Numerical Properties of a Gaussian Translation Operator for the 2D FMM

A diagonal Gaussian translation operator for the time-harmonic fast multipole method (FMM) in two dimensions is examined numerically. The Gaussian translation operator depends on a beam parameter that determines its sharpness. When the beam parameter is set to zero, the Gaussian translation operator reduces to the standard FMM translation operator. The sampling rate can be determined straightforwardly to achieve any desired accuracy. The directionality of the Gaussian translation operator makes it possible to reduce the number of plane waves required to achieve a given accuracy. The required number of plane waves depends strongly on the actual source-receiver locations, not just on the diameter of the source and receiver regions.