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NONLINEAR APPROXIMATION BY SUMS OF EXPONENTIALS AND TRANSLATES
"... In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Let h be a linear combination of exponentials with real frequencies. Determine all frequ ..."
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In this paper, we discuss the numerical solution of two nonlinear approximation problems. Many applications in electrical engineering, signal processing, and mathematical physics lead to the following problem: Let h be a linear combination of exponentials with real frequencies. Determine all frequencies, all coefficients, and the number of summands, if finitely many perturbed, uniformly sampled data of h are given. We solve this problem by an approximate Prony method (APM) and prove the stability of the solution in the square and uniform norm. Further, an APM for nonuniformly sampled data is proposed too. The second approximation problem is related to the first one and reads as follows: Let ϕ be a given 1–periodic window function as defined in Section 4. Further let f be a linear combination of translates of ϕ. Determine all shift parameters, all coefficients, and the number of translates, if finitely many perturbed, uniformly sampled data of f are given. Using Fourier technique, this problem is transferred into the above parameter estimation problem for an exponential sum which is solved by APM. The stability of the solution is discussed in the square and uniform norm too. Numerical experiments show the performance of our approximation methods.
Nonequispaced hyperbolic cross fast Fourier transform
"... A straightforward discretisation 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 decre ..."
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A straightforward discretisation 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 fast Fourier transform 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. Key words and phrases: trigonometric approximation, hyperbolic cross, sparse grid, fast Fourier transform, nonequispaced FFT
Particle Simulation Based on Nonequispaced Fast Fourier Transforms
"... The fast calculation of longrange interactions is a demanding problem in particle simulation. The main focus of our approach is the decomposition of the problem in building blocks and present efficient numerical realizations for these blocks. For that reason we recapitulate the fast Fourier transfo ..."
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The fast calculation of longrange interactions is a demanding problem in particle simulation. The main focus of our approach is the decomposition of the problem in building blocks and present efficient numerical realizations for these blocks. For that reason we recapitulate the fast Fourier transform at nonequispaced nodes and the fast summation method. We describe the application of these algorithms to the evaluation of longrange potentials and compare our methods with the existing fast multipole method.
OpenMP parallelization in the NFFT software library
, 2012
"... software library and present the used parallelization approaches. Besides the NFFT kernel, the NFFT on the twosphere and the fast summation based on NFFT are also parallelized. Thereby, the parallelization is based on OpenMP and the multithreaded FFTW library. Furthermore, benchmarks for various c ..."
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software library and present the used parallelization approaches. Besides the NFFT kernel, the NFFT on the twosphere and the fast summation based on NFFT are also parallelized. Thereby, the parallelization is based on OpenMP and the multithreaded 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. 1 Overview The NFFT3 library [3] and its MATLAB interface were parallelized using OpenMP [7]. Both the nonparallel version and the multithread OpenMP version of the NFFT3 library provide an identical Application Programming Interface (API). This is realized by using distinct library files for both versions. The nonparallel version of the NFFT3 library can be found in libnfft3.so and libnfft3.a, the multithread OpenMP version in libnfft3_threads.so and libnfft3_threads.a. For the MATLAB interface, the user has to specifiy at compile time whether the nonparallel or multithread OpenMP version should be built. The following kernels of the NFFT3 library were parallelized using OpenMP: • kernel/nfft: – NDFT (nonequidistant discrete Fourier transform)
PARALLEL THREEDIMENSIONAL NONEQUISPACED FAST FOURIER TRANSFORMS AND THEIR APPLICATION TO PARTICLE SIMULATION
"... Abstract. 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 fo ..."
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Abstract. 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. Key words and phrases: parallel nonequispaced fast Fourier transform, parallel fast summation, parallel particle mesh methods, NFFT
OpenMP parallelization in the NFFT3 library version 3.2
"... The NFFT3 library [3] and its MATLAB interface were parallelized using OpenMP [7]. Both the nonparallel version and the multithread OpenMP version of the NFFT3 library provide an identical Application Programming Interface (API). This is realized by using distinct library files for both versions. ..."
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The NFFT3 library [3] and its MATLAB interface were parallelized using OpenMP [7]. Both the nonparallel version and the multithread OpenMP version of the NFFT3 library provide an identical Application Programming Interface (API). This is realized by using distinct library files for both versions. The nonparallel version of the NFFT3 library can be found in libnfft3.so and libnfft3.a, the multithread OpenMP version in libnfft3_threads.so and libnfft3_threads.a. For the MATLAB interface, the user has to specifiy at compile time whether the nonparallel or multithread OpenMP version should be built. The following kernels of the NFFT3 library were parallelized using OpenMP: • kernel/nfft: – NDFT (nonequidistant discrete Fourier transform) – NDFT ⊢ ⊣ (adjoint nonequidistant discrete Fourier transform) – NFFT (nonequidistant fast Fourier transform) – NFFT ⊢ ⊣ (adjoint nonequidistant fast Fourier transform) • kernel/nfsft: – NDSFT (nonequidistant discrete spherical Fourier transform, NDFT on the sphere
Reconstructing
"... hyperbolic cross trigonometric polynomials by sampling along generated sets ..."
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hyperbolic cross trigonometric polynomials by sampling along generated sets
Quadrature Nodes Meet Stippling Dots
"... Abstract. The stippling technique places black dots such that their density gives the impression of tone. This is the first paper that relates the distribution of stippling dots to the classical mathematical question of finding ’optimal ’ nodes for quadrature rules. More precisely, we consider quadr ..."
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Abstract. The stippling technique places black dots such that their density gives the impression of tone. This is the first paper that relates the distribution of stippling dots to the classical mathematical question of finding ’optimal ’ nodes for quadrature rules. More precisely, we consider quadrature error functionals on reproducing kernel Hilbert spaces (RKHSs) with respect to the quadrature nodes and suggest to use optimal distributions of these nodes as stippling dot positions. Interestingly, in special cases, our quadrature errors coincide with discrepancy functionals and with recently proposed attractionrepulsion functionals. Our framework enables us to consider point distributions not only in R 2 but also on the torus T 2 and the sphere S 2. For a large number of dots the computation of their distribution is a serious challenge and requires fast algorithms. To this end, we work in RKHSs of bandlimited functions, where the quadrature error can be replaced by a least squares functional. We apply a nonlinear conjugate gradient (CG) method on manifolds to compute a minimizer of this functional and show that each step can be efficiently realized by nonequispaced fast Fourier transforms. We present numerical stippling results on S 2. 1
Fast Ewald summation under 2d and 1dperiodic boundary conditions based on NFFTs
"... Abstract—Ewald summation has established as basic element of fast algorithms evaluating the Coulomb interaction energy of charged particle systems in three dimensions subject to periodic boundary conditions. In this context particle mesh routines, as the P3M method, and the P2NFFT, which is based on ..."
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Abstract—Ewald summation has established as basic element of fast algorithms evaluating the Coulomb interaction energy of charged particle systems in three dimensions subject to periodic boundary conditions. In this context particle mesh routines, as the P3M method, and the P2NFFT, which is based on nonequispaced fast Fourier transforms (NFFT), should be mentioned. These methods treat the problem efficiently in case that periodic boundary conditions in all three dimensions are assumed. In this paper we present a new approach for the efficient calculation of the Coulomb interaction energy subject to mixed boundary conditions based on NFFTs. I.