Truncation Parameter Research Articles

Continuous window functions for NFFT

In this paper, we study the error behavior of the nonequispaced fast Fourier transform (NFFT). This approximate algorithm is mainly based on the convenient choice of a compactly supported window function. Here, we consider the continuous Kaiser–Bessel, continuous exp-type, sinh-type, and continuous cosh-type window functions with the same support and same shape parameter. We present novel explicit error estimates for NFFT with such a window function and derive rules for the optimal choice of the parameters involved in NFFT. The error constant of a window function depends mainly on the oversampling factor and the truncation parameter. For the considered continuous window functions, the error constants have an exponential decay with respect to the truncation parameter.

Sparse grid-based adaptive noise reduction strategy for particle-in-cell schemes

We propose a sparse grid-based adaptive noise reduction strategy for electrostatic particle-in-cell (PIC) simulations. By projecting the charge density onto sparse grids we reduce the high-frequency particle noise. Thus, we exploit the ability of sparse grids to act as a multidimensional low-pass filter in our approach. Thanks to the truncated combination technique [1–3], we can reduce the larger grid-based error of the standard sparse grid approach for non-aligned and non-smooth functions. The truncated approach also provides a natural framework for minimizing the sum of grid-based and particle-based errors in the charge density. We show that our approach is, in fact, a filtering perspective for the noise reduction obtained with the sparse PIC schemes first introduced in [4]. This enables us to propose a heuristic based on the formal error analysis in [4] for selecting the optimal truncation parameter that minimizes the total error in charge density at each time step. Hence, unlike the physical and Fourier domain filters typically used in PIC codes for noise reduction, our approach automatically adapts to the mesh size, number of particles per cell, smoothness of the density profile and the initial sampling technique. It can also be easily integrated into high performance large-scale PIC code bases, because we only use sparse grids for filtering the charge density. All other operations remain on the regular grid, as in typical PIC codes. We demonstrate the efficiency and performance of our approach with two test cases: the diocotron instability in two dimensions and the three-dimensional electron dynamics in a Penning trap. Our run-time performance studies indicate that our approach can provide significant speedup and memory reduction to PIC simulations for achieving comparable accuracy in the charge density.

12 Bioinformatic analysis of a large-scale equine microbiome study

As sequencing becomes more cost effective, large-scale microbiome studies will become commonplace. Therefore, methodology for analyzing large amounts of microbiome data would benefit the equine science community. The MQ project (Microbiome Quotient®; Purina Animal Nutrition, Gray Summit, MO) is a large-scale, nation-wide, analysis of the equine microbiome. The aim of this project is to characterize the healthy or normal microbiome of the horse and relate that enterotype with population demographics to build predictive algorithms. Horse owners are sent custom microbiome kits by request and are instructed to collect and send in fecal swabs after completing a survey containing relevant metadata. To date, there are over 2,500 participants and 1,358 of those samples have been sequenced, amounting to 54.7 GB of data over 6 sequencing runs. Due to the scale of the project, a protocol was developed to efficiently process these data. Samples were sequenced on an Illumina MiSeq (Illumina, San Diego, CA). Data were analyzed using QIIME2 (ver. 2020.11) on a dedicated cloud server. Each run was imported separately as a QIIME2 artifact. Read quality was evaluated using the interactive quality plot produced by the q2-demux plugin. Each run was then denoised and filtered through DADA2 separately but with the same truncation parameters based on the average base length where quality dropped below 30. Feature tables and representative sequence files for each run were then merged. The amplicon sequence variants of the combined table were then aligned and used to construct a phylogenetic tree using the q2-alignment (mafft) and q2-phylogny (fasttree2) plugins, respectively. Taxonomic classification was performed with the q2-feature-classifier using a classifier based on the SILVA genomic database. Relative abundance values were calculated from the taxonomic bar plot produced by the taxa visualizer plugin. Diversity metrics of within sample (α-diversity; Shannon's Index) and between samples (β-diversity; Weighted and Unweighted UniFrac) were calculated using the q2-diversity plugin at a sampling depth at 10,000 features. Statistical significance of diversity metrics was also assessed using QIIME2. Shannon's Index and UniFrac measurements were analyzed with Kruskal-Wallis test and PERMANOVA, respectively. Pairwise comparisons were adjusted with the Benjamini & Hotchberg procedure. Relative abundance was assessed using one-way ANOVA in R for each metadata category (ver. 3.6.0). This protocol was effective at producing high quality microbiome data and demonstrates an applicable method for bioinformatic analysis for large-scale equine microbiome studies.

Robust Interference Cancellation Using Bi-Unknown Vectors Equations for User-Centric C-RANs

The user-centric cloud radio access network (C-RAN) is promising for significantly reducing the channel training overhead because only the intra-cluster channel state information (CSI) is required. However, the inter-cluster interference may degrade the network performance. To address this problem, we present a novel framework for the uplink of user-centric C-RANs where the interference cancellation is posed as a system of bi-unknown vectors linear equations. To solve this unusual system of equations, we propose a quasi-least squares (QLS) algorithm and analyze its robustness by exploiting the random matrix theory. We reveal the fact that QLS is very sensitive to the channel estimation error due to involving the inverse of an ill-conditioned matrix. It is well known that truncated singular value decomposition (TSVD) is an effective regularization scheme that can mitigate this ill-conditioning effect. Accordingly, we employ TSVD to further improve the robustness of QLS against the imperfect channel estimation. In addition, since the performance of the TSVD based algorithm strongly depends on the truncation parameter, a parameter-choice method using the constant modulus (CM) feature is also provided. Finally, simulation results are presented to examine the effectiveness and robustness of the proposed method.

Intensity Distribution of Partially Coherent Array Finite Airy Beams Propagating in Atmospheric Turbulence

Based on the extended Huygens–Fresnel integral and the Rytov phase structure function, the closed-form expression of a partially coherent array finite Airy beams (PCAFABs) cutting through the Kolmogorov atmospheric turbulence is derived in the space domain under the paraxial approximation. The characteristics of the PCAFABs evoluting in the atmospheric environment are investigated in detail on the basis of the derived wave propagation formulae. We mainly illustrate the intensity profile of this beam changed with the truncation parameter, coherence length, and turbulence factor at several cross sections of the atmospheric space by means of numerical figures. It is convinced that the present concept and derived conclusions will provide useful exploration for learning the optical properties of the PCAFABs transmitting in the atmospheric turbulence, especially for free-space optical communication area.

Tight focusing properties and focal field tailoring of cylindrical vector beams generated from a linearly polarized coherent beam array.

We investigate the focusing properties of cylindrical vector beams (CVBs) generated from the combination of an array of beams, each with sub-apertures and controllable polarization. The analytical expression of the tight focusing field of the combined CVBs has been derived based on the Richard-Wolf vector diffraction integral. To obtain a desired focal spot size which includes efficient sidelobe suppression, the required parameters, such as the exit sub-aperture, numerical aperture and truncation parameter, have been studied in detail. The result shows that the combined CVB distribution has a good match with the theoretical ideal CVB distribution. However, compared with the ideal CVBs, the focal spot width produced by the combined radially polarized beams is smaller. With the increase of initial polarization rotation of sub-aperture, the focal spot width increases, and the focal shape shifts from Gaussian-like to a flat-topped distribution and then to an annular distribution. Furthermore, flexible focal field tailoring can also be realized by adjusting the initial polarization rotation of each sub-aperture. These results might provide a valuable reference for material processing, microlithography and multi-particle manipulation.

A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization.

This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements.

Spectral computation of low probability tails for the homogeneous Boltzmann equation

We apply the spectral-Lagrangian method of Gamba and Tharkabhushanam for solving the homogeneous Boltzmann equation to compute the low probability tails of the velocity distribution function, f, of a particle species. This method is based on a truncation, Qtr(f,f), of the Boltzmann collision operator, Q(f,f), whose Fourier transform is given by a weighted convolution. The truncated collision operator models the situation in which two colliding particles ignore each other if their relative speed exceeds a threshold, gtr. We demonstrate that the choice of truncation parameter plays a critical role in the accuracy of the numerical computation of Q. Significantly, if gtr is too large, then accurate numerical computation of the weighted convolution integral is not feasible, since the degree of oscillation of the convolution weighting function increases as gtr increases. We derive an upper bound on the pointwise error between Q and Qtr, assuming that both operators are computed exactly. This bound provides some additional theoretical justification for the spectral-Lagrangian method, and can be used to guide the choice of gtr in numerical computations. We then demonstrate how to choose gtr and the numerical discretization parameters so that the computation of the truncated collision operator is a good approximation to Q in the low probability tails. Finally, for several different initial conditions, we demonstrate the feasibility of accurately computing the time evolution of the velocity pdf down to probability density levels ranging from 10−5 to 10−9.

Propagation properties of Hollow higher-order cosh-Gaussian beams in quadratic index medium and Fractional Fourier transform

Hollow higher-order cosh-Gaussian (HhCG) beam is a new mathematical model to describe the dark-hollow beam. An analytical expression for HhCG beams propagating through a paraxail ABCD optical system is derived and used to investigate its propagation properties in a quadratic index medium (QIM) and a Fractional Fourier transform (FRFT) optical system. By using the derived expressions, the properties of HhCG beams of both systems are graphically illustrated with numerical example. Several influence parameters on the evolution of HhCG in the both systems QIM and FRFT, respectively, are discussed in detail. Results show that the intensity distribution of HhCG evolves periodically during propagation in the QIM. In FRFT system, it is found that the normalized intensity distribution of HhCG beams plane depends not only on the fractional order but also on the beam order and the truncation parameter. The results obtained in this work are valuable to provide a convenient way for HhCG beams shaping, optical fibers, lenses, and other fields.

The Size‐Power Tradeoff in HAR Inference

Heteroskedasticity‐ and autocorrelation‐robust (HAR) inference in time series regression typically involves kernel estimation of the long‐run variance. Conventional wisdom holds that, for a given kernel, the choice of truncation parameter trades off a test's null rejection rate and power, and that this tradeoff differs across kernels. We formalize this intuition: using higher‐order expansions, we provide a unified size‐power frontier for both kernel and weighted orthonormal series tests using nonstandard “fixed‐b” critical values. We also provide a frontier for the subset of these tests for which the fixed‐bdistribution istorF. These frontiers are respectively achieved by the QS kernel and equal‐weighted periodogram. The frontiers have simple closed‐form expressions, which show that the price paid for restricting attention to tests withtandFcritical values is small. The frontiers are derived for the Gaussian multivariate location model, but simulations suggest the qualitative findings extend to stochastic regressors.

A Robust Wavenumber-Domain Superdirective Beamforming for Endfire Arrays

Among various array configurations, superdirective beamforming based on linear differential arrays has attracted much attention. It is well known that its directivity performance degrades at higher frequencies due to the limitation of finite difference approximation, and its robustness deteriorates due to the mismatch of practical systems. In this paper, we develop a wavenumber-domain superdirective beamforming for endfire arrays, an alternative way of phase mode beamforming that cannot be applied directly to linear arrays. Here, “wavenumber-domain” means the spatial derivatives are calculated in the wavenumber domain through multiplication operations. An important feature of proposed method is that it provides a novel way to obtain the higher order derivatives from the reconstructed sound field utilizing all sensors of array, which is valid when the Nyquist criterion is satisfied, resulting in a broadband frequency-independent beamformer. On the other hand, since the proposed $N\rm {th}$ -order superdirective beamformer has no upper limit of number of sensors, its robustness at lower frequencies can be enhanced through increasing the number of sensors and using an optimum truncation parameter which plays a role to remove small singular values sensitive to the mismatch of array. Although it is demonstrated that the proposed method and the weighted least-squares optimization method lead to almost identical performances, the main advantage of the proposed method is that an optimum truncation parameter determined by the experiment is adopted to increase the robustness of beamformers, where the high sensitivity of the higher eigenbeams to random errors is exploited.

Maximum likelihood estimation for a one-sided truncated family of distributions

For a one-sided truncated family of distributions with an interest parameter $$\theta $$ and a truncation parameter $$\gamma $$ as a nuisance parameter, we consider the maximum likelihood estimators (MLEs) $${\hat{\theta }}_\mathrm{ML}^{\gamma} $$ and $${\hat{\theta }}_{\mathrm{ML}}$$ of $$\theta $$ for known $$\gamma $$ and unknown $$\gamma $$ , respectively. In this paper, the stochastic expansions of $${\hat{\theta }}_{\mathrm{ML}}^{\gamma} $$ and $${\hat{\theta }}_\mathrm{ML}$$ are derived, and their second-order asymptotic variances are obtained. The second-order asymptotic loss of a bias-adjusted $${\hat{\theta }}_{\mathrm{ML}^*}$$ relative to $${\hat{\theta }}_{\mathrm{ML}}^{\gamma} $$ is also given. The results are a generalization of those for a one-sided truncated exponential family of distributions. Examples on a one-sided truncated Cauchy distribution, a general truncated exponential family, etc. are also given.

An adaptive finite element DtN method for the three-dimensional acoustic scattering problem

<p style='text-indent:20px;'>This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

An Exact Realization of a Modified Hilbert Transformation for Space-Time Methods for Parabolic Evolution Equations

Abstract We present different possibilities of realizing a modified Hilbert type transformation as it is used for Galerkin–Bubnov discretizations of space-time variational formulations for parabolic evolution equations in anisotropic Sobolev spaces of spatial order 1 and temporal order 1 2 \frac{1}{2} . First, we investigate the series expansion of the definition of the modified Hilbert transformation, where the truncation parameter has to be adapted to the mesh size. Second, we introduce a new series expansion based on the Legendre chi function to calculate the corresponding matrices for piecewise polynomial functions. With this new procedure, the matrix entries for a space-time finite element method for parabolic evolution equations are computable to machine precision independently of the mesh size. Numerical results conclude this work.

Fast Sparse-TSVD Super-Resolution Method of Real Aperture Radar Forward-Looking Imaging

Most existing super-resolution imaging methods fail to work in low signal-to-noise ratio (SNR) condition due to the ill-posed antenna measurement matrix, but the sparse-truncated singular value decomposition (TSVD) method can effectively suppress noise and improve azimuth resolution in low SNR condition. However, the current sparse-TSVD method encounters large computation cost, resulting in a slow algorithm speed. In this work, a fast sparse-TSVD super-resolution imaging method of real aperture radar is proposed. First, the proposed method is based on the results of TSVD, using the truncated unitary matrix and diagonal matrix to reconstruct the signal convolution model. The dimension of the reconstructed antenna measurement matrix reduces from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N \times N$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k \times N$ </tex-math></inline-formula> , and the dimension of the reconstructed echo matrix reduces from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N \times 1$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k \times 1$ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> is azimuth sampling points and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$ </tex-math></inline-formula> is truncation parameter, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N \gg k$ </tex-math></inline-formula> . Much of the expensive matrix– multiplication computation can then be performed on the smaller matrices, thereby accelerating the algorithm. Second, an objective function is established as the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${l_{1}}$ </tex-math></inline-formula> constraint based on the regularization strategy. Lastly, this article employs iterative reweighted least square (IRLS) method to solve the objective function, and the dimension of the reversed matrix is lessened from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N \times N$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k \times k$ </tex-math></inline-formula> , speeding up the algorithm further. The simulation and real data verify that the proposed algorithm not only improves the azimuth resolution in low SNR condition but also increases computational efficiency compared with the sparse-TSVD method.

A Two-Stage Three-Machine Flow Shop Assembly Problem Mixed with a Controllable Number and Sum-of-Processing Times-Based Learning Effect by Simulated Annealing Algorithms

The two-stage assembly scheduling problem is widely used in industrial and service industries. This study focuses on the two-stage three-machine flow shop assembly problem mixed with a controllable number and sum-of-processing times-based learning effect, in which the job processing time is considered to be a function of the control of the truncation parameters and learning based on the sum of the processing time. However, the truncation function is very limited in the two-stage flow shop assembly scheduling settings. Thus, this study explores a two-stage three-machine flow shop assembly problem with truncated learning to minimize the makespan criterion. To solve the proposed model, we derive several dominance rules, lemmas, and lower bounds applied in the branch-and-bound method. On the other hand, three simulated annealing algorithms are proposed for finding approximate solutions. In both the small and large size number of job situations, the SA algorithm is better than the JS algorithm in this study. All the experimental results of the proposed algorithm are presented on small and large job sizes, respectively.

Bayesian inference for hidden truncation Pareto (IV) models

In this paper, we consider observations arising out from a hidden truncation Pareto (IV) distribution and to be used to make inferences about the inequality, precision, shape and truncation parameter(s). Two different types of dependent prior analyses are reviewed and compared with each other. We conjecture that mathematical tractability should be, perhaps, a minor consideration in choosing the appropriate choices of the hyper-parameters for the prior densities. Some illustrative examples are provided. A simulation study is conducted to illustrate the applicability of the proposed model.

Metaheuristics for two-stage flow-shop assembly problem with a truncation learning function

ABSTRACT This study examines a two-stage three-machine flow-shop assembly scheduling model in which job processing time is considered as a mixed function of a controlled truncation parameter with a sum-of-processing-times-based learning effect. However, the truncation function is very limited in the two-stage flow-shop assembly scheduling settings. To overcome this limitation, this study investigates a two-stage three-machine flow-shop assembly problem with a truncation learning function where the makespan criterion (completion of the last job) is minimized. Given that the proposed model is NP hard, dominance rules, lemmas and a lower bound are derived and applied to the branch-and-bound method. A dynamic differential evolution algorithm, a hybrid greedy iterated algorithm and a genetic algorithm are also proposed for searching approximate solutions. Results obtained from test experiments validate the performance of all the proposed algorithms.

Toward a less costly but accurate calculation of the CCSD(T)/CBS noncovalent interaction energy

The popular method of calculating the noncovalent interaction energies at the coupled-cluster single-, double-, and perturbative triple-excitations [CCSD(T)] theory level in the complete basis set (CBS) limit was to add a CCSD(T) correction term to the CBS second-order Møller-Plesset perturbation theory (MP2). The CCSD(T) correction term is the difference between the CCSD(T) and MP2 interaction energies evaluated in a medium basis set. However, the CCSD(T) calculations with the medium basis sets are still very expensive for systems with more than 30 atoms. Comparatively, the domain-based local pair natural orbital coupled-cluster method [DLPNO-CCSD(T)] can be applied to large systems with over 1,000 atoms. Considering both the computational accuracy and efficiency, in this work, we propose a new scheme to calculate the CCSD(T)/CBS interaction energies. In this scheme, the MP2/CBS term keeps intact and the CCSD(T) correction term is replaced by a DLPNO-CCSD(T) correction term which is the difference between the DLPNO-CCSD(T) and DLPNO-MP2 interaction energies evaluated in a medium basis set. The interaction energies of the noncovalent systems in the S22, HSG, HBC6, NBC10, and S66 databases were recalculated employing this new scheme. The consistent and tight settings of the truncation parameters for DLPNO-CCSD(T) and DLPNO-MP2 in this noncanonical CCSD(T)/CBS calculations lead to the maximum absolute deviation and root-mean-square deviation from the canonical CCSD(T)/CBS interaction energies of less than or equal to 0.28 kcal/mol and 0.09 kcal/mol, respectively. The high accuracy and low cost of this new computational scheme make it an excellent candidate for the study of large noncovalent systems.

DEBLURRING STUDY OF DMSP/OLS NIGHTTIME LIGHT DATA BY RTSVD

Abstract. DMSP/OLS, as the earliest Nighttime light remote sensing data, has great application value and can greatly improve the data quality by solving the blurring problem existing in the data. The blur reason is analyzed, and a new algorithm of regularization truncated singular value decomposition (RTSVD) combining with Pct image luminescence frequency filtering is proposed, which can effectively eliminate the blurring phenomenon and retain the real information of the image. Firstly, considering that the luminescence frequency of the light source pixel must be higher than that of the non-light source pixel, the luminescence frequency of the pixel in the Pct image is used to exclude the non-light source pixel in the average light image, and then the truncation parameter of the regularized truncation singular value decomposition (RTSVD) is obtained by using the L curve, so as to decompose and recombine the image. The experiments show that the regularized truncation singular value decomposition method combined with Pct image luminescence frequency filtering can remove the blurring phenomenon on the basis of preserving the image information.