Recent progress on nonuniform fast Fourier transform algorithms and their applications

Abstract

Recently, nonuniform fast Fourier transform (NUFFT) algorithms have received significant attention [Dutt and Rokhlin, SIAM J. Sci. Stat. Comput. 14, 1368–1393 (1993)]. Unlike the regular fast Fourier transform (FFT) algorithms, the NUFFT algorithms allow the data to be sampled nonuniformly. The leading order of the number of arithmetic operations for these NUFFT algorithms is O(N<th>log2<th>N). Here, we review the recent progress of the NUFFT algorithms using the regular Fourier matrices and conjugate-gradient method for the forward and inverse NUFFT algorithms [Liu and Nguyen, IEEE Microwave Guid. Wave Lett. 8, 18–20 (1998); Liu and Tang, Electron. Lett. 34, 1913–1914 (1998)]. Because of their least-square errors, these NUFFT algorithms are about one order of magnitude more accurate than the previous algorithms. These NUFFT algorithms have been applied to develop the nonuniform fast Hankel transform (NUFHT) and nonuniform fast cosine transform (NUFCT) algorithms. Both NUFFT and NUFHT algorithms have been used to solve integral equations in computational electromagnetics and acoustics; The NUFCT has been used to solve time-dependent wave equations. Numerical examples will demonstrate the efficiency of the fast transform algorithms, and the applications in computational electromagnetics and computational acoustics.