Algebraic and Fast Nested Construction Method for Generating Rank-Minimized ${\mathcal H}^{2}$-Matrix for Solving Electrically Large Surface Integral Equations

In this work, we develop a kernel-independent and purely algebraic method, Nested Construction Method, which can construct a rank-minimized ${\mathcal H}^{2}$ -matrix with low complexity based on prescribed accuracy. The time cost of this method in generating each cluster basis and coupling matrix is of $O(k n \log {n})$ , while the memory consumption scales as $O(k^{2})$ , where $k$ is the rank of the cluster basis, and $n$ is cluster size. The accuracy and efficiency of the proposed method are demonstrated by extensive numerical experiments. In addition to surface integral equations, the proposed algorithms can also be applied to solving other electrically large integral equations.