A Nonuniform Fast Fourier Transform Based on Low Rank Approximation

Ruiz-Antolín, Diego; Townsend, Alex

doi:10.1137/17m1134822

Cited by 43 publications

(57 citation statements)

References 27 publications

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“…In the univariate (1D) case, there are several variants of the second approach: Dutt-Rokhlin [13] proposed spectral Lagrange interpolation (using the cotangent kernel applied via the fast multipole method), combined with a single FFT. Recently, Ruiz-Antolín and Townsend [53] proposed a stable Chebyshev approximation in intervals centered about each uniform point, which needs an independent N -point FFT for each of the O(log 1/ε) coefficients, but is embarrassingly parallelizable. This improves upon earlier work [2] using Taylor approximation that was numerically unstable without upsampling [33, Ex.…”

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confidence: 99%

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

Barnett¹,

Magland²,

Klinteberg³

2019

SIAM J. Sci. Comput.

156

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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confidence: 99%

“…This improves upon earlier work [2] using Taylor approximation that was numerically unstable without upsampling [33, Ex. 3.10] [53].…”

mentioning

confidence: 99%

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

Barnett¹,

Magland²,

Klinteberg³

2019

SIAM J. Sci. Comput.

156

163

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“…Generally speaking, r is related to n and it was conjectured in [40] that r = O( log n log log n ) via excessive numerical experiments. In practice, inspired by the NUFFT in [33], we can replace the Hadamard product of a low-rank matrix and a NUFFT matrix in (42) with a Hadamard product of a low-rank matrix and a DFT matrix, the matvec of which can be carried out more efficiently with a few numbers of FFTs.…”

Section: One-dimensional Transform and Its Inversementioning

confidence: 99%

“…In [40], it was observed that the second summation in (8) could be interpreted as the application of a structured matrix that is the real part of a Hadamard product of a NUFFT matrix and a numerically low-rank matrix. This leads to a method for its computation which takes quasi-linear time [33,44,45]. Expansions in terms of the functions of the second kind {Q (a,b) j } ∞ j=0 can be handled similarly.…”

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Fast algorithms for the multi-dimensional Jacobi polynomial transform

Bremer

Pang

Yang

2021

Applied and Computational Harmonic Analysis

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We use the well-known observation that the solutions of Jacobi's differential equation can be represented via the non-oscillatory phase and amplitude functions to develop a fast algorithm for computing multi-dimensional Jacobi polynomial transforms. More explicitly, it follows from this observation that the matrix corresponding to the discrete Jacobi transform is the Hadamard product of a numerically low-rank matrix and a multi-dimensional discrete Fourier transform (DFT) matrix. The application of the Hadamard product can be carried out via r d fast Fourier transforms (FFTs), where r = O( log n log log n ) and d is the dimension, resulting in a nearly optimal algorithm to compute the multidimensional Jacobi polynomial transform.

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“…The numerical cost of this step then retains the O (N log (N )) scaling of FFT techniques, where N is here N l , N k or n t . Efficient algorithms also exist in case of non-uniform sampling, e.g., Ruiz-Antolín & Townsend (2018), and retain the O (N log (N )) scaling. The remaining step involves a POD in the non-homogeneous dimension for each atom of triad (l, k, f ).…”

Section: Requirements and Cost Of The Decompositionmentioning

confidence: 99%

Spatio-temporal proper orthogonal decomposition of turbulent channel flow

et al. 2019

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An extension of Proper Orthogonal Decomposition is applied to the wall layer of a turbulent channel flow (Re τ = 590), so that empirical eigenfunctions are defined in both space and time. Due to the statistical symmetries of the flow, the eigenfunctions are associated with individual wavenumbers and frequencies. Self-similarity of the dominant eigenfunctions, consistent with wall-attached structures transferring energy into the core region, is established. The most energetic modes are characterized by a fundamental time scale in the range 200-300 viscous wall units. The full spatio-temporal decomposition provides a natural measure of the convection velocity of structures, with a characteristic value of 12u τ in the wall layer. Finally, we show that the energy budget can be split into specific contributions for each mode, which provides a closed-form expression for nonlinear effects. Key words:where a n is stochastic and χ n are orthogonal, square integrable, functions. In all that follows the superscript refers to the mode index. It is important to note that U (t) represents a stochastic variable and not a sample. Realizations of U (t) will be noted u(t). The functions χ n (t) are the eigenfunctions of the covariance function K U (t, t ) = E[U t U t ], where the operator E refers to expectation with respect to the measure of U . In the Karhunen-Loève derivation, the variable t corresponds to time, but it could indicate any other variable -such as space.Lumley (1967) (see also Berkooz et al. (1993)) adapted the decomposition to Fluid Mechanics: the samples were constituted by flow realizations, and the ergodicity assumption was used to replace the covariance function corresponding to an ensemble average with † Email address for correspondence: Berengere.Podvin@limsi.fr arXiv:1805.01494v3 [physics.flu-dyn]

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A Nonuniform Fast Fourier Transform Based on Low Rank Approximation

Cited by 43 publications

References 27 publications

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

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

Fast algorithms for the multi-dimensional Jacobi polynomial transform

Spatio-temporal proper orthogonal decomposition of turbulent channel flow

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