In this paper we describe a parallel algorithm for calculating nonequispaced fast Fourier transforms on massively parallel distributed memory architectures. These algorithms are implemented in an open source software library called PNFFT. Furthermore, we derive a parallel fast algorithm for the computation of the Coulomb potentials and forces in a charged particle system, which is based on the parallel nonequispaced fast Fourier transform. To prove the high scalability of our algorithms we provide performance results on a BlueGene/P system using up to 65536 cores.

The present thesis is dedicated to the efficient computation of electrostatic interactions in particle systems, which is of great importance in the field of molecular dynamics simulations. In order to compute the therefor required physical quantities with only O(N log N) arithmetic operations, so called particle-mesh methods make use of the well-known Ewald summation approach and the fast Fourier transform (FFT). Typically, such methods are able to handle systems of point charges subject to periodic boundary conditions in all spatial directions. However, periodicity is not always desired in all three dimensions and, moreover, also interactions to dipoles play an important role in many applications. Within the scope of the present work, we consider the particle-particle NFFT method (P²NFFT), a particle-mesh approach based on the fast Fourier transform for nonequispaced data (NFFT). An extension of this method for mixed periodic as well as open boundary conditions is presented. Furthermore, the method is appropriately modified in order to treat particle systems containing both charges and dipoles. Consequently, an efficient algorithm for mixed charge-dipole systems, that additionally allows a unified handling of various types of periodic boundary conditions, is presented for the first time. Appropriate error estimates as well as parameter tuning strategies are developed and verified by numerical examples. / Die vorliegende Arbeit widmet sich der Berechnung elektrostatischer Wechselwirkungen in Partikelsystemen, was beispielsweise im Bereich der molekulardynamischen Simulationen eine zentrale Rolle spielt. Um die dafür benötigten physikalischen Größen mit lediglich O(N log N) arithmetischen Operationen zu berechnen, nutzen sogenannte Teilchen-Gitter-Methoden die Ewald-Summation sowie die schnelle Fourier-Transformation (FFT). Typischerweise können derartige Verfahren Systeme von Punktladungen unter periodischen Randbedingungen in allen Raumrichtungen handhaben. Periodizität ist jedoch nicht immer bezüglich aller drei Dimensionen erwünscht. Des Weiteren spielen auch Wechselwirkungen zu Dipolen in vielen Anwendungen eine wichtige Rolle. Zentraler Gegenstand dieser Arbeit ist die Partikel-Partikel-NFFT Methode (P²NFFT), ein Teilchen-Gitter-Verfahren, welches auf der schnellen Fouriertransformation für nichtäquidistante Daten (NFFT) basiert. Eine Erweiterung dieses Verfahrens auf gemischt periodische sowie offene Randbedingungen wird vorgestellt. Außerdem wird die Methode für die Behandlung von Partikelsystemen, in denen sowohl Ladungen als auch Dipole vorliegen, angepasst. Somit wird erstmalig ein effizienter Algorithmus für gemischte Ladungs-Dipol-Systeme präsentiert, der zusätzlich die Behandlung sämtlicher Arten von Randbedingungen mit einem einheitlichen Zugang erlaubt. Entsprechende Fehlerabschätzungen sowie Strategien für die Parameterwahl werden entwickelt und anhand numerischer Beispiele verifiziert.

By the help of several examples the interactive work with the ClassPad330 is considered. The student can solve difficult exercises of practical applications step by step using the symbolic calculation and the graphic possibilities of the calculator. Sometimes several fields of mathematics are combined to solve a problem. Let us consider the ClassPad330 (with the actual operating system OS 03.03) and discuss on some new exercises in analysis, e.g. solving a linear differential equation by the help of the Laplace transformation and using the inverse Laplace transformation or considering the Fourier transformation in discrete time (the Fast Fourier Transformation FFT and the inverse FFT). We use the FFT- and IFFT-function to study periodic signals, if we only have a sequence generated by sampling the time signal. We know several ways to get a solution. The techniques for studying practical applications fall into the following three categories: analytic, graphic and numeric. We can use the Classpad software in the handheld or in the PC (ClassPad emulator version of the handheld).

The present thesis provides a modularized view on the structure of fast numerical methods for computing Coulomb interactions between charged particles in three-dimensional space. Thereby, the common structure is given in terms of three self-contained algorithmic frameworks that are built on top of each other, namely fast Fourier transform (FFT), nonequispaced fast Fourier transform (NFFT) and NFFT based particle-mesh methods (P²NFFT). For each of these frameworks algorithmic enhancement and parallel implementations are presented with special emphasis on scalability up to hundreds of thousands of parallel processes. In the context of FFT massively parallel algorithms are composed from hardware adaptive low level modules provided by the FFTW software library. The new algorithmic NFFT concepts include pruned NFFT, interlacing, analytic differentiation, and optimized deconvolution in Fourier space with respect to a mean square aliasing error. Enabled by these generalized concepts it is shown that NFFT provides a unified access to particle-mesh methods. Especially, mixed-periodic boundary conditions are handled in a consistent way and interlacing can be incorporated more efficiently. Heuristic approaches for parameter tuning are presented on the basis of thorough error estimates. / Die vorliegende Dissertation beschreibt einen modularisierten Blick auf die Struktur schneller numerischer Methoden für die Berechnung der Coulomb-Wechselwirkungen zwischen Ladungen im dreidimensionalen Raum. Die gemeinsame Struktur ist geprägt durch drei selbstständige und auf einander aufbauenden Algorithmen, nämlich der schnellen Fourier-Transformation (FFT), der nicht äquidistanten schnellen Fourier-Transformation (NFFT) und der NFFT-basierten Teilchen-Gitter-Methode (P²NFFT). Für jeden dieser Algorithmen werden Verbesserungen und parallele Implementierungen vorgestellt mit besonderem Augenmerk auf massiv paralleler Skalierbarkeit. Im Kontext der FFT werden parallele Algorithmen aus den Hardware adaptiven Modulen der FFTW Softwarebibliothek zusammengesetzt. Die neuen NFFT-Konzepte beinhalten abgeschnittene NFFT, Versatz, analytische Differentiation und optimierte Entfaltung im Fourier-Raum bezüglich des mittleren quadratischen Aliasfehlers. Mit Hilfe dieser Verallgemeinerungen bietet die NFFT einen vereinheitlichten Zugang zu Teilchen-Gitter-Methoden. Insbesondere gemischt periodische Randbedingungen werden einheitlich behandelt und Versatz wird effizienter umgesetzt. Heuristiken für die Parameterwahl werden auf Basis sorgfältiger Fehlerabschätzungen angegeben.

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. Typically B-splines are applied in the scope of particle mesh methods, as for instance within the well-known particle-particle particle-mesh (P3M) algorithm. The publicly available P2NFFT algorithm allows the application of an oversampled FFT as well as the usage of different window functions. We consider for the first time also an approximation by Bessel functions and 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 is in many cases even the better choice in terms of computational costs. Moreover, the results indicate that it is often advantageous in terms of efficiency to spend some oversampling within the NFFT while using a window function with a smaller support.

We consider multivariate trigonometric polynomials with frequencies supported on a fixed but arbitrary frequency index set I, which is a finite set of integer vectors of length d. Naturally, one is interested in spatial discretizations in the d-dimensional torus such that - the sampling values of the trigonometric polynomial at the nodes of this spatial discretization uniquely determines the trigonometric polynomial, - the corresponding discrete Fourier transform is fast realizable, and - the corresponding fast Fourier transform is stable. An algorithm that computes the discrete Fourier transform and that needs a computational complexity that is bounded from above by terms that are linear in the maximum of the number of input and output data up to some logarithmic factors is called fast Fourier transform. We call the fast Fourier transform stable if the Fourier matrix of the discrete Fourier transform has a condition number near one and the fast algorithm does not corrupt this theoretical stability. We suggest to use rank-1 lattices and a generalization as spatial discretizations in order to sample multivariate trigonometric polynomials and we develop construction methods in order to determine reconstructing sampling sets, i.e., sets of sampling nodes that allow for the unique, fast, and stable reconstruction of trigonometric polynomials. The methods for determining reconstructing rank-1 lattices are component{by{component constructions, similar to the seminal methods that are developed in the field of numerical integration. During this thesis we identify a component{by{component construction of reconstructing rank-1 lattices that allows for an estimate of the number of sampling nodes M |I|\le M\le \max\left(\frac{2}{3}|I|^2,\max\{3\|\mathbf{k}\|_\infty\colon\mathbf{k}\in I\}\right) that is sufficient in order to uniquely reconstruct each multivariate trigonometric polynomial with frequencies supported on the frequency index set I. We observe that the bounds on the number M only depends on the number of frequency indices contained in I and the expansion of I, but not on the spatial dimension d. Hence, rank-1 lattices are suitable spatial discretizations in arbitrarily high dimensional problems. Furthermore, we consider a generalization of the concept of rank-1 lattices, which we call generated sets. We use a quite different approach in order to determine suitable reconstructing generated sets. The corresponding construction method is based on a continuous optimization method. Besides the theoretical considerations, we focus on the practicability of the presented algorithms and illustrate the theoretical findings by means of several examples. In addition, we investigate the approximation properties of the considered sampling schemes. We apply the results to the most important structures of frequency indices in higher dimensions, so-called hyperbolic crosses and demonstrate the approximation properties by the means of several examples that include the solution of Poisson's equation as one representative of partial differential equations.

In this work, the fast evaluation and reconstruction of multivariate trigonometric polynomials with frequencies supported on arbitrary index sets of finite cardinality is considered, where rank-1 lattices are used as spatial discretizations. The approximation of multivariate smooth periodic functions by trigonometric polynomials is studied, based on a one-dimensional FFT applied to function samples. The smoothness of the functions is characterized via the decay of their Fourier coefficients, and various estimates for sampling errors are shown, complemented by numerical tests for up to 25 dimensions. In addition, the special case of perturbed rank-1 lattice nodes is considered, and a fast Taylor expansion based approximation method is developed. One main contribution is the transfer of the methods to the non-periodic case. Multivariate algebraic polynomials in Chebyshev form are used as ansatz functions and rank-1 Chebyshev lattices as spatial discretizations. This strategy allows for using fast algorithms based on a one-dimensional DCT. The smoothness of a function can be characterized via the decay of its Chebyshev coefficients. From this point of view, estimates for sampling errors are shown as well as numerical tests for up to 25 dimensions. A further main contribution is the development of a high-dimensional sparse FFT method based on rank-1 lattice sampling, which allows for determining unknown frequency locations belonging to the approximately largest Fourier or Chebyshev coefficients of a function. / In dieser Arbeit wird die schnelle Auswertung und Rekonstruktion multivariater trigonometrischer Polynome mit Frequenzen aus beliebigen Indexmengen endlicher Kardinalität betrachtet, wobei Rang-1-Gitter (rank-1 lattices) als Diskretisierung im Ortsbereich verwendet werden. Die Approximation multivariater glatter periodischer Funktionen durch trigonometrische Polynome wird untersucht, wobei Approximanten mittels einer eindimensionalen FFT (schnellen Fourier-Transformation) angewandt auf Funktionswerte ermittelt werden. Die Glattheit von Funktionen wird durch den Abfall ihrer Fourier-Koeffizienten charakterisiert und mehrere Abschätzungen für den Abtastfehler werden gezeigt, ergänzt durch numerische Tests für bis zu 25 Raumdimensionen. Zusätzlich wird der Spezialfall gestörter Rang-1-Gitter-Knoten betrachtet, und es wird eine schnelle Approximationsmethode basierend auf Taylorentwicklung vorgestellt. Ein wichtiger Beitrag dieser Arbeit ist die Übertragung der Methoden vom periodischen auf den nicht-periodischen Fall. Multivariate algebraische Polynome in Chebyshev-Form werden als Ansatzfunktionen verwendet und sogenannte Rang-1-Chebyshev-Gitter als Diskretisierungen im Ortsbereich. Diese Strategie ermöglicht die Verwendung schneller Algorithmen basierend auf einer eindimensionalen DCT (diskreten Kosinustransformation). Die Glattheit von Funktionen kann durch den Abfall ihrer Chebyshev-Koeffizienten charakterisiert werden. Unter diesem Gesichtspunkt werden Abschätzungen für Abtastfehler gezeigt sowie numerische Tests für bis zu 25 Raumdimensionen. Ein weiterer wichtiger Beitrag ist die Entwicklung einer Methode zur Berechnung einer hochdimensionalen dünnbesetzten FFT basierend auf Abtastwerten an Rang-1-Gittern, wobei diese Methode die Bestimmung unbekannter Frequenzen ermöglicht, welche zu den näherungsweise größten Fourier- oder Chebyshev-Koeffizienten einer Funktion gehören.

We consider multivariate trigonometric polynomials with frequencies supported on a fixed but arbitrary frequency index set I, which is a finite set of integer vectors of length d. Naturally, one is interested in spatial discretizations in the d-dimensional torus such that - the sampling values of the trigonometric polynomial at the nodes of this spatial discretization uniquely determines the trigonometric polynomial, - the corresponding discrete Fourier transform is fast realizable, and - the corresponding fast Fourier transform is stable. An algorithm that computes the discrete Fourier transform and that needs a computational complexity that is bounded from above by terms that are linear in the maximum of the number of input and output data up to some logarithmic factors is called fast Fourier transform. We call the fast Fourier transform stable if the Fourier matrix of the discrete Fourier transform has a condition number near one and the fast algorithm does not corrupt this theoretical stability. We suggest to use rank-1 lattices and a generalization as spatial discretizations in order to sample multivariate trigonometric polynomials and we develop construction methods in order to determine reconstructing sampling sets, i.e., sets of sampling nodes that allow for the unique, fast, and stable reconstruction of trigonometric polynomials. The methods for determining reconstructing rank-1 lattices are component{by{component constructions, similar to the seminal methods that are developed in the field of numerical integration. During this thesis we identify a component{by{component construction of reconstructing rank-1 lattices that allows for an estimate of the number of sampling nodes M |I|\le M\le \max\left(\frac{2}{3}|I|^2,\max\{3\|\mathbf{k}\|_\infty\colon\mathbf{k}\in I\}\right) that is sufficient in order to uniquely reconstruct each multivariate trigonometric polynomial with frequencies supported on the frequency index set I. We observe that the bounds on the number M only depends on the number of frequency indices contained in I and the expansion of I, but not on the spatial dimension d. Hence, rank-1 lattices are suitable spatial discretizations in arbitrarily high dimensional problems. Furthermore, we consider a generalization of the concept of rank-1 lattices, which we call generated sets. We use a quite different approach in order to determine suitable reconstructing generated sets. The corresponding construction method is based on a continuous optimization method. Besides the theoretical considerations, we focus on the practicability of the presented algorithms and illustrate the theoretical findings by means of several examples. In addition, we investigate the approximation properties of the considered sampling schemes. We apply the results to the most important structures of frequency indices in higher dimensions, so-called hyperbolic crosses and demonstrate the approximation properties by the means of several examples that include the solution of Poisson's equation as one representative of partial differential equations.

In this work, the fast evaluation and reconstruction of multivariate trigonometric polynomials with frequencies supported on arbitrary index sets of finite cardinality is considered, where rank-1 lattices are used as spatial discretizations. The approximation of multivariate smooth periodic functions by trigonometric polynomials is studied, based on a one-dimensional FFT applied to function samples. The smoothness of the functions is characterized via the decay of their Fourier coefficients, and various estimates for sampling errors are shown, complemented by numerical tests for up to 25 dimensions. In addition, the special case of perturbed rank-1 lattice nodes is considered, and a fast Taylor expansion based approximation method is developed. One main contribution is the transfer of the methods to the non-periodic case. Multivariate algebraic polynomials in Chebyshev form are used as ansatz functions and rank-1 Chebyshev lattices as spatial discretizations. This strategy allows for using fast algorithms based on a one-dimensional DCT. The smoothness of a function can be characterized via the decay of its Chebyshev coefficients. From this point of view, estimates for sampling errors are shown as well as numerical tests for up to 25 dimensions. A further main contribution is the development of a high-dimensional sparse FFT method based on rank-1 lattice sampling, which allows for determining unknown frequency locations belonging to the approximately largest Fourier or Chebyshev coefficients of a function. / In dieser Arbeit wird die schnelle Auswertung und Rekonstruktion multivariater trigonometrischer Polynome mit Frequenzen aus beliebigen Indexmengen endlicher Kardinalität betrachtet, wobei Rang-1-Gitter (rank-1 lattices) als Diskretisierung im Ortsbereich verwendet werden. Die Approximation multivariater glatter periodischer Funktionen durch trigonometrische Polynome wird untersucht, wobei Approximanten mittels einer eindimensionalen FFT (schnellen Fourier-Transformation) angewandt auf Funktionswerte ermittelt werden. Die Glattheit von Funktionen wird durch den Abfall ihrer Fourier-Koeffizienten charakterisiert und mehrere Abschätzungen für den Abtastfehler werden gezeigt, ergänzt durch numerische Tests für bis zu 25 Raumdimensionen. Zusätzlich wird der Spezialfall gestörter Rang-1-Gitter-Knoten betrachtet, und es wird eine schnelle Approximationsmethode basierend auf Taylorentwicklung vorgestellt. Ein wichtiger Beitrag dieser Arbeit ist die Übertragung der Methoden vom periodischen auf den nicht-periodischen Fall. Multivariate algebraische Polynome in Chebyshev-Form werden als Ansatzfunktionen verwendet und sogenannte Rang-1-Chebyshev-Gitter als Diskretisierungen im Ortsbereich. Diese Strategie ermöglicht die Verwendung schneller Algorithmen basierend auf einer eindimensionalen DCT (diskreten Kosinustransformation). Die Glattheit von Funktionen kann durch den Abfall ihrer Chebyshev-Koeffizienten charakterisiert werden. Unter diesem Gesichtspunkt werden Abschätzungen für Abtastfehler gezeigt sowie numerische Tests für bis zu 25 Raumdimensionen. Ein weiterer wichtiger Beitrag ist die Entwicklung einer Methode zur Berechnung einer hochdimensionalen dünnbesetzten FFT basierend auf Abtastwerten an Rang-1-Gittern, wobei diese Methode die Bestimmung unbekannter Frequenzen ermöglicht, welche zu den näherungsweise größten Fourier- oder Chebyshev-Koeffizienten einer Funktion gehören.