Fast directional algorithms for the Helmholtz kernel

This paper presents a new directional multilevel algorithm for solving NN-body or NN-point problems with highly oscillatory kernels. We address the problem by first proving that the interaction between a ball of radius rr 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)O(1/r) and is at a distance which is at least O(r2)O(r2) 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(NlogN)O(NlogN) computational complexity for any given accuracy when the points are sampled from a two-dimensional surface.