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tree code for collision detection of large numbers of particles applied to the Breit-Wheeler process

JOURNAL OF COMPUTATIONAL PHYSICS 2018年 355卷 582-596页

作者： Jansen, O. d'Humieres, E. Ribeyre, X. Jequier, S. Tikhonchuk, V. T. Univ Bordeaux CNRS CEA Ctr Lasers Intenses & ApplicatUMR 5107 F-33405 Talence France

Collision detection of a large number N of particles can be challenging. Directly testing N particles for collisions among each other leads to N-2 queries. Especially in scenarios, where fast, densely packed particles interact, challenges arise for classical methods like Particle-in-Cell or Monte-Carlo. Modern collision detection methods utilising bounding volume hierarchies are suitable to overcome these challenges and allow a detailed analysis of the interaction of large number of particles. This approach is applied to the analysis of the collision of two photon beams leading to the creation of electron-positron pairs. (c) 2017 Elsevier Inc. All rights reserved.

关键词： tree code Collision detection QED Breit-Wheeler process Pair creation Astrophysics

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Towards a petascale tree code: Scaling and efficiency of the PEPC library

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JOURNAL OF COMPUTATIONAL SCIENCE 2011年第2期2卷 138-143页

作者： Speck, R. Arnold, L. Gibbon, P. Forschungszentrum Julich Julich Supercomp Ctr Inst Adv Simulat D-52425 Julich Germany

The highly scalable parallel tree code PEPC for rapid computation of long-range (1/r) Coulomb forces is presented. It can be used as a library for applications involving electrostatics or Newtonian gravity in 3D. The code is based on the hashed oct-tree algorithm, in which particle coordinates are projected onto a space-filling curve prior to sorting and construction of multipole moments. However, standard particle sorting techniques can ultimately limit the scalability of such algorithms for thousands of cores, a bottleneck which can be alleviated by a recursive sort scheme specially adapted to the Morton curve. More serious limitations of the original locally essential tree concept of Salmon and Warren, which ultimately lead to a failure in memory scaling, are identified and analyzed rigorously. Benchmarks for the code on the IBM Blue Gene/P Jugene are presented which demonstrate scaling for multi-million particle systems on up to 8192 cores. (C) 2011 Elsevier B.V. All rights reserved.

关键词： tree code HPC N-body problem Petascaling Hashed oct-tree IBM Blue Gene/P

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A FAST SUMMATION tree code FOR MATERN KERNEL

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2014年第1期36卷 A289-A309页

作者： Chen, Jie Wang, Lei Anitescu, Mihai Argonne Natl Lab Div Math & Comp Sci Argonne IL 60439 USA Univ Wisconsin Dept Math Sci Milwaukee WI 53211 USA

The Matern family of functions is a widely used covariance kernel in spatial statistics for Gaussian process modeling, which in many instances requires calculations with a covariance matrix. In this paper, we design a fast summation algorithm for the Matern kernel in order to efficiently perform matrix-vector multiplications. This algorithm is based on the Barnes-Hut tree code framework and addresses several practical issues: the anisotropy of the kernel, the nonuniform distribution of the point set, and a tight error estimate of the approximation. Even though the algorithmic details differ from the standard tree code in several aspects, empirically the computational cost of our algorithm scales as O(n log n) for n points. Comprehensive numerical experiments are shown to demonstrate the practicality of the design.

关键词： Matern kernel Gaussian process fast summation tree code

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Gravitational lensing: numerical simulations with a hierarchical tree code

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1999年第1-2期109卷 353-372页

作者： Wambsganss, J Astrophys Inst Potsdam D-14482 Potsdam Germany

The mathematical formulation of gravitational lensing - the lens equation - is a very simple mapping R-2 --> R-2, between the lens (or sky) plane and the source plane. This approximation assumes that all the deflecting matter is in one plane. In this case the deflection angle a is just the sum over all mass elements in the lens plane. For certain problems - like the determination of the magnification of sources over a large number of source positions (up to 10(10)) for very many lenses (up to 10(6)) - straightforward techniques for the determination of the deflection angle are far too slow. We implemented an algorithm that includes a two-dimensional tree-code plus a multipole expansion in order to make such microlensing simulations "inexpensive". Subsequently we modified this algorithm such that it could be applied to a three-dimensional mass distribution that fills the universe (approximated by many lens planes), in order to determine the imaging properties of cosmological lens simulations. Here we describe the techniques and the numerical methods, and we mention a few astrophysical results obtained with these methods. (C) 1999 Elsevier Science B.V. All rights reserved.

关键词： astrophysics gravitational lensing tree code

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DEM simulations of toner particles with an O (N log N) hierarchical tree code algorithm

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GRANULAR MATTER 2006年第3-4期8卷 151-157页

作者： Hoffmann, Rainer Oce Printing Syst D-85581 Poing Germany

The computation time for a particle simulation where long-reaching forces, such as electrostatic forces for charged particles, have to be taken into account, rises asymptotically with O (N(2)). A significant reduction of computation time can be achieved by applying a tree code algorithm. The algorithm exploits the idea of grouping suitable particles to macroparticles thus reducing the number of necessary calculations within a reasonably limited error bound. Although this algorithm was developed for astronomical galaxy simulations, it can be also used for simulations of microscopical particle with finite radii. The tree code can be used for an efficient collision routine if some additional boundary conditions are taken into account. This results in an asymptotic computation time dependence of O (N log N).

关键词： discrete element method Barnes-Hut tree code collision criteria print process

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GOTHIC: Gravitational oct-tree code accelerated by hierarchical time step controlling

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NEW ASTRONOMY 2017年 52卷 65-81页

作者： Miki, Yohei Umemura, Masayuki Univ Tsukuba Ctr Computat Sci 1-1-1 Tennodai Tsukuba Ibaraki 3058577 Japan JST CREST 1-1-1 Tennodai Tsukuba Ibaraki 3058577 Japan

The tree method is a widely implemented algorithm for collisionless N-body simulations in astrophysics well suited for GPU(s). Adopting hierarchical time stepping can accelerate N-body simulations;however, it is infrequently implemented and its potential remains untested in GPU implementations. We have developed a Gravitational Oct-tree code accelerated by Hierarchical time step Controlling named GOTHIC, which adopts both the tree method and the hierarchical time step. The code adopts some adaptive optimizations by monitoring the execution time of each function on-the-fly and minimizes the time-to solution by balancing the measured time of multiple functions. Results of performance measurements with realistic particle distribution performed on NVIDIA Tesla M2090, K20X, and GeForce GTX TITAN X, which are representative GPUs of the Fermi, Kepler, and Maxwell generation of GPUs, show that the hierarchical time step achieves a speedup by a factor of around 3-5 times compared to the shared time step. The measured elapsed time per step of GOTHIC is 0.30 s or 0.44 s on GTX TITAN X when the particle distribution represents the Andromeda galaxy or the NFW sphere, respectively, with 2(24) = 16,777,216 particles. The averaged performance of the code corresponds to 10-30% of the theoretical single precision peak performance of the GPU. (C) 2016 The Authors. Published by Elsevier B.V.

关键词： N-body simulation tree code Block time step GPU computing

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Linear tree codes and the problem of explicit constructions

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LINEAR ALGEBRA AND ITS APPLICATIONS 2016年 490卷 124-144页

作者： Pudlak, Pavel

We reduce the problem of constructing asymptotically good tree codes to the construction of triangular totally nonsingular matrices over fields with polynomially many elements. We show a connection of this problem to ... 详细信息

关键词： tree code Error correcting code Triangular totally nonsingular matrix

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On matrices potentially useful for tree codes

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INFORMATION PROCESSING LETTERS 2022年 174卷

作者： Pudlak, Pavel Czech Acad Sci Inst Math Prague Czech Republic

Motivated by a concept studied in [1], we consider a property of matrices over finite fields that generalizes triangular totally nonsingular matrices to block matrices. We show that (1) matrices with this property suffice to construct asymptotically good tree codes and (2) a random block-triangular matrix over a field of quadratic size satisfies this property. We will also show that a generalization of this randomized construction yields codes over quadratic size fields for which the sum of the rate and minimum relative distance gets arbitrarily close to 1. (C) 2021 Elsevier B.V. All rights reserved.

关键词： Theory of computation Matrix tree code Finite field Singleton bound

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A hierarchical O(N) force calculation algorithm

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JOURNAL OF COMPUTATIONAL PHYSICS 2002年第1期179卷 27-42页

作者： Dehnen, W Max Planck Inst Astron D-69117 Heidelberg Germany

A novel code for the approximate computation of long-range forces between N mutually interacting bodies is presented. The code is based on a hierarchical tree of cubic cells and features mutual cell-cell interactions which are calculated via a Cartesian Taylor expansion in a symmetric way. such that total momentum is conserved. The code benefits from an improved and simple multipole acceptance criterion that reduces the force error and the computational effort. For N greater than or equal to 10(4) the computational costs are found empirically to rise sublinearly with N. For applications in stellar dynamics, this is the first competitive code with complexity O(N): it is faster than the standard tree code by a factor of 10 or more. (C) 2002 Elsevier Science (USA).

关键词： n-body simulations tree code fast multipole method adaptive algorithms

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Coding for interactive communication

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IEEE TRANSACTIONS ON INFORMATION THEORY 1996年第6期42卷 1745-1756页

作者： Schulman, LJ UNIV CALIF BERKELEY DIV COMP SCI BERKELEY CA 94720 USA

Let the input to a computation problem be split between two processors connected by a communication link;and let an interactive protocol pi be known by which, on any input, the processors can solve the problem using no more than T transmissions of bits between them, provided the channel is noiseless in each direction, We study the following question: If in fact the channel is noisy, what is the effect upon the number of transmissions needed in order to solve the computation problem reliably? Technologically this concern is motivated by the increasing importance of communication as a resource in computing, and by the tradeoff in communications equipment between bandwidth, reliability, and expense. We treat a model with random channel noise. We describe a deterministic method for simulating noiseless-channel protocols on noisy channels, with only a constant slowdown. This is an analog for general interactive protocols of Shannon's coding theorem, which deals only with data transmission, i.e., one-way protocols. We cannot use Shannon's block coding method because the bits exchanged in the protocol are determined only one at a time, dynamically, in the course of the interaction, Instead, we describe a simulation protocol using a new kind of code, explicit tree codes.

关键词： interactive communication coding theorem tree code distributed computing reliable communication error correction

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