L. Greengard and V. Rokhlin. On the Efficient Implementation of the Fast Multipole Algorithm. Technical Report RR-602, Yale University Department of Computer Science, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....through 6 x 6 conductors. The conductor surfaces are discretized by first cutting each conductor into sections based on where pairs of conductors overlap. In the 2 x 2 bus example, each conductor is cut into five sections (see Fig. 5(a) and in the 6 x 6 ex ample each conductor is divided into 13 sections. The discrctization is then completed by dividing each face of each section into nine panels, as demonstrated in Fig. 5(b) The edge panels have widths that are 10 of the inner panel widths to accurately discretize the expectcd in creased charge density near conductor edges 114] In ....

....GCR, and multipole accelerated GCR (MGCR) The MGCR algorithm s CPU times are Fig, 4. Bus structure test problem with 2 x 2 conductors. a) b) Fig. 5. Conduclor sclions are divided into panels. TABLE I COMPARISON OF EXTRACTION METHODS (CPU TIMES IN IBM 6000 SECONVS) Test Problem 2x2 3x3 4x4 5x5 6x6 Panels 792 1620 2736 4140 5832 Direct time 275 2700 12969 44345 (141603) GCR time 121 570 2115 4881 (14877) MGCR time (l = 2) 55 218 378 790 1412 MGCR time (l = 1) 29 108 245 436 775 GCR iters 48 78 120 150 (180) MGCR item (I = 2) 48 82 120 150 180 MGCR iters (l = 1) 54 88 120 150 ....

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L. Greengard and V. Rohklin, "On the efficient implementation of the fast multipole algorithm," Yale University Research Report YALEU/DCS/RR-602, Feb. 1988.

....CM 5 [10] promise to achieve Tera FLOPS performance and allow one to carry out simulations currently beyond reach. It is essential, however, not only to demand faster hardware, but also to optimize the software for optimal performance. This optimization is two fold: on the one hand, new algorithms [11, 12, 13, 14] can bring considerable performance improvements, while on the other hand a careful analysis and tuning of the code at hand avoids a wastage of the available processing resources. This motivated us to conduct a performance analysis of the communication computation structure of the parallel ....

L. Greengard and V. Rokhlin, On the efficient implementation of the fast multipole algorithm (1988).

....as well as the multipole translation needed to compute the initial expansions, M i , can be formulated in terms of a convolution operation on polynomials. In 2D the O(p 2 ) cost of the multipole to local conversion can be reduced to an O(p log p) operation via a 1D FFT like convolution [9]. The cost of computing the initial expansions, M i for i = 0 : k Gamma 2 is proportional to np kp log p as it takes np steps to compute the initial multipole expansion, M 0 , and 9p log p steps each time 9 multipole expansions are shifted and added to create successive expansions M 1 : ....

.... i = 0 : k Gamma 2 is proportional to (np 2 kp 2 log p) as it takes np 2 steps to compute the initial multipole expansion, M 0 , and 27p 2 log p steps each time 27 multipole expansions are shifted and added to create successive expansions M 1 : M k Gamma2 via FFT techniques [6, 9]. Doing the multipole to local conversion on all macroscopic cubes takes time proportional to kp 2 log p np 2 steps. It costs O(p 2 log p) operations to do a multipole to local conversion, which has to be done for a constant 702 cubes in (k Gamma 2) shells. It costs a final np 2 steps ....

L. Greengard and V. Rokhlin. On the efficient implementation of the fast multipole algorithm. Technical Report RR-602, Yale University Dept. of Computer Science, 1988.

....Gerasoulis algorithm turned out to be numerically unstable for large n. Looking at the n body problem at about the same time, Greengard and Rokhlin gave provably good approximation algorithms (known as the fast multipole algorithm) for the n body problem in both two and three dimensions [12, 13, 14]. In order to understand the complexity of their algorithm, we define A = P n i=1 jq i j to be the sum of the absolute particle charges, and if we desire the output to meet error bound ffl we define p = dlog(A=ffl)e. We refer to a problem with these input output constraints as an accuracy p ....

....the exact algorithm of Gerasoulis [9] gives asymptotic complexity at least as good as the approximation algorithms of this paper. For the two dimensional problem with accuracy p, Greengard and Rokhlin s initial algorithm had running time O(np 2 ) which was later improved to O(np log p) time [14]. The fast multipole algorithm attracted a lot of attention, and numerous studies and implementations have been made (see, for example, 5, 20, 23, 25, 24, 40] The running time s dependence on p can be considerable, since n body simulations for molecular dynamics typically require values of p of ....

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Greengard, L., and Rokhlin, V. On the efficient implementation of the fast multipole algorithm. Tech. Rep. Technical Report RR-602, Yale University, Department of Computer Science, 1988.

....precision for any binomial coefficient. 2.4.1 Exploiting the Symmetry of a Quad tree The following scheme exploits the symmetry inherent to the process of translating local expansions, and reduces the overall execution time for the 2DFMM. This idea was introduced by Greengard and Rokhlin in [43] where it was described as a precursor to how the Fast Fourier Transform can be employed to reduce the computational time to compute the local expansions, in terms of multipole coefficients, from O(p 2 ) to O(p log p) The latter step was not carried out for this thesis however, because the ....

....coefficients in a 1 dimensional array, without storing coefficients M m n , such that m n. 3.4.1 Exploiting the Symmetry of an Oct tree The following scheme exploits the symmetry inherent in the process of translating the local expansions. This idea was introduced by Greengard and Rokhlin in [43], where it was described as a precursor to how the Fast Fourier Transform can be employed to reduce the computational time to form the local expansions O(p 4 ) to O(p 2 log p) Unfortunately this technique does not work well for large p as it involves large factorials which will exceed the ....

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Greengard, L., Rokhlin, V., "On the Efficient Implementation of the Fast Multipole Algorithm", Research Report 602, Yale University, Feb. 1988.

.... Init potential O(g(p) Delta N) Upward pass O(f(p) Delta 8 h ) Downward pass O( N int Delta f(p) f(p) Delta 8 h ) Far field O(g(p) Delta N) Near field O(Nnear Delta N 2 =8 h ) Minimized total O( g(p) p Nnear Delta N int Delta f(p) Delta N) The use of supernodes [14, 6] reduces the effective value of the number of computational elements, boxes , in the interactive field. This field is defined as the part of the far field that is in the parent box s near field. In [14] Zhao observes that for a near field extending two boxes away in each direction from a ....

L. F. Greengard and V. Rokhlin. On the efficient implementation of the fast multipole algorithm. Technical Report YALEU/DCS/RR--602, Yale Univ., February 1988.

....methods [1, 2, 6] for the N body problem partition the potentials 4 Y. C. HU AND S. L. JOHNSSON Table 1.1 Characteristics of some previous implementations of multipole like O(N) N body methods. Author Method Degree of Use of Hierarchy separation supernodes depth Greengard Rokhlin [7, 9] GR 2 No blog N 8 c Zhao [21] Zhao 2 Yes blog N 8 c Schmidt Lee [16] GR 1 No 3,4,5 Shimada et al. 17] GR 1,2 No log N 8 Leathrum [13] GR 1,2,mixed 1 and 2 Yes and No blog N 8 c Esselink [5] GR 2 Yes 2,3,4 into two parts: OE total = OE near Gammafield OE far Gammaf ield ; 2.1) ....

....329 2:8 Gammap by a factor of 4:6 by changing the separateness from two separation to one separation. The reduced separation will lower the base in the error estimate ( 2d 2) p 3 Gamma 1) Gammap and therefore increase the error for a fixed p. 2.3. Supernodes. The use of supernodes [9, 21] reduces the effective value of N int for a given separation, which brings about a dramatic improvement in the overall performance. In [21] Zhao observes that for two separations in three dimensions, there are many groups of eight sibling nodes of common parents in the the 567 interactive ....

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L. F. Greengard and V. Rokhlin. On the efficient implementation of the fast multipole algorithm. Technical Report YALEU/DCS/RR--602, Dept. of Computer Science, Yale Univ., February 1988.

....the n body problem is as follows. Given n point charges and their associated charge strengths, compute the value of the potential field (or the induced force) generated by these charges at each of the point charge locations. A breakthrough in this problem was made by Greengard and Rokhlin [4, 2, 5] who gave an algorithm for the two dimensional n body problem that ran in time O(np log p) when the input points met certain uniform distribution constraints and with the condition that p log n. Callahan and Kosaraju [1] gave an algorithm for this same problem that had time complexity O(n log ....

....n) be the time required to compute the n body potential to p bits of accuracy given a fair split tree for the point charges. Then we can compute all the potentials required by the n body problem in time O(n log log n T (p; n) Using direct application of the algorithms of Greengard and Rokhlin [4, 2, 5] and Callahan and Kosaraju [1] gives T (p; n) O(np log p) for an overall complexity of O(n log log n np log p) However, this is not the best possible result for the n body problem under our limited input precision restrictions Using improved techniques for the actual potential field ....

Greengard, L., and Rokhlin, V. On the efficient implementation of the fast multipole algorithm. Tech. Rep. Technical Report RR-602, Yale University, Department of Computer Science, 1988.

....and are especially unsuitable for highly nonuniform particle distributions. The class of so called heirarchical N Body methods were first developed almost a decade ago. Of these, the most promising are those based on the use of multipole expansions developed by Greengard and Rokhlin [CGR88, GR87b, GR87a] and some variants developed by Zhao [Zha87] and Anderson [And92] These methods are also approximation methods but can be used to achieve any fixed accuracy up to the machine precision. Using these methods, it is possible to calculate the forces (or the potential) using O(N) ....

....provided. The two methods were later extended to be of O(N) with analytical error bounds and by combining with the idea of multipole expansions [Ess92, WS95] The methods of Appel, and Barnes and Hut, are readily extended to nonuniformly distributed particles. Carrier, Greengard, and Rokhlin [CGR88] presented an adaptive version of the Greengard Rokhlin method. Similar extensions can be made to Anderson s and Zhao s methods. The proof in [CGR88] that the adaptive method retains O(N) complexity uses the fact that the depth of the hierarchy in computer simulations is limited by the machine ....

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Leslie F. Greengard and Vladimir Rokhlin. On the efficient implementation of the fast multipole algorithm. Technical Report YALEU/DCS/RR--602, Dept. of Computer Science, Yale Univ., February 1988.

....potential field due to all other bodies, at the current location of each body. The naive algorithm for this potential computation requires quadratic work; however, the potential can be approximated to p bits of accuracy in O(np 2 log p) time using the multipole method of Greengard and Rokhlin [4, 5], or by the recent modified multipole method of Reif and Tate [11] that has time complexity O(np 2 ) 1.1 The n body Reachability Problem In this paper, we consider the complexity of simulating a set of n charged particles in three dimensions, where the particles interact under the induced ....

L. Greengard and V. Rokhlin. On the Efficient Implementation of the Fast Multipole Algorithm. Technical Report RR-602, Yale University Department of Computer Science, 1988.

....expansions in Cartesian coordinates, which yields more costly multipole expansion calculations than polar coordinates. Leathrum and Board [23, 5] and Elliott and Board [9] achieved efficiencies in the range 14 20 in implementing Fast Fourier Transform accelerated Greengard Rokhlin s method [15] on the KSR 1. Schmidt and Lee [28] vectorized this method for the Cray Y MP and achieved an efficiency of 39 on a single processor. For comparison, we have also included the results reported in this paper. Little progress has been made in the implementation of adaptive O(N) methods in ....

L. F. Greengard and V. Rokhlin. On the efficient implementation of the fast multipole algorithm. Technical Report YALEU/DCS/RR--602, Dept. of Computer Science, Yale Univ., February 1988.

....as well as the multipole translation needed to compute the initial expansions, M i , can be formulated in terms of a convolution operation on polynomials. In 2D the O(p 2 ) cost of the multipole to local conversion can be reduced to an O(p log p) operation via a 1D FFT like convolution [9]. The cost of computing the initial expansions, M i for i = 0 : k Gamma 2 is proportional to np kp log p as it takes np steps to compute the initial multipole expansion, M 0 , and 9p log p steps each time 9 multipole expansions are shifted and added to create successive expansions M 1 : ....

.... i = 0 : k Gamma 2 is proportional to (np 2 kp 2 log p) as it takes np 2 steps to compute the initial multipole expansion, M 0 , and 27p 2 log p steps each time 27 multipole expansions are shifted and added to create successive expansions M 1 : M k Gamma2 via FFT techniques [6, 9]. Doing the multipole to local conversion on all macroscopic cubes takes time proportional to kp 2 log p np 2 steps. It costs O(p 2 log p) operations to do a multipole to local conversion, which has to be done for a constant 702 cubes in (k Gamma 2) shells. It costs a final np 2 steps ....

L. Greengard and V. Rokhlin. On the efficient implementation of the fast multipole algorithm. Technical Report RR-602, Yale University Dept. of Computer Science, 1988.

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L. Greengard and V. Rokhlin. On the Efficient Implementation of the Fast Multipole Algorithm. Technical Report RR-602, Yale University Department of Computer Science, 1988.

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L. Greengard and V. Rokhlin, On the efficient implementation of the fast multipole algorithm, Tech. Rep. RR-602, Department of Computer Science, Yale University, 1988.

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Greengard, L., Rokhlin, V., "On the Efficient Implementation of the Fast Multipole Algorithm", Research Report 602, Yale Uni. (Feb 1988).

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