Abstract:
The Taylor Interpolation through FFT (TI-FFT) algorithm for the computation of the electromagnetic wave propagation in the quasi-planar geometry within the half-space is proposed in this article. There are two types of TI-FFT algorithm, i.e., the spatial TI-FFT and the spectral TI-FFT. The former works in the spatial domain and the latter works in the spectral domain. It has been shown that the optimized computational complexity is the same for both types of TI-FFT algorithm, which is N_r^{opt} N_o^{opt} O (N log_2 N) for an N = N_x \times N_y computational grid, where N_r^{opt} is the optimized number of slicing reference planes and N_o^{opt} is the optimized order of Taylor series. Detailed analysis shows that N_o^{opt} is closely related to the algorithm's computational accuracy \gamma_{TI}, which is given as N_o^{opt} ~ - ln(\gamma_{TI}) and the optimized spatial slicing spacing between two adjacent spatial reference planes \delta_z^{opt} only depends on the characteristic wavelength \lambda_c of the electromagnetic wave, which is given as \delta_z^{opt} ~ 1/17 \lambda_c. The planar TI-FFT algorithm allows a large sampling spacing required by the sampling theorem. What's more, the algorithm is free of singularities and it works particularly well for the narrow-band beam and the quasi-planar geometry.

Abstract:
Fourier reconstruction algorithms significantly outperform conventional back-projection algorithms in terms of computation time. In photoacoustic imaging, these methods require interpolation in the Fourier space domain, which creates artifacts in reconstructed images. We propose a novel reconstruction algorithm that applies the one-dimensional nonuniform fast Fourier transform to photoacoustic imaging. It is shown theoretically and numerically that our algorithm avoids artifacts while preserving the computational effectiveness of Fourier reconstruction.

Abstract:
The integral equation fast Fourier transform (IE-FFT) is a fast algorithm for 3D electromagnetic scattering and radiation problems based on the interpolation of the Green's function. In this paper, a novel floating interpolation stencil topology is used to improve the IE-FFT algorithm. Compared to the traditional interpolation stencil topology, it can further reduce the storage and CPU time for the IE-FFT algorithm. The reduction is especially significant for volume integral equations. Furthermore, the accuracy of the algorithm is still good though the near-interaction element numbers are reduced. Finally, some numerical results including perfectly electric conductors, dielectric objects, composite conducting and dielectric objects are given to demonstrate the performance of the present method.

Abstract:
We propose an efficient and accurate solver for the nonlocal potential in the Davey-Stewartson equation using nonuniform FFT (NUFFT). A discontinuity in the Fourier transform of the nonlocal potential causes accuracy locking if the potential is solved by standard FFT with periodic boundary conditions on a truncated domain. Using the fact that the discontinuity disappears in polar coordinates, we reformulate the potential integral and split it into high and low frequency parts. The high frequency part can be approximated by the standard FFT method, while the low frequency part is evaluated with a high order Gauss quadrature accelerated by nonuniform FFT. The NUFFT solver has $O(N\log N)$ complexity, where $N$ is the total number of discretization points, and achieves higher accuracy than standard FFT solver, which makes it a good alternative in simulation. Extensive numerical results show the efficiency and accuracy of the proposed method.

Abstract:
We present a fast and accurate algorithm for the evaluation of nonlocal (long-range) Coulomb and dipole-dipole interactions in free space. The governing potential is simply the convolution of an interaction kernel $U(\bx)$ and a density function $\rho(\bx)=|\psi(\bx)|^2$, for some complex-valued wave function $\psi(\bx)$, permitting the formal use of Fourier methods. These are hampered by the fact that the Fourier transform of the interaction kernel $\widehat{U}(\bk)$ has a singularity at the origin $\bk={\bf 0}$ in Fourier (phase) space. Thus, accuracy is lost when using a uniform Cartesian grid in $\bk$ which would otherwise permit the use of the FFT for evaluating the convolution. Here, we make use of a high-order discretization of the Fourier integral, accelerated by the nonuniform fast Fourier transform (NUFFT). By adopting spherical and polar phase-space discretizations in three and two dimensions, respectively, the singularity in $\hat{U}(\bk)$ at the origin is canceled, so that only a modest number of degrees of freedom are required to evaluate the Fourier integral, assuming that the density function $\rho(\bx)$ is smooth and decays sufficiently fast as $\bx \rightarrow \infty$. More precisely, the calculation requires $O(N\log N)$ operations, where $N$ is the total number of discretization points in the computational domain. Numerical examples are presented to demonstrate the performance of the algorithm.

Abstract:
The fast Fourier transform (FFT) based matrix-free ansatz interpolatory approximations of periodic functions are fundamental for efficient realization in several applications.In this work we design, analyze, and implement similar constructive interpolatory approximations of spherical functions, using samples of the unknown functions at the poles and at the uniform spherical-polar grid locations. The spherical matrix-free interpolation operator range space consists of a selective subspace of two dimensional trigonometric polynomials which are rich enough to contain all spherical polynomials of degree less than $N$. The spherical interpolatory approximation is efficiently constructed by applying the FFT techniques with only ${\mathcal{O}}(N^2 \log N)$ complexity. We describe the construction details using the FFT operators and provide complete convergence analysis of the interpolatory approximation in the Sobolev space framework. We prove that the rate of spectrally accurate convergence of the interpolatory approximations in Sobolev norms (of order zero and one) are similar (up to a log term) to that of the best approximation in the finite dimensional ansatz space. Efficient interpolatory quadratures on the sphere are important for several applications including radiation transport and wave propagation computer models. We use our matrix-free interpolatory approximations to construct robust FFT-based quadrature rules for a wide class of non-, mildly-, and strongly-oscillatory integrands on the sphere. We provide numerical experiments to demonstrate fast evaluation of the algorithm and various theoretical results presented in the article.

Abstract:
Finite-Difference Time-Domain (FDTD) subgridding schemes can significantly improve efficiency of various electromagnetic circuit simulations. However, numerous subgridding schemes suffer from issues associated with stability, efficiency, and material traverse capability. These issues limit general applicability of FDTD subgridding schemes to realistic problems. Herein, a robust nonuniform subgridding scheme is presented that overcomes those weaknesses. The scheme improves simulation accuracy with the aid of greatly increased stability margin and an optimal interpolation technique. It also improves simulation efficiency by allowing the use of time step factors as close as the Courant-Friedrichs-Lewy (CFL) limit. In addition, latetime stability and general applicability are verified through practical microstrip circuit simulation examples.

Abstract:
A nonuniform grid-based coordinated routing design in wireless sensor networks is presented. The conditions leading to network partition and analysis of energy consumption that prolongs the network lifetime are studied. We focus on implementing routing in densely populated sensor networks. By maintaining constant values for parameters such as path loss exponent, receiver sensitivity and transmit power, and varying between uniform and non-uniform grids, we observe energy consumption patterns for each of the grid structures and infer from the network lifetime the better suited grids for uniformly and randomly deployed sensor nodes.

Abstract:
In this paper a general theory for interpolation methods on a rectangular grid is introduced. By the use of this theory an efficient B-spline based interpolation method for spectral codes is presented. The theory links the order of the interpolation method with its spectral properties. In this way many properties like order of continuity, order of convergence and magnitude of errors can be explained. Furthermore, a fast implementation of the interpolation methods is given. We show that the B-spline based interpolation method has several advantages compared to other methods. First, the order of continuity of the interpolated field is higher than for other methods. Second, only one FFT is needed whereas e.g. Hermite interpolation needs multiple FFTs for computing the derivatives. Third, the interpolation error almost matches the one of Hermite interpolation, a property not reached by other methods investigated.

Abstract:
In this paper we present a locally and dimension-adaptive sparse grid method for interpolation and integration of high-dimensional functions with discontinuities. The proposed algorithm combines the strengths of the generalised sparse grid algorithm and hierarchical surplus-guided local adaptivity. A high-degree basis is used to obtain a high-order method which, given sufficient smoothness, performs significantly better than the piecewise-linear basis. The underlying generalised sparse grid algorithm greedily selects the dimensions and variable interactions that contribute most to the variability of a function. The hierarchical surplus of points within the sparse grid is used as an error criterion for local refinement with the aim of concentrating computational effort within rapidly varying or discontinuous regions. This approach limits the number of points that are invested in `unimportant' dimensions and regions within the high-dimensional domain. We show the utility of the proposed method for non-smooth functions with hundreds of variables.