A genetic programming based learning system to derive multipole and local expansions for the fast multipole method

This paper introduces an automatic learning algorithm based on genetic programming to derive local and multipole expansions required by the Fast Multipole Method (FMM). FMM is a well-known approximation method widely used in the field of computational physics, which was first developed to approximately evaluate the product of particular N x N dense matrices with a vector in O(N log N) operations, while direct multiplication requires O(N-2) operations. Soon after its invention, the FMM algorithm was applied successfully in many scientific fields such as simulation of physical systems (Electromagnetic, Stellar clusters, Turbulence), Computer Graphics and Vision (Light scattering) and Molecular dynamics. However, FMM relies on the analytical expansions of the underlying kernel function defining the interactions between particles, which are not obvious to derive. This is a major factor that severely limits the application of the FMM to many interesting problems. Thus, the proposed automatic technique in this article can be regarded as a very useful tool helping practitioners to apply FMM to their own problems. Here, we have implemented a prototype system and tested it on various types of kernels. The preliminary results are very promising, and so we hope that the proposed method can be applied successfully to other problems in different application domains.