Interpreting Moment Matrix Blocks Spectra using Mutual Shadow Area

TL;DR

This paper introduces a novel approach using mutual shadow area to predict spectral components and rank behavior of wave interactions in surface regions, enhancing understanding of solver performance for complex scatterers.

Contribution

It presents a new geometric predictor for the spectral knee and rank scaling in wave interaction matrices, linking shadow area to wave phenomena and solver efficiency.

Findings

Mutual shadow area accurately predicts the spectral knee location.

Diffraction effects dominate rank scaling in 3D interactions with small aspect angles.

Shadow area vanishing in endfire interactions leads to lower rank and better solver performance.

Abstract

The mutual shadow area of pairs of surface regions is used for guiding the study of the spectral components and 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 dimension that scales with the subdomains area (or length in 2-D) and a remainder "diffraction" subspace of dimension that scales much slower with the electrical length, depending on the geometric configuration. For interactions between open surface domains typical for the common hierarchical partitioning in most fast solvers, the latter can be attributed to the domain edges visible by its interacting counterpart. For interactions in 3-D with a small aspect angles between the source and observers, 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 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 governs also the asymptotic rank, which translates to superior asymptotic solver performance.