# Preconditioning the discrete dipole approximation

## TL;DR

This paper introduces a preconditioning method for the discrete dipole approximation that significantly accelerates convergence in scattering simulations of atmospheric ice crystals, enabling faster computations.

## Contribution

It proposes a multi-level circulant preconditioner for the DDA system matrix, improving iterative solution convergence for large and complex scattering problems.

## Key findings

- Reduces simulation times by orders of magnitude
- Effective for scattering by hexagonal ice prisms
- Available MATLAB implementation online

## 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.