High‐frequency asymptotic acceleration of the fast multipole method

The plane wave translation operator of the fast multipole method (FMM) is evaluated asymptotically in the high‐frequency limit. This operator relates the plane wave components of a source region to a plane wave expansion for the radiated fields over a region away from the source. The asymptotic evaluation exploits the source‐to‐receiver directivity of the operator, leading to terms analogous to the geometrical optics (GO) and diffracted fields in the uniform theory of diffraction. Most importantly, the GO term identifies a “lit” region which may be used to define a windowing function for filtering out weak plane wave translations in the “shadow” region. The reduction in plane waves lowers the operational count of the FMM from O(N3/2) to O(N4/3) without increasing the complexity of the implementation.