Multilevel FMA for the discrete dipole approximation

The discrete dipole approximation (DDA) developed by Purcell and Pennypacker (1973) is a powerful and quite general method to calculate the scattering from arbitrary particles and has been applied to a variety of problems such as calculations of the scattering from graphite grains and porous dust particles. In the DDA, a continuum target is replaced by an array of point dipoles which interact with each other and a consistent solution is sought. Direct inversion of the matrix is not feasible for most problems due to the huge number of unknowns and iterative solutions become inevitable. In this work, the multilevel fast multipole algorithm (MLFMA) is used. The fast multipole algorithm (FMA) was successfully used for different problems, and the complexity of the MLFMA is O(N) for densely packed particles, and O(NlogN) for sparse and/or nonuniform distribution of particles, for any prescribed degree of accuracy. This is clearly an improvement over the FFT method.