Applications of nonuniform fast transform algorithms in numerical solutions of differential and integral equations

Author

Liu, Qing Huo ; Xu, Xue Min ; Tian, Bo ; Zhang, Zhong Qing

Author_Institution

Dept. of Electr. & Comput. Eng., Duke Univ., Durham, NC, USA

Volume

38

Issue

4

fYear

2000

fDate

7/1/2000 12:00:00 AM

Firstpage

1551

Lastpage

1560

Abstract

We review our efforts to apply the nonuniform fast Fourier transform (NUFFT) and related fast transform algorithms to numerical solutions of Maxwell´s equations in the time and frequency domains. The NUFFT is a fast algorithm to perform the discrete Fourier transform of data sampled nonuniformly (NUDFT). Through oversampling and fast interpolation, the forward and inverse NUFFTs can be achieved with O(N log₂ N) arithmetic operations, asymptotically the same as the regular fast Fourier transform (FFT) algorithms. Using the NUFFT scheme, we develop nonuniform fast cosine transform (NUFCT) and fast Hankel transform (NUFHT) algorithms. These algorithms provide an efficient tool for numerical differentiation and integration, the key in the solutions to differential equations and volume integral equations. We present sample applications of these nonuniform fast transform algorithms in the numerical solution to Maxwell´s equations

Keywords

Hankel transforms; Maxwell equations; differential equations; differentiation; discrete Fourier transforms; fast Fourier transforms; frequency-domain analysis; integral equations; integration; time-domain analysis; Maxwell´s equations; NUDFT; NUFCT; NUFFT; NUFHT algorithms; arithmetic operations; differential equations; discrete Fourier transform of data sampled nonuniformly; fast Fourier transform; fast Hankel transform; fast transform algorithms; frequency domain; integral equations; nonuniform fast Fourier transform; nonuniform fast cosine transform; nonuniform fast transform algorithms; numerical differentiation; numerical integration; numerical solutions; time domain; volume integral equations; Chebyshev approximation; Convolution; Differential equations; Discrete Fourier transforms; Fast Fourier transforms; Finite difference methods; Integral equations; Maxwell equations; Partial differential equations; Time domain analysis;

fLanguage

English

Journal_Title

Geoscience and Remote Sensing, IEEE Transactions on

Publisher

ieee

ISSN

0196-2892

Type

jour

DOI

10.1109/36.851955

Filename

851955