Nested Pseudo Skeleton Approximation Algorithm for Generating ${\mathcal H}^{2}$-Matrix Representations of Electrically Large Surface Integral Equations

In this paper, we develop a kernel-independent and purely algebraic method, Nested Pseudo-Skeleton Approximation (NPSA) algorithm, to generate a low-rank ${\mathcal H}^{2}$-matrix representation of electrically large surface integral equations (SIEs). The algorithm only uses $O(NlogN)$ entries of the original dense SIE matrix of size $N$ to generate the ${\mathcal H}^{2}$-representation. It also provides a closed-form expression of the cluster bases and coupling matrices with respect to original matrix entries. The resultant ${\mathcal H}^{2}$-matrix is then directly solved for electrically large scattering analysis. Numerical experiments have demonstrated the accuracy and efficiency of the proposed algorithm. In addition to surface integral equations, the proposed algorithms can also be applied to solving other electrically large integral equations.