  Linear complexity iterative and log-linear complexity direct solvers are
developed for the volume integral equation (VIE) based general large-scale
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 ${\cal H}^2$-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. Taking into account the rank's growth with
electrical size, we develop linear-complexity ${\cal H}^2$-matrix-based storage
and matrix-vector multiplication, and thereby an $O(N)$ iterative VIE solver
regardless of electrical size. Moreover, we develop an $O(NlogN)$ matrix
inversion, and hence a fast $O(NlogN)$ \emph{direct} VIE solver for large-scale
electrodynamic analysis. Both theoretical analysis and numerical simulations of
large-scale $1$-, $2$- and $3$-D 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. %The algorithms
developed in this work are kernel-independent, and hence applicable to other IE
operators as well.
