Fast Ewald summation based on NFFT with mixed periodicity

Nestler, Franziska; Pippig, Michael; Potts, Daniel

doi:10.1016/j.jcp.2014.12.052

Cited by 31 publications

(40 citation statements)

References 43 publications

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“…The integral can however be evaluated to obtain the closed form of the formula, see [27,Appendix D] and references therein. Recently, Nestler et al [22] developed a fast algorithm which employs non-equispaced FFTs (NFFT). To the best of our knowledge, this is the only method with O(N log(N )) complexity for 1d-periodic problems.…”

Section: Introductionmentioning

confidence: 99%

The Spectral Ewald method for singly periodic domains

Shamshirgar

Tornberg

2017

Journal of Computational Physics

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We present a fast and spectrally accurate method for efficient computation of the three dimensional Coulomb potential with periodicity in one direction. The algorithm is FFT-based and uses the so-called Ewald decomposition, which is naturally most efficient for the triply periodic case. In this paper, we show how to extend the triply periodic Spectral Ewald method to the singly periodic case, such that the cost of computing the singly periodic potential is only marginally larger than the cost of computing the potential for the corresponding triply periodic system. In the Fourier space contribution of the Ewald decomposition, a Fourier series is obtained in the periodic direction with a Fourier integral over the non periodic directions for each discrete wave number. We show that upsampling to resolve the integral is only needed for modes with small wave numbers. For the zero wave number, this Fourier integral has a singularity. For this mode, we effectively need to solve a free-space Poisson equation in two dimensions. A very recent idea by Vico et al. makes it possible to use FFTs to solve this problem, allowing us to unify the treatment of all modes. An adaptive 3D FFT can be established to apply different upsampling rates locally. The computational cost for other parts of the algorithm is essentially unchanged as compared to the triply periodic case, in total yielding only a small increase in both computational cost and memory usage for this singly periodic case.Comment: 36 pages, 19 figure

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Section: Introductionmentioning

confidence: 99%

The Spectral Ewald method for singly periodic domains

Shamshirgar

Tornberg

2017

Journal of Computational Physics

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“…Real-time Fourier-domain optical coherence tomography (OCT) relies on rapid one-dimensional NUFFTs [63]. Periodic electrostatic and Stokes problems are commonly solved by fast "particle-mesh Ewald" summation, whose spectral part is equivalent to a pair of NUFFTs [37,42]. Spectrally-accurate function interpolation may be efficiently performed with the NUFFT [30,Sec.…”

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confidence: 99%

A Parallel Nonuniform Fast Fourier Transform Library Based on an “Exponential of Semicircle" Kernel

Barnett¹,

Magland²,

Klinteberg³

2019

SIAM J. Sci. Comput.

154

163

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The nonuniform fast Fourier transform (NUFFT) generalizes the FFT to off-grid data. Its many applications include image reconstruction, data analysis, and the numerical solution of differential equations. We present FINUFFT, an efficient parallel library for type 1 (nonuniform to uniform), type 2 (uniform to nonuniform), or type 3 (nonuniform to nonuniform) transforms, in dimensions 1, 2, or 3. It uses minimal RAM, requires no precomputation or plan steps, and has a simple interface to several languages. We perform the expensive spreading/interpolation between nonuniform points and the fine grid via a simple new kernel-the "exponential of semicircle" e β √ 1−x 2 in x ∈ [−1, 1]-in a cache-aware load-balanced multithreaded implementation. The deconvolution step requires the Fourier transform of the kernel, for which we propose efficient numerical quadrature. For types 1 and 2, rigorous error bounds asymptotic in the kernel width approach the fastest known exponential rate, namely that of the Kaiser-Bessel kernel. We benchmark against several popular CPU-based libraries, showing favorable speed and memory footprint, especially in three dimensions when high accuracy and/or clustered point distributions are desired.

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“…For the B-spline window the upper bounds s j (k), as introduced by Equation (42), are given in Equation (12). Example 4.6.…”

Section: B-spline Windowmentioning

confidence: 99%

“…It can be seen that the B-Spline performs best in terms of accuracy only for very small values of α. In this case the coefficientsψ(k) tend to zero very rapidly, i.e., the error is strongly dominated by the terms k ≈ 0 and thus the B-spline profits from the fact that s j (k) → 0 as k → 0, see Equation (12).…”

Section: Comparisonmentioning

confidence: 99%

“…The results show that the P 3 M and P 2 NFFT solvers rank among the best methods in this field. We further remark that the P 2 NFFT method has already been generalized to mixed periodic as well as open boundary conditions, see Nestler et al [12] and Nestler et al [13]. The fast multipole method (FMM) [14][15][16][17], which scales like O(N), can also handle all types of periodic as well as open boundary conditions very efficiently, see Kudin and Scuseria [18].…”

Section: Introductionmentioning

confidence: 99%

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Parameter Tuning for the NFFT Based Fast Ewald Summation

Nestler

2016

Front. Phys.

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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 (P 2 NFFT) 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 (P 3 M) algorithm. The publicly available P 2 NFFT 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.

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Fast Ewald summation based on NFFT with mixed periodicity

Cited by 31 publications

References 43 publications

The Spectral Ewald method for singly periodic domains

The Spectral Ewald method for singly periodic domains

A Parallel Nonuniform Fast Fourier Transform Library Based on an “Exponential of Semicircle" Kernel

Parameter Tuning for the NFFT Based Fast Ewald Summation

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