Rapid Rank Estimation and Low-Rank Approximation of Impedance Matrix Blocks Using Proxy Grids

A physics-based rank-revealing multilevel algorithm to more efficiently compute low-rank (LR) approximations of the method of moments matrix blocks ${\mathbf{Z}}^{\textrm {os}}$ is presented. Using surface subsets of volumetric nonuniform spherical grids (proxy grids), an LR approximation ${\mathbf{Z}}^{\textrm {os}}\approx {\mathbf{AB}}^{\dagger }$ is obtained in two stages: 1) an upward pass and a downward pass on a multilevel tree, via truncated singular value decomposition-based analyses of interactions between source subclusters and their attendant proxy grids, rapidly estimates the rank and yields a domain of ${\mathbf{Z}}^{\textrm {os}}$ , in the form of a matrix ${\mathbf{B}}$ with orthonormal columns and 2) a fast matrix-matrix multiplication step yields ${\mathbf{A}}$. The algorithm reduces the ${\mathcal{ O}}(N^{3})$ computational costs of revealing the rank to ${\mathcal{ O}}(N^{2})$ or ${\mathcal{ O}}(N^{3/2})$ operations and that of finding an LR approximation to ${\mathcal{ O}}(N^{2})$ or ${\mathcal{ O}}(N^{3/2}\log N)$ operations by adopting enclosing or belt-like proxy grids for electromagnetically large basis/testing function distributions that are either densely packed in a volume or quasi-planar, respectively.