Nonequispaced Fast Fourier Transform Research Articles

Nonequispaced Hyperbolic Cross Fast Fourier Transform

A straightforward discretization of problems in $d$ spatial dimensions often leads to an exponential growth in the number of degrees of freedom. Thus, even efficient algorithms like the fast Fourier transform (FFT) have high computational costs. Hyperbolic cross approximations allow for a severe decrease in the number of used Fourier coefficients to represent functions with bounded mixed derivatives. We propose a nonequispaced hyperbolic cross FFT based on one hyperbolic cross FFT and a dedicated interpolation by splines on sparse grids. Analogously to the nonequispaced FFT for trigonometric polynomials with Fourier coefficients supported on the full grid, this allows for the efficient evaluation of trigonometric polynomials with Fourier coefficients supported on the hyperbolic cross at arbitrary spatial sampling nodes.

FAST MOLECULAR SOLVATION ENERGETICS AND FORCE COMPUTATION.

The total free energy of a molecule includes the classical molecular mechanical energy (which is understood as the free energy in vacuum) and the solvation energy which is caused by the change of the environment of the molecule (solute) from vacuum to solvent. The solvation energy is important to the study of the inter-molecular interactions. In this paper we develop a fast surface-based generalized Born method to compute the electrostatic solvation energy along with the energy derivatives for the solvation forces. The most time-consuming computation is the evaluation of the surface integrals over an algebraic spline molecular surface (ASMS) and the fast computation is achieved by the use of the nonequispaced fast Fourier transform (NFFT) algorithm. The main results of this paper involve (a) an efficient sampling of quadrature points over the molecular surface by using nonlinear patches, (b) fast linear time estimation of energy and inter-molecular forces, (c) error analysis, and (d) efficient implementation combining fast pairwise summation and the continuum integration using nonlinear patches.

Parameter estimation for exponential sums by approximate Prony method

The recovery of signal parameters from noisy sampled data is a fundamental problem in digital signal processing. In this paper, we consider the following spectral analysis problem: Let f be a real-valued sum of complex exponentials. Determine all parameters of f, i.e., all different frequencies, all coefficients, and the number of exponentials from finitely many equispaced sampled data of f. This is a nonlinear inverse problem. In this paper, we present new results on an approximate Prony method (APM) which is based on [1]. In contrast to [1], we apply matrix perturbation theory such that we can describe the properties and the numerical behavior of the APM in detail. The number of sampled data acts as regularization parameter. The first part of APM estimates the frequencies and the second part solves an overdetermined linear Vandermonde-type system in a stable way. We compare the first part of APM also with the known ESPRIT method. The second part is related to the nonequispaced fast Fourier transform (NFFT). Numerical experiments show the performance of our method.

Simulating weak lensing of CMB maps

<i>Aims. <i/>Accurate predictions of cosmic microwave background (CMB) anisotropies and polarization are required for analyzing future CMB data sets, which ultimately require accurately simulated lensed maps.<i>Methods. <i/>We present a fast, arbitrarily accurate method to simulate the effect of gravitational lensing of the CMB anisotropies and polarization fields by large-scale structures on arbitrarily spaced grid points over a unit sphere using a non-equispaced fast Fourier transform (NFFT).<i>Results. <i/>The angular power spectrum of the simulated lensed CMB map, particularly the B-mode of polarization, agrees extremely well with analytical predictions. The analytical derivation of CMB-lensed spectra is based on non-trivial, partially resummed perturbative expansions of the correlation functions, for which our simulations therefore provide an accurate numerical validation. We demonstrate the efficiency and accuracy of the method and exhibit their dependence on the algorithm parameters. Lensed CMB maps simulated in this method are a useful tool for the analysis and interpretation of upcoming CMB experiments, such as PLANCK and ACT. Our code is available on request.

Using NFFT 3---A Software Library for Various Nonequispaced Fast Fourier Transforms

NFFT 3 is a software library that implements the nonequispaced fast Fourier transform (NFFT) and a number of related algorithms, for example, nonequispaced fast Fourier transforms on the sphere and iterative schemes for inversion. This article provides a survey on the mathematical concepts behind the NFFT and its variants, as well as a general guideline for using the library. Numerical examples for a number of applications are given.

Nonequispaced fast Fourier transforms without oversampling

AbstractRecently, the fast Fourier transform (FFT) has been generalised for arbitrary sampling nodes by the use of approximation schemes. We show that such nonequispaced FFTs can be implemented without oversampling, i.e., no extra memory besides the input and output array is needed. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Combined Complex Ridgelet Shrinkage and Total Variation Minimization

A new algorithm for the characterization of engineering surface topographies with line singularities is proposed. It is based on thresholding complex ridgelet coefficients combined with total variation (TV) minimization. The discrete ridgelet transform is designed by first using a discrete Radon transform based on the nonequispaced fast Fourier transform (NFFT) and then applying a dual-tree complex wavelet transform (DT CWT). The NFFT-based approach of the Radon transform completely avoids linear interpolations of the Cartesian-to-polar grid and requires only O(n2 log n) arithmetic operations for n by n arrays, while its inverse preserves the good reconstruction quality of the filtered backprojection. The DT CWT in the second step of the ridgelet transform provides approximate shift invariance on the projections of the Radon transform. After hard thresholding the ridgelet coefficients, they are restored using TV minimization to eliminate the pseudo-Gibbs artifacts near the discontinuities. Numerical experiments demonstrate the remarkable ability of the methodology to extract line scratches.