A Fast Algorithm for Fourier Continuation

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.

[1]  Mark Lyon,et al.  High-order unconditionally stable FC-AD solvers for general smooth domains I. Basic elements , 2010, J. Comput. Phys..

[2]  Gilbert Strang,et al.  The Discrete Cosine Transform , 1999, SIAM Rev..

[3]  H. Whitney.  Analytic Extensions of Differentiable Functions Defined in Closed Sets , 1934 .

[4]  J. Boyd.  A Comparison of Numerical Algorithms for Fourier Extension of the First, Second, and Third Kinds , 2002 .

[5]  V. Rokhlin,et al.  A fast randomized algorithm for the approximation of matrices ✩ , 2007 .

[6]  John P. Boyd,et al.  Exponentially-convergent strategies for defeating the Runge Phenomenon for the approximation of non-periodic functions, part two: Multi-interval polynomial schemes and multidomain Chebyshev interpolation , 2011 .

[7]  Anne Gelb,et al.  Robust reprojection methods for the resolution of the Gibbs phenomenon , 2006 .

[8]  C. Fefferman,et al.  Fitting a Cm-Smooth Function to Data , 2005 .

[9]  Petros Drineas,et al.  FAST MONTE CARLO ALGORITHMS FOR MATRICES II: COMPUTING A LOW-RANK APPROXIMATION TO A MATRIX∗ , 2004 .

[10]  Peter D. Lax,et al.  The computation of discontinuous solutions of linear hyperbolic equations , 1978 .

[11]  Alex Solomonoff,et al.  On the Gibbs phenomenon I: recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function , 1992 .

[12]  A. Duijndam,et al.  Nonuniform Fast Fourier Transform , 1997 .

[13]  Alan M. Frieze,et al.  Fast Monte-Carlo algorithms for finding low-rank approximations , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[14]  A.,et al.  FAST FOURIER TRANSFORMS FOR NONEQUISPACED DATA * , .

[15]  Nathan Albin,et al.  A spectral FC solver for the compressible Navier-Stokes equations in general domains I: Explicit time-stepping , 2011, J. Comput. Phys..

[16]  Tobin A. Driscoll,et al.  A Padé-based algorithm for overcoming the Gibbs phenomenon , 2004, Numerical Algorithms.

[17]  M. Hestenes,et al.  Extension of the range of a differentiable function , 1941 .

[18]  J. Mixter.  Fast , 2012 .

[19]  Daan Huybrechs,et al.  On the Fourier Extension of Nonperiodic Functions , 2010, SIAM J. Numer. Anal..

[20]  C. Fefferman.  Fitting a Cm-smooth function to data, III , 2009 .

[21]  Philipp Birken,et al.  Numerical Linear Algebra , 2011, Encyclopedia of Parallel Computing.

[22]  Mark Lyon,et al.  High-order unconditionally stable FC-AD solvers for general smooth domains II. Elliptic, parabolic and hyperbolic PDEs; theoretical considerations , 2010, J. Comput. Phys..

[23]  Jared Tanner,et al.  Optimal filter and mollifier for piecewise smooth spectral data , 2006, Math. Comput..

[24]  Leslie Greengard,et al.  Accelerating the Nonuniform Fast Fourier Transform , 2004, SIAM Rev..

[25]  Jeffrey A. Fessler,et al.  Nonuniform fast Fourier transforms using min-max interpolation , 2003, IEEE Trans. Signal Process..

[26]  Eitan Tadmor,et al.  Recovering Pointwise Values of Discontinuous Data within Spectral Accuracy , 1985 .

[27]  Oscar P. Bruno,et al.  Accurate, high-order representation of complex three-dimensional surfaces via Fourier continuation analysis , 2007, J. Comput. Phys..

[28]  Steven G. Johnson,et al.  The Design and Implementation of FFTW3 , 2005, Proceedings of the IEEE.

[29]  Petros Drineas,et al.  FAST MONTE CARLO ALGORITHMS FOR MATRICES III: COMPUTING A COMPRESSED APPROXIMATE MATRIX DECOMPOSITION∗ , 2004 .

[30]  C. Fefferman.  Whitney’s extension problems and interpolation of data , 2008 .

[31]  Andrew J. Majda,et al.  The Fourier method for nonsmooth initial data , 1978 .

[32]  Antony Ware,et al.  Fast Approximate Fourier Transforms for Irregularly Spaced Data , 1998, SIAM Rev..

[33]  Oscar P. Bruno,et al.  Fast, High-Order, High-Frequency Integral Methods for Computational Acoustics and Electromagnetics , 2003 .

[34]  Qing Huo Liu,et al.  The Regular Fourier Matrices and Nonuniform Fast Fourier Transforms , 1999, SIAM J. Sci. Comput..

[35]  Per-Gunnar Martinsson,et al.  Randomized algorithms for the low-rank approximation of matrices , 2007, Proceedings of the National Academy of Sciences.