Fast direct solution of large-scale volume integral equations with exact arithmetic

A new HSS (Hierarchically Semiseparable) direct matrix solution algorithm, including both factorization and inversion, is developed for solving the volume integral equation (VIE) for large-scale electrodynamic analysis. This direct solution algorithm is exact in the sense that the matrix factorization, inversion, and solution do not involve any approximation. The HSS-representation of the VIE dense system has a controlled accuracy, by choosing the rank of admissible blocks to satisfy the prescribed accuracy. Taking into account the rank's dependence with electrical size, the resultant direct VIE solver is shown to have O(N) complexity in memory, O(NlogN) complexity in factorization as well as inversion, and O(N) complexity in solution, 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 factorization and inversion algorithms are also applicable to other integral operators.