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Interpreting Moment Matrix Blocks Spectra using Mutual Shadow Area

Yaniv Brick, Francesco P. Andriulli, Mats Gustafsson

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.

Interpreting Moment Matrix Blocks Spectra using Mutual Shadow Area

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.
Paper Structure (5 sections, 10 equations, 15 figures)

This paper contains 5 sections, 10 equations, 15 figures.

Table of Contents

  1. Introduction
  2. Moment Matrix Block Spectrum and Rank
  3. Mutual Shadow and the Asymptotic Scaling of Moment Matrix Blocks Ranks
  4. Mutual Shadow and the Moment Matrix Block Rank and Spectrum
  5. Conclusion and Discussion

Figures (15)

  • Figure 1: Source $S_\mathrm{s}$ and observer $S_\mathrm{o}$ subdomains of a scatterer $S$.
  • Figure 2: Admissible blocks and corresponding interactions for rank and compressive-bases revealing: (a) Strong admissibility interactions in non-nested bases settings. (b) Non-nested bases hierarchical matrix block structure. (c) Interaction for strong admissibility nested-bases compression. (d) Nested-bases compression of a hierarchical matrix bormDatasparseApproximationNonlocal2007.
  • Figure 3: Mutual shadow areas $A_\mathrm{os}(\hat{\boldsymbol k})$ for two cases of the illumination direction $\hat{\boldsymbol k}$.
  • Figure 4: Normalized SVs for two discs evaluated using four different models and three electrical sizes $a/\lambda\in\{2.5,5,10\}$. The asymptotic DoF \ref{['eq:NAdef']} based on \ref{['eq:Area_discs']} is given in the legend.
  • Figure 5: Normalized SVs on scaled axis for parallel circular discs of various configurations of $d/a$ and $a/\lambda$.
  • ...and 10 more figures