Exact-arithmetic HSS-inversion algorithm for fast direct solution of electrically large volume integral equations

A new HSS (Hierarchically Semiseparable) matrix inversion algorithm is developed for solving the volume integral equation (VIE) for large-scale electrodynamic analysis. The HSS-matrix representation of the VIE dense system has a controlled accuracy. The inversion of the HSS matrix is exact in the sense that the underlying multiplications and additions are exact involving no approximations. Taking into account the rank's dependence with electrical size, the resultant direct VIE solver is shown to have O(N) complexity in memory and O(NlogN) complexity in inversion, irrespective of electrical size. Numerical experiments on large-scale scattering problems involving millions of unknowns have validated the accuracy and complexity of the proposed new direct solver. The proposed inversion algorithm is also applicable to other integral operators.