Fast computation of 3D inhomogeneous scattered field using a discrete BCG-FFT algorithm

In this paper, we present a discrete formulation for solving the 3D scattering problems. The scattered fields from dielectric scatterers with arbitrary geometry are modeled using integral equation with equivalent sources. The inhomogeneity of the parameter of the scatterer is approximated by a set of 3D simple functions. The total field is represented by a set of local basis functions. A Galerkin testing formulation is applied and no approximation is made to the differential operators involved in the integral equation except for the projection of the unknown field and the operators onto the subspace spanned by the basis functions. It will be shown that the testing formulation can be represented by a multi-input and multi-output system with known linear kernels. The kernels are discretized using the rooftop basis functions and they are independent of the scattering configuration and the incident waves. Consequently, for a given sampling rate per wavelength, one need only evaluate once the kernels whose storage is of order N. The problem of solving the integral equation is similar to the deconvolution problem where the unknown fields inside the scatterers are modulated with the dielectric permittivity distribution.