Accelerating the discrete dipole approximation via circulant preconditioning

The discrete dipole approximation (DDA) is a popular numerical method for electromagnetic scattering calculations. The standard DDA formulation involves the uniform discretization of the underlying volume integral equation, leading to a linear system of convolution form. This permits a matrix-vector product to be performed with O(nlogn) complexity via the fast-Fourier transform (FFT). Thus, in principle, the system can be solved rapidly using an iterative method. However, it is well known that the convergence of iterative methods becomes increasing slow as the optical size and refractive index of the scattering obstacle are increased. In this paper, we present a preconditioning strategy based on the multi-level circulant preconditioner of Chan and Olkin [Numer. Algorithms 6, 89 (1994)] and assess its performance for improving this rate of convergence. In particular, we approximate the system matrix by a circulant matrix which can be inverted efficiently using the FFT. We present numerical experiments for scattering by highly oblate non-absorbing hexagonal plates, demonstrating that this approach serves as an effective preconditioning strategy, reducing simulation times by orders of magnitude in many cases. A Matlab implementation of this work is freely available online.