Interpreting Moment Matrix Blocks Spectra Using Mutual Shadow Area

The mutual shadow area of pairs of surface regions is used for guiding the study of the spectral components and the rank of their wave interaction, as captured by the corresponding moment-matrix blocks. It is demonstrated that the mutual shadow area provides an asymptotically accurate predictor of the location of the singular value curve knee. This predicted knee index is shown to partition the interacting parts of the range and domain of blocks into two subspaces that can be associated with different wave phenomena: an 'aperture' subspace of a dimension that scales with the subdomains area (or length in 2-D) and a remainder 'diffraction' subspace of a dimension that scales much more slowly with the electrical length, depending on the geometric configuration. For interactions between open surface domains typical of the common hierarchical partitioning in most fast solvers, the latter can be attributed to the domain edges visible by its interacting counterparts. For interactions in 3-D with small aspect angles between the source and observer, the diffraction subspace dimension is dominant in determining the rank until fairly large electrical lengths are reached. This explains the delayed asymptotic scaling of ranks and the impressive fast solver performance observed in recent literature for seemingly arbitrary scatterers with no special geometric characteristics. In the extreme cases of 'endfire' reduced dimensionality interactions, where the shadow area vanishes, the diffraction also governs the asymptotic rank, which translates to superior asymptotic solver performance.