The ocean and acoustic modeling community has specifically asked for roughness from bathymetry. An effort has been undertaken to provide what can be thought of as the high frequency content of bathymetry. By contrast, the low frequency content of bathymetry is the set of contours. The two-dimensional amplitude spectrum calculated with the nonequispaced fast Fourier transform (Kunis, 2006) is exploited as the statistic to provide several parameters of roughness following the method of Fox (1996). When an area is uniformly rough, it is termed isotropically rough. When an area exhibits lineation effects (like in a trough or a ridge line in the bathymetry), the term anisotropically rough is used. A predominant spatial azimuth of lineation summarizes anisotropic roughness. The power law model fit produces a roll-off parameter that also provides insight into the roughness of the area. These four parameters give rise to several derived parameters. Algorithmic accomplishments include reviving Fox’s method (1985, 1996) and improving the method with the possibly geophysically more appropriate nonequispaced fast Fourier transform. A new composite parameter, simply the overall integral length of the nonlinear parameterizing function, is used to make within-dataset comparisons. A synthetic dataset and six multibeam datasets covering practically all depth regimes have been analyzed with the tools that have been developed. Data specific contributions include possibly discovering an aspect ratio isotropic cutoff level (less than 1.2), showing a range of spectral fall-off values between about -0.5 for a sandy- bottomed Gulf of Mexico area, to about -1.8 for a coral reef area just outside of the Saipan harbor. We also rank the targeted type of dataset, the best resolution gridded datasets, from smoothest to roughest using a factor based on the kernel dimensions, a percentage from the windowing operation, all multiplied by the overall integration length.

In this paper we study the error behavior of the well known fast Fourier transform for nonequispaced data (NFFT) with respect to the L2-norm. We compare the arising errors for different window functions and show that the accuracy of the algorithm can be significantly improved by modifying the shape of the window function. Based on the considered error estimates for different window functions we are able to state an easy and efficient method to tune the involved parameters automatically. The numerical examples show that the optimal parameters depend on the given Fourier coefficients, which are assumed not to be of a random structure or roughly of the same magnitude but rather subject to a certain decrease.

In this paper we study the error behavior of the well known fast Fourier transform for nonequispaced data (NFFT) with respect to the L2-norm. We compare the arising errors for different window functions and show that the accuracy of the algorithm can be significantly improved by modifying the shape of the window function. Based on the considered error estimates for different window functions we are able to state an easy and efficient method to tune the involved parameters automatically. The numerical examples show that the optimal parameters depend on the given Fourier coefficients, which are assumed not to be of a random structure or roughly of the same magnitude but rather subject to a certain decrease.

We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.

The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P2NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. While typically B-splines are applied in the scope of particle mesh methods, we consider for the first time also an approximation by Bessel functions. We show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that if the parameters are tuned appropriately the Bessel window function can keep up with the B-spline window and is in many cases even the better choice with respect to computational costs.

A prestudy indicated superior accuracy and convergence properties of apseudo-spectral method compared to a spline-based method implemented byCòrdoba et al. in 2005 when solving the α-patches problem. In this thesis wefurther investigate the numerical properties of the pseudo-spectral method and makeit more robust by implementing the Nonequispaced Fast Fourier Transform. Wepresent a more detailed overview and analysis of the pseudo-spectral method and theα-patches problem in general and conclude that the pseudo-spectral method issuperior in regards to accuracy in periodic settings.

The nonequispaced fast Fourier transform (NFFT) allows the fast approximate evaluation of trigonometric polynomials with frequencies supported on full box-shaped grids at arbitrary sampling nodes. Due to the curse of dimensionality, the total number of frequencies and thus, the total arithmetic complexity can already be very large for small refinements at medium dimensions. In this paper, we present an approach for the fast approximate evaluation of trigonometric polynomials with frequencies supported on an arbitrary subset of the full grid at arbitrary sampling nodes, which is based on Taylor expansion and rank-1 lattice methods. For the special case of symmetric hyperbolic cross index sets in frequency domain, we present error estimates and numerical results.

We describe an implementation of a multi-threaded NFFT (nonequispaced fast Fourier transform) software library and present the used parallelization approaches. Besides the NFFT kernel, the NFFT on the two-sphere and the fast summation based on NFFT are also parallelized. Thereby, the parallelization is based on OpenMP and the multi-threaded FFTW library. Furthermore, benchmarks for various cases are performed. The results show that an efficiency higher than 0.50 and up to 0.79 can still be achieved at 12 threads.

We present an efficient method to compute the electrostatic fields, torques and forces in dipolar systems, which is based on the fast Fourier transform for nonequispaced data (NFFT). We consider 3d-periodic, 2d-periodic, 1d-periodic as well as 0d-periodic (open) boundary conditions. The method is based on the corresponding Ewald formulas, which immediately lead to an efficient algorithm only in the 3d-periodic case. In the other cases we apply the NFFT based fast summation in order to approximate the contributions of the nonperiodic dimensions in Fourier space. This is done by regularizing or periodizing the involved functions, which depend on the distances of the particles regarding the nonperiodic dimensions. The final algorithm enables a unified treatment of all types of periodic boundary conditions, for which only the precomputation step has to be adjusted.

The computation of the Coulomb potentials and forces in charged particle systems under 3d-periodic boundary conditions is possible in an efficient way by utilizing the Ewald summation formulas and applying the fast Fourier transform (FFT). In this paper we consider the particle-particle NFFT (P2NFFT) approach, which is based on the fast Fourier transform for nonequispaced data (NFFT) and compare the error behaviors regarding different window functions, which are used in order to approximate the given continuous charge distribution by a mesh based charge density. While typically B-splines are applied in the scope of particle mesh methods, we consider for the first time also an approximation by Bessel functions. We show how the resulting root mean square errors in the forces can be predicted precisely and efficiently. The results show that if the parameters are tuned appropriately the Bessel window function can keep up with the B-spline window and is in many cases even the better choice with respect to computational costs.