The Fast Multipole Method allows the rapid evaluation of sums of radial basis functions centered at points distributed inside a computational domain at a large number of evaluation points to a specified accuracy . The method scales as O (N ) in both time and memory compared to the direct method with complexity O(N 2 ), which allows the solution of larger problems with given resources. Graphical processing units (GPU) are now increasingly viewed as data parallel compute coprocessors that can provide significant computational performance at low price. We describe acceleration of the FMM using the data parallel GPU architecture.The FMM has a complex hierarchical (adaptive) structure, which is not easily implemented on dataparallel processors. We described strategies for parallelization of all components of the FMM, develop a model to explain the performance of the algorithm on the GPU architecture; and determined optimal settings for the FMM on the GPU, which are different from those on usual CPUs. Some innovations in the FMM algorithm, including the use of modified stencils, real polynomial basis functions for the Laplace kernel, and decompositions of the translation operators, are also described.We obtained accelerations of the Laplace kernel FMM on a single NVIDIA GeForce 8800 GTX GPU in the range of 30-60 compared to a serial CPU FMM implementation. For a problem with a million sources, the summations involved are performed in approximately one second. This performance is equivalent to solving of the same problem at a 43 Teraflop rate if we use straightforward summation.

The head-related transfer function (HRTF) for distant sources is a complicated function of azimuth, elevation and frequency. This paper presents simple geometric models of the head and torso that provide insight into its low-frequency behavior, especially at low elevations. The head-and-torso models are obtained by adding both spherical and ellipsoidal models of the torso to a classical spherical-head model. Two different numerical techniques--multipole reexpansion and boundary element methods--are used to compute the HRTF of the models in both the frequency domain and the time domain. These computed HRTFs quantify the characteristics of elevation-dependent torso reflections for sources above the torso-shadow cone, and reveal the qualitatively different effects of torso shadow for sources within the torso-shadow cone. These effects include a torso bright spot that is prominent for the spherical torso, and significant attenuation of frequencies above 1 kHz in a range of elevations. Both torso reflections and torso shadow provide potentially significant elevation cues. Comparisons of the model HRTF with acoustic measurements in the horizontal, median, and frontal planes confirm the basic validity of the computational methods and establish that the geometric models provide good approximations of the HRTF for the KEMAR mannequin with its pinnae removed.

Abstract. We develop exact expressions for the coef cients of series representations of translations and rotations of local and multipole fundamental solutions of the Helmholtz equation in spherical coordinates. These expressions are based on the derivation of recurrence relations, some of which, to our knowledge are presented here for the rst time. The symmetry and other properties of the coef cients are also examined, and based on these, ef cient procedures for calculating them are presented. Our expressions are direct, and do not use the Clebsch-Gordan coef cients or the Wigner 3-symbols, though we compare our results with methods that use these, to prove their accuracy. For evaluating a term truncation of the translated series (involving multipoles), compared to previous exact expressions that require operations, our expressions require evaluations.Key words. Helmholtz equation, multipole solutions, translation and rotation coef cients, fast evaluation.AMS subject classi cations. 33C55, 33C10, 35J05, 65N38, 65N99, 65Y201. Introduction. In several scienti c computing applications, the solution to the Helmholtz or Maxwell Equations is expressed in terms of the singular (multipole) and regular solutions of the Helmholtz equation in spherical coordinates, centered at various points. Series of such solutions (see Eq. (2.16)) in one coordinate system must be expressed, in terms of series of singular or regular solutions in another coordinate system. Such expressions are guaranteed to exist by the completeness of the functions on a sphere. Addition theorems [5], [15] provide the expressions for the coef cients of the series in the shifted coordinates, in terms of the original coef cients. The paper by Epton and Dembart [8] provides an introduction to expressions of the coef cients. Chew [22] applied differentiation theorems for spherical functions similar to those in this paper, to obtain recursions for the translation coef cients.One important scienti c computing area where there is a need for such expressions is in the Fast Multipole Method (FMM) solution of the Helmholtz and Maxwell equations [9,27,10]. The FMM algorithm was referred in [1] as one of the top algorithms of the 20th century. Here the complexity of the translation expressions on the one hand, and the numerical accuracy achievable on the other, are key barriers to use of these methods to more complicated problems that are of interest, and these are thus an area of active research. Other scienti c computing areas where there is a need for such translation theorems are in the solution of boundary value problems of scatterings from many spheres [24], and in the use of the T-matrix method for solution of scattering problems from many scatterers [14]. Note that in some multipole methods (e.g. [24]) computation of each entry of the translation matrix is needed. In this case the recursive computation of the matrix elements provides the algorithm with theoretical minimum of asymptotic complexity. For speci c applications we refer the reader to these papers....