Hybrid Dealiasing of Complex Convolutions
Abstract.
Efficient algorithms based on the fast Fourier transform are developed for computing linear convolutions. A hybrid approach is described that combines the conventional practice of explicit dealiasing (explicitly padding the input data with zeros) and implicit dealiasing (mathematically accounting for these zero values). The new approach generalizes implicit dealiasing to arbitrary padding ratios and includes explicit dealiasing as a special case. Unlike existing implementations of implicit dealiasing, hybrid dealiasing tailors its subtransform sizes to the convolution geometry. Multidimensional convolutions are implemented with hybrid dealiasing by decomposing them into lower-dimensional convolutions. Convolutions of complex-valued and Hermitian inputs of equal length are illustrated with pseudocode and implemented in the open-source FFTW++ library. Hybrid dealiasing is shown to outperform explicit dealiasing in one, two, and three dimensions.
Reproducibility of computational results.
This paper has been awarded the “SIAM Reproducibility Badge: Code and Data Available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available from https://github.com/dealias/fftwpp and in the supplementary materials.
Keywords
MSC codes
Get full access to this article
View all available purchase options and get full access to this article.
Supplementary Materials
PLEASE NOTE: These supplementary files have not been peer-reviewed.
Index of Supplementary Materials
Title of paper: Hybrid Dealiasing of Complex Convolutions
Authors: Noel Murasko and John C. Bowman
File: fftwpp-master.zip
Type: X-ZIP-COMPRESSED
Contents: zipped snapshot of the git repository required for reproducibility badge
- Download
- 294.91 KB
References
1.
C. Basdevant, Technical improvements for direct numerical simulation of homogeneous three-dimensional turbulence, J. Comput. Phys., 50 (1983), pp. 209–214.
2.
J. C. Bowman, Casimir cascades in two-dimensional turbulence, J. Fluid Mech., 729 (2013), pp. 364–376.
3.
J. C. Bowman and M. Roberts, Efficient dealiased convolutions without padding, SIAM J. Sci. Comput., 33 (2011), pp. 386–406, https://doi.org/10.1137/100787933.
4.
J. C. Bowman, M. Roberts, and N. Murasko, FFTW++: A Fast Fourier Transform \(\textrm{C}^{++}\) Header Class for the FFTW3 Library, http://fftwpp.sourceforge.net, 2023.
5.
K. Chitsaz, M. Hajabdollahi, N. Karimi, S. Samavi, and S. Shirani, Acceleration of Convolutional Neural Network Using FFT-Based Split Convolutions, preprint, https://arxiv.org/abs/2003.12621, 2020.
6.
J. W. Cooley and J. W. Tukey, An algorithm for the machine calculation of complex Fourier series, Math. Comp., 19 (1965), pp. 297–301.
7.
M. Frigo and S. G. Johnson, FFTW, http://www.fftw.org.
8.
M. Frigo and S. G. Johnson, The design and implementation of FFTW3, Proc. IEEE, 93 (2005), pp. 216–231.
9.
D. Gottlieb and S. A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conf. Ser. in Appl. Math. 26, SIAM, Philadelphia, 1977, https://doi.org/10.1137/1.9781611970425.
10.
T. Highlander and A. Rodriguez, Very efficient training of convolutional neural networks using fast Fourier transform and overlap-and-add, in Proc. British Machine Vision Conference, X. H. Xie, M. W. Jones, and G. K. L. Tam, eds., 2015, pp. 160.1–160.9, https://doi.org/10.48550/arXiv.1601.06815.
11.
J. Lin and Y. Yao, A fast algorithm for convolutional neural networks using tile-based fast Fourier transforms, Neural Process. Lett., 50 (2019), pp. 1951–1967, https://doi.org/10.1007/s11063-019-09981-z.
12.
M. Mathieu, M. Henaff, and Y. LeCun, Fast training of convolutional networks through FFTs, in 2nd International Conference on Learning Representations, Y. Bengio and Y. LeCun, eds., Conference Track Proceedings, 2014, https://doi.org/10.48550/arXiv.1312.5851.
13.
S. A. Orszag, On the elimination of aliasing in finite difference schemes by filtering high-wavenumber components, J. Atmos. Sci., 28 (1971), 1074.
14.
G. S. Patterson, Jr., and S. A. Orszag, Spectral calculations of isotropic turbulence: Efficient removal of aliasing interactions, Phys. Fluids, 14 (1971), pp. 2538–2541.
15.
M. Roberts and J. C. Bowman, Multithreaded implicitly dealiased convolutions, J. Comput. Phys., 356 (2018), pp. 98–114.
16.
H. V. Sorensen, D. L. Jones, M. T. Heideman, and C. S. Burrus, Real-valued fast Fourier transform algorithms, IEEE Trans. Acoust. Speech Signal Process., 35 (1987), pp. 849–863.
Information & Authors
Information
Published In
SIAM Journal on Scientific Computing
Pages: B159 - B178
DOI: 10.1137/23M1552073
ISSN (online): 1095-7197
Copyright
© 2024 Society for Industrial and Applied Mathematics.
History
Submitted: 8 February 2023
Accepted: 12 December 2023
Published online: 2 May 2024
Keywords
MSC codes
Reproducibility of computational results.
This paper has been awarded the “SIAM Reproducibility Badge: Code and Data Available” as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available from https://github.com/dealias/fftwpp and in the supplementary materials.
Authors
Funding Information
Natural Sciences and Engineering Research Council of Canada (NSERC): RES0043585, RES0046040
Funding: This work was supported by Natural Sciences and Engineering Research Council of Canada grants RES0043585 and RES0046040.
Metrics & Citations
Metrics
Citations
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.
Cited By
View Options
- Access via your Institution
- Questions about how to access this content? Contact SIAM at [email protected].