Unlike electrically small problems, the rank of an electrodynamic IE operator increases with electric size for achieving a prescribed accuracy, which results in a higher computational complexity if no advanced algorithms are conceived to effectively manage the rank's growth with electric size. The true indicator of this rank's growth is singular value decomposition (SVD), which is computationally O(N^3) and is thus not used explicitly.
Prevailing source-observer separated representations of the IE operator adopted by existing fast IE solvers do not lead to a minimal rank representation required by accuracy. These representations include interpolation-based, Taylor series expansion based, plane wave expansion based methods, etc. In fact, these representations result in a full-rank representation of the IE operator for electrically large problems, which makes the compression of the IE operator sub-optimal for general electrodynamic analysis.
In view of the pivotal importance of the rank of the off-diagonal blocks in an IE operator, we have carried out a theoretical study on its growth with electric size in integral equations. The significance of this study lies in the fact that it derives a closed-form analytical expression of the rank of the coupling Green's function, which has the same scaling as that depicted by SVD-based rank revealing. The findings on the rank are summarized as follows:
Based on the above findings, we have developed fast direct integral equation solvers for electrically large analysis. In our direct surface IE solvers, a dense matrix for a 96-wavelength problem with over 1 million unknowns was directly factorized in 20 hours with 85-seconds LU solution time on a single 3 GHz CPU core. In our electrically large volume IE solver, we have achieved O(N) complexity in matrix-vector multiplication, and O(NlogN) complexity in inverse irrespective of electrical size.