Fast Nested Cross Approximation Algorithm for Solving Large-Scale Electromagnetic Problems

TL;DR: A fast nested cross approximation (NCA) algorithm for solving large-scale electromagnetic problems that does not rely on the projection of the basis functions onto the dummy interpolation points to select pivots of each cluster and has a reduced complexity compared to that reported in the mathematical literature.

Abstract: A fast nested cross approximation (NCA) algorithm is developed in this paper for solving large-scale electromagnetic problems. Different from the existing NCA, the proposed method does not rely on the projection of the basis functions onto the dummy interpolation points to select pivots of each cluster. Instead, a purely algebraic and kernel-independent algorithm is developed to find the row and column pivots of all clusters in $\mathcal {O}(N\log {}N)$ complexity for constant-rank cases with controlled accuracy. This algorithm is then further extended to an $\mathcal {O}(N)$ NCA algorithm, which includes a bottom-up tree traversal for finding the local pivots of each cluster, followed by a top-down procedure to take into account the far field of each cluster. The proposed method has a reduced complexity compared to that reported in the mathematical literature. The resultant nested representation constitutes an $\mathcal {H}^{2}$ -matrix representation of the original dense system of equations, whose solution can be obtained in linear complexity in both iterative and direct solvers. The method is also applicable to variable rank cases, but the complexity therein depends on the rank’s relationship with $N$ . Various numerical experiments have demonstrated the accuracy and computational performance of the proposed algorithms.