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38
Fast Fourier transforms for nonequispaced data: A tutorial
, 2000
"... In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity o ..."
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Cited by 111 (33 self)
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In this section, we consider approximative methods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particular, we are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTalgorithms with respect to roundoff errors and apply NDFTalgorithms for the fast computation of Bessel transforms.
Multilevel Fast Multipole Algorithm for Solving Combined Field Integral Equation of Electromagnetic Scattering
, 1995
"... The fast multipole method (FMM) has been implemented to speed up the matrixvector multiply when an iterative method is used to solve combined field integral equation (CFIE). FMM reduces the complexity from O(N 2 ) to O(N 1:5 ). With a multilevel fast multipole algorithm (MLFMA), it is further re ..."
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Cited by 82 (10 self)
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The fast multipole method (FMM) has been implemented to speed up the matrixvector multiply when an iterative method is used to solve combined field integral equation (CFIE). FMM reduces the complexity from O(N 2 ) to O(N 1:5 ). With a multilevel fast multipole algorithm (MLFMA), it is further reduced to O(NlogN ). A 110,592 unknown problem can be solved within 24 hours on a SUN Sparc10. 1. Introduction The electromagnetic (EM) field scattering by threedimensional (3D) arbitrarily shaped conductor can be obtained by finding the solution of an integral equation where the unknown function is the induced current distribution. The integral equation is discretized into a matrix equation by the method of moments (MOM). The resultant matrix equation y The authors would like to thank L. Hernquist, J.E. Barnes and P. Hut for providing us with copies of their codes, and thank M.B. Woodworth, M.G. Cot'e, and A.D. Yaghjian for providing us with their numerical and experimental data. This wor...
A PrecorrectedFFT Method for Electrostatic Analysis of Complicated 3D Structures
 IEEE Transactions on ComputerAided Design of Integrated Circuits and Systems
, 1997
"... In this paper we present a new algorithm for accelerating the potential calculation which occurs in the inner loop of iterative algorithms for solving electromagnetic boundary integral equations. Such integral equations arise, for example, in the extraction of coupling capacitances in threedimensio ..."
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Cited by 68 (26 self)
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In this paper we present a new algorithm for accelerating the potential calculation which occurs in the inner loop of iterative algorithms for solving electromagnetic boundary integral equations. Such integral equations arise, for example, in the extraction of coupling capacitances in threedimensional (3D) geometries. We present extensive experimental comparisons with the capacitance extraction code FASTCAP [1] and demonstrate that, for a wide variety of geometries commonly encountered in integrated circuit packaging, onchip interconnect and microelectromechanical systems, the new "precorrectedFFT " algorithm is superior to the fast multipole algorithm used in FASTCAP in terms of execution time and memory use. At engineering accuracies, in terms of a speedmemory product, the new algorithm can be superior to the fast multipole based schemes by more than an order of magnitude.
Completion energies and scale
, 2000
"... The detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping. In this study, we examine the notion of completion energy of curve elements, showing, and exploiting its intrinsic dependence on length and width scales. We introduce a fast met ..."
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Cited by 49 (6 self)
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The detection of smooth curves in images and their completion over gaps are two important problems in perceptual grouping. In this study, we examine the notion of completion energy of curve elements, showing, and exploiting its intrinsic dependence on length and width scales. We introduce a fast method for computing the most likelycompletion between two elements, by developing novel analytic approximations and a fast numerical procedure for computing the curve of least energy. We then use our newlydeveloped energies to find the most likelycompletions in images through a generalized summation of induction fields. This is done through multiscale procedures, i.e., separate processing at different scales with some interscale interactions. Such procedures allow the summation of all induction fields to be done in a total of only O(N log N) operations, where N is the number of pixels in the image. More important, such procedures yield a more realistic dependence of the induction field on the length and width scales: The field of a long element is verydifferent from the sum of the fields of its composing short segments.
Multiscale scientific computation: Review 2001
 Multiscale and Multiresolution Methods
, 2001
"... ..."
A Multidimensional Approach to ForceDirected Layouts of Large Graphs
, 2000
"... Abstract. We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or thr ..."
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Cited by 36 (5 self)
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Abstract. We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or three dimensional subspace of E. Projecting highdimensional drawings onto two or three dimensions often results in drawings that are “smoother ” and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, efficient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a midrange PC. 1
A short course on fast multipole methods
 Wavelets, Multilevel Methods and Elliptic PDEs
, 1997
"... In this series of lectures, we describe the analytic and computational foundations of fast multipole methods, as well as some of their applications. They are most easily understood, perhaps, in the case of particle simulations, where they reduce the cost of computing all pairwise interactions in a s ..."
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Cited by 31 (2 self)
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In this series of lectures, we describe the analytic and computational foundations of fast multipole methods, as well as some of their applications. They are most easily understood, perhaps, in the case of particle simulations, where they reduce the cost of computing all pairwise interactions in a system of N particles from O(N 2)toO(N)orO(N log N) operations. They are equally useful, however, in solving certain partial differential equations by first recasting them as integral equations. We will draw heavily from the existing literature, especially Greengard [23, 24, 25]; Greengard and Rokhlin [29, 32]; Greengard and Strain [34].
A Fast MultiDimensional Algorithm for Drawing Large Graphs
 In Graph Drawing’00 Conference Proceedings
, 2000
"... We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or three dimensi ..."
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Cited by 28 (4 self)
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We present a novel hierarchical forcedirected method for drawing large graphs. The algorithm produces a graph embedding in an Euclidean space E of any dimension. A two or three dimensional drawing of the graph is then obtained by projecting a higherdimensional embedding into a two or three dimensional subspace of E. Projecting highdimensional drawings onto two or three dimensions often results in drawings that are "smoother" and more symmetric. Among the other notable features of our approach are the utilization of a maximal independent set filtration of the set of vertices of a graph, a fast energy function minimization strategy, e#cient memory management, and an intelligent initial placement of vertices. Our implementation of the algorithm can draw graphs with tens of thousands of vertices using a negligible amount of memory in less than one minute on a midrange PC. 1 Introduction Graphs are common in many applications, from data structures to networks, from software engineering...
A Multilevel Algorithm For Solving Boundary Integral Equation
 Micro. Opt. Tech. Lett
, 1994
"... In the solution of an integral equation using the Conjugate Gradient (CG) method, the most expansive part is the matrixvector multiplication, requiring O(N 2 ) floating point operations. The Fast Multipole Method (FMM) reduced the operation to N 1:5 . In this paper, we apply a multilevel algor ..."
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Cited by 23 (12 self)
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In the solution of an integral equation using the Conjugate Gradient (CG) method, the most expansive part is the matrixvector multiplication, requiring O(N 2 ) floating point operations. The Fast Multipole Method (FMM) reduced the operation to N 1:5 . In this paper, we apply a multilevel algorithm to this problem and show that the complexity of a matrixvector multiplication is proportional to N(log(N)) 2 . y This work was supported by NASA under contract NASA NAG 2871, Office of Naval Research under grant N0001489J1286, and the Army Research Office under contract DAAL0391G0339, and the National Science Foundation under grant NSFECS9224466. The computer time was provided by the National Center for Supercomputing Applications (NCSA) at the University of Illinois, UrbanaChampaign. Published in Micro. Opt. Tech. Lett., Vol. 7, No. 10, pp. 466470, July, 1994. File:mlfma1.tex, January 13, 1995 1. Introduction Multilevel algorithms have been used to generate fast algorit...
A PrecorrectedFFT method for Capacitance Extraction of Complicated 3D Structures
 Proc. ICCAD
, 1994
"... In this paper we present a new approach to threedimensional capacitance extraction based on a precorrected FFT scheme. The approach is compared to the now commonly used multipoleaccelerated algorithms for a variety of structures, and the new method is shown to have substantial performance and memor ..."
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Cited by 23 (6 self)
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In this paper we present a new approach to threedimensional capacitance extraction based on a precorrected FFT scheme. The approach is compared to the now commonly used multipoleaccelerated algorithms for a variety of structures, and the new method is shown to have substantial performance and memory advantages.