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The Fourier approximation of smooth but non-periodic functions from unevenly spaced data

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Abstract

We develop an algorithm to extend, to the nonequispaced case, a recently-introduced fast algorithm for constructing spectrally-accurate Fourier approximations of smooth, but nonperiodic, data. Fast Fourier continuation algorithms, which allow for the Fourier approximation to be periodic in an extended domain, are combined with the underlying ideas behind nonequispaced fast Fourier transform (NFFT) algorithms. The result is a method which allows for the fast and accurate approximation of unevenly sampled nonperiodic multivariate data by Fourier series. A particular contribution of the proposed method is that its formulation avoids the difficulties related to the conditioning of the linear systems that must be solved in order to construct a Fourier continuation. The efficiency, essentially equivalent to that of an NFFT, and accuracy of the algorithm is shown through a number of numerical examples. Numerical results demonstrate the spectral rate of convergence of this method for sufficiently smooth functions. The accuracy, for sufficiently large data sets, is shown to be improved by several orders of magnitudes over previously published techniques for scattered data interpolation.

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References

  1. Adcock, B., Huybrechs, D.: On the resolution power of Fourier extensions for oscillatory functions. In Press (2013)

  2. Adcock, B., Huybrechs, D., Martin-Vaquero, J.: On the numerical stability of Fourier extensions. In Press (2013)

  3. Albin, N., Bruno, O.P.: A spectral FC solver for the compressible Navier-Stokes equations in general domains I: explicit time-stepping. J. Comput. Phys. 230(16), 6248–6270 (2011). doi:10.1016/j.jcp.2011.04.023

    Article  MATH  MathSciNet  Google Scholar 

  4. Albin, N., Bruno, O.P., Cheung, T.Y., Cleveland, R.O.: Fourier continuation methods for high-fidelity simulation of nonlinear acoustic beams. J. Accoust. Soc. Am. 132(4, Part 1), 2371–2387 (2012). doi:10.1121/1.4742722

    Article  Google Scholar 

  5. Allasia, G., Besenghi, R., Cavoretto, R., De Rossi, A.: Scattered and track data interpolation using an efficient strip searching procedure. Appl. Math. Comput. 217, 5949–5966 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  6. Anderson, C., Dahleh, M.D.: Rapid computation of the discrete Fourier transform. SIAM J. Sci. Comput. 17(4), 913–919 (1996). doi:10.1137/0917059

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Boyd, J.P.: A fast algorithm for Chebyshev, Fourier, and sinc interpolation onto an irregular grid. J. Comput. Phys. 103(2), 243–257 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  9. Boyd, J.P.: A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds. J. Comput. Phys. 178(1), 118–160 (2002). doi:10.1006/jcph.2002.7023

    Article  MATH  MathSciNet  Google Scholar 

  10. Boyd, J.P., Ong, J.R.: Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions, Part I: single-interval schemes. Commun. Comput. Phys. 5(2–4), 484–497 (2009)

    MathSciNet  Google Scholar 

  11. Bruno, O.P.: Fast, high-order, high-frequency integral methods for computational acoustics and electromagnetics. In: Ainsworth, M., Davies, P., Duncan, D., Martin, P., Rynne B. (eds.) Topics in Computational Wave Propagation Direct and Inverse Problems Series, Lecture Notes in Computational Science and Engineering, vol. 31, pp. 43–82 (2003)

  12. Bruno, O.P., Han, Y., Pohlman, M.M.: Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis. J. Comput. Phys. 227(2), 1094–1125 (2007). doi:10.1016/j.jcp.2007.08.029

    Article  MATH  MathSciNet  Google Scholar 

  13. Bruno, O.P., Lyon, M.: High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements. J. Comput. Phys. 229(6), 2009–2033 (2010). doi:10.1016/j.jcp.2009.11.020

    Article  MATH  MathSciNet  Google Scholar 

  14. Cooley, J.W., Tukey, J.W.: An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301 (1965)

    Article  MATH  MathSciNet  Google Scholar 

  15. Davydov, O., Morandi, R., Sestini, A.: Local hybrid approximation for scattered data fitting with bivariate splines. Comput. Aided Geom. D 23, 703–721 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  16. Drineas, P., Kannan, R., Mahoney, M.W.: Fast Monte- Carlo algorithms for matrices II: computing a low-rank approximation to a matrix. SIAM J. Comput. 36(1), 158–183 (2006). doi:10.1137/S0097539704442696

    Article  MATH  MathSciNet  Google Scholar 

  17. Drineas, P., Kannan, R., Mahoney, M.W.: Fast Monte-Carlo algorithms for matrices III: computing a compressed approximate matrix decomposition. SIAM J. Comput. 36(1), 184–206 (2006). doi:10.1137/S0097539704442702

    Article  MATH  MathSciNet  Google Scholar 

  18. Duijndam, A.JW., Schonewille, M.A.: Nonuniform fast Fourier transform. Geophysics 64(2), 539–551 (1999)

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Fessler, J.A., Sutton, B.P.: Nonuniform fast Fourier transforms using min-max interpolation. IEEE Trans. Signal Process. 51(2), 560–574 (2003). doi:10.1109/TSP.2002.807005

    Article  MathSciNet  Google Scholar 

  22. Floater, M.S., Iske, A.: Multistep scattered data using compactly supported radial basis functions. J. Comput. Appl. Math. 73(5), 65–78 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  23. Franke, R., Nielson, G.: Smooth interpolation of large sets of scattered data. Int. J. Numer. Methods Eng. 15, 1691–1704 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  24. Franke, R.: Scattered data interpolation: test of some methods. Math. Comput. 38(157), 181–200 (1982)

    MATH  MathSciNet  Google Scholar 

  25. Frieze, A., Kannan, R., Vempala, S.: Fast Monte-Carlo algorithms for finding low-rank approximations. J. ACM 51(6), 1025–1041 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  26. Greengard, L., Lee, J.Y.: Accelerating the nonuniform fast Fourier transform. SIAM Rev. 46(3), 443–454 (2004). doi:10.1137/S003614450343200X

    Article  MATH  MathSciNet  Google Scholar 

  27. Huybrechs, D.: On the Fourier extension of nonperiodic functions. SIAM J. Numer. Anal. 47(6), 4326–4355 (2010). doi:10.1137/090752456

    Article  MATH  MathSciNet  Google Scholar 

  28. Keiner, J., Kunis, S., Potts, D.: Using NFFT 3-A software library for various nonequispaced fast Fourier transforms. ACM Trans. Math. Softw. 36(4) (2009). doi:10.1145/1555386.1555388

  29. Lazzaro, D., Montefusco, L.B.: Radial basis functions for the multivariate interpolation of large scattered data sets. J. Comput. Appl. Math. 140, 521–536 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  30. Liberty, E., Woolfe, F., Martinsson, P.G., Rokhlin, V., Tyger, M.: Randomized algorithms for the low-rank approximation of matrices. Proc. Natl. Acad. Sci. USA 104(51), 20167–20172 (2007). doi:10.1073/pnas.0709640104

    Article  MATH  MathSciNet  Google Scholar 

  31. Lyon, M.: A fast algorithm for Fourier continuation. SIAM J. Sci. Comput. 33, 3241–3260 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  32. Lyon, M.: Approximation error in regularized SVD-based Fourier continuations. Appl. Numer. Math. 62(12), 1790–1803 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  33. Lyon, M.: Sobolev smoothing of SVD-based Fourier continuations. Appl. Math. Lett. 25(12), 2227–2231 (2012). doi:10.1016/j.aml.2012.06.008

    Article  MATH  MathSciNet  Google Scholar 

  34. Lyon, M., Bruno, O.P.: High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations. J. Comput. Phys. 229(9), 3358–3381 (2010). doi:10.1016/j.jcp.2010.01.006

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  36. Paige, C.C., Saunders, M.A.: LSQR—an algorithm for sparse linear-equations and sparse least-squares. ACM Trans. Math. Softw. 8(1), 43–71 (1982). doi:10.1145/355984.355989

    Article  MATH  MathSciNet  Google Scholar 

  37. Potts, D., Tasche, M.: Numerical stability of nonequispaced fast Fourier transforms. J. Comput. Appl. Math. 222(2), 655–674 (2008). doi:10.1016/j.cam.2007.12.025

    Article  MATH  MathSciNet  Google Scholar 

  38. Renka, R.J.: Multivariate interpolation of large sets of scattered data. ACM Trans. Math. Softw. 14(2), 139–148 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  39. Shahbazi, K., Albin, N., Bruno, O.P., Hesthaven, J.S.: Multi-domain Fourier-continuation/WENO hybrid solver for conservation laws. J. Comput. Phys. 230(24), 8779–8796 (2011). doi:10.1016/j.jcp.2011.08.024

    Article  MATH  MathSciNet  Google Scholar 

  40. Steidl, G.: A note on fast Fourier transforms for nonequispaced grids. Adv. Comput. Math. 9(3–4), 337–352 (1998). doi:10.1023/A:1018901926283

    Article  MATH  MathSciNet  Google Scholar 

  41. Strang, G.: The discrete cosine transform. SIAM Rev. 41(1), 135–147 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  42. Ware, A.F.: Fast approximate Fourier transforms for irregularly spaced data. SIAM Rev. 40(4), 838–856 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  43. Woolfe, F., Liberty, E., Rokhlin, V., Tygert, M.: A fast randomized algorithm for the approximation of matrices. Appl. Comput. Harmon. Anal. 25(3), 335–366 (2008). doi:10.1016/j.acha.2007.12.002

    Article  MATH  MathSciNet  Google Scholar 

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Communicated by: Alexander Barnett

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Lyon, M., Picard, J. The Fourier approximation of smooth but non-periodic functions from unevenly spaced data. Adv Comput Math 40, 1073–1092 (2014). https://doi.org/10.1007/s10444-014-9342-7

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  • DOI: https://doi.org/10.1007/s10444-014-9342-7

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