A Block Diagonal Preconditioned and Tensor Butterfly Accelerated Volume Integral Equation Solver for 3D High-Frequency Scattering Analysis

Electromagnetic analyses involving 3D inhomogeneous dielectric objects can be simulated with differential equation (DE) or volume integral equation (VIE)-based numerical algorithms. Compared to DE-based algorithms such as finite difference or finite volume, VIE algorithms do not suffer from numerical dispersion and require no artificial absorbing boundary conditions. That said, the VIE solvers can be computationally more expensive than DE solvers as they typically form a N×N dense matrix system in terms of discretized background Green's functions. In the past, analytical and algebraic fast algorithms have been developed to accelerate both the iterative and direct solutions of the VIE system. Most notable efforts include fast multipole methods (K. Sertel and J. L. Volakis, IEEE Trans. Antennas Propag., 2004), and fast Fourier transform algorithms (X. Nie, et al., IEEE Trans. Antennas Propag., 2005) to accelerate the matrix vector multiplication at each Krylov iteration, as well as low-rank compression (S. Omar and D. Jiao, IEEE Trans. Microwave Theory Tech., 2015; Y.-N. Liu, et al., Int. J. Numer. Model.: Electron. Netw. Devices Fields, 2020), butterfly compression (S. B. Sayed, et al., IEEE Trans. Antennas Propag., 2022) and equivalence principle (W. C. Chew and C.-C. Lu, Microw. Opt. Tech. Lett., 1993)-based fast direct solvers.