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 in...

The approximate computation of all gravitational forces between N interacting particles via the fast multipole method (FMM) can be made as accurate as direct summation, but requires less thanO (N) operations. FMM groups particles into spatially bounded cells and uses cell-cell interactions to approximate the force at any position within the sink cell by a Taylor expansion obtained from the mult...

We present a fast multipole algorithm for the evaluation of pairwise interaction through the radial basis functions such as 1/r, ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ a2 p and 1= ffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2 þ a2 p (with a > 0) in both two and three dimensions. Our algorithm is an extension of the kernel independent fast multipole method presented in Ying et al. [L. Y...

The fast multipole method (FMM) has been implemented to speed up the matrix-vector multiply when an iterative method is used to solve combined eld integral equation (CFIE). FMM reduces the complexity from O(N 2) to O(N 1:5). With a multilevel fast multipole algorithm (MLFMA), it is further reduced to O(NlogN). A 110,592 unknown problem can be solved within 24 hours on a SUN Sparc10.

A diagonal form fast multipole boundary element method (BEM) is presented in this paper for solving 3-D acoustic wave problems based on the Burton–Miller boundary integral equation (BIE) formulation. Analytical expressions of the moments in the diagonal fast multipole BEM are derived for constant elements, which are shown to be more accurate, stable and efficient than those using direct numeric...

In the analysis of an acoustic radiation mode of a baffled plate, Rayleigh integral with free space Green’s function is involved. The boundary element method (BEM) is one of the approaches to compute its modes and radiation efficiencies. In this paper a fast multipole BEM in conjunction with an iterative solver based on the implicit restart Arnold method is proposed to efficiently and accuratel...

The evaluation of sums (matrix-vector products) of the solutions of the three-dimensional biharmonic equation can be accelerated using the fast multipole method, while memory requirements can also be significantly reduced. We develop a complete translation theory for these equations. It is shown that translations of elementary solutions of the biharmonic equation can be achieved by considering ...

In this paper we present an implementation of the well known “fast multipole” method for the efficient calculation of dipole fields, that uses polynomials in the Cartesian coordinates rather than spherical harmonics. This has considerable efficiency and simplicity advantages. We have implemented it in the context of an arbitrary hierarchical system of cells – no periodic mesh is required. The i...

Despite its wide applicability in scientific computing, the linear time fast multipole method (FMM) with provable error bounds has not been used extensively in computer graphics. This paper presents – to our knowledge – the first application of FMM to the problem of global illumination. The light transport kernel is broken into multipole and local expansions and required transformations for the...

The solution of the Helmholtz and Maxwell equations using integral formulations requires to solve large complex linear systems. A direct solution of those problems using a Gauss elimination is practical only for very small systems with few unknowns. The use of an iterative method such as GMRES can reduce the computational expense. Most of the expense is then computing large complex matrix vecto...