Fast directional algorithms for the Helmholtz kernel

This paper presents a new directional multilevel algorithm for solving N -body or N -point problems with highly oscillatory kernels. We address the problem by first proving that the interaction between a ball of radius r and a well-separated region has an approximate low rank representation, as long as the well-separated region belongs to a cone with a spanning angle of O ( 1 / r ) and is at a distance which is at least O ( r 2 ) away from the ball. Based on this representation, our algorithm organizes the high frequency computation using a multidirectional and multiscale strategy. Our algorithm is proved to have an optimal O ( N log N ) computational complexity for any given accuracy when the points are sampled from a two-dimensional surface.