The application of far-field approximation to accelerate the fast multipole method

The fast multipole method (FMM) is an efficient method in the iterative solution of the matrix equation that is associated with the integral equation of wave scattering. The key idea of the FMM is to divide the interactions between groups of scatterers into near-field interactions and non-near-field interactions, and perform the non-near-field interactions efficiently with the aid of the multipole expansion of the fields. The far-field approximation is introduced in FMM to calculate the wave interactions between groups that are separated by a very large distance. The advantage of this approach is that it automatically switches back to the original FMM for small problems. Under the far-field approximation, the translation operator is much simpler than that used in the FMM. Numerical results show speed up of the modified FMM over the original FMM for problems as small as several thousands.