Low-Complexity Direct and Iterative Volume Integral Equation Solvers With a Minimal-Rank ${\mathcal H}^2$-Representation for Large-Scale Three-Dimensional Electrodynamic Analysis

Low-complexity iterative and direct solvers are developed for the volume integral equation (VIE) based general largescale electrodynamic analysis. The dense VIE system matrix is first represented by a new cluster-based multilevel low-rank representation. In this representation, all the admissible blocks associated with a single cluster are grouped together and represented by a single low-rank block, whose rank is minimized based on prescribed accuracy. From such an initial representation, an efficient algorithm is developed to generate a minimal-rank H
²
-matrix representation. This representation facilitates faster computation, and ensures the same minimal rank's growth rate with electrical size as evaluated from singular value decomposition. For the minimal-rank representation whose rank grows with electrical size linearly for a prescribed accuracy, we develop linear-complexity H
²
-matrix-based storage and matrix-vector multiplication, and thereby an O(N) iterative VIE solver. Moreover, we develop an O(NlogN) matrix inversion, and hence a fast O(NlogN) direct VIE solver for large-scale electrodynamic analysis. Both theoretical analysis and numerical simulations of large-scale one-, two-, and three-dimensional structures on a single-core CPU, resulting in millions of unknowns, have demonstrated the low complexity and superior performance of the proposed VIE electrodynamic solvers.