A discrete BCG-FFT algorithm for solving 3D inhomogeneous scatterer problems

In this paper, an algorithm for computation of the scattered fields from three dimensional inhomogeneous dielectric scatterers is presented. In this method, the Galerkin's testing formulation of an integral equation for 3D electromagnetic (EM) fields is represented by a multi-input and multi-output linear system with known kernels. On a regular grid with rooftop basis functions, the kernels are discretized and accurately evaluated. Furthermore they are represented by Toeplitz matrices which dramatically reduces the storage and computational complexity in solving for the scattered fields. Also, the kernels are independent of the scattering configuration and the incident waves. For a given frequency, they can be evaluated once and for all. The biconjugate gradient (BCG) algorithm combined with fast Fourier transform (FFT) is applied to solve the discrete linear system iteratively. The memory required for this algorithm is of order N, and the computational complexity of the BCG process costs order N log N operations per iteration, where N is the number of 3D unknowns. Unlike previous approaches, no approximation is made when FFT is used to accelerate the matrix-vector multiplication.