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This research aims to make a comparison between the analytical and numerical methods of simulation of the Gaussian beam propagation in free space. This comparison brings evaluation of the accuracy and applicability limitations of the angular spectrum diffraction algorithm. In this approach, planar harmonic waves are assumed as basic components of a propagating wave, so the algorithm combines the angular spectrum technique and the 2D fast Fourier transformation procedures. It means a quite different approach than the well-established Huygens–Fresnel–Rayleigh–Sommerfeld one, based on spherical waves. The Gaussian beam is a rare case of the exact analytical 3D solution of a specific wave diffraction problem, in paraxial scalar approximation. Gaussian beam form has been chosen here as an evaluation reference. It allows to avoid comparisons between two numerical procedures, both of which may suffer from imperfections. In this comparison, the respective 2D cross-sections of calculated 3D complex valued wave fields distributions are presented. Also values of specific Gaussian beam parameters like full phase and Gouy phase shift and wave-front curvature radius are presented. The comparison between that form and the results of the angular spectrum algorithm brings information about the performance quality of that algorithm. [ABSTRACT FROM AUTHOR]

In this paper, we present algorithms for the approximation of multivariate periodic functions by trigonometric polynomials. The approximation is based on sampling of multivariate functions on rank-1 lattices. To this end, we study the approximation of periodic functions of a certain smoothness. Our considerations include functions from periodic Sobolev spaces of generalized mixed smoothness. Recently an algorithm for the trigonometric interpolation on generalized sparse grids for this class of functions was investigated by Griebel and Hamaekers (2014). The main advantage of our method is that the algorithm is based mainly on a single one-dimensional fast Fourier transform, and that the arithmetic complexity of the algorithm depends only on the cardinality of the support of the trigonometric polynomial in the frequency domain. Therefore, we investigate trigonometric polynomials with frequencies supported on hyperbolic crosses and energy norm based hyperbolic crosses in more detail. Furthermore, we present an algorithm for sampling multivariate functions on perturbed rank-1 lattices and show the numerical stability of the suggested method. Numerical results are presented up to dimension d = 10 , which confirm the theoretical findings. [ABSTRACT FROM AUTHOR]

Pricing Asian options is a long-standing hard problem; there is no analytical formula for the probability density of its payoff even when the process of the underlying asset follows the simple lognormal diffusion process. It is known that the density function of a discretely-sampled Asian option’s payoff can be efficiently approximated by the Fast Fourier Transform (FFT). As a result, we can accurately price the option under more general Lévy processes. This paper shows that the pricing error of this approach, called the FFT approach, can be decomposed into the truncation error, the integration error, and the interpolation error. We prove that previous algorithms that follow the FFT approach converge quadratically. To improve the error convergence rate, our proposed algorithms reduce the integration error by the higher-order Newton–Cotes formulas and new integration rules derived from the Lagrange interpolating polynomial. The interpolation error is reduced by the higher-order Newton divided-difference interpolation formula. Consequently, our algorithms can be sped up by the FFT to achieve the same time complexity as previous algorithms, but with a faster error convergence rate. Numerical results are given to verify the efficiency and the fast convergence of our algorithms. [ABSTRACT FROM AUTHOR]

In this paper, motivated by Lin, Ching and Ng [Theoretical Computer Science, 315:511- 523 (2004)], a scaling version of Bini's algorithm [SIAM J. Comput., 13:268-276 (1984)] for an approximate inversion of a triangular Toeplitz matrix is proposed. The scaling algorithm introduces a new scale parameter and is mathematically equivalent to the original Bini's. Its computational cost is about two fast Fourier transforms of n-vectors (FFTs(n)), equal to that of Bini's. We also improve the accuracy of the proposed approach by embedding the n-by-n triangular Toeplitz matrix into an (n+n0)-by-(n+n0) triangular (banded) Toeplitz matrix, where n0 is a positive integer. The complexity of the resulting revised scaling Bini's algorithm is about two FFTs(2n). Several numerical examples are given to illustrate the effectiveness and stability of the proposed methods. [ABSTRACT FROM AUTHOR]

Abstract: This paper introduces a fast algorithm for computing multilinear integrals which are defined through Fourier multipliers. The algorithm is based on generating a hierarchical decomposition of the summation domain into squares, constructing a low-rank approximation for the multiplier function within each square, and applying an FFT based fast convolution algorithm for the computation associated with each square. The resulting algorithm is accurate and has a linear complexity, up to logarithmic factors, with respect to the number of the unknowns in the input functions. Numerical results are presented to demonstrate the properties of this algorithm. [Copyright &y& Elsevier]

The methodological difficulties of estimating Fourier integrals using the fast Fourier transform (FFT) algorithm have intensified the interest in an alternative approach based on the Filon’s method of computing the trigonometric integrals. Following this approach, we introduce in this paper a similar basis function (SBF) algorithm that decomposes the function to be transformed into the sum of finite elements termed “similar basis functions”. Due to a simple analytical form of SBF, the reassignment of the SBFs’ similarity relationships into the transformation domain reduces the estimation of the Fourier integrals to a number of standard computational procedures. The SBF algorithm is capable to deal with both uniform and non-uniform samples of the function under analysis. Using this opportunity, we extend a general SBF algorithm by a fast SBF algorithm which deals with exponentially increasing sampling intervals. The efficiency and the accuracy of the method are illustrated by computer experiments with frequency characteristics and transient responses of a typical dynamic system. [ABSTRACT FROM AUTHOR]

This paper is concerned with the evaluation of the partial Fourier transform ux = Σ|k|