No abstract

A large‐scale three‐dimensional volume integral equation solution for electromagnetic radiation and scattering problems remains a great challenge in spite of many ongoing research efforts. The conventional method of moments, although accurate and flexible, is limited to small‐scale problems because of its large requirement of computer memory and computation time. In this paper, we develop two fast methods, the weak‐form conjugate‐ and biconjugate‐gradient FFT methods, to solve the Fredholm integral equation of the second kind arising from Maxwell's equations in three dimensions. The weak form is a modified version of the Zwamborn–van den Berg formulation, where the singularity is circumvented by employing the weak‐form discretization by rooftop vectorial basis and testing functions. Both weak‐ form CG–FFT and BCG–FFT methods require O(N log2 N) CPU time, and O(N) computer memory, but the latter converges three–six times faster than the CG–FFT method. We validate the numerical results by comparing them with analytical solutions to multilayer spherical media, and with other published results. © 2001 John Wiley & Sons, Inc. Microwave Opt Technol Lett 29: 350–356, 2001.

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.