Galerkin-based reduced-order models (G-ROMs) have provided efficient and accurate approximations of laminar flows. In order to capture the complex dynamics of the turbulent flows, standard G-ROMs require a relatively large number of reduced basis functions (on the order of hundreds and even thousands). Although the resulting G-ROM is still relatively low-dimensional compared to the full-order model (FOM), its computational cost becomes prohibitive due to the 3rd-order convection tensor contraction. The tensor requires storage of $N^3$ entries with a corresponding work of $2N^3$ operations per timestep, which makes such ROMs impossible to use in realistic applications, such as control of turbulent flows. In this paper, we focus on the scenario where the G-ROM requires large $N$ values and propose a novel approach that utilizes the CANDECOMC/PARAFAC decomposition (CPD), a tensor decomposition technique, to accelerate the G-ROM by approximating the 3rd-order convection tensor by a sum of $R$ rank-1 tensors. In addition, we show that the tensor is partially skew-symmetric and derive two conditions for the CP decomposition for preserving the skew-symmetry. Moreover, we investigate the G-ROM with the singular value decomposition (SVD). The G-ROM with CP decomposition is investigated in several flow configurations from 2D periodic flow to 3D turbulent flows. Our numerical investigation shows CPD-ROM achieves at least a factor of 10 speedup. Additionally, the skew-symmetry preserving CPD-ROM is more stable and allows the usage of smaller rank $R$. Moreover, from the singular value behavior, the advection tensor formed using the $H^1_0$-POD basis has a low-rank structure, and is preserved even in higher Reynolds numbers. Furthermore, for a given level of accuracy, the CP decomposition is more efficient in size and cost than the SVD.

翻译：Galerkin降阶模型（G-ROM）已为层流流动提供了高效且精确的近似。为捕捉湍流流动的复杂动力学特性，标准G-ROM需要数量较多（数百甚至数千阶）的约化基函数。尽管生成的G-ROM相对于全阶模型（FOM）仍具有较低维数，但由于三阶对流张量收缩运算的存在，其计算成本变得难以承受。该张量需存储$N^3$个元素，且每时间步对应的计算量为$2N^3$次操作，这使得此类ROM无法应用于湍流控制等实际场景。本文聚焦于G-ROM需要较大$N$值的场景，提出一种利用CANDECOMC/PARAFAC分解（CPD）——一种张量分解技术——来加速G-ROM的新方法，通过将三阶对流张量近似为$R$个秩-1张量的和。此外，我们揭示了该张量具有部分斜对称性，并推导了CP分解保持斜对称性的两个条件。同时研究了基于奇异值分解（SVD）的G-ROM。我们在从二维周期流动到三维湍流的多种流场构型中验证了CP分解G-ROM的性能。数值研究表明，CPD-ROM实现了至少10倍的加速比。此外，保持斜对称性的CPD-ROM具有更好的稳定性，并允许使用更小的秩$R$。从奇异值特性来看，基于$H^1_0$-POD基构建的平流张量具有低秩结构，且该结构在高雷诺数下仍能保持。而在给定精度水平下，CP分解在规模和计算代价上均优于SVD。