This paper introduces a parallel directional fast multipole method (FMM) for solving N-body problems with highly oscillatory kernels, with a focus on the Helmholtz kernel in three dimensions. This class of oscillatory kernels requires a more restrictive low-rank criterion than that of the low-frequency regime, and thus effective parallelizations must adapt to the modified data dependencies. We ...

We give a fast numerical algorithm to evaluate a class of multivariable integrals. A direct numerical evaluation of these integrals costs Nm, where m is the number of variables and N is the number of the quadrature points for each variable. For m = 2 and m = 3 and for only one-dimensional variables, we present an algorithm which is able to reduce this cost from Nm to a cost of the order of O((−...

An implementation of the multilevel Fast Multipole method for time harmonic electromagnetic computations is presented. The method is parallelized for shared memory machines. A new parallelization scheme which is a hybrid between previous methods is proposed. Several symmetries and other methods that reduce the storage requirement are proposed. The most important symmetry is for the translation ...

In this paper we discuss the development and implementation of volumetric integral equations for both dielectric and magnetically permeable materials using curvilinear hexahedral elements. Both piecewise constant and higher order basis functions will be examined in the context of volumetric multilevel fast multipole method implementation. Comparisons with corresponding finite element– boundary ...

In this paper, we propose a novel method to extend the fast multipole method (FMM) to calculate the electrostatic potential due to charges inside or outside a dielectric sphere. The key result which allows such an extension is the construction of a small number (two for a 10 2 relative error in reaction potentials inside the sphere) of image point charges for source point charges inside or outs...

We describe a wideband version of the Fast Multipole Method for the Helmholtz equation in three dimensions. It unifies previously existing versions of the FMM for high and low frequencies into an algorithm which is accurate and efficient for any frequency, having a CPU time of O(N) if low-frequency computations dominate, or O(N logN) if high-frequency computations dominate. The performance of t...

The Fast Multipole Method (FMM) designed by V. Rokhlin rapidly computes the field scattered from an obstacle. This computation consists of solving an integral equation on the boundary of the obstacle. The main result of this paper shows the convergence of the FMM for the two dimensional Helmholtz equation. Before giving the theorem, we give an overview of the main ideas of the FMM. This is done...

An image method due to Allen and Berkley (1979) is often used to simulate the effect of reverberation in rooms. This method is relatively expensive computationally. We present a fast method for conducting such simulations using multipole expansions. For M real and image sources and N evaluation points, while the image method requiresO(MN) operations, our method achieves the calculations in O(M ...

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 ...