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On the Numerical Stability of Fourier Extensions

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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.

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Notes

  1. The constant of growth was obtained in private communication with A. Kuijlaars. A closed expression (up to several integrals involving the potential function ϕ for the nodes z n ) can be found for c(γ;T). We omit the full argument as it is rather lengthy, but note that it is based on standard results in potential theory. A general reference is [29].

Abbreviations

T :

Extension parameter

N :

Truncation parameter

M, γ :

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

ϕ n (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

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

Exact continuous, discrete and equispaced FEs

G N , \(\tilde{G}_{N}(f)\), G N,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]

f e(x), f o(x):

Even and odd parts of the function f(x)

g 1(y), g 2(y), g 1,N (y), g 2,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

h i (z), h i,N (z):

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

m(x):

The mapping x↦z

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

N 0, N 1, N 2 :

Breakpoints in convergence

{u n ,σ n ,v n }:

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

Φ n :

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}

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

References

  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.

  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).

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Bateman, Higher Transcendental Functions, vol. 2 (McGraw–Hill, New York, 1953).

    Google Scholar 

  4. J.P. Boyd, Chebyshev and Fourier Spectral Methods (Springer, Berlin, 1989).

    Book  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Chapter  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  13. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer, Berlin, 2006).

    Google Scholar 

  14. O. Christensen, An Introduction to Frames and Riesz Bases (Birkhauser, Basel, 2003).

    Book  MATH  Google Scholar 

  15. D. Coppersmith, T. Rivlin, The growth of polynomials bounded at equally spaced points, SIAM J. Math. Anal. 23, 970–983 (1992).

    Article  MATH  MathSciNet  Google Scholar 

  16. R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Am. Math. Soc. 72, 341–366 (1952).

    Article  MATH  MathSciNet  Google Scholar 

  17. A. Edelman, P. McCorquodale, S. Toledo, The future Fast Fourier Transform? SIAM J. Sci. Comput. 20(3), 1094–1114 (1999).

    Article  MATH  MathSciNet  Google Scholar 

  18. B. Fornberg, A Practical Guide to Pseudospectral Methods (Cambridge University Press, Cambridge, 1996).

    MATH  Google Scholar 

  19. D. Gottlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, 1st edn. (SIAM, Philadelphia, 1977).

    Book  MATH  Google Scholar 

  20. D. Gottlieb, C.-W. Shu, On the Gibbs’ phenomenon and its resolution, SIAM Rev. 39(4), 644–668 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    MathSciNet  Google Scholar 

  22. D. Huybrechs, On the Fourier extension of non-periodic functions, SIAM J. Numer. Anal. 47(6), 4326–4355 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. M. Lyon, A fast algorithm for Fourier continuation, SIAM J. Sci. Comput. 33(6), 3241–3260 (2012).

    Article  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  27. R. Pasquetti, M. Elghaoui, A spectral embedding method applied to the advection–diffusion equation, J. Comput. Phys. 125, 464–476 (1996).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  29. T. Ransford, Potential Theory in the Complex Plane (Cambridge Univ. Press, Cambridge, 1995).

    Book  MATH  Google Scholar 

  30. T.J. Rivlin, Chebyshev Polynomials: From Approximation Theory to Algebra and Number Theory (Wiley, New York, 1990).

    MATH  Google Scholar 

  31. D. Slepian, Prolate spheroidal wave functions. Fourier analysis, and uncertainty. V: The discrete case, Bell Syst. Tech. J. 57, 1371–1430 (1978).

    Article  MATH  Google Scholar 

  32. L.N. Trefethen, D. Bau, Numerical Linear Algebra (SIAM, Philadelphia, 1997).

    Book  MATH  Google Scholar 

  33. J. Varah, The prolate matrix, Linear Algebra Appl. 187(1), 269–278 (1993).

    Article  MATH  MathSciNet  Google Scholar 

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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.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Ben Adcock

  2. Department of Computer Science, Katholieke Universiteit Leuven, Leuven, Belgium

    Daan Huybrechs

  3. Department of Applied Mathematics, E.T.S.I.I. Béjar, University of Salamanca, Salamanca, Spain

    Jesús Martín-Vaquero

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  1. Ben Adcock
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  2. Daan Huybrechs
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Correspondence to Ben Adcock.

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Communicated by Nira Dyn.

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Adcock, B., Huybrechs, D. & Martín-Vaquero, J. On the Numerical Stability of Fourier Extensions. Found Comput Math 14, 635–687 (2014). https://doi.org/10.1007/s10208-013-9158-8

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  • DOI: https://doi.org/10.1007/s10208-013-9158-8

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