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Fast Fourier Transforms for Nonequispaced Data

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Numerical Fourier Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

Abstract

In this chapter, we describe fast algorithms for the computation of the DFT for d-variate nonequispaced data, since in a variety of applications the restriction to equispaced data is a serious drawback. These algorithms are called nonequispaced fast Fourier transforms and abbreviated by NFFT.

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References

  1. A. Arnold, M. Bolten, H. Dachsel, F. Fahrenberger, F. Gähler, R. Halver, F. Heber, M. Hofmann, J. Iseringhausen, I. Kabadshow, O. Lenz, M. Pippig, ScaFaCoS - Scalable fast Coloumb solvers (2013). http://www.scafacos.de

  2. R.F. Bass, K. Gröchenig, Random sampling of multivariate trigonometric polynomials. SIAM J. Math. Anal. 36(3), 773–795 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. R.K. Beatson, W.A. Light, Fast evaluation of radial basis functions: methods for two–dimensional polyharmonic splines. IMA J. Numer. Anal. 17(3), 343–372 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. J.-P. Berrut, L.N. Trefethen, Barycentric Lagrange interpolation. SIAM Rev. 46(3), 501–517 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Beylkin, On the fast Fourier transform of functions with singularities. Appl. Comput. Harmon. Anal. 2(4), 363–381 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Beylkin, R. Cramer, A multiresolution approach to regularization of singular operators and fast summation. SIAM J. Sci. Comput. 24(1), 81–117 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Å. Björck, Numerical Methods for Least Squares Problems (SIAM, Philadelphia, 1996)

    Book  MATH  Google Scholar 

  8. M. Broadie, Y. Yamamoto, Application of the fast Gauss transform to option pricing. Manag. Sci. 49, 1071–1088 (2003)

    Article  MATH  Google Scholar 

  9. E.J. Candès, The restricted isometry property and its implications for compressed sensing. C. R. Acad. Sci. Paris 346(9–10), 589–592 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Duchon, Fonctions splines et vecteurs aleatoires. Technical report, Seminaire d’Analyse Numerique, Universite Scientifique et Medicale, Grenoble, 1975

    Google Scholar 

  11. A.J.W. Duijndam, M.A. Schonewille. Nonuniform fast Fourier transform. Geophysics 64, 539–551 (1999)

    Article  Google Scholar 

  12. A. Dutt, V. Rokhlin, Fast Fourier transforms for nonequispaced data. SIAM J. Sci. Stat. Comput. 14(6), 1368–1393 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Dutt, V. Rokhlin, Fast Fourier transforms for nonequispaced data II. Appl. Comput. Harmon. Anal. 2(1), 85–100 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. B. Elbel, Mehrdimensionale Fouriertransformation für nichtäquidistante Daten. Diplomarbeit, Technische Hochschule Darmstadt, 1998

    Google Scholar 

  15. B. Elbel, G. Steidl, Fast Fourier transform for nonequispaced data, in Approximation Theory IX (Vanderbilt University Press, Nashville, 1998), pp. 39–46

    MATH  Google Scholar 

  16. A. Elgammal, R. Duraiswami, L.S. Davis, Efficient non-parametric adaptive color modeling using fast Gauss transform. Technical report, University of Maryland, 2001

    Book  Google Scholar 

  17. H.G. Feichtinger, K. Gröchenig, T. Strohmer, Efficient numerical methods in non-uniform sampling theory. Numer. Math. 69(4), 423–440 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  18. J.A. Fessler, B.P. Sutton, Nonuniform fast Fourier transforms using min-max interpolation. IEEE Trans. Signal Process. 51(2), 560–574 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  19. S. Foucart, A note guaranteed sparse recovery via 1-minimization. Appl. Comput. Harmon. Anal. 29(1), 97–103 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. S. Foucart, H. Rauhut, A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, New York, 2013)

    Book  MATH  Google Scholar 

  21. K. Fourmont, Schnelle Fourier–Transformation bei nichtäquidistanten Gittern und tomographische Anwendungen. Dissertation, Universität Münster, 1999

    MATH  Google Scholar 

  22. M. Frigo, S.G. Johnson, The design and implementation of FFTW3. Proc. IEEE 93, 216–231 (2005)

    Article  Google Scholar 

  23. M. Frigo, S.G. Johnson, FFTW, C subroutine library (2009). http://www.fftw.org

  24. L. Greengard, The Rapid Evaluation of Potential Fields in Particle Systems (MIT Press, Cambridge, 1988)

    MATH  Google Scholar 

  25. L. Greengard, P. Lin, Spectral approximation of the free-space heat kernel. Appl. Comput. Harmon. Anal. 9(1), 83–97 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  26. L. Greengard, J. Strain, The fast Gauss transform. SIAM J. Sci. Stat. Comput. 12(1), 79–94 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  27. L. Greengard, X. Sun, A new version of the fast Gauss transform, in Proceedings of the International Congress of Mathematicians (Berlin, 1998), Documenta Mathematica, vol. 3 (1998), pp. 575–584

    Google Scholar 

  28. J.I. Jackson, C.H. Meyer, D.G. Nishimura, A. Macovski, Selection of a convolution function for Fourier inversion using gridding. IEEE Trans. Med. Imag. 10, 473–478 (1991)

    Article  Google Scholar 

  29. M. Jacob, Optimized least-square nonuniform fast Fourier transform. IEEE Trans. Signal Process. 57(6), 2165–2177 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. J. Keiner, S. Kunis, D. Potts, Using NFFT3 - a software library for various nonequispaced fast Fourier transforms. ACM Trans. Math. Softw. 36, Article 19, 1–30 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  31. J. Keiner, S. Kunis, D. Potts, NFFT 3.4, C subroutine library. http://www.tu-chemnitz.de/~potts/nfft. Contributor: F. Bartel, M. Fenn, T. Görner, M. Kircheis, T. Knopp, M. Quellmalz, T. Volkmer, A. Vollrath

  32. S. Kunis, S. Kunis, The nonequispaced FFT on graphics processing units. Proc. Appl. Math. Mech. 12, 7–10 (2012)

    Article  Google Scholar 

  33. S. Kunis, D. Potts, Stability results for scattered data interpolation by trigonometric polynomials. SIAM J. Sci. Comput. 29(4), 1403–1419 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Kunis, D. Potts, G. Steidl, Fast Gauss transforms with complex parameters using NFFTs. J. Numer. Math. 14(4), 295–303 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  35. J.-Y. Lee, L. Greengard, The type 3 nonuniform FFT and its applications. J. Comput. Phys. 206(1), 1–5 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  36. F. Nestler, Automated parameter tuning based on RMS errors for nonequispaced FFTs. Adv. Comput. Math. 42(4), 889–919 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  37. F. Nestler, Parameter tuning for the NFFT based fast Ewald summation. Front. Phys. 4(28), 1–28 (2016)

    Google Scholar 

  38. N. Nguyen, Q.H. Liu, The regular Fourier matrices and nonuniform fast Fourier transforms. SIAM J. Sci. Comput. 21(1), 283–293 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Nieslony, G. Steidl, Approximate factorizations of Fourier matrices with nonequispaced knots. Linear Algebra Appl. 366, 337–351 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Pippig, PNFFT, Parallel Nonequispaced FFT subroutine library (2011). http://www.tu-chemnitz.de/~potts/workgroup/pippig/software.php.en

  41. M. Pippig, D. Potts, Parallel three-dimensional nonequispaced fast Fourier transforms and their application to particle simulation. SIAM J. Sci. Comput. 35(4), C411–C437 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  42. D. Potts, Fast algorithms for discrete polynomial transforms on arbitrary grids. Linear Algebra Appl. 366, 353–370 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  43. D. Potts, Schnelle Fourier-Transformationen für nichtäquidistante Daten und Anwendungen. Habilitation, Universität zu Lübeck, 2003

    Google Scholar 

  44. D. Potts, G. Steidl, Fast summation at nonequispaced knots by NFFTs. SIAM J. Sci. Comput. 24(6), 2013–2037 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  45. D. Potts, G. Steidl, M. Tasche, Trigonometric preconditioners for block Toeplitz systems, in Multivariate Approximation and Splines (Mannheim, 1996) (Birkhäuser, Basel, 1997), pp. 219–234

    Chapter  Google Scholar 

  46. D. Potts, G. Steidl, M. Tasche, Fast Fourier transforms for nonequispaced data. A tutorial, in Modern Sampling Theory: Mathematics and Applications (Birkhäuser, Boston, 2001), pp. 247–270

    Chapter  Google Scholar 

  47. D. Potts, G. Steidl, A. Nieslony, Fast convolution with radial kernels at nonequispaced knots. Numer. Math. 98(2), 329–351 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  48. H. Rauhut, R. Ward, Sparse Legendre expansions via 1-minimization. J. Approx. Theory 164(5), 517–533 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  49. G. Steidl, A note on fast Fourier transforms for nonequispaced grids. Adv. Comput. Math. 9(3–4), 337–353 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  50. T. Volkmer, OpenMP parallelization in the NFFT software library. Preprint 2012-07, Faculty of Mathematics, Technische Universität Chemnitz (2012)

    Google Scholar 

  51. S.-C. Yang, H.-J. Qian, Z.-Y. Lu, A new theoretical derivation of NFFT and its implementation on GPU. Appl. Comput. Harmon. Anal. 44(2), 273–293 (2018)

    Article  MathSciNet  MATH  Google Scholar 

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Plonka, G., Potts, D., Steidl, G., Tasche, M. (2018). Fast Fourier Transforms for Nonequispaced Data. In: Numerical Fourier Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04306-3_7

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