Accelerated Cartesian expansions – A fast method for computing of potentials of the form R−ν for all real ν

Shanker, B.; Huang, He

doi:10.1016/j.jcp.2007.04.033

Cited by 67 publications

(77 citation statements)

References 27 publications

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“…This type of Cartesian far-field Taylor series treecode was previously used in computational vortex methods [11,27,29,32,33] and molecular dynamics [13,26], but not yet in the context of multiquadric RBFs. Alternative tree-based methods using Cartesian coordinates include an FMM for the Newtonian potential [39] and an FMM using Cartesian tensors for general power law potentials [36].…”

Section: The Present Workmentioning

confidence: 99%

Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode

Krasny¹,

Wang²

2011

SIAM J. Sci. Comput.

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Abstract.A treecode is presented for evaluating sums defined in terms of the multiquadric radial basis function (RBF), φ(x) = (|x| 2 + c 2 ) 1/2 , where x ∈ R 3 and c ≥ 0. Given a set of N nodes, evaluating an RBF sum directly requires CPU time that scales like O(N 2 ). For a given level of accuracy, the treecode reduces the CPU time to O(N log N ) using a far-field expansion of φ(x). We consider two options for the far-field expansion: (1) a Laurent series previously used in applications of the Fast Multipole Method to multiquadric RBFs, and (2) a certain Taylor series previously used in treecode particle simulations, but not yet in the context of multiquadric RBFs. It is known that the Laurent series converges when the RBF parameter c lies in an interval 0 ≤ c ≤c, wherec is proportional to the minimum node spacing, but here we show that the Taylor series converges uniformly for c ≥ 0. We implement the treecode in Cartesian coordinates and use a recurrence relation to compute the Taylor coefficients. Numerical results exhibit the treecode error, CPU time, and memory usage in two test cases, random nodes in a cube and on the surface of a sphere. The treecode approach presented here is applicable to generalized multiquadrics in any dimension.

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Section: The Present Workmentioning

confidence: 99%

Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode

Krasny¹,

Wang²

2011

SIAM J. Sci. Comput.

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“…For such kinds of simulations, the so-called Fast Multipole Methods (FMM) have gained prominence [27][28][29][30][31]. They attempt to capitalize on the fact that distant charge distributions influence local behavior less than nearby ones, and suitably grouping those distant ones into collective representations with an error that scales with a high power of distance.…”

Section: Topic 38b De-sc0001232 Advanced Simulation and Optimization mentioning

confidence: 99%

Advanced Simulation and Optimization Tools for Dynamic Aperture of Non-scaling FFAGs and Accelerators including Modern User Interfaces

Mills¹,

Makino²,

Berz³

et al. 2010

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TECHNICAL ABSTRACT (Limit to space provided)Statement of the problem or situation that is being addressed -typically, one to three sentences.With the U.S. experimental effort in HEP largely located at laboratories supporting the operations of large, highly specialized accelerators, the understanding and prediction of high energy particle accelerators becomes critical to the overall success of the DOE HEP program. One area in which small businesses can contribute to the ongoing success of the U.S. program in HEP is through innovations in computer techniques and sophistication in the modeling of high-energy accelerators. A specific newly identified problem lies in the simulation and optimization of FFAGs and related devices, for which currently available tools originally developed for other purposes provide only approximate and inefficient simulation. We propose to develop a set of tools for this purpose based on modern techniques and simulation approaches.General statement of how this problem is being addressed. This is the overall objective of the combined Phase I and Phase II projects.The simulation framework of the code COSY INFINITY is highly accurate, efficient and well-developed with a large user base, and provides powerful tools for global, non-local optimization. For the specific problem at hand, new tools need to be developed that can describe the complex specific electromagnetic fields, including high-order fringe fields, edge effects, and general field profiles, and that can accommodate the necessary very large emittance of the beam by dynamic subdivision of the phase space. In order to simplify design needs, modern global, non-local optimization techniques will be connected to the tools to allow efficient probing of the usually very high dimensional parameter space. User-friendly interfaces will also be developed to enhance the audience and lessen the expertise typically required in the use of accelerator codes.What will be done in Phase I -typically, two to three sentences. Phase I will consist of the development of a bare-bones prototype of the core numerical engines necessary for the task, including the ability to treat arbitrary field arrangements, computation of local closed orbits, tunes, tune shifts, and parameter optimization, as well as analysis and benchmarking of their performance characteristics.COMMERCIAL APPLICATIONS AND OTHER BENEFITS as described by the applicant. (Limit to space provided).The development of broad, highly-accurate accelerator models with powerful optimization tools and user-friendly interfaces and graphics will not only aid in the critical development of new accelerators for the core mission of the DOE HEP program, but also promote the transfer of accelerator technology into commercial and medical applications such as hadrontherapy, neutron and light sources, and other potential markets for accelerators. The development of broad, highly-accurate accelerator models with powerful optimization tools and user-friendly interfaces will enhance not only the HEP program but al...

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“…Despite their advantages, their formulation is more difficult than that of their differential equation counterparts, and as a result this method has seen sporadic development in the past 10, 11 , and a more concerted effort recently 12,13,14,15,16,17,18,19 . The recent development of fast solvers that ameliorate the CPU and memory complexity of surface integral equation based solvers, i.e., reduce the scaling from O(N 2 s ) to O(N s log 2 N s ) where N s is the number of spatial degrees of freedom, has made these techniques more appealing 20,21 . However, when compared to their differential equation based counterparts, the analysis here has been more or less restricted to simple basis functions (piecewise constant) and linear tessellations of the geometry.…”

Section: Introductionmentioning

confidence: 99%

“…The method, called the Generalized Method of Moments (GMM), aims to introduce this range of functionality. Specifically, in this paper we will 1. present the GMM computational framework that will (a) introduce a framework to develop local analytical surface representations on overlapping domains starting from either a tesselated object or a point cloud Note, while it is not a direct focus of this work, the GMM framework introduced here can be easily accelerated using the fast multipole method 20,21 to permit the analysis of very large objects. The framework constructed here also permits easy integration with the fast multipole method The rest of this paper proceeds as follows: In the next section, we will formally state the scattering problem.…”

Section: Introductionmentioning

confidence: 99%

Generalized Method of Moments

2007

Encyclopedia of Measurement and Statistics

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The analysis of scattering from complex objects using surface integral equations is a challenging problem. Its resolution has wide ranging applications-from crack propagation to diagnostic medicine. The two ingredients of any integral equation methodology is the representation of the domain and the design of approximation spaces to represent physical quantities on the domain. The order of convergence depends on both the surface and geometry representation. For instance, most surface models are restricted to piecewise flat or second order tessellations. Similarly, the most commonly known basis spaces for acoustics are piecewise constant functions. What is desirable is a framework that permits adaptivity (of size and order) in both geometry and function representations. Unlike volumetric, differential equation solvers, such as the finite element method, developing an hpadaptive framework for surface integral equations is very difficult. This papers proposes a resolution to this problem by developing a novel framework that relies on reconstruction of the surface using locally smooth parameterizations, and defining partition of unity functions and higher order basis spaces on overlapping domains. This permits easy refinement of both the geometry and function representation. This capabilities of the proposed framework are shown via a number of numerical examples.

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Accelerated Cartesian expansions – A fast method for computing of potentials of the form R−ν for all real ν

Cited by 67 publications

References 27 publications

Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode

Fast Evaluation of Multiquadric RBF Sums by a Cartesian Treecode

Advanced Simulation and Optimization Tools for Dynamic Aperture of Non-scaling FFAGs and Accelerators including Modern User Interfaces

Generalized Method of Moments

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