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

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 log/sub 2/ 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.