Parallelization of 2D Fast Multipole Method

for each i = 1, 2, .., n, where K is a given (usually smooth away from origin) kernel. For example, for gravitaional interactions in 3D, K(x, y) = 1 |x−y| ; for 2D Laplacian equation, K(x, y) = log |x− y|. The direct computation has a cost of O(n). In recent decades, many algorithm has been invented and developed to accelerate the quadratic time complexity. Fast multipole method (FMM), first introduced by V. Rokhlin and L. Greengard in 1985[1], is probably the most famous and prominent one – it achieves the linear complexity O(n), with constant factor depends on precision. This fact makes FMM one of the first choice in simulations of large-scale physical problems, and thus, an efficient parallelization of FMM is crucial in many applications. Many papers[2][3] have explored different methods/platforms to parallel different versions of FMM. In this project, we will explore the parallelization of 2D Laplacian FMM using OpenMP on a shared memory computing structure. We will first make a brief introduction to the FMM algorithm (section 2); after that we will discuss about the fork-join model (section 3.1) – a natural but clever way to parallel FMM, and then we will explore yet another parallelization of FMM – the interleaving model (section 3.2). Finally we will show some numerical result about the scaling of the parallel FMM (section 4). The code we use in this project is based on FMM2D Libraries of Courant Mathematics and Computing Laboratory (CMCL). The original version of code is available at http://www.cims.nyu.edu/cmcl/ fmm2dlib/fmm2dlib.html. Our parallel FMM code is available on the project github https://github. com/tsl665/fmm2d_omp.