A Fast Algorithm for Fourier Continuation
Abstract
A new algorithm is presented which provides a fast method for the computation of recently developed Fourier continuations (a particular type of Fourier extension method) that yield superalgebraically convergent Fourier series approximations of nonperiodic functions. Previously, the coefficients of an approximating Fourier series have been obtained by means of a regularized singular value decomposition (SVD)-based least-squares solution to an overdetermined linear system of equations. These SVD methods are effective when the size of the system does not become too large, but they quickly become unwieldy as the number of unknowns in the system grows. We demonstrate a novel decoupling of the least-squares problem which results in two systems of equations, one of which may be solved quickly by means of fast Fourier transforms (FFTs) and another that is demonstrated to be well approximated by a low-rank system. Utilizing randomized algorithms, the low-rank system is reduced to a significantly smaller system of equations. This new system is then efficiently solved with drastically reduced computational cost and memory requirements while still benefiting from the advantages of using a regularized SVD. The computational cost of the new algorithm in on the order of the cost of a single FFT multiplied by a slowly increasing factor that grows only logarithmically with the size of the system.
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References
1.
N. Albin and O. P. Bruno, A spectral FC solver for the compressible Navier-Stokes equations in general domains I: Explicit time-stepping, J. Comput. Phys., 230 (2011), pp. 6248–6270.
2.
J. P. Boyd, A fast algorithm for Chebyshev, Fourier, and sinc interpolation onto an irregular grid, J. Comput. Phys., 103 (1992), pp. 243–257.
3.
J. P. Boyd, A comparison of numerical algorithms for Fourier extension of the first, second, and third kinds, J. Comput. Phys., 178 (2002), pp. 118–160.
4.
J. P. Boyd and J. R. Ong, Exponentially-convergent strategies for defeating the Runge phenomenon for the approximation of non-periodic functions, part I: Single-interval schemes, Commun. Comput. Phys., 5 (2009), pp. 484–497.
5.
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 Series, Lect. Notes Comput. Sci. Eng. 31, M. Ainsworth, P. Davies, D. Duncan, P. Martin, and B. Rynne, eds., springer, Berlin, 2003, pp. 43–82.
6.
O. P. Bruno, Y. Han, and M. M. Pohlman, Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis, J. Comput. Phys., 227 (2007), pp. 1094–1125.
7.
O. P. Bruno and M. Lyon, High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements, J. Comput. Phys., 229 (2010), pp. 2009–2033.
8.
P. Drineas, R. Kannan, and M. W. Mahoney, Fast Monte Carlo algorithms for matrices II: Computing a low-rank approximation to a matrix, SIAM J. Comput., 36 (2006), pp. 158–183.
9.
P. Drineas, R. Kannan, and M. W. Mahoney, Fast Monte Carlo algorithms for matrices III: Computing a compressed approximate matrix decomposition, SIAM J. Comput., 36 (2006), pp. 184–206.
10.
T. A. Driscoll and B. Fornberg, A Padé-based algorithm for overcoming the Gibbs phenomenon, Numer. Algorithms, 26 (2001), pp. 77–92.
11.
A. J. W. Duijndam and M. A. Schonewille, Nonuniform fast Fourier transform, Geophys., 64 (1999), pp. 539–551.
12.
A. Dutt and V. Rokhlin, Fast Fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), pp. 1368–1393.
13.
A. Dutt and V. Rokhlin, Fast Fourier transforms for nonequispaced data, II, Appl. Comput. Harmon. Anal., 2 (1995), pp. 85–100.
14.
C. Fefferman, Fitting a $C^m$-smooth function to data, III, Ann. of Math. (2), 170 (2009), pp. 427–441.
15.
C. Fefferman, Whitney's extension problems and the interpolation of data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), pp. 207–220.
16.
J. A. Fessler and B. P. Sutton, Nonuniform fast Fourier transforms using min-max interpolation, IEEE Trans. Signal Process., 51 (2003), pp. 560–574.
17.
A. Frieze, R. Kannan, and S. Vempala, Fast Monte-Carlo algorithms for finding low-rank approximations, J. ACM, 51 (2004), pp. 1025–1041.
18.
M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proc. IEEE, 93 (2005), pp. 216–231.
19.
J. Geer, Rational trigonometric approximations to piece-wise smooth periodic functions, J. Sci. Comput., 10 (1995), pp. 325–356.
20.
A. Gelb and J. Tanner, Robust reprojection methods for the resolution of the Gibbs phenomenon, Appl. Comput. Harmon. Anal., 20 (2006), pp. 3–25.
21.
D. Gottlieb, C.-W. Shu, A. Solomonoff, and 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 (1992), pp. 81–98.
22.
D. Gottlieb and E. Tadmor, Recovering pointwise values of discontinuous data within spectral accuracy, in Progress and Supercomputing in Computational Fluid Dynamics, E. Murman and S. Abarbanel, eds., Birkhäuser, Boston, 1985, pp. 357–375.
23.
L. Greengard and J.-Y. Lee, Accelerating the nonuniform fast Fourier transform, SIAM Rev., 46 (2004), pp. 443–454.
24.
M. R. Hestenes, Extension of the range of a differentiable function, Duke Math. J., 8 (1941), pp. 183–192.
25.
D. Huybrechs, On the Fourier extension of nonperiodic functions, SIAM J. Numer. Anal., 47 (2010), pp. 4326–4355.
26.
E. Liberty, F. Woolfe, P.-G. Martinsson, V. Rokhlin, and M. Tyger, Randomized algorithms for the low-rank approximation of matrices, Proc. Natl. Acad. Sci. USA, 104 (2007), pp. 20167–20172.
27.
M. Lyon and O. P. Bruno, High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations, J. Comput. Phys., 229 (2010), pp. 3358–3381.
28.
A. Majda, J. McDonough, and S. Osher, The Fourier method for nonsmooth initial data, Math. Comp., 32 (1978), pp. 1041–1081.
29.
M. S. Mock and P. D. Lax, The computational of discontinuous solutions of linear hyperbolic equations, Comm. Pure Appl. Math., 31 (1978), pp. 423–430.
30.
N. Nguyen and Q. H. Liu, The regular Fourier matrices and nonuniform fast Fourier transforms, SIAM J. Sci. Comput., 21 (1999), pp. 283–293.
31.
G. Strang, The discrete cosine transform, SIAM Rev., 41 (1999), pp. 135–147.
32.
J. Tanner, Optimal filter and mollifier for piecewise smooth spectral data, Math. Comp., 75 (2006), pp. 767–790.
33.
L. N. Trefethen and D. Bau III, Numerical Linear Algebra, SIAM, Philadelphia, 1997.
34.
A. F. Ware, Fast approximate Fourier transforms for irregularly spaced data, SIAM Rev., 40 (1998), pp. 838–856.
35.
H. Whitney, Analytic extensions of differentiable functions defined in closed sets, Trans. Amer. Math. Soc., 36 (1934), pp. 63–89.
36.
F. Woolfe, E. Liberty, V. Rokhlin, and M. Tygert, A fast randomized algorithm for the approximation of matrices, Appl. Comput. Harmon. Anal., 25 (2008), pp. 335–366.
Information & Authors
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Published In
SIAM Journal on Scientific Computing
Pages: 3241 - 3260
DOI: 10.1137/11082436X
ISSN (online): 1095-7197
Copyright
Copyright © 2011 Society for Industrial and Applied Mathematics.
History
Accepted: 25 August 2011
Published online: 10 November 2011
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