Frequency- and Time-domain Green function methods for electromagnetic simulation, optimization, and design

We present a family of numerical methods for the solution of Maxwell's equations, with application to simulation, optimization, and design. In particular, a novel rectangular-polar integral equation solver is mentioned which can produce solutions to the time harmonic Maxwell's equations, with high order accuracy, for general2D and 3D structures, with an extension to time domain problems on the basis of a time re-centering synthesis technique. An effective integral equation acceleration method, the IFGF method (Interpolated Factored Green Function), is used, which evaluates the action of Green function-based integral operators for an $N$ - point surface discretization at a computational cost of $O(N\log N)$ operations without recourse to the FFT-thus, lending itself to effective distributed memory parallelization. Computational illustrations include applications to photonic optimization and design.