Advertisement

Foundations of Computational Mathematics

, Volume 14, Issue 4, pp 635–687 | Cite as

On the Numerical Stability of Fourier Extensions

  • Ben Adcock
  • Daan Huybrechs
  • Jesús Martín-Vaquero
Article

Abstract

An effective means to approximate an analytic, nonperiodic function on a bounded interval is by using a Fourier series on a larger domain. When constructed appropriately, this so-called Fourier extension is known to converge geometrically fast in the truncation parameter. Unfortunately, computing a Fourier extension requires solving an ill-conditioned linear system, and hence one might expect such rapid convergence to be destroyed when carrying out computations in finite precision. The purpose of this paper is to show that this is not the case. Specifically, we show that Fourier extensions are actually numerically stable when implemented in finite arithmetic, and achieve a convergence rate that is at least superalgebraic. Thus, in this instance, ill-conditioning of the linear system does not prohibit a good approximation.

In the second part of this paper we consider the issue of computing Fourier extensions from equispaced data. A result of Platte et al. (SIAM Rev. 53(2):308–318, 2011) states that no method for this problem can be both numerically stable and exponentially convergent. We explain how Fourier extensions relate to this theoretical barrier, and demonstrate that they are particularly well suited for this problem: namely, they obtain at least superalgebraic convergence in a numerically stable manner.

Keywords

Fourier series Fourier extension Convergence Stability Equispaced data 

Symbols

T

Extension parameter

N

Truncation parameter

M, γ

Number of equispaced nodes of the equispaced FE, and the oversampling parameter γ=M/N

ϕn(x)

The exponential \(\frac{1}{\sqrt{2T}} \mathrm{e}^{{\mathrm{i}}\frac{n \pi}{T} x } \)

\(\mathcal {G}_{N} \), \(\mathcal {S}_{N}\), \(\mathcal {C}_{N}\)

Finite-dimensional spaces of exponentials, sines and cosines

FN, \(\tilde{F}_{N}(f)\), FN,M(f)

Exact continuous, discrete and equispaced FEs

GN, \(\tilde{G}_{N}(f)\), GN,M(f)

Numerical continuous, discrete and equispaced FEs

a

Vector of coefficients of an FE

A, \(\tilde{A}\), \(\bar{A}\)

Matrices of the continuous, discrete and equispaced FE’s

b, \(\tilde{b}\), \(\bar{b}\)

Data vectors for the continuous, discrete and equispaced FEs

x, y, z

Physical domain variable x∈[−1,1], and the mapped variables y∈[c(T),1] and z∈[−1,1]

fe(x), fo(x)

Even and odd parts of the function f(x)

g1(y), g2(y), g1,N(y), g2,N(y)

Images of f e(x) and \(f_{\mathrm{o}}(x) / \sin \frac{\pi}{T} x\) in the y-domain and their polynomial approximations

hi(z), hi,N(z)

Images of g i and g i,N in the z-domain

m(x)

The mapping xz

c(T), E(T)

FE constants \(\cos \frac{\pi}{T}\) and \(\cot^{2} ( \frac{\pi}{4 T} )\).

\(\mathcal {B}(\rho) \), \(\mathcal {D}(\rho)\)

Bernstein ellipse in the z-domain and its image in the x-domain

κ(F)

Condition number of a mapping F

N0, N1, N2

Breakpoints in convergence

{un,σn,vn}

Singular system of A, \(\tilde{A}\) or \(\bar{A}\)

Φn

Fourier series corresponding to v n

\(\mathcal {G}_{N,\epsilon} \), \(\mathcal {G}'_{N,\epsilon} \), \(\mathcal {G}_{N,M,\epsilon}\)

The subspace span{Φ n :σ n >ϵ}

HN,ϵ(f), \(\tilde{H}_{N,\epsilon}(f)\), HN,M,ϵ(f)

Truncated SVD FEs corresponding to the continuous, discrete and equispaced cases

a(γ;T)

Quantity determining the maximal achievable accuracy of the equispaced FE

L2(I), 〈⋅,⋅〉I, ∥⋅∥I

Space of square-integral functions on a domain I and corresponding inner product and norm

〈⋅,⋅〉, ∥⋅∥

Inner product and norm on L2(−1,1)

\(\mathrm {L}^{2}_{w}(I)\), 〈⋅,⋅〉w,I, ∥⋅∥w,I

Space of square integrable functions with respect to a weight function w and corresponding inner product and norm

∥⋅∥∞,I, ∥⋅∥

Uniform norms on an arbitrary domain I and the interval [−1,1] respectively

Mathematics Subject Classification (2010)

42A10 65T40 42C15 

Notes

Acknowledgements

The authors would like to thank John Boyd, Doug Cochran, Laurent Demanet, Anne Gelb, Anders Hansen, Arieh Iserles, Arno Kuijlaars, Mark Lyon, Nilima Nigam, Sheehan Olver, Rodrigo Platte, Jie Shen and Nick Trefethen for useful discussions and comments. They would also like to thank the anonymous referees for their constructive and helpful remarks.

References

  1. 1.
    B. Adcock, D. Huybrechs, On the resolution power of Fourier extensions for oscillatory functions. Technical Report TW597, Dept. Computer Science, K.U. Leuven, 2011. Google Scholar
  2. 2.
    N. Albin, O.P. Bruno, 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). CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    H. Bateman, Higher Transcendental Functions, vol. 2 (McGraw–Hill, New York, 1953). Google Scholar
  4. 4.
    J.P. Boyd, Chebyshev and Fourier Spectral Methods (Springer, Berlin, 1989). CrossRefGoogle Scholar
  5. 5.
    J.P. Boyd, A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds, J. Comput. Phys. 178, 118–160 (2002). CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    J. Boyd, Fourier embedded domain methods: extending a function defined on an irregular region to a rectangle so that the extension is spatially periodic and C , Appl. Math. Comput. 161(2), 591–597 (2005). CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    J.P. Boyd, Trouble with Gegenbauer reconstruction for defeating Gibbs phenomenon: Runge phenomenon in the diagonal limit of Gegenbauer polynomial approximations, J. Comput. Phys. 204(1), 253–264 (2005). CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    J.P. Boyd, J.R. Ong, Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions. I. Single-interval schemes, Commun. Comput. Phys. 5(2–4), 484–497 (2009). MathSciNetGoogle Scholar
  9. 9.
    J. Boyd, F. Xu, Divergence (Runge phenomenon) for least-squares polynomial approximation on an equispaced grid and Mock–Chebyshev subset interpolation, Appl. Math. Comput. 210(1), 158–168 (2009). CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    O.P. Bruno, Fast, high-order, high-frequency integral methods for computational acoustics and electromagnetics, in Topics in Computational Wave Propagation: Direct and Inverse Problems, ed. by M. Ainsworth et al. Lecture Notes in Computational Science and Engineering, vol. 31 (Springer, Berlin, 2003), pp. 43–82. CrossRefGoogle Scholar
  11. 11.
    O. Bruno, M. Lyon, High-order unconditionally stable FC–AD solvers for general smooth domains. I. Basic elements, J. Comput. Phys. 229(6), 2009–2033 (2010). CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    O.P. Bruno, Y. Han, M.M. Pohlman, Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis, J. Comput. Phys. 227(2), 1094–1125 (2007). CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer, Berlin, 2006). Google Scholar
  14. 14.
    O. Christensen, An Introduction to Frames and Riesz Bases (Birkhauser, Basel, 2003). CrossRefzbMATHGoogle Scholar
  15. 15.
    D. Coppersmith, T. Rivlin, The growth of polynomials bounded at equally spaced points, SIAM J. Math. Anal. 23, 970–983 (1992). CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Am. Math. Soc. 72, 341–366 (1952). CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    A. Edelman, P. McCorquodale, S. Toledo, The future Fast Fourier Transform? SIAM J. Sci. Comput. 20(3), 1094–1114 (1999). CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge University Press, Cambridge, 1996). zbMATHGoogle Scholar
  19. 19.
    D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, 1st edn. (SIAM, Philadelphia, 1977). CrossRefzbMATHGoogle Scholar
  20. 20.
    D. Gottlieb, C.-W. Shu, On the Gibbs’ phenomenon and its resolution, SIAM Rev. 39(4), 644–668 (1997). CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    D. Gottlieb, C.-W. Shu, A. Solomonoff, H. Vandeven, On the Gibbs phenomenon. I: Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math. 43(1–2), 91–98 (1992). MathSciNetGoogle Scholar
  22. 22.
    D. Huybrechs, On the Fourier extension of non-periodic functions, SIAM J. Numer. Anal. 47(6), 4326–4355 (2010). CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    D. Kosloff, H. Tal-Ezer, A modified Chebyshev pseudospectral method with an \(\mathcal {O}(N^{-1})\) time step restriction, J. Comput. Phys. 104, 457–469 (1993). CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    M. Lyon, Approximation error in regularized SVD-based Fourier continuations, Appl. Numer. Math. 62, 1790–1803 (2012). CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    M. Lyon, A fast algorithm for Fourier continuation, SIAM J. Sci. Comput. 33(6), 3241–3260 (2012). CrossRefMathSciNetGoogle Scholar
  26. 26.
    M. Lyon, O. Bruno, 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). CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    R. Pasquetti, M. Elghaoui, A spectral embedding method applied to the advection–diffusion equation, J. Comput. Phys. 125, 464–476 (1996). CrossRefzbMATHMathSciNetGoogle Scholar
  28. 28.
    R. Platte, L.N. Trefethen, A. Kuijlaars, Impossibility of fast stable approximation of analytic functions from equispaced samples, SIAM Rev. 53(2), 308–318 (2011). CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    T. Ransford, Potential Theory in the Complex Plane (Cambridge Univ. Press, Cambridge, 1995). CrossRefzbMATHGoogle Scholar
  30. 30.
    T.J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory (Wiley, New York, 1990). zbMATHGoogle Scholar
  31. 31.
    D. Slepian, Prolate spheroidal wave functions. Fourier analysis, and uncertainty. V: The discrete case, Bell Syst. Tech. J. 57, 1371–1430 (1978). CrossRefzbMATHGoogle Scholar
  32. 32.
    L.N. Trefethen, D. Bau, Numerical Linear Algebra (SIAM, Philadelphia, 1997). CrossRefzbMATHGoogle Scholar
  33. 33.
    J. Varah, The prolate matrix, Linear Algebra Appl. 187(1), 269–278 (1993). CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© SFoCM 2014

Authors and Affiliations

  • Ben Adcock
    • 1
  • Daan Huybrechs
    • 2
  • Jesús Martín-Vaquero
    • 3
  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA
  2. 2.Department of Computer ScienceKatholieke Universiteit LeuvenLeuvenBelgium
  3. 3.Department of Applied Mathematics, E.T.S.I.I. BéjarUniversity of SalamancaSalamancaSpain

Personalised recommendations