Fast Summation at Nonequispaced Knots by NFFT
Abstract
We develop a new algorithm for the fast computation of discrete sums $f(y_j) := \sum_{k=1}^N \alpha_k K(y_j-x_k)$ (j =1, . . ., M) based on the recently developed fast Fourier transform (FFT) at nonequispaced knots. Our algorithm, in particular our regularization procedure, is simply structured and can be easily adapted to different kernels K. Our method utilizes the widely known FFT and can consequently incorporate advanced FFT implementations. In summary, it requires ${\cal O} (N \log N +M)$ arithmetic operations. We prove error estimates to obtain clues about the choice of the involved parameters and present numerical examples in one and two dimensions.
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References
1.
B. Alpert, G. Beylkin, R. Coifman, V. Rokhlin, Wavelet‐like bases for the fast solution of second‐kind integral equations, SIAM J. Sci. Comput., 14 (1993), 159–184
2.
Christopher Anderson, A method of local corrections for computing the velocity field due to a distribution of vortex blobs, J. Comput. Phys., 62 (1986), 111–123
3.
Rick Beatson, Leslie Greengard, A short course on fast multipole methods, Numer. Math. Sci. Comput., Oxford Univ. Press, New York, 1997, 1–37
4.
R. Beatson, W. Light, Fast evaluation of radial basis functions: methods for two‐dimensional polyharmonic splines, IMA J. Numer. Anal., 17 (1997), 343–372
5.
R. Beatson, G. Newsam, Fast evaluation of radial basis functions. I, Comput. Math. Appl., 24 (1992), 7–19, Advances in the theory and applications of radial basis functions
6.
G. Beylkin, On the fast Fourier transform of functions with singularities, Appl. Comput. Harmon. Anal., 2 (1995), 363–381
7.
G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms. I, Comm. Pure Appl. Math., 44 (1991), 141–183
8.
Gregory Beylkin, Robert Cramer, A multiresolution approach to regularization of singular operators and fast summation, SIAM J. Sci. Comput., 24 (2002), 81–117
9.
R. Chan, F. Lin, C. Chan, A fast solver for Fredholm equations of the second kind with weakly singular kernels, J. Numer. Math., 10 (2002), 13–36
10.
Raymond Chan, Michael Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), 427–482
11.
A. J. W. Duijndam and M. A. Schonewille, Nonuniform fast Fourier transform, Geophysics, 64 (1999), pp. 539–551.
12.
A. Dutt, M. Gu, V. Rokhlin, Fast algorithms for polynomial interpolation, integration, and differentiation, SIAM J. Numer. Anal., 33 (1996), 1689–1711
13.
A. Dutt, V. Rokhlin, Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), 1368–1393
14.
J. A. Fessler and B. P. Sutton, Nonuniform fast Fourier transforms using min‐max interpolation, IEEE Trans. Signal Process., to appear.
15.
K. Fourmont, Schnelle Fourier–Transformation bei nichtäquidistanten Gittern und tomographische Anwendungen, Dissertation, Universität Münster, Münster, Germany, 1999.
16.
S. A. Goreinov, E. E. Tyrtyshnikov, and E. E. Yeremin, Matrix–free iterative solution strategies for large dense systems, Numer. Linear Algebra Appl., 4 (1997), pp. 273–294.
17.
Leslie Greengard, The rapid evaluation of potential fields in particle systems, ACM Distinguished Dissertations, MIT Press, 1988xiv+91
18.
L. Greengard, V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), 325–348
19.
W. Hackbusch, A sparse matrix arithmetic based on ‐matrices. I. Introduction to ‐matrices, Computing, 62 (1999), 89–108
20.
W. Hackbusch, Z. Nowak, On the fast matrix multiplication in the boundary element method by panel clustering, Numer. Math., 54 (1989), 463–491
21.
Ami Harten, Itai Yad‐Shalom, Fast multiresolution algorithms for matrix‐vector multiplication, SIAM J. Numer. Anal., 31 (1994), 1191–1218
22.
J. I. Jackson, Selection of a convolution function for Fourier inversion using gridding, IEEE Trans. Med. Imag., 10 (1991), pp. 473–478.
23.
M. Konik, R. Schneider, and G. Steidl, Matrix sparsification by discrete multiscale methods, in Approximation Theory VIII: Approximation and Decomposition, C. K. Chui and L. L. Schumaker, eds., World Scientific, River Edge, NJ, 1995.
24.
S. Kunis and D. Potts, NFFT, Softwarepackage, C subroutine library, http://www.math.uni‐luebeck.de/potts/nfft (2002).
25.
Nhu Nguyen, Qing Liu, The regular Fourier matrices and nonuniform fast Fourier transforms, SIAM J. Sci. Comput., 21 (1999), 283–293
26.
A. Nieslony and G. Steidl, Approximate factorizations of Fourier matrices with nonequispaced knots, Linear Algebra Appl., to appear.
27.
Daniel Potts, Gabriele Steidl, Optimal trigonometric preconditioners for nonsymmetric Toeplitz systems, Linear Algebra Appl., 281 (1998), 265–292
28.
Daniel Potts, Gabriele Steidl, Manfred Tasche, Trigonometric preconditioners for block Toeplitz systems, Internat. Ser. Numer. Math., Vol. 125, Birkhäuser, Basel, 1997, 219–234
29.
Daniel Potts, Gabriele Steidl, Manfred Tasche, Fast Fourier transforms for nonequispaced data: a tutorial, Appl. Numer. Harmon. Anal., Birkhäuser Boston, Boston, MA, 2001, 247–270
30.
W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, Cambridge University Press, Cambridge, UK, 1992.
31.
Gabriele Steidl, A note on fast Fourier transforms for nonequispaced grids, Adv. Comput. Math., 9 (1998), 337–352
32.
Xiaobai Sun, Nikos Pitsianis, A matrix version of the fast multipole method, SIAM Rev., 43 (2001), 289–300
33.
E. E. Tyrtyshnikov, Mosaic‐skeleton approximations, Calcolo, 33 (1996), pp. 47–57.
34.
Norman Yarvin, Vladimir Rokhlin, An improved fast multipole algorithm for potential fields on the line, SIAM J. Numer. Anal., 36 (1999), 629–666
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SIAM Journal on Scientific Computing
Pages: 2013 - 2037
ISSN (online): 1095-7197
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Copyright © 2003 Society for Industrial and Applied Mathematics.
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Published online: 25 July 2006
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