Equispaced Grid Research Articles

Uniform approximation of common Gaussian process kernels using equispaced Fourier grids

The high efficiency of a recently proposed method for computing with Gaussian processes relies on expanding a (translationally invariant) covariance kernel into complex exponentials, with frequencies lying on a Cartesian equispaced grid. Here we provide rigorous error bounds for this approximation for two popular kernels—Matérn and squared exponential—in terms of the grid spacing and size. The kernel error bounds are uniform over a hypercube centered at the origin. Our tools include a split into aliasing and truncation errors, and bounds on sums of Gaussians or modified Bessel functions over various lattices. For the Matérn case, motivated by numerical study, we conjecture a stronger Frobenius-norm bound on the covariance matrix error for randomly-distributed data points. Lastly, we prove bounds on, and study numerically, the ill-conditioning of the linear systems arising in such regression problems.

Embracing off-the-grid samples

Many empirical studies suggest that samples of continuous-time signals taken at locations randomly deviated from an equispaced grid (i.e., off-the-grid) can benefit signal acquisition, e.g., undersampling and anti-aliasing. However, explicit statements of such advantages and their respective conditions are scarce in the literature. This paper provides some insight on this topic when the sampling positions are known, with grid deviations generated i.i.d. from a variety distributions. By solving a square-root LASSO decoder with an interpolation kernel we demonstrate the capabilities of nonuniform samples for compressive sampling, an effective paradigm for undersampling and anti-aliasing. For functions in the Wiener algebra that admit a discrete s-sparse representation in some transform domain, we show that O(spolylogN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(s{{\\,\\mathrm{poly\\,\\hspace{-2pt}log}\\,}}N)$$\\end{document} random off-the-grid samples are sufficient to recover an accurate N2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{N}{2}$$\\end{document}-bandlimited approximation of the signal. For sparse signals (i.e., s≪N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$s \\ll N$$\\end{document}), this sampling complexity is a great reduction in comparison to equispaced sampling where O(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(N)$$\\end{document} measurements are needed for the same quality of reconstruction (Nyquist–Shannon sampling theorem). We further consider noise attenuation via oversampling (relative to a desired bandwidth), a standard technique with limited theoretical understanding when the sampling positions are non-equispaced. By solving a least squares problem, we show that O(NlogN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(N\\log N)$$\\end{document} i.i.d. randomly deviated samples provide an accurate N2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{N}{2}$$\\end{document}-bandlimited approximation of the signal with suppression of the noise energy by a factor ∼1log(N).\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sim \\frac{1}{\\sqrt{\\log (N)}}.$$\\end{document}

Enhanced trapezoidal rule for discontinuous functions

In many applications, data to be integrated is available only in the form of function values at predetermined equispaced points. For smooth periodic functions, the trapezoidal rule gives very accurate approximations to the integral, but is only second order accurate in more commonly arising non-periodic cases. The Gregory approach for end corrections can improve the accuracy order to 9 before the onset of negative weights, and of rapidly increasing ill-conditioning. In recent work, this has been extended to accuracy orders around 20.This present study focuses on the more general case when one or both end points of the integration interval do not coincide with any of the equispaced grid points. We show that accuracy orders up to around 10 can also then be achieved, with all quadrature weights still remaining non-negative. This method can be utilized for example when functions to be integrated feature discontinuities at locations that can be separately determined (at or in between the equispaced grid points).

СУБДИФУЗIЙНИЙ ЧАС ПЕРШОГО ВИХОДУ НАДИСКРЕТНIЙ СIТЦI

In the work considered process of continuous-time random walk, that has fat-tailed jump waiting time, on an equispaced grid of one-dimensional domain with absorbing boundary. Deduced fractional equation w.r.t. cumulative distribution function of first passage time. Obtained asymptotic of density of this variable and shown that it has fat tail.

Seismic Data Regularization on Nonequispaced Grid via a Joint Sparsity-Promotion Method

The seismic data regularization problem is vital to seismic data processing. We propose a joint sparsity-promotion method based on the compressive sensing theory named the curvelet-data-driven-tight-frame-based sparsity-promoting (CDSP) method. The CDSP method regularizes the seismic data directly on the nonequispaced grid along the spatial dimension. The joint sparsity is exploited in the curvelet and data-driven tight frame transform simultaneously on the projected regular spatial grid. The projection from the nonequispaced grid to the equispaced grid is achieved by the nonequispaced discretized Fourier transform. Comparing with the curvelet-sparsity-promotion-based (CSP) regularization method, CDSP combines the advantage of the predefined curvelet transform and adaptive-learned sparse transform into one single optimization model. An alternative directional method of multipliers (ADMM) is applied to solve the optimization problem. One synthetic and two field examples show that the CDSP method performs better on the preservation of the continuity of events and produces less artifacts than the CSP method.

Testing for changes in the tail behavior of Brown–Resnick Pareto processes

We consider the class of r-Pareto processes defined on [0,1] whose max-stable counterparts are Brown–Resnick processes. The aim of this paper is to propose a test whether the extreme value index function of a r-Pareto process of this class remains constant over [0,1]. We assume that we observe several independent r-Pareto processes over equispaced grids of [0,1] but with possibly heterogeneous grid resolutions. We build a test based on the normalized approximate total variations of the paths of these processes. We provide its properties under infill asymptotics and assess its finite sample performance through simulation experiments. We discuss how to proceed for random processes that belong to the domain of attraction of Brown–Resnick r-Pareto processes and illustrate the approach with an application to wind speed data from Germany.

Fast vertex-based graph convolutional neural network and its application to brain images

This paper proposes a vertex-based graph convolutional neural network (vertex-CNN) for analyzing structured data on graphs. We represent graphs using semi-regular triangulated meshes in which each vertex has 6 connected neighbors. We generalize classical CNN defined on equi-spaced grids to that defined on semi-regular triangulated meshes by introducing main building blocks of the CNN, including convolution, down-sampling, and pooling, on a vertex domain. By exploiting the regularity of semi-regular meshes in terms of vertex connections, the proposed vertex-CNN keeps the inherent properties of classical CNN in a Euclidean space, such as shift-invariance and down-sampling at a rate of 2, 4, etc. We employ brain images from Alzheimer’s Disease Neuroimaging Initiative (ADNI) (n = 6767) and extract cortical features (e.g., cortical thickness, surface area, curvature, Jacobian, sulcal depth, and volume) for the classification of healthy controls (CON), patients with mild cognitive impairment (MCI) and Alzheimer’s disease (AD). Based on cortical thickness, we show that the proposed vertex-CNN is near 3 times faster and performs significantly better in the classification performance of CON, MCI, and AD than an existing graph CNN defined on the graph spectral domain given in Defferrard (2016). Moreover, we examine the robustness of a multi-channel implementation of vertex-CNN on 6 cortical measures for the MCI and AD classification. Finally, we show a promising finding of the prediction accuracy from MCI to AD as a function of years before the onset of AD. Our experiments demonstrate the fast computation and promising classification performance of the vertex-CNN.

Accelerating the evaluation of inspiral–merger–ringdown waveforms with adapted grids

This paper presents an algorithm to accelerate the evaluation of inspiral–merger–ringdown waveform models for gravitational wave data analysis. While the idea can also be applied in the time domain, here we focus on the frequency domain, which is most typically used to reduce computational cost in gravitational wave data analysis. Our work extends the idea of multibanding Vinciguerra S et al (2017 Class. Quantum Grav. 34 115006), which has been developed to accelerate frequency domain waveforms, to include the merger and ringdown and spherical harmonics beyond the dominant quadrupole spherical harmonic. The original method of Vinciguerra S et al (2017 Class. Quantum Grav. 34 115006) is based on a heuristic algorithm based on the inspiral to de-refine the equi-spaced frequency grid used for data analysis where a coarser grid is sufficient for accurate evaluation of a waveform model. Here we use a different criterion, based on the local interpolation error, which is more flexible and can easily be adapted to general waveforms, if their phenomenology is understood. We discuss our implementation in the LIGO Algorithms Library (The LIGO Scientific Collaboration 2015 https://lsc-group.phys.uwm.edu/daswg/projects/lalsuite.html) for the IMRPhenomXHM García Quirós C et al (2020 arXiv:2001.10914) frequency domain model, and report the acceleration in different parts of the parameter space of compact binary systems.

Contour integrals of analytic functions given on a grid in the complex plane

Abstract Ability to evaluate contour integrals is central to both the theory and the utilization of analytic functions. We present here a complex plane realization of the Euler–Maclaurin formula that includes weights also at some grid points adjacent to each end of a line segment (made up of equispaced grid points, along which we use the trapezoidal rule). For example, with a $5\times 5$ ‘correction stencil’ (with weights about two orders of magnitude smaller than those of the trapezoidal rule), the accuracy is increased from $2$nd to $26$th order.

Fast evaluation of three-dimensional gamma dose rate fields on non-equispaced grids for complex atmospheric radionuclide distributions

The gamma dose rate caused by airborne radionuclides is a major concern in the mitigation of nuclear accidents. Unfortunately, there is no fast method for calculating the three-dimensional (3D) gamma dose rate field near the source, because the corresponding airborne radionuclide distribution is usually calculated on non-equispaced grids and existing fast methods are only suitable for equispaced grids. This paper presents a method that accurately calculates the 3D dose rate field on non-equispaced grids, accelerating the computation by around two orders of magnitude. This method splits the time-consuming 3D integral in the dose rate model into a large convolution with a regularized smooth function and a small correction term. A nonuniform fast Fourier transform (NFFT) is used to rapidly calculate the convolution, which significantly enhances the computational speed. Our approach is applied to different grids and is compared with the FFT-based convolution method in two complex air dispersion simulations and a field experiment. The results show that the proposed method is in good agreement with the original 3D integral method and avoids grid-dependent interpolation errors in the FFT-based convolution method. This method enables a coupled analysis of wind, radioactivity, and dose rate on arbitrary grids, which is important for simplifying the emergency response in the case of small modular reactors.

Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

We explore two classes of exponential integrators in this letter to design nonlinear Fourier transform (NFT) algorithms with a desired accuracy-complexity trade-off and a convergence order of $4$ on an equispaced grid. The integrating factor based method in the class of Runge-Kutta methods yield algorithms with complexity $O(N\log^2N)$ (where $N$ is the number of samples of the signal) which have superior accuracy-complexity trade-off than any of the fast methods known currently. The integrators based on Magnus series expansion, namely, standard and commutator-free Magnus methods yield algorithms of complexity $O(N^2)$ that have superior error behavior even for moderately small step-sizes and higher signal strengths.

A fast algorithm for computing the mock-Chebyshev nodes

Runge Phenomenon which is a very well-known example and published by C. Runge in 1901 is as follows: polynomial interpolation of a function f, using equidistant interpolation points on [−1,1] could diverge on certain parts of this interval even if f is analytic anywhere on the interval. Among all the techniques that have been proposed to defeat this phenomenon in the literature of approximation theory, there is the mock-Chebyshev interpolation on a grid: a subset of (n+1) points from an equispaced grid with O(n2) points chosen to mimic the non-uniform (n+1)-point Chebyshev–Lobatto grid [1]. This study suggests a fast algorithm for computing the mock-Chebyshev nodes using the distance between each pair of consecutive points. The complexity of the algorithm is O(n), where n+1 is the number of the Chebyshev–Lobatto nodes on the interval [−1,1]. A discussion of bivariate generalization of the mock-Chebyshev nodes to the Padua interpolation points in [−1,1]2 is given and numerical results are also provided.

Sixth-order schemes for laser-matter interaction in the Schrödinger equation.

Control of quantum systems via lasers has numerous applications that require fast and accurate numerical solution of the Schrödinger equation. In this paper, we present three strategies for extending any sixth-order scheme for the Schrödinger equation with time-independent potential to a sixth-order method for the Schrödinger equation with laser potential. As demonstrated via numerical examples, these schemes prove effective in the atomic regime as well as the semiclassical regime and are a particularly appealing alternative to time-ordered exponential splittings when the laser potential is highly oscillatory or known only at specific points in time (on an equispaced grid, for instance). These schemes are derived by exploiting the linear in space form of the time dependent potential under the dipole approximation (whereby commutators in the Magnus expansion reduce to a simpler form), separating the time step of numerical propagation from the issue of adequate time-resolution of the laser field by keeping integrals intact in the Magnus expansion and eliminating terms with unfavorable structure via carefully designed splittings.

Omnipolarity applied to equi-spaced electrode array for ventricular tachycardia substrate mapping.

Aims Bipolar electrogram (BiEGM)-based substrate maps are heavily influenced by direction of a wavefront to the mapping bipole. In this study, we evaluate high-resolution, orientation-independent peak-to-peak voltage (Vpp) maps obtained with an equi-spaced electrode array and omnipolar EGMs (OTEGMs), measure its beat-to-beat consistency, and assess its ability to delineate diseased areas within the myocardium compared against traditional BiEGMs on two orientations: along (AL) and across (AC) array splines.Methods and resultsThe endocardium of the left ventricle of 10 pigs (three healthy and seven infarcted) were each mapped using an Advisor™ HD grid with a research EnSite Precision™ system. Cardiac magnetic resonance images with late gadolinium enhancement were registered with electroanatomical maps and were used for gross scar delineation. Over healthy areas, OTEGM Vpp values are larger than AL bipoles by 27% and AC bipoles by 26%, and over infarcted areas OTEGM Vpp values are 23% larger than AL bipoles and 27% larger than AC bipoles (P < 0.05). Omnipolar EGM voltage maps were 37% denser than BiEGM maps. In addition, OTEGM Vpp values are more consistent than bipolar Vpps showing less beat-by-beat variation than BiEGM by 39% and 47% over both infarcted and healthy areas, respectively (P < 0.01). Omnipolar EGM better delineate infarcted areas than traditional BiEGMs from both orientations.ConclusionAn equi-spaced electrode grid when combined with omnipolar methodology yielded the largest detectable bipolar-like voltage and is void of directional influences, providing reliable voltage assessment within infarcted and non-infarcted regions of the heart.

BlackNUFFT: Modular customizable black box hybrid parallelization of type 3 NUFFT in 3D

Many applications benefit from an efficient Discrete Fourier Transform (DFT) between arbitrarily spaced points. The Non Uniform Fast Fourier Transform reduces the computational cost of such operation from O(N2) to O(NlogN) exploiting gridding algorithms and a standard Fast Fourier Transform on an equi-spaced grid. The parallelization of the NUFFT of type 3 (between arbitrary points in space and frequency) still poses some challenges: we present a novel and flexible hybrid parallelization in a MPI-multithreaded environment exploiting existing HPC libraries on modern architectures. To ensure the reliability of the developed library, we exploit continuous integration strategies using Travis CI. We present performance analyses to prove the effectiveness of our implementation, possible extensions to the existing library, and an application of NUFFT type 3 to MRI image processing. Program summaryProgram Title: BlackNUFFTProgram Files doi:http://dx.doi.org/10.17632/vxfj6x2p8x.1Licensing provisions: LGPLProgramming language: C++External routines/libraries:deal.II , FFTW, PFFTNature of problem: Provide a modular and extensible implementation of a parallel Non Uniform Fast Fourier Transform of type 3.Solution method: Use of hybrid shared distributed memory paradigm to achieve high level of efficiency. We exploit existing HPC library following best practices in scientific computing (as continuous integration via TravisCI) to reach higher complexities and guarantee the accuracy of the solution proposed.

Optimal sampling rates for approximating analytic functions from pointwise samples

AbstractWe consider the problem of approximating an analytic function on a compact interval from its values at $M+1$ distinct points. When the points are equispaced, a recent result (the so-called impossibility theorem) has shown that the best possible convergence rate of a stable method is root-exponential in M, and that any method with faster exponential convergence must also be exponentially ill conditioned at a certain rate. This result hinges on a classical theorem of Coppersmith &amp; Rivlin concerning the maximal behavior of polynomials bounded on an equispaced grid. In this paper, we first generalize this theorem to arbitrary point distributions. We then present an extension of the impossibility theorem valid for general nonequispaced points and apply it to the case of points that are equidistributed with respect to (modified) Jacobi weight functions. This leads to a necessary sampling rate for stable approximation from such points. We prove that this rate is also sufficient, and therefore exactly quantify (up to constants) the precise sampling rate for approximating analytic functions from such node distributions with stable methods. Numerical results—based on computing the maximal polynomial via a variant of the classical Remez algorithm—confirm our main theorems. Finally, we discuss the implications of our results for polynomial least-squares approximations. In particular, we theoretically confirm the well-known heuristic that stable least-squares approximation using polynomials of degree N &amp;lt; M is possible only once M is sufficiently large for there to be a subset of N of the nodes that mimic the behavior of the $N$th set of Chebyshev nodes.

Non-Equispaced System Matrix Acquisition for Magnetic Particle Imaging Based on Lissajous Node Points.

Magnetic Particle Imaging (MPI) is an emerging technology in the field of (pre)clinical imaging. The acquisition of a particle signal is realized along specific sampling trajectories covering a defined field of view (FOV). In a system matrix (SM) based reconstruction procedure, the commonly used acquisition path in MPI is a Lissajous trajectory. Such a trajectory features an inhomogeneous coverage of the FOV, i.e. the center region is sampled less dense than the regions towards the edges of the FOV. Conventionally, the respective SM acquisition and the subsequent reconstruction do not reflect this inhomogeneous coverage. Instead, they are performed on an equispaced grid. The objective of this work is to introduce a sampling grid that inherently features the aforementioned inhomogeneity by using node points of Lissajous trajectories. Paired with a tailored polynomial interpolation of the reconstructed MPI signal, the entire image can be recovered. It is the first time that such a trajectory related non-equispaced grid is used for image reconstruction on simulated and measured MPI data and it is shown that the number of sampling positions can be reduced, while the spatial resolution remains constant.

Bicubic splines and biquartic polynomials

Abstract The paper proposes a new efficient approach to computation of interpolating spline surfaces. The generalization of an unexpected property, noticed while approximating polynomials of degree four by cubic ones, confirmed that a similar interrelation property exists between biquartic and bicubic polynomial surfaces as well. We prove that a 2×2-component C1 -class bicubic Hermite spline will be of class C2 if an equispaced grid is used and the coefficients of the spline components are computed from a corresponding biquartic polynomial. It implies that a 2×2 uniform clamped spline surface can be constructed without solving any equation. The applicability of this biquartic polynomials based approach to reducing dimensionalitywhile computing spline surfaces is demonstrated on an example.

An extended Prony’s interpolation scheme on an equispaced grid

Abstract An interpolation scheme on an equispaced grid based on the concept of the minimal order of the linear recurrent sequence is proposed in this paper. This interpolation scheme is exact when the number of nodes corresponds to the order of the linear recurrent function. It is shown that it is still possible to construct a nearest mimicking algebraic interpolant if the order of the linear recurrent function does not exist. The proposed interpolation technique can be considered as the extension of the Prony method and can be useful for describing noisy and defected signals.

A Fourier Continuation Method for the Solution of Elliptic Eigenvalue Problems in General Domains

We present a new computational method for the solution of elliptic eigenvalue problems with variable coefficients in general two-dimensional domains. The proposed approach is based on use of the novel Fourier continuation method (which enables fast and highly accurate Fourier approximation of nonperiodic functions in equispaced grids without the limitations arising from the Gibbs phenomenon) in conjunction with an overlapping patch domain decomposition strategy and Arnoldi iteration. A variety of examples demonstrate the versatility, accuracy, and generality of the proposed methodology.