrate the efficiency of the proposed method through a comparison to existing techniques on classical benchmark problems such as the two- and three-dimensional Burgers equation.
Structure-Preserving Galerkin POD-DEIM Reduced-Order Modeling of Hamiltonian Systems
[Gong et al. 2017]中针对哈密顿系统开发了一种结构保持适当正交分解降阶建模方法，该方法使用基于Galerkin投影的模型降阶的传统框架，但修改了降阶模型，以便保留适当的哈密尔顿结构。然而，它的计算复杂性的在线模拟仍然是高的，如果哈密尔顿涉及非多项式非线性。在本文中，我们应用离散经验插值方法，以提高在线效率的结构保持降阶模拟。由于缩减基截断会降低哈密顿近似，我们建议使用从移位快拍得到的基。一个非线性波动方程被用作测试床，数值结果表明所提出的方法的有效性。
Global energy preserving model reduction for multi-symplectic PDEs
许多哈密顿系统可以重铸为多辛形式。本文建立了多辛哈密顿偏微分方程的降阶模型，该模型保持了系统的整体能量。通过空间有限差分离散和全局能量保持平均向量场（AVF）方法得到了全阶解。ROM的构造方式与全阶模型（FOM）相同，采用Galerkin投影的本征正交分解（POD）。降阶系统具有与FOM相同的结构，并保留了离散的约化全局能量。在在线阶段，应用离散经验插值方法（DEIM）高效地计算降阶解。先验误差界推导出的DEIM近似的非线性哈密顿。对多辛形式的Korteweg de Vries（KdV）方程、Zakharov-Kuznetzov（ZK）方程和非线性Schr{<$}dinger（NLS）方程的计算结果表明了ROM的精度和计算效率.约化能量的保持表明，降阶解保证了解的长期稳定性。
Energy stable model order reduction for the Allen-Cahn equation
University of Turkish Aeronautical Association（土耳其航空协会大学）；Institute of Applied Mathematics, Middle East Technical University（中东理工大学应用数学研究所）；Department of Mathematics, Middle East Technical University（中东理工大学数学系）
