An Automatic Learning System to Derive Multipole and Local Expansions for the Fast Multipole Method

This paper introduces an automatic learning method 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 × N dense matrices with a vector in O(N log N) operations. Later, it was applied successfully in many scientific fields such as simulation of physical systems, Computer Graphics and Molecular dynamics. However, FMM relies on the analytical expansions of the underlying kernel function defining the interactions between particles, which are not always obvious to derive. This is a major factor limiting the application of the FMM to many interesting problems. Thus, the proposed method here can be regarded as a useful tool helping practitioners to apply FMM to their own problems such as agent-based simulation of large complex systems. The preliminary results of the implemented system are very promising, and so we hope that the proposed method can be applied to other problems in different application domains.