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(N-3/2) to O(N-4/3) without increasing the complexity of the implementation.