Efficient Dealiased Convolutions without Padding

• Published in: SIAM journal on scientific computing Vol. 33; no. 1; pp. 386 - 406
• Main Authors: Bowman, John C., Roberts, Malcolm
• Format: Journal Article
• Language: English
• Published: Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2011
• Subjects: Algorithms; Arrays; Computation; Computer programs; Convolution; Decoupling; Exact sciences and technology; Fourier analysis; Fourier transforms; Mathematical analysis; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Sciences and techniques of general use; Studies; zero padding; 65R99; 65T50; implicit padding; Fast Fourier transformation; Fourier analysis; Algorithm; in-place transform; Implementation; fast Fourier transform; dealiasing; 2/3 padding rule; Computation time; Numerical analysis; Convolution; ternary convolution; One-dimensional calculations; Scientific computation; Pseudospectral method; Software; bit reversal
• ISSN: 1064-8275; 1095-7197
• DOI: 10.1137/100787933

Algorithms are developed for calculating dealiased linear convolution sums without the expense of conventional zero-padding or phase-shift techniques. For one-dimensional in-place convolutions, the memory requirements are identical with the zero-padding technique, with the important distinction that the additional work memory need not be contiguous with the input data. This decoupling of data and work arrays dramatically reduces the memory and computation time required to evaluate higher-dimensional in-place convolutions. The technique also allows one to dealias the higher-order convolutions that arise from Fourier transforming cubic and higher powers. Implicitly dealiased convolutions can be built on top of state-of-the-art fast Fourier transform libraries: vectorized multidimensional implementations for the complex and centered Hermitian (pseudospectral) cases have been implemented in the open-source software FFTW++. [PUBLICATION ABSTRACT]