Low-Rank Matrix Factorization Method for Multiscale Simulations: A Review

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.