The conjugate-gradient nonuniform fast Fourier transform (CG-NUFFT) method for one- and two-dimensional media

Abstract

A novel method combining the conjugate‐gradient and nonuniform fast Fourier transform (CG–NUFFT) algorithms for solving integral equations in electromagnetics is presented. The CG–NUFFT method has been applied to one‐ and two‐dimensional inhomogeneous media. Numerical results show the advantages of the CG–NUFFT method over the CG–FFT method for problems with both electrically large and small regions. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 24: 385–389, 2000.

Three-Dimensional Weak-Form DRGMRES-NUFFT for Volume Integral Equations

“…In order to validate our 3D NUFFT algorithms, the cuboid is discretized with 16 16 32 × × grids, nonuniformly distributed in both x and y directions and uniformly in z direction. Some data for the RCS of the cuboid are given in Figure 1, and the solutions from the weak-form CG-FFT [2] are also displayed. As compared in Figure 1(a), our numerical results have an excellent agreement with the CG-FFT.…”

“…To overcome this limitation, the nonuniform fast Fourier transform (NUFFT) algorithm is introduced. In [2], the NUFFT is combined with the CG method for one-and two-dimensional problems. With NUFFT algorithms, the CG-NUFFT method gives the important flexibility of nonuniform sampling and maintains the computational efficiency of the conventional CG-FFT method.…”

“…Using the weak-form formulation of the Green's function defined in [2], and expanding and testing the volume integral equation with 3-D vector volumetric rooftop functions, a linear algebraic equation is derived as: = Ld e . This can be iteratively solved with any Krylov-subspace methods, where the matrix-vector product operation is efficiently evaluated by using NUFFT algorithms.…”

On NUFFT-based gridding for non-Cartesian MRI

“…The generalization to 2D and 3D is straightforward using separable interpolators as described previously, e.g., [2][3][4]. Given samples {F(ν m )}, m = 1, …, M of the Fourier transform of an object f(x), one can estimate f by an approximate inverse Fourier transform as follows:…”

“…Defining ω m ≜ 2πν m Δ x and the phase modulated spectrum (3) the essence of the gridding reconstruction method is to evaluate summations of the following form: (4) This is a "type 1" NUFFT as defined for example in [6].…”