Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. MIT Press, Cambridge, MA, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....context of gravitation, hierarchical N body methods based on it have found increasing applicability in various problem domains. To demonstrate the wide ranging applicability of these methods and their consequent importance for high performance computing, we list some of the problem domains below [14]: 1. Astrophysics: The bodies in the system are stars or planets in a galaxy, and the governing interaction law is gravitational. 2. Plasma Physics: The bodies are ions or electrons, and the governing law is electrostatic. 1We use the term body and particle interchangeably in this paper. ....

....more complex version that adapts to arbitrary distributions. We use the adaptive FMM in this paper. While the mathematics of the Barnes Hut algorithm are the same in three dimensions as they are in two, the FMM uses a different, more complex mathematical foundation in three dimensions [14]. The new mathematics substantially increases the constant factors in the time complexity expression, so that the three dimensional FMM is far more expensive than the two dimensional FMM. The structures of the twoand three dimensional algorithms are the same, however, as are the issues in ....

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Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. ACM Press, 1987.

....in the number of particles is surpassed, approximating the interaction of particles with interactions between sufficiently separated particle clusters allows the computational effort to be substantially reduced. The best known of these fast summation approaches is the fast multipole method [10], which, under certain assumptions, gives a method of . The fast multipole method is a typical divide and conquer algorithm. A cubic computational domain is recursively subdivided into octants. At the finest level the influence of the particles within a cell onto sufficiently separated cells is ....

Leslie Greengard, The rapid evaluation of potential fields in particle systems, 1987.

....memory in the SPLASH 2 version. FMM: Like Barnes, the FMM application also simulates a system of bodies over a number of time steps. FMM differs from Barnes in that it simulates interactions in two dimensions using a different hierarchical N body method called the adaptive Fast Multipole Method [Gre87]. As in Barnes Hut, the major data structures are body and tree cells, which again are records of body attributes. Like Barnes, FMM also allows multiple particles per leaf cell. However, there are two main differences between them. First, the tree is not traversed once per body, but only in a ....

Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. ACM Press, 1987.

....above require further work in a NUMA architecture. Our strategy is to first try to solve the compiler portability issue in a NUMA machine, and then re examine the implemented algorithms in that environment. In addition, we plan to examine a sufficiently large application, such as the N body problem[24], to study data placement problems which will arise. There has been some work done on the data allocation problem on distributed memory machines( 31] 36] 12] 25] 47] 32] but the studies have been limited to the allocation of arrays and or SIMD machines (such as the Connection Machine) ....

Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. ACM Distinguished Dissertations. The MIT Press, Cambridge, Massachusetts, 1988.

....method uses a new Hybrid Fast Multipole (HFM) technique that was developed for Accsim in order to ad dress beam distribution and halo issues that may arise in short term or long term injection simulations. The HFM technique utilizes the DAPIP2 package of routines developed by L. Greengard[1]. They are a robust 2D implementation of his Fast Multipole Method (FMM) field solver, which is designed to solve the field for an arbitrary collection of discrete charges. The FMM method does not use a grid, but rather subdivides the solution domain into a heirarchical tree of square or cubic ....

Leslie F. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, Cambridge, Mass: MIT Press, 1988.

....of parallel Grobner basis program w.r.t. pSather V3. 240 Level 0 Level 1 Level 2 C0 C2 X F X F C3 C1 X X F F X F F F A0 B0 B2 B1 B3 A0 B0 B1 B2 B3 C3 C2 C1 C0 Figure 4.40: A region and two levels of refinement, and a corresponding quadtree representation. algorithm [114] takes time O(N ) We implemented two versions of the algorithm, for when the particles are evenly or unevenly distributed throughout the region. A non adaptive algorithm is used when the particles are evenly distributed and an adaptive algorithm when they are unevenly distributed. Section 4.8.1 ....

....distributed throughout the region. A non adaptive algorithm is used when the particles are evenly distributed and an adaptive algorithm when they are unevenly distributed. Section 4.8. 1 describes the general outline of the algorithm, but does not go into the detailed mathematics described in [114]. Section 4.8.2 describes our parallel implementation and data structures used. Section 4.8.3 gives our timing measurements. 4.8.1 Outline of Greengard Rohklin s N Body Algorithm The crucial idea of the algorithm is that the potential on a particle p due to other N Gamma 1 particles can be ....

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Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. ACM Distinguished Dissertations. The MIT Press, Cambridge, Massachusetts, 1988.

....for image forces, by the placement of an appropriate set of fictitious charges. In the few GeV regime of typical Accsim applications these forces are of reduced significance [7] and in the present study are not included. Fast Multipole Method The adaptive Fast Multipole Method (FMM) of Greengard [6] and Rokhlin is one of the best performing and most accurate tree code algorithms. It uses adaptive subdivision to break down the problem domain into a quad tree in which the leaf regions each contain a small number (e.g. 40) of charges and uses the properties of multipole expansions and of ....

Leslie F. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, Cambridge, Mass: MIT Press, 1988.

....of directed graphs with a distinguished root and k vertices, each having out degree at most 2. This program makes extensive use of mutation and vectors. Boyer is a term rewriting theorem prover that allocates heavily. N Body is a Scheme implementation [23] of the Greengard multipole algorithm [6] for computing gravitational forces on point masses distributed uniformly in a cube. Dynamic is an implementation of a tagging optimization algorithm [8] for Scheme. Nucleic is a constraint satisfaction algorithm used to determine the three dimensional structure of nucleic acids [5] It is ....

Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. ACM Press, 1987.

....decomposition and approximations that treat many bodies as one. The Barnes Hut algorithm is very popular, as it is not too complex to implement, and tends to have fast execution. In 1987, a new algorithm was developed for solving the N body problem by Greengard with a complexity of O(n) [Gre87]. It is called the Fast Multipole Algorithm (FMA) and relies on rather complex field mathematics to solve the problem. The mathematics are substantially more complex in three dimensions than those used for two. The extreme complexity of the algorithm leads to both a large constant hidden by the ....

....can be used to calculate the force from all bodies outside of a given radius. Its value is dependent on its parent s local expansion and the multipole expansions of some close, but not adjacent, neighbors. Similar translation formulae exist for translating local expansions and can be found in [Gre87]. The third phase is to compute the force and potential for each body in the simulation. The force on a body is the summation of the forces directly calculated from bodies in adjacent regions and the single evaluation of the local expansion for the region in which the body is contained. The ....

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Leslie Greengard. The rapid evaluation of potential fields in particle systems. MIT Press, 1987.

....in a medium under electrostatic forces. Many N body problems have the properties that long range interactions between bodies cannot be ignored, but that the magnitude of interactions falls off with distance between the interacting bodies. The hierarchically structured Fast Multipole Method (FMM) [7] is an efficient, accurate, and hence very promising algorithm for solving such problems. Besides being very efficient and applicable to a wide range of problem domains including both classical Nbody problems as well as others that can be formulated as such [7] the FMM is also highly parallel ....

....Fast Multipole Method (FMM) 7] is an efficient, accurate, and hence very promising algorithm for solving such problems. Besides being very efficient and applicable to a wide range of problem domains including both classical Nbody problems as well as others that can be formulated as such [7] the FMM is also highly parallel in structure. It is therefore likely to find substantial use in applications for highperformance multiprocessors. There are two versions of the FMM: The uniform FMM works very well when the particles in the domain are uniformly distributed, while the adaptive FMM ....

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Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. ACM Press, 1987.

....is considered, its parallel implementation is described, its execution time performance is modeled and compared with measured results, and two alternative load balancing algorithms for enhancing performance are investigated. Parallel N body techniques are widely applied in a number of fields[9] ranging from astrophysics, to fluid dynamics, to computational geometry[7] They require dynamically changing, non uniform, intensive computation and irregular, unstructured communication. They are therefore good candidates for use as parallel computing benchmarks[8] and the results presented in ....

Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. ACM Press, 1987.

....as a point or spherical geometry (atom) with an associated charge and mass distribution. The complexity in determining the evolution of the system resides in reducing the long range all to all coupling between bodies. This can be accomplished using clustering techniques or multipole expansions [3]. A review of contact detection schemes for discrete elements with more complex shapes is contained in Williams and O Connor, 1995 [4] and Hogue, 1996 [5] We note there are two distinct algorithmic stages in contact detection, namely spatial search and contact resolution. Spatial search ....

Leslie F. Greengard. The Rapid Evaluation of Potential Fields in Particle Systems, ACM Distinguished Dissertation 1987. MIT Press, 1988. ACM Distinguished Dissertation Series 1987.

....[WSH94] for more details. FMM: Like Barnes, the FMM application also simulates a system p n n n n n n p n p of bodies over a number of timesteps. However, it simulates interactions in two dimensions using a different hierarchical N body method called the adaptive Fast Multipole Method [Gre87]. As in Barnes, the major data structures are body and tree cells, with multiple particles per leaf cell. FMM differs from Barnes in two respects: i) the tree is not traversed once per body, but only in a single upward and downward pass (per timestep) that computes interactions among cells and ....

Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. ACM Press, 1987.

.... of equidistributed particles reveals that the complexity for this case is only O(N ) while the algorithm of Barnes and Hut remains O(N log N ) Rokhlin [53] independently of Appel, introduced very similar ideas for the solution of integral equations of potential theory, which he and Greengard [28, 29, 9, 30] extended rigorously to what is now known as the Fast Multipole Method. In his dissertation Greengard describes the non adaptive Fast Multipole Method for two and three space dimensions and also presents the adaptive Fast Multipole Method in two space dimensions. Due to the applicability of ....

....boundary conditions, where the case anm 6= 0 is commonly eschewed in order to force the potential to zero at infinity. However, on occasion it is of advantage to expand the potential locally into a series with b nm = 0. 3.1. 2 Translation and Conversion Operators The ensuing theorems (Greengard [28]) describe how these expansions can be evaluated (theorem 3.1.4) translated to different expansion centers (theorems 3.1.2 and 3.1.4) and how the different expansions can be transformed into each other (theorem 3.1.3) Our representations of the involved formulas are better adapted to the memory ....

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Leslie Greengard. The rapid evaluation of potential fields in particle systems. PhD thesis, The Massachusetts Institute of Technology, 1987. Association for Computing Machinery distingished dissertations ISBN 0-262-07110-X.

....of directed graphs with a distinguished root and k vertices, each having out degree at most 2. This program makes extensive use of mutation and vectors. Boyer is a term rewriting theorem prover that allocates heavily. N Body is a Scheme implementation [24] of the Greengard multipole algorithm [6] for computing gravitational forces on point masses Benchmark Lines Sites Analysis Time (in seconds) Polymorphic Soft Typing 0CFA 1CFA Splitting Lattice 215 252 .26 .46 .13 .49 Browse 233 283 .21 .96 .18 .50 Check 278 376 1.94 1.76 12.42 10.97 Graphs 621 413 .30 .73 .22 1.10 Boyer 632 212 ....

Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. ACM Press, 1987.

....N X i=1;i6=j A i z i Gamma z j ; j = 1 : N where A i is the charge of particle i located at z i and N is the total number of particles. The direct summation for the above expressions costs O(N 2 ) Two fast sequential algorithms are available: BarnesHut [2] or Fast Multipole Method (FMM)[23]. Since FMM has a lower complexity, we consider the 2D FMM for computing F j . The basic idea of the algorithm is to subdivide the space into boxes and the evaluation of particle positions between far away boxes can be approximated by a series of expansions. The coefficients of the series are ....

....than the previous level boxes and the series expansions are used to evaluate a larger constant number of far away boxes, which is always less or equal to 27 for a 2D space. At the root of the tree the entire 2D space particle evaluation has been covered. For more details see Greengard s thesis [23]. o x x x x x x x x x x x x x x x x x x x x x x x x x x x Figure 13: A 3 level non adaptive subdivision. X are the boxes in the interaction list of box O. Any box has most 27 boxes in its interaction list and at most 8 neighbors Since n body simulation involves the iterative updating of particle ....

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Leslie Greengrad, The Rapid Evaluation of Potential Fields in Particle Systems Ph.D thesis, Yale University, 1987.

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Leslie Greengard. The Rapid Evaluation of Potential Fields in Particle Systems. MIT Press, Cambridge, MA, 1988.

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Leslie Greengard, The Rapid Evaluation of Potential Fields in Particle Systems, MIT Press, Cambridge, MA, 1988.

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