Abstract
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.
Highlights
I NTEGRAL equation methods are the preferred methods when modeling and simulating large and multiscale problems due to their high computation precision and small number of unknowns
The computation time and memory requirements have complexity O(N3) and O(N2) respectively, where N is the number of unknowns, when a straightforward direct solver is used
To enhance the computational performance of MoM, there are mainly three kinds of fast solvers based on different computation acceleration techniques: (1) Sparsification of the impedance matrix by using specialized basis functions such as wavelet expansions [3]
Summary
I NTEGRAL equation methods are the preferred methods when modeling and simulating large and multiscale problems due to their high computation precision and small number of unknowns. The typical use of low-rank factorization techniques in the MoM consists of a block decomposition of the impedance matrix, followed by compression of those blocks that represent interactions between well-separated regions of the target geometry. LI et al.: LOW-RANK MATRIX FACTORIZATION METHOD FOR MULTISCALE SIMULATIONS: A REVIEW respect to these two parameters, yielding O(N3/2) complexity both for computation time and memory requirements, as shown in [49]. The post compression techniques reduce the memory requirements and matrix-vector product time significantly due to a relative smaller rank with respect to standard low-rank factorization methods. The hierarchically off-diagonal low rank method has been proposed, where the dense matrix is decomposed as the product of several block diagonal matrices, direct inversion or highly efficient preconditioners can be constructed for multiscale simulations [87]–[91].
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