The use of Huygens' equivalence principle for solving the volume integral equation of scattering

A new algorithm is introduced to solve the volume integral equation. A volume scatterer is first divided into $N$ subscatterers. Then the subscatterers are divided into four groups, and the groups are in turn divided into four subgroups and so on. By using the idea used in many fast algorithms, a smaller problem can hence be nested within a larger problem. Moreover, by way of Huygens' equivalence principle, the scattering properties of a group of subscatterers in a volume can be replaced by a group of subscatterers distributed on a surface enclosing the volume. Based on this idea, we present an algorithm which solves the scattering problem by several stages, where at each stage, the interaction matrix algorithm is first used to find the scattering solution of each subgroup of subscatterers. Subscatterers are then replaced by equivalent surface subscatterers which are used in the next stage. Consequently, this results in a reduction in the number of subscatterers at every stage. This algorithm can be shown to have a CPU time asymptotically proportional to N*sqrt(N) for N subscatterers.