One-Stage $ O(N \log N)$ Algorithm for Generating Nested Rank-Minimized Representation of Electrically Large Volume Integral Equations

In this paper, we develop a new one-stage $ O(N \log N)$ algorithm to generate a rank-minimized $\mathcal {H}^{2}$-representation of electrically large volume integral equations (VIEs), which significantly reduces the CPU run time of state-of-the-art algorithms for completing the same task. Unlike existing two-stage algorithms, this new algorithm requires only one stage to build nested cluster bases. The cluster basis is obtained directly from the interaction between a cluster and its admissible clusters composed of real or auxiliary ones that cover all interaction directions. Furthermore, the row and column pivots of the resultant low-rank representation are chosen from the source and observer points in an analytical way without the need for numerically finding them. This further speeds up the computation. Numerical experiments on a suite of electrically large 3D scattering problems have demonstrated the efficiency and accuracy of the proposed new algorithm.