Fast discrete algorithms for sparse Fourier expansions of high dimensional functions

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 s > 0 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(n log2d−1 n) 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. © 2009 Elsevier Inc. All rights reserved.