Fast discrete algorithms for sparse Fourier expansions of high dimensional functions

  • Authors:

  • Ying Jiang;Yuesheng Xu

  • Affiliations:

  • Department of Scientific Computing and Computer Applications, Sun Yat-sen University, Guangzhou 510275, PR China;Department of Scientific Computing and Computer Applications, Sun Yat-sen University, Guangzhou 510275, PR China and Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA

  • Venue:
  • Journal of Complexity
  • Year:
  • 2010

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Abstract

We develop a fast discrete algorithm for computing the sparse Fourier expansion of a function of d dimension. For this purpose, we introduce a sparse multiscale Lagrange interpolation method for the function. Using this interpolation method, we then design a quadrature scheme for evaluating the Fourier coefficients of the sparse Fourier expansion. This leads to a fast discrete algorithm for computing the sparse Fourier expansion. We prove that this method gives the optimal approximation order O(n^-^s) for the sparse Fourier expansion, where s0 is the order of the Sobolev regularity of the function to be approximated and where n is the order of the univariate trigonometric polynomial used to construct the sparse multivariate approximation, and requires only O(nlog^2^d^-^1n) number of multiplications to compute all of its Fourier coefficients. We present several numerical examples with d=2,3 and 4 that confirm the theoretical estimates of approximation order and computational complexity and compare the numerical performance of the proposed method with that of a well-known existing algorithm. We also have a numerical example for d=8 to test the efficiency of the propose algorithm for functions of a higher dimension.