Adaptive FMM for fractal sets

N 2 interactions amongN points orN variables appears in the boundary element methods, problems involving radial basis functions or in probability theory to describe dense covariance matrices. The fast multipole formulation introduced by Greengard and Rokhlin approximates a matrixvector multiplication of the above form with desired accuracy in O(N) time. Several works have extended the algorithm by studying di↵erent kernels, analyzing the approximation error, introducing parallel implementation techniques, etc. [1] The adaptive FMM refers to the case where the particle distribution, and the corresponding hierarchical tree, are not uniform. The adaptive FMM and various aspects of its parallel implementation on di↵erent machines are an ongoing topic of research. [2] It is known that the adaptive FMM algorithm maintains the O(N) complexity irrespective of the point distribution [3]. This requires a modification to the original FMM. We will present a new proof for the linear complexity of the adaptive FMM for any distribution of the points. This also will make it apparent what modifications to the original FMM are required to ensure O(N) complexity for general particle distributions. Previous works have limited their analysis to very specific point distributions. The key point essentially is the manner in which points are distributed, in a non-uniform adaptive setting, as N goes to infinity. In the uniform case, the issue of increasing N presents no particular di culty. We can simply increase the density of points uniformly, and study how accuracy and parameters in the FMM are adjusted as a function of N . However, the non-uniform case is more di cult. One essential point is describing the process of adding points so that N !1. The adaptive test cases considered by most previous works fall broadly into the following categories: