In this paper, a review of the low-rank factorization method is presented, with emphasis on their application to multiscale problems. Low-rank matrix factorization methods exploit the rankdeficient nature of coupling impedance matrix blocks between two separated groups. They are widely used, because they are purely algebraic and kernel free. To improve the computation precision and efficiency of low-rank based methods, the improved sampling technologies of adaptive cross approximation (ACA), post compression methods, and the nested low-rank factorizations are introduced. ${\mathrm {O}}(N)$ and ${\mathrm {O}}(N \log N)$ computation complexity of the nested equivalence source approximation can be achieved in low and high frequency regime, which is parallel to the multilevel fast multipole algorithm, N is the number of unknowns. Efficient direct solution and high efficiency preconditioning techniques can be achieved with the low-rank factorization matrices. The trade-off between computation efficiency and time are discussed with respect to the number of levels for low-rank factorizations.

中文翻译：

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用于多尺度仿真的低秩矩阵分解方法：综述

本文对低阶分解法进行了综述，重点介绍了它们在多尺度问题中的应用。低秩矩阵分解方法利用了两个分离的组之间耦合阻抗矩阵块的秩不足性质。它们被广泛使用，因为它们是纯代数的，并且没有内核。为了提高基于低秩的方法的计算精度和效率，介绍了改进的自适应交叉逼近（ACA）采样技术，后压缩方法以及嵌套的低秩分解。 $ {\ mathrm {O}}（N）$ 和 $ {\ mathrm {O}}（N \ log N）$ 嵌套等价源近似的计算复杂度可以在低频和高频状态下实现，这与多级快速多极子算法并行， ñ是未知数。使用低秩分解矩阵可以实现有效的直接解决方案和高效的预处理技术。关于低秩分解的级别数，讨论了计算效率和时间之间的权衡。