Minimal-rank ℋ2-matrix-based iterative and direct volume integral equation solvers for large-scale scattering analysis

It can be shown that the matrix structure resulting from a fast multipole method (FMM)-based algorithm is an ℋ²-matrix, but with a full-rank representation for electrically large analysis. We compare the computational complexity of a volume integral equation (VIE) solver having a minimal-rank ℋ²-representation with that of a VIE solver using an FMM-based ℋ²-representation. The former is shown to be strict O(N) in storage and matrix-vector multiplication, and O(NlogN) in inverse irrespective of electric size. Such a complexity has been demonstrated by the analysis of large-scale dielectric scattering problems involving millions of unknowns.