Multidimensional Fourier Research Articles

GPU Parallelization of a Hybrid Pseudospectral Geophysical Turbulence Framework Using CUDA

An existing hybrid MPI-OpenMP scheme is augmented with a CUDA-based fine grain parallelization approach for multidimensional distributed Fourier transforms, in a well-characterized pseudospectral fluid turbulence code. Basics of the hybrid scheme are reviewed, and heuristics provided to show a potential benefit of the CUDA implementation. The method draws heavily on the CUDA runtime library to handle memory management and on the cuFFT library for computing local FFTs. The manner in which the interfaces to these libraries are constructed, and ISO bindings utilized to facilitate platform portability, are discussed. CUDA streams are implemented to overlap data transfer with cuFFT computation. Testing with a baseline solver demonstrated significant aggregate speed-up over the hybrid MPI-OpenMP solver by offloading to GPUs on an NVLink-based test system. While the batch streamed approach provided little benefit with NVLink, we saw a performance gain of 30 % when tuned for the optimal number of streams on a PCIe-based system. It was found that strong GPU scaling is nearly ideal, in all cases. Profiling of the CUDA kernels shows that the transform computation achieves 15% of the attainable peak FlOp-rate based on a roofline model for the system. In addition to speed-up measurements for the fiducial solver, we also considered several other solvers with different numbers of transform operations and found that aggregate speed-ups are nearly constant for all solvers.

Fast algorithms for the multi-dimensional Jacobi polynomial transform

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.

Image reconstruction algorithm based on frequency-wavenumber decoupling for three-dimensional MIMO-SAR imaging.

In this paper, a frequency-wavenumber decoupling algorithm with high-efficiency and high-precise for three-dimensional (3-D) multiple-input-multiple-output synthetic aperture radar (MIMO-SAR) imaging is proposed. Based on one-dimensional (1-D) MIMO array combined with synthetic aperture scan along another dimension, MIMO-SAR imaging scheme allows the number of array elements to be greatly reduced compared with the two-dimensional (2-D) MIMO arrays. By multi-dimensional Fourier transforming and Method of Stationary Phase (MSP), analytical expression of the object function in the frequency-wavenumber domain was derived. By further expanding the range Fourier transform factor to its Taylor series form, the range compression can be realized by a simple fast Fourier transform (FFT) without multi-dimensional interpolation. After that, a decoupling factor was multiplied to compensate for the cross-range and range coupling in frequency domain. Finally, 2-D IFFT is carried out after rearrangement in the MIMO spatial frequency to get a fully focused 3-D image. Simulation and experimental results demonstrated that the algorithm can obtain the same high-precision images as back projection (BP) algorithm, and has the same high efficiency as range migration algorithm (RMA) while avoiding cumbersome multi-dimensional interpolation. A bistatic prototype imaging system in 0.1 THz band was designed for the proof-of-principle experiments. The 3-D reconstruction results of different targets were presented to verify the theoretical results and effectiveness of the proposed algorithm for MIMO-SAR imaging.

A Flexible Framework for Multidimensional DFTs

Multidimensional discrete Fourier transforms (DFTs) are typically decomposed into multiple one-dimensional (1D) transforms. Hence, parallel implementations of any multidimentional DFT focus on para...

Multidimensional fractional Fourier transform and generalized fractional convolution

ABSTRACTIn this paper, we prove inversion theorems and Parseval identity for the multidimensional fractional Fourier transform. Analogous to the existing fractional convolutions on functions of single variable, we also introduce a generalized fractional convolution on functions of several variables and we derive their properties including convolution theorem and product theorem for the multidimensional fractional Fourier transform.

Multiscale high-dimensional sparse Fourier algorithms for noisy data

We develop an efficient and robust high-dimensional sparse Fourier for noisy samples. Earlier in the paper ``Multi-dimensional sublinear sparse Fourier algorithm (2016), an efficient sparse Fourier with $\Theta(ds \log s)$ average-case runtime and $\Theta(ds)$ sampling complexity under certain assumptions was developed for signals that are $s$-sparse and bandlimited in the $d$-dimensional Fourier domain, i.e. there are at most $s$ energetic frequencies and they are in $ \left[-N/2, N/2\right)^d\cap \mathbb{Z}^d$. However, in practice the measurements of signals often contain noise, and in some cases may only be nearly sparse in the sense that they are well approximated by the best $s$ Fourier modes. In this paper, we propose a multiscale sparse Fourier for noisy samples that proves to be both robust against noise and efficient.

Some Studies on Multidimensional Fourier Theory for Hilbert Transform, Analytic Signal and AM–FM Representation

In this paper, we propose the Fourier frequency vector (FFV), inherently, associated with multidimensional Fourier transform. With the help of FFV, we are able to provide physical meaning of so called negative frequencies in multidimensional Fourier transform (MDFT), which in turn provide multidimensional spatial and space-time series analysis. The complex exponential representation of sinusoidal function always yields two frequencies, negative frequency corresponding to positive frequency and vice versa, in the multidimensional Fourier spectrum. Thus, using the MDFT, we propose multidimensional Hilbert transform (MDHT) and associated multidimensional analytic signal (MDAS) with following properties: (a) the extra and redundant positive, negative, or both frequencies, introduced due to complex exponential representation of multidimensional Fourier spectrum, are suppressed, (b) real part of MDAS is original signal, (c) real and imaginary part of MDAS are orthogonal, and (d) the magnitude envelope of a original signal is obtained as the magnitude of its associated MDAS, which is the instantaneous amplitude of the MDAS. The proposed MDHT and associated DMAS are generalization of the 1D HT and AS, respectively. We also provide the decomposition of an image into the AM-FM image model by the Fourier method and obtain explicit expression for the analytic image computation by 2DDFT.

Multidimensional Sparse Fourier Transform Based on the Fourier Projection-Slice Theorem

We propose Multidimensional Random Slice-based Sparse Fourier Transform (MARS-SFT), a sparse Fourier transform for multidimensional, frequency-domain sparse signals, inspired by the idea of the Fourier projection-slice theorem. MARS-SFT identifies frequencies by operating on one-dimensional slices of the discrete-time domain data, taken along specially designed lines; these lines are parametrized by slopes that are randomly generated from a set at runtime. The discrete Fourier transforms (DFTs) of data slices represent DFT projections onto the lines along which the slices were taken. On designing the line lengths and slopes so that they allow for orthogonal and uniform projections of the sparse frequencies, frequency collisions are avoided with high probability, and the multidimensional frequencies can be recovered from their projections with low sample and computational complexity. We show analytically that the large number of degrees of freedom of frequency projections allows for the recovery of less sparse signals. Although the theoretical results are obtained for uniformly distributed frequencies, empirical evidence suggests that MARS-SFT is also effective in recovering clustered frequencies. We also propose an extension of MARS-SFT to address noisy signals that contain off-grid frequencies and demonstrate its performance in digital beamforming automotive radar signal processing. In that context, the robust MARS-SFT is used to identify range, velocity, and angular parameters of targets with low sample and computational complexity.

Inverse modeling approach for parametric frequency domain analysis of an electrohydraulic system

Abstract This paper presents parametric frequency domain analys.isf an electrohydraulic system, in particular, with respect to the hydraulic supply pressure. The supply pressure is a design parameter of significance such that it affects control bandwidth, load capability and system durability. However, intensive analysis on this parameter has not been reported yet due to the nonlinear dynamic features of an electrohydraulic system. In this paper, the inverse generalized frequency response functions are derived and used to conduct multi-dimensional Fourier analysis of the control input and to investigate its relation to the supply pressure. In addition, the hydraulic energy consumption is analyzed with respect to the supply pressure. The analysis reveals that the supply pressure is a critical design parameter that determines the degree of nonlinearity and trade-off between the control bandwidth and energy consumption.

Variable Hardy and Hardy-Lorentz spaces and applications in Fourier analysis

We summarize some results about the variable Hardy and Hardy-Lorentz spaces Hp()(Rd) and Hp();q(Rd) and about the -summability of multi-dimensional Fourier transforms. We prove that the maximal operator of the -means is bounded from Hp()(Rd) to Lp()(Rd) and from Hp();q(Rd) to Lp();q(Rd). This implies some norm and almost everywhere convergence results for the Riesz, Bochner-Riesz, Weierstrass, Picard and Bessel summations.

Modeling of the random texture surface based on self-similar structures

The construction realized of random texture surfaces based on self-similar ones with the addition of certain rules of the accidental element in the construction process. An example of a random fractal widespread in nature is obtained in the process of so-called diffusion-precise aggregation. I investigated spatial spectrum of the formed surfaces on the basis of the multidimensional Fourier transform. I used software package ParaView to visualize 3D textures.

Free-Energy Landscapes for Macromolecules that Form a Unique 3D Structure

Abstract—The entropic effects for the energy landscapes of macromolecules are considered. We use the ideas and methods of multidimensional geometry and topology, expansion into a multidimensional Fourier series for a potential-energy surface, and the Gaussian distribution function for the expansion coefficients to describe the interaction between conformational degrees of freedom to investigate the free energy surface in the space of dihedral (torsion) angles. We show that the free-energy surface for a macromolecule that forms a unique 3D structure in the framework of the proposed approach has the following features. The main properties of the free-energy surface can be represented in terms of a two-dimensional surface, which depends on two generalized variables. At low temperatures, the multidimensional space of torsion angles can be divided into region that belongs to the deepest central energy funnel and the region of satellite funnels. The funnels are separated by relatively high energy barriers, whose height decreases as the representative point moves up further away from the bottom of the funnel. The depths of satellite funnels decrease while the distances from the central funnel becomes larger. The funnels are surrounded by smooth entropic barriers. The height of entropic barrier is at a minimum for the central funnel, while the entropic barriers heights of satellite funnels increase as funnels move further away from the central funnel. As the temperature increases, the satellite funnels are destroyed, starting from the most distant central one. When the temperature is higher than the critical point (proportional to the specific gain in the potential energy during the formation of a unique 3D structure) the compact state becomes unstable and the global minimum of free energy disappears. As the number of interacting torsion angles (the length of the polymer chain) increases, the critical temperature becomes higher.

Energy Landscapes of Macromolecules with Unique 3D Structures

Abstract—A paradox of the convergence of the potential energy surface (PES) presented as a sum of pairwise interactions was formulated. Regularities of the formation of a multidimensional PES in the space of torsion (dihedral) angles were considered for the case of the macromolecules that form unique 3D structures. The presence of a single global minimum on the PES was shown to be impossible for nonchiral macromolecules. The chirality of the spatial structure of a macromolecule creates conditions for the formation of a single global minimum on the PES. The structure was studied for a model PES such that components of the multidimensional Fourier series exponentially damped with the increasing number of harmonics. It was proposed to describe the interaction between conformational degrees of freedom in the space of torsion angles by assigning distribution functions to linear combinations of harmonic numbers. A mathematical tool was developed for this purpose. The structure of the PES was studied for the cases of the Gaussian and Lorentzian distribution functions for the linear combinations of the Fourier series harmonic numbers. It was shown that the properties of such a PES can be described by introducing two generalized variables. A feature of the PES is the existence of a central funnel, which leads to a global energy minimum, and satellite funnels, which acts as traps during the folding process. Relatively rapid folding events (the achievement of the global energy minimum) may take place in the configuration space region that corresponds to the central funnel. This structure of the PES makes it possible to identify the configuration space areas that are important for the folding and to understand the basic difference between reversible (in solution) and irreversible (using an atomic force microscope) unfolding of unique 3D structures of biopolymers.

Local stress in periodic composites via the Riesz summability method

Abstract Different summation methods of multidimensional trigonometric Fourier series are adopted for approximating the local strain and stress in elastic periodic composites. In fact, approximating the exact solution of a problem with the original partial sums of Fourier series can produce unwanted effects, such as the Gibbs phenomenon at the jump discontinuities that may occur at the interface between the constituents of the composites. Nevertheless, the Fourier coefficients of the original partial sums contain enough information to provide better estimates via the summability methods. In the means provided by the summability methods, the Fourier coefficients are suitably weighted by reducing windows, involving a regularization of the numerical solution. First, a complete solution method is used in order to determine some Fourier coefficients of the local strain in periodic composites. Next, these Fourier coefficients are used to construct suitable summations and means exhibiting better convergence properties than those of the original partial sums of Fourier series. The proposed method, valid for a great variety of the geometry of the constituents embedded in the matrix, is here applied to unidirectional composites with cylindrical fibers. The behavior of the iterated Fejer partial sums, Fejer and Riesz means is investigated in order to improve the convergence, observing that the Riesz means have better convergence properties than those of the original partial sums of Fourier series.

Approximation in L2 by Partial Integrals of the Multidimensional Fourier Transform over the Eigenfunctions of the Sturm–Liouville Operator

For approximations in the space L2(ℝ+ d ) by partial integrals of the multidimensional Fourier transform over the eigenfunctions of the Sturm–Liouville operator, we prove the Jackson inequality with sharp constant and optimal argument in the modulus of continuity. The multidimensional weight that defines the Sturm–Liouville operator is the product of onedimensional weights. The one-dimensional weights can be, in particular, power and hyperbolic weights with various parameters. The optimality of the argument in the modulus of continuity is established by means of the multidimensional Gauss quadrature formula over zeros of an eigenfunction of the Sturm–Liouville operator. The obtained results are complete; they generalize a number of known results.

Dialectical Logic K-model: On Multidimensional Discrete Dynamical Sampling System and Further Properties of Kirchhoff Matrices

In order to solve the problem of multidimensional logic variable true value function remarked in the paper(Yaozhi Jiang., 2017), now author has used discrete multiple Fourier transform to deal with the problem remarked above, and obtained an theoretical formulations of discrete multidimensional Fourier transform for that multidimensional logic variable true value function is unknown or we need the frequency properties of multidimensional logic variable true value function. Another problem is about further and deeper properties of Kirchhoff matrices defined in author’s paper(Yaozhi Jiang., 2017), author has established a series of matrix expression for Kirchhoff laws and some new properties of Kirchhoff matrices. These results are all compute-able and complicated.

Gas storage valuation under multifactor Lévy processes

A practical 1 The author's views are his own and do not necessarily reflect those of Gazprom Marketing & Trading Limited or any of its affiliates. problem for energy companies is instituting a consistent framework across its supply and trading activities to deliver on all-important P&L and at-Risk reporting requirements. With a focus on storage assets and wider natural gas market exposures, we present a gas storage valuation methodology, which uniquely uses a flexible multifactor Lévy process setting that allows for consistent valuation and risk management reporting across a general derivative book. Our approach is capable of replicating the complex covariance structure of the natural gas forward curve and capturing time spread volatility, a key driver of extrinsic storage value, while being simultaneously capable of accurately calibrating to market traded options. We begin by extending a single factor Mean Reverting Variance Gamma process to an arbitrary number of dimensions and, by way of specific examples, show how the traditional Principal Component Analysis based view of gas forward curve dynamics can be incorporated into a primarily market based valuation. We develop in the process an innovative implied moments based calibration technique, which allows for efficient calibration of general multifactor forward curve models to delivery period options common in energy and commodity markets. Furthermore, to accommodate the forward curve and traded options market consistency, we propose an appropriate joint market based calibration and historical estimation methodology. Through a formal model specification analysis, we provide evidence that the multifactor Lévy models we propose provide a better joint fit to NBP natural gas options-forward market data, relative to comparative benchmark models. Finally, we develop a novel multidimensional fast Fourier transform based storage valuation algorithm and provide empirical evidence that the multifactor Lévy model suite is better specified to more accurately capture extrinsic value.

FFT-based evaluation of multivariate aggregation integrals in population balance equations on uniform tensor grids

We consider the numerical solution of the multivariate aggregation population balance equation on a uniform tensor grid. We introduce a multidimensional fast Fourier transformation for the efficient evaluation of the aggregation integrals leading to a reduction in the complexity order of the algorithm compared to the direct evaluation approach. We illustrate the new evaluation algorithm for two discretizations, an FEM approach as well as the sectional method. We discuss the conservation of moments for these methods and provide numerical comparisons illustrating the superior performance of FFT-based algorithms. We also discuss and numerically illustrate their potential for parallelization.

Addendum to foundations of multidimensional wave field signal theory: Gaussian source function

Many important physical phenomena are described by wave or diffusion-wave type equations. Recent work has shown that a transform domain signal description from linear system theory can give meaningful insight to multi-dimensional wave fields. In N. Baddour [AIP Adv. 1, 022120 (2011)], certain results were derived that are mathematically useful for the inversion of multi-dimensional Fourier transforms, but more importantly provide useful insight into how source functions are related to the resulting wave field. In this short addendum to that work, it is shown that these results can be applied with a Gaussian source function, which is often useful for modelling various physical phenomena.

A Modified Omega-K Algorithm for Near-Field MIMO Array-Based 3-D Reconstruction

A modified omega-K algorithm for planar multiple-input-multiple-output (MIMO) array-based 3-D image reconstruction is proposed. By applying several proper approximations in wavenumber domain, the complex and time-consuming bistatic Stolt transformation within the traditional MIMO- $\omega \text{K}$ is changed into a relatively simple monostatic one, and therefore the amount of the interpolation given for the data is greatly reduced. Combining with multidimensional fast Fourier transformation, extremely high computation efficiency can be achieved. Implementation details are well described. The imaging accuracy and the high efficiency of the algorithm are demonstrated by a near-field imaging experiment with a flat mannequin target based on a planar MIMO array radar.