Preconditioning the discrete dipole approximation

Abstract

The discrete dipole approximation (DDA) is a popular numerical method for calculating the scattering properties of atmospheric ice crystals. The standard DDA formulation involves the uniform discretization of the underlying volume integral equation, leading to a linear system with a block-Toeplitz Toeplitz-block matrix. This structure permits a matrix-vector product to be performed with $\mathcal{O}(n\log n)$ 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 and assess its performance for improving this rate of convergence. In particular, we approximate the system matrix by a block-circulant circulant-block matrix which can be inverted rapidly using the FFT. We present numerical results for scattering by hexagonal ice prisms demonstrating that this 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.

References in corpus (3)

• The discrete dipole approximation for simulation of light scattering by particles much larger than the wavelength
• Experimental and theoretical study of light scattering by individual mature red blood cells by use of scanning flow cytometry and discrete dipole approximation
• Light scattering by hexagonal columns in the discrete dipole approximation