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Journal articles on the topic 'High–Order Spectral Methods'

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Author: Grafiati

Published: 10 December 2022

Last updated: 28 January 2023

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1

Mercader, I., M. Net, and A. Falques. "Spectral methods for high order equations." Computer Methods in Applied Mechanics and Engineering 91, no. 1-3 (October 1991): 1245–51. http://dx.doi.org/10.1016/0045-7825(91)90076-i.

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2

Vanharen, Julien, Guillaume Puigt, Xavier Vasseur, Jean-François Boussuge, and Pierre Sagaut. "Revisiting the spectral analysis for high-order spectral discontinuous methods." Journal of Computational Physics 337 (May 2017): 379–402. http://dx.doi.org/10.1016/j.jcp.2017.02.043.

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3

Thalhammer, Mechthild, Marco Caliari, and Christof Neuhauser. "High-order time-splitting Hermite and Fourier spectral methods." Journal of Computational Physics 228, no. 3 (February 2009): 822–32. http://dx.doi.org/10.1016/j.jcp.2008.10.008.

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4

Guo, Benyu. "Spectral and Spectral Element Methods for High Order Problems with Mixed Boundary Conditions." Journal of Computational Mathematics 32, no. 4 (June 2014): 392–411. http://dx.doi.org/10.4208/jcm.1403-m4373.

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5

Zhang, Chao, Hanfeng Yao, and Huiyuan Li. "New space–time spectral and structured spectral element methods for high order problems." Journal of Computational and Applied Mathematics 351 (May 2019): 153–66. http://dx.doi.org/10.1016/j.cam.2018.08.038.

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6

Aleksic, Branislav. "High order spectral symplectic methods for solving PDEs on GPU." Qatar Foundation Annual Research Forum Proceedings, no. 2013 (November 2013): ICTP 043. http://dx.doi.org/10.5339/qfarf.2013.ictp-043.

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7

Yoon, Hyun C., and Jihoon Kim. "Spectral deferred correction methods for high‐order accuracy in poroelastic problems." International Journal for Numerical and Analytical Methods in Geomechanics 45, no. 18 (October 19, 2021): 2709–31. http://dx.doi.org/10.1002/nag.3283.

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8

Gray, M. "Reconstruction of Spectra obtained with PYTHEAS through Photometric Methods." International Astronomical Union Colloquium 149 (1995): 311–13. http://dx.doi.org/10.1017/s0252921100023228.

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PYTHEAS is an imaging spectrophotometer giving high spectral and spatial resolutions in joining together several optical concepts (Le Coarer et al 1992, and Georgelin et al. in these Proceedings). It will be shown in this paper that, contrary to other spectrographs using somewhat complex methods of data reduction, the PYTHEAS imaging spectrometer requires simple photometric methods in order to reconstruct the spectra of astrophysical objects.The CCD image of a continuous spectral energy distribution for a permanent gap between the layers of the Fabry-Perot consists in a series of channelled spectra: each elementary beam, spectrally filtered by the FP interferometer and sampled by a microlens in the frame, gives light to the whole surface of the grism which separates the various F-P orders by displaying them in a line on the detector. Each line or channelled spectrum consists of a series of Fabry pupils (spectral elements), each of them containing the luminous flux emitted on a certain wavelength by the object under investigation. After scanning the FP interferometer across its free spectral range, we obtain a series of shifted channelled spectra whose set provides us with a chart showing the photometric values of flux according to the wavelength. Consequently, some forms of calibration (continuum lamp, spectral lamp) allow the reconstruction of the spectra of the astrophysical object through simple photometric measures.

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Bezrukov, Andrei, and Igor Zarubin. "METHODS FOR IMPROVEMENT OF HIGH-RESOLUTION SPECTROMETER CHARACTERISTICS." Interexpo GEO-Siberia 8 (2019): 226–37. http://dx.doi.org/10.33764/2618-981x-2019-8-226-237.

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The present paper demonstrates results of high-resolution spectrometer characteristics improvement methods. Increasing resolution, spectral range extending and illumination efficiency for the spectrometer were investigated. Obtained results will be found useful in atomic spectroscopy applications such as atomic absorption, atomic emission spectroscopy, mass-spectroscopy, chromatography and others. In order to increase spectrometer resolution it was suggested to use higher diffractive grating curvature radius. Experimentally, characteristics of both spectrometer prototypes assembled using diffractive gratings R1000 and R2000 were obtained and compared. New approach for polychromatous displacement was developed in order to extend operational spectrum range. The main feature is single spectrometer entrance slit for both UV and visible range Paschen-Runge polychromatous with beam splitting by coupled flat folding mirrors placed behind slit. Diffractive grating illumination monitor system was designed in order to provide this spectrometer by alignment control for lightning system. New spectrometer “Grand-2” was fabricated. It includes coupled Paschen-Runge polychromators for UV and visible spectral range providing 12 and 30 pm spectral resolution respectively with single entrance slit equipped with diffractive grating illumination monitoring system. This spectrometer can be used for various atomic spectroscopy applications.

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10

Tang, Tao, Hehu Xie, and Xiaobo Yin. "High-Order Convergence of Spectral Deferred Correction Methods on General Quadrature Nodes." Journal of Scientific Computing 56, no. 1 (October 25, 2012): 1–13. http://dx.doi.org/10.1007/s10915-012-9657-9.

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11

Cai, Wei, and Chi-Wang Shu. "Uniform high order spectral methods for one- and two-dimensional Euler equations." Journal of Computational Physics 102, no. 2 (October 1992): 425. http://dx.doi.org/10.1016/0021-9991(92)90392-c.

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12

Cai, Wei, and Chi-Wang Shu. "Uniform High-Order Spectral Methods for One- and Two-Dimensional Euler Equations." Journal of Computational Physics 104, no. 2 (February 1993): 427–43. http://dx.doi.org/10.1006/jcph.1993.1041.

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13

Opschoor, Joost A. A., Philipp C. Petersen, and Christoph Schwab. "Deep ReLU networks and high-order finite element methods." Analysis and Applications 18, no. 05 (February 21, 2020): 715–70. http://dx.doi.org/10.1142/s0219530519410136.

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Approximation rate bounds for emulations of real-valued functions on intervals by deep neural networks (DNNs) are established. The approximation results are given for DNNs based on ReLU activation functions. The approximation error is measured with respect to Sobolev norms. It is shown that ReLU DNNs allow for essentially the same approximation rates as nonlinear, variable-order, free-knot (or so-called “[Formula: see text]-adaptive”) spline approximations and spectral approximations, for a wide range of Sobolev and Besov spaces. In particular, exponential convergence rates in terms of the DNN size for univariate, piecewise Gevrey functions with point singularities are established. Combined with recent results on ReLU DNN approximation of rational, oscillatory, and high-dimensional functions, this corroborates that continuous, piecewise affine ReLU DNNs afford algebraic and exponential convergence rate bounds which are comparable to “best in class” schemes for several important function classes of high and infinite smoothness. Using composition of DNNs, we also prove that radial-like functions obtained as compositions of the above with the Euclidean norm and, possibly, anisotropic affine changes of co-ordinates can be emulated at exponential rate in terms of the DNN size and depth without the curse of dimensionality.

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14

Breviglieri, Carlos, and Joao Luiz F. Azevedo. "Further Development and Application of High-Order Spectral Volume Methods for Compressible Flows." Journal of Aerospace Technology and Management 9, no. 3 (August 3, 2017): 301–27. http://dx.doi.org/10.5028/jatm.v9i3.707.

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15

Bourlioux, Anne, Anita T. Layton, and Michael L. Minion. "High-order multi-implicit spectral deferred correction methods for problems of reactive flow." Journal of Computational Physics 189, no. 2 (August 2003): 651–75. http://dx.doi.org/10.1016/s0021-9991(03)00251-1.

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16

Mengaldo, G., D. De Grazia, D. Moxey, P. E. Vincent, and S. J. Sherwin. "Dealiasing techniques for high-order spectral element methods on regular and irregular grids." Journal of Computational Physics 299 (October 2015): 56–81. http://dx.doi.org/10.1016/j.jcp.2015.06.032.

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17

Lodato, Guido, Patrice Castonguay, and Antony Jameson. "Discrete filter operators for large-eddy simulation using high-order spectral difference methods." International Journal for Numerical Methods in Fluids 72, no. 2 (October 24, 2012): 231–58. http://dx.doi.org/10.1002/fld.3740.

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18

Beck, Andrea D., Thomas Bolemann, David Flad, Hannes Frank, Gregor J. Gassner, Florian Hindenlang, and Claus-Dieter Munz. "High-order discontinuous Galerkin spectral element methods for transitional and turbulent flow simulations." International Journal for Numerical Methods in Fluids 76, no. 8 (August 8, 2014): 522–48. http://dx.doi.org/10.1002/fld.3943.

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19

Yi, Tae-Hyeong, and Francis X. Giraldo. "Vertical Discretization for a Nonhydrostatic Atmospheric Model Based on High-Order Spectral Elements." Monthly Weather Review 148, no. 1 (December 27, 2019): 415–36. http://dx.doi.org/10.1175/mwr-d-18-0283.1.

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Abstract This study addresses the treatment of vertical discretization for a high-order, spectral element model of a nonhydrostatic atmosphere in which the governing equations of the model are separated into horizontal and vertical components by introducing a coordinate transformation, so that one can use different orders and types of approximations in both directions. The vertical terms of the decoupled governing equations are discretized using finite elements based on either Lagrange or basis-spline polynomial functions in the sigma coordinate, while maintaining the high-order spectral elements for the discretization of the horizontal terms. This leads to the fact that the high-order model of spectral elements with a nonuniform grid, interpolated within an element, can be easily accommodated with existing physical parameterizations. Idealized tests are performed to compare the accuracy and efficiency of the vertical discretization methods, in addition to the central finite differences, with those of the standard high-order spectral element approach. Our results show, through all the test cases, that the finite element with the cubic basis-spline function is more accurate than the other vertical discretization methods at moderate computational cost. Furthermore, grid dependency studies in the tests with and without orography indicate that the convergence rate of the vertical discretization methods is lower than the expected level of discretization accuracy, especially in the Schär mountain test, which yields approximately first-order convergence.

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20

Xu, Mohammad Tanzil Hasan &. Chuanju. "High Order Finite Difference/Spectral Methods to a Water Wave Model with Nonlocal Viscosity." Journal of Computational Mathematics 38, no. 4 (June 2020): 580–605. http://dx.doi.org/10.4208/jcm.1902-m2017-0280.

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21

Zudrop, J., and J. S. Hesthaven. "Accuracy of High Order and Spectral Methods for Hyperbolic Conservation Laws with Discontinuous Solutions." SIAM Journal on Numerical Analysis 53, no. 4 (January 2015): 1857–75. http://dx.doi.org/10.1137/140992758.

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22

Wahlbin, Lars B., Joseph D. Ward, Lars B. Wahlbin, and S. S. Wagstaff Jr. "Book Review: Proceedings of the Third International Conference on Spectral and High Order Methods." Mathematics of Computation 67, no. 223 (July 1, 1998): 1331–35. http://dx.doi.org/10.1090/s0025-5718-98-01012-6.

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23

Zhang, Nan, Zhiping Mao null, and Tao Xiong. "High Order Conservative Finite Difference/Fourier Spectral Methods for Inviscid Surface Quasi-Geostrophic Flows." Communications in Computational Physics 32, no. 5 (June 2022): 1474–509. http://dx.doi.org/10.4208/cicp.oa-2022-0111.

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24

Ekaterinaris, John A. "Aeroacoustic Predictions Using High-Order Shock-Capturing Schemes." International Journal of Aeroacoustics 2, no. 2 (April 2003): 175–92. http://dx.doi.org/10.1260/147547203322775524.

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High-order accurate, finite-difference methods, such as the compact centered schemes with spectral-type or characteristic-based filters and the weighted essentially non-oscillatory (WENO) schemes, which are used in high resolution CFD solutions and for DNS or LES of compressible turbulence, are applied to aeroacoustics. Implicit and explicit schemes are used for time marching. The accuracy of the numerical solutions is evaluated for test problems. It is found that these methods are appropriate for sound propagation in complex flows that require use of curvilinear coordinates. Therefore they are applicable for the prediction of sound generation from both smooth subsonic flows, and transonic or supersonic flows with discontinuities.

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25

FENG, QUANDONG, JINGFANG HUANG, NINGMING NIE, ZAIJIU SHANG, and YIFA TANG. "IMPLEMENTING ARBITRARILY HIGH-ORDER SYMPLECTIC METHODS VIA KRYLOV DEFERRED CORRECTION TECHNIQUE." International Journal of Modeling, Simulation, and Scientific Computing 01, no. 02 (June 2010): 277–301. http://dx.doi.org/10.1142/s1793962310000171.

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In this paper, an efficient numerical procedure is presented to implement the Gaussian Runge–Kutta (GRK) methods (also called Gauss methods). The GRK technique first discretizes each marching step of the initial value problem using collocation formulations based on Gaussian quadrature. As is well known, it preserves the geometric structures of Hamiltonian systems. Existing analysis shows that the GRK discretization with s nodes is of order 2s, A-stable, B-stable, symplectic and symmetric, and hence "optimal" for solving initial value problems of general ordinary differential equations (ODEs). However, as the unknowns at different collocation points are coupled in the discretized system, direct solution of the resulting algebraic equations is in general inefficient. Instead, we use the Krylov deferred correction (KDC) method in which the spectral deferred correction (SDC) scheme is applied as a preconditioner to decouple the original system, and the resulting preconditioned nonlinear system is solved efficiently using Newton–Krylov schemes such as Newton–GMRES method. The KDC accelerated GRK methods have been applied to several Hamiltonian systems and preliminary numerical results are presented to show the accuracy, stability, and efficiency features of these methods for different accuracy requirements in short- and long-time simulations.

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26

Gheorghiu, Călin-Ioan. "Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations." Computation 9, no. 1 (December 29, 2020): 2. http://dx.doi.org/10.3390/computation9010002.

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We are concerned with the study of some classical spectral collocation methods, mainly Chebyshev and sinc as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrödinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them in conjunction with the usual ones. In order to resolve a boundary singularity, we use Chebfun with domain truncation. Although it is applicable with spectral collocation, a special technique to introduce boundary conditions as well as a coordinate transform, which maps an unbounded domain to a finite one, are the special ingredients. A challenging set of “hard”benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting, etc.) fail, were analyzed. In order to separate “good”and “bad”eigenvalues, we have estimated the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as to problems with a mixed spectrum.

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27

Gao, Yu Ling, Qing Huang, Bao Ming Yu, Xiao Tao Kang, and Yao Wu Shi. "False Peaks Suppression based on Cross-High-Order Spectral QR Decomposition Approach." Advanced Materials Research 403-408 (November 2011): 177–81. http://dx.doi.org/10.4028/www.scientific.net/amr.403-408.177.

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In signal processing, a frequently encountered problem is harmonic retrieval in additive colored noise, especially false peaks existence in harmonic signal peaks. The purpose of this paper is to develop an efficient approach to clear the false peaks based on cross-high-order spectral QR decomposition approach. Simulation results indicate that spectral density curve is smooth without false peaks existence. The methods have better in resolving power and performance than previous MUSIC approach. Thus, this approach is ideally suited for harmonic retrieval in additive colored noise and short data conditions, and is also accurate to estimation signal parameter in hybrid colored noises.

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28

Hofstätter, Harald, Othmar Koch, and Mechthild Thalhammer. "Convergence analysis of high-order time-splitting pseudo-spectral methods for rotational Gross–Pitaevskii equations." Numerische Mathematik 127, no. 2 (October 12, 2013): 315–64. http://dx.doi.org/10.1007/s00211-013-0586-9.

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29

Pech, Jan. "Advances in simulations of heat exchange in fluid flows by high-order methods." EPJ Web of Conferences 264 (2022): 01029. http://dx.doi.org/10.1051/epjconf/202226401029.

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The article presents high-order simulations of heat transfer in fluid flow. Heat induced changes in material properties (density, viscosity and heat conductivity) substantially influence flow structures. The change of fluid density implies naturally the thermal expansion what results in non-homogeneous divergence of velocity and the model of evolutionary Navier-Stokes-Fourier equations is not strictly incompressible in this case. Proposed algorithm treats the model as a fully coupled system of equations. Applied method of operator splitting is promissing to become a more general concept for numerical modelling of active quantities advected by incompressible fluids. This work follows the 2D results presented during EFM 2019 and extends to 3D together with advances of the code reimplementation to current version of the Nektar++ library (open-source framework for spectral/hp finite elements). Results for various settings for both the forced/mixed and natural convection are shown.

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30

Herrington, Adam R., Peter H. Lauritzen, Mark A. Taylor, Steve Goldhaber, Brian E. Eaton, Julio T. Bacmeister, Kevin A. Reed, and Paul A. Ullrich. "Physics–Dynamics Coupling with Element-Based High-Order Galerkin Methods: Quasi-Equal-Area Physics Grid." Monthly Weather Review 147, no. 1 (January 2019): 69–84. http://dx.doi.org/10.1175/mwr-d-18-0136.1.

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Atmospheric modeling with element-based high-order Galerkin methods presents a unique challenge to the conventional physics–dynamics coupling paradigm, due to the highly irregular distribution of nodes within an element and the distinct numerical characteristics of the Galerkin method. The conventional coupling procedure is to evaluate the physical parameterizations ( physics) on the dynamical core grid. Evaluating the physics at the nodal points exacerbates numerical noise from the Galerkin method, enabling and amplifying local extrema at element boundaries. Grid imprinting may be substantially reduced through the introduction of an entirely separate, approximately isotropic finite-volume grid for evaluating the physics forcing. Integration of the spectral basis over the control volumes provides an area-average state to the physics, which is more representative of the state in the vicinity of the nodal points rather than the nodal point itself and is more consistent with the notion of a “large-scale state” required by conventional physics packages. This study documents the implementation of a quasi-equal-area physics grid into NCAR’s Community Atmosphere Model Spectral Element and is shown to be effective at mitigating grid imprinting in the solution. The physics grid is also appropriate for coupling to other components within the Community Earth System Model, since the coupler requires component fluxes to be defined on a finite-volume grid, and one can be certain that the fluxes on the physics grid are, indeed, volume averaged.

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31

Ivanov, Valentin D., Lodovico Coccato, Mark J. Neeser, Fernando Selman, Alessandro Pizzella, Elena Dalla Bontà, Enrico M. Corsini, and Lorenzo Morelli. "MUSE library of stellar spectra." Astronomy & Astrophysics 629 (September 2019): A100. http://dx.doi.org/10.1051/0004-6361/201936178.

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Context. Empirical stellar spectral libraries have applications in both extragalactic and stellar studies, and they confer an advantage over theoretical libraries because they naturally include all relevant chemical species and physical processes. In recent years we have seen a stream of new sets of high-quality spectra, but increasing the spectral resolution and widening the wavelength coverage means resorting to multi-order echelle spectrographs. Assembling the spectra from many pieces results in lower fidelity of their shapes. Aims. We aim to offer the community a library of high-signal-to-noise spectra with reliable continuum shapes. Furthermore, the use of an integral field unit (IFU) alleviates the issue of slit losses. Methods. Our library was built with the MUSE (Multi-Unit Spectroscopic Explorer) IFU instrument. We obtained spectra over nearly the entire visual band (λ ∼ 4800–9300 Å). Results. We assembled a library of 35 high-quality MUSE spectra for a subset of the stars from the X-shooter Spectral Library. We verified the continuum shape of these spectra with synthetic broadband colors derived from the spectra. We also report some spectral indices from the Lick system, derived from the new observations. Conclusions. We offer a high-fidelity set of stellar spectra covering the Hertzsprung-Russell diagram. These can be used for both extragalactic and stellar studies and demonstrate that the IFUs are excellent tools for building reliable spectral libraries.

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32

Adibekyan, V., S. G. Sousa, N. C. Santos, P. Figueira, C. Allende Prieto, E. Delgado Mena, J. I. González Hernández, et al. "Benchmark stars, benchmark spectrographs." Astronomy & Astrophysics 642 (October 2020): A182. http://dx.doi.org/10.1051/0004-6361/202038793.

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Context. Gaia benchmark stars are selected to be calibration stars for different spectroscopic surveys. Very high-quality and homogeneous spectroscopic data for these stars are therefore required. We collected ultrahigh-resolution ESPRESSO spectra for 30 of the 34 Gaia benchmark stars and made them public. Aims. We quantify the consistency of the results that are obtained with different high- (R ~ 115 000), and ultrahigh- (R ~ 220 000) resolution spectrographs. We also comprehensively studied the effect of using different spectral reduction products of ESPRESSO on the final spectroscopic results. Methods. We used ultrahigh- and high-resolution spectra obtained with the ESPRESSO, PEPSI, and HARPS spectrographs to measure spectral line characteristics (line depth; line width; and equivalent width, EW) and determined stellar parameters and abundances for a subset of 11 Gaia benchmark stars. We used the ARES code for automatic measurements of the spectral line parameters. Results. Our measurements reveal that the same individual spectral lines measured from adjacent 2D (spectrum in the wavelength-order space) echelle orders of ESPRESSO spectra differ slightly in line depth and line width. When a long list of spectral lines is considered, the EW measurements based on the 2D and 1D (the final spectral product) ESPRESSO spectra agree very well. The EW spectral line measurements based on the ESPRESSO, PEPSI, and HARPS spectra also agree to within a few percent. However, we note that the lines appear deeper in the ESPRESSO spectra than in PEPSI and HARPS. The stellar parameters derived from each spectrograph by combining the several available spectra agree well overall. Conclusions. We conclude that the ESPRESSO, PEPSI, and HARPS spectrographs can deliver spectroscopic results that are sufficiently consistent for most of the science cases in stellar spectroscopy. However, we found small but important differences in the performance of the three spectrographs that can be crucial for specific science cases.

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33

Ai, Changfa, Youxuan Zhao, Peng Cao, Enhui Yang, and Yanjun Qiu. "High-Order Spectral Finite Elements in Analysis of Collinear Wave Mixing." Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/260641.

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Implementing collinear wave mixing techniques with numerical methods to detect acoustic nonlinearity due to damage and defects is of vital importance in nondestructive examination engineering. However, numerical simulations in existing literatures are often limited due to the compromise between computational efficiency and accuracy. In order to balance the contradiction, spectral finite element (abbreviated as SFE) with 3 × 3 and 8 × 6 nodes is developed to simulate collinear wave mixing for 1D and 2D cases in this study. The comparisons among analytical solutions, experiments, finite element method (FEM), and spectral finite element method are presented to validate the feasibility, efficiency, and accuracy of the proposed SFEs. The results demonstrate that the proposed SFEs are capable of increasing computational efficiency by as much as 14 times while maintaining the same accuracy in comparison with FEM. In addition, five 3 × 3 nodes’ SFEs or one 8 × 6 nodes’ SFE per the shortest wavelength is sufficient to capture mixing waves. Finally, the proposed 8 × 6 nodes’ SFE is recommended for collinear wave mixing to detect damage, which can offer more accuracy with similar efficiency compared to 3 × 3 nodes’ SFE.

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34

Katkovsky, Leonid, Boris Beliaev, Volha Siliuk, Mikhail Beliaev, Erik Sarmin, and Yurii Davidovich. "Remote spectral methods for detecting stress coniferous." E3S Web of Conferences 223 (2020): 02004. http://dx.doi.org/10.1051/e3sconf/202022302004.

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The article presents investigation of the possibility of drying coniferous forest areas detecting by multispectral satellite data in the visible and NIR spectral range with low spatial resolution, obtained by the imaging systems of three satellites - the Belarusian spacecraft (BS), Landsat 8 and Sentinel 2. A forest area in the south of Belarus was considered as a test site. High-resolution multispectral airborne data and, in part, ground measurements were used as reference ground data by which training samples were formed. Most of the known classical methods of supervised classification have been tested, the maximum likelihood method turned out to be the best for this task. In order to improve the accuracy of identifying the drying areas of coniferous forests on multispectral images, parametric transformations of images in the spectral space are proposed, which should lead to an increase in initial small spectral differences. The methodological issues of assessing the accuracy of the satellite images classification are considered using the result of the classification of airborne image with high spatial resolution as a ground truth image. The assessment of the classification accuracy, both visually and using the obtained confusion matrices, allows us to conclude that the images of the BS, Landsat 8 and Sentinel 2 can be used to detect drying area of coniferous forests as well as the expediency of carrying out the proposed transformations of the original images.

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35

Alsayyed, Omar, Fadi Awawdeh, Safwan Al-Shara’, and Edris Rawashdeh. "High-Order Schemes for Nonlinear Fractional Differential Equations." Fractal and Fractional 6, no. 12 (December 19, 2022): 748. http://dx.doi.org/10.3390/fractalfract6120748.

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We propose high-order schemes for nonlinear fractional initial value problems. We split the fractional integral into a history term and a local term. We take advantage of the sum of exponentials (SOE) scheme in order to approximate the history term. We also use a low-order quadrature scheme to approximate the fractional integral appearing in the local term and then apply a spectral deferred correction (SDC) method for the approximation of the local term. The resulting one-step time-stepping methods have high orders of convergence, which make adaptive implementation and accuracy control relatively simple. We prove the convergence and stability of the proposed schemes. Finally, we provide numerical examples to demonstrate the high-order convergence and adaptive implementation.

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36

Qiu, Zihua, Min Xu, Bin Zhang, and Chunlei Liang. "High-Order Spectral Difference Method on 3D Unstructured Grids via Mixed Elements." Xibei Gongye Daxue Xuebao/Journal of Northwestern Polytechnical University 37, no. 5 (October 2019): 968–76. http://dx.doi.org/10.1051/jnwpu/20193750968.

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The high-order methods is difficultly applied in various elements. The development of a 3D solver by using the spectral difference method of unstructured grids via mixed elements is presented. A mixed tri-prism and tetrahedral grid is firstly refined using one-level h-refinement to generate a hexahedral grid while keeping the curvature of wall boundaries. The SD method designed for hexahedral elements can subsequently be applied for refining the unstructured grid. Through a series of numerical tests, the present method is high-order accurate for both inviscid and viscous flows is demonstrated; the results obtained for inviscid and viscous compressible flows compare well with other published results.

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37

Sucharitha, B., and Dr K. Anitha Sheela. "Compression of Hyper Spectral Images using Tensor Decomposition Methods." International Journal of Circuits, Systems and Signal Processing 16 (October 7, 2022): 1148–55. http://dx.doi.org/10.46300/9106.2022.16.138.

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Tensor decomposition methods have beenrecently identified as an effective approach for compressing high-dimensional data. Tensors have a wide range of applications in numerical linear algebra, chemo metrics, data mining, signal processing, statics, and data mining and machine learning. Due to the huge amount of information that the hyper spectral images carry, they require more memory to store, process and send. We need to compress the hyper spectral images in order to reduce storage and processing costs. Tensor decomposition techniques can be used to compress the hyper spectral data. The primary objective of this work is to utilize tensor decomposition methods to compress the hyper spectral images. This paper explores three types of tensor decompositions: Tucker Decomposition (TD_ALS), CANDECOMP/PARAFAC (CP) and Tucker_HOSVD (Higher order singular value Decomposition) and comparison of these methods experimented on two real hyper spectral images: the Salinas image (512 x 217 x 224) and Indian Pines corrected (145 x 145 x 200). The PSNR and SSIM are used to evaluate how well these techniques work. When compared to the iterative approximation methods employed in the CP and Tucker_ALS methods, the Tucker_HOSVD method decomposes the hyper spectral image into core and component matrices more quickly. According to experimental analysis, Tucker HOSVD's reconstruction of the image preserves image quality while having a higher compression ratio than the other two techniques.

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Bohm, Marvin, Sven Schermeng, Andrew R. Winters, Gregor J. Gassner, and Gustaaf B. Jacobs. "Multi-element SIAC Filter for Shock Capturing Applied to High-Order Discontinuous Galerkin Spectral Element Methods." Journal of Scientific Computing 81, no. 2 (August 26, 2019): 820–44. http://dx.doi.org/10.1007/s10915-019-01036-8.

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Sevilla, Ruben, and Régis Cottereau. "Influence of periodically fluctuating material parameters on the stability of explicit high-order spectral element methods." Journal of Computational Physics 373 (November 2018): 304–23. http://dx.doi.org/10.1016/j.jcp.2018.07.002.

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Williams, Christopher R., Maximilian Maahn, Joseph C. Hardin, and Gijs de Boer. "Clutter mitigation, multiple peaks, and high-order spectral moments in 35 GHz vertically pointing radar velocity spectra." Atmospheric Measurement Techniques 11, no. 9 (September 3, 2018): 4963–80. http://dx.doi.org/10.5194/amt-11-4963-2018.

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Abstract. This study presents and applies three separate processing methods to improve high-order moments estimated from 35 GHz (Ka band) vertically pointing radar Doppler velocity spectra. The first processing method removes Doppler-shifted ground clutter from spectra collected by a US Department of Energy (DOE) Atmospheric Radiation Measurement (ARM) program Ka-band zenith pointing radar (KAZR) deployed at Oliktok Point (OLI), Alaska. Ground clutter resulted from multiple pathways through antenna side lobes and reflections off a rotating scanning radar antenna located 2 m away from KAZR, which caused Doppler shifts in ground clutter returns from stationary targets 2.5 km away. After removing clutter in the recorded velocity spectra, the second processing method identifies multiple separate and sub-peaks in the spectra and estimates high-order moments for each peak. Multiple peaks and high-order moments were estimated for both original 2 and 15 s averaged spectra. The third processing step improves the spectrum variance, skewness, and kurtosis estimates by removing velocity variability due to turbulent broadening during 15 s averaging intervals. Applying the multiple peak processing to Doppler velocity spectra during liquid-only clouds can identify cloud and drizzle particles and during mixed-phase clouds can identify liquid cloud and frozen hydrometeors. Consistent with previous studies, this work found that spectrum skewness assuming only a single spectral peak was a good indicator of two hydrometeor populations (for example, cloud and drizzle particles) being present in the radar pulse volume. Yet, after dividing the spectrum into multiple peaks, velocity spectrum skewness for individual peaks is near zero, indicating nearly symmetric peaks. This suggests that future studies should use velocity skewness of single-peak spectra as an indicator of possible multiple hydrometeor populations and then use multiple-peak moments for quantitative studies. Three future activities will continue this work. First, KAZR spectra from several ARM sites have been processed and are available in the ARM archive as a principal investigator (PI) product. ARM programmers are evaluating these processing methods as part of future multiple-peak products generated by ARM. Third, MATLAB code generating the Oliktok Point products has been uploaded as supplemental material for public dissemination.

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Guerra, Jorge E., and Paul A. Ullrich. "A high-order staggered finite-element vertical discretization for non-hydrostatic atmospheric models." Geoscientific Model Development 9, no. 5 (June 1, 2016): 2007–29. http://dx.doi.org/10.5194/gmd-9-2007-2016.

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Abstract. Atmospheric modeling systems require economical methods to solve the non-hydrostatic Euler equations. Two major differences between hydrostatic models and a full non-hydrostatic description lies in the vertical velocity tendency and numerical stiffness associated with sound waves. In this work we introduce a new arbitrary-order vertical discretization entitled the staggered nodal finite-element method (SNFEM). Our method uses a generalized discrete derivative that consistently combines the discontinuous Galerkin and spectral element methods on a staggered grid. Our combined method leverages the accurate wave propagation and conservation properties of spectral elements with staggered methods that eliminate stationary (2Δx) modes. Furthermore, high-order accuracy also eliminates the need for a reference state to maintain hydrostatic balance. In this work we demonstrate the use of high vertical order as a means of improving simulation quality at relatively coarse resolution. We choose a test case suite that spans the range of atmospheric flows from predominantly hydrostatic to nonlinear in the large-eddy regime. Our results show that there is a distinct benefit in using the high-order vertical coordinate at low resolutions with the same robust properties as the low-order alternative.

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WELLS, J. C., V. E. OBERACKER, M. R. STRAYER, and A. S. UMAR. "SPECTRAL PROPERTIES OF DERIVATIVE OPERATORS IN THE BASIS-SPLINE COLLOCATION METHOD." International Journal of Modern Physics C 06, no. 01 (February 1995): 143–67. http://dx.doi.org/10.1142/s0129183195000125.

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We discuss the basis-spline collocation method for the lattice solution of boundary-value differential equations, drawing particular attention to the difference between lattice and continuous collocation methods. Spectral properties of the basis-spline lattice representation of the first and second spatial derivatives are studied for the case of periodic boundary conditions with homogeneous lattice spacing and compared to spectra obtained using traditional finite-difference schemes. Basis-spline representations are shown to give excellent resolution on small-length scales and to satisfy the chain rule with good fidelity for the lattice-derivative operators using high-order splines. Application to the one-dimensional Dirac equation shows that very high-order spline representations of the Hamiltonian on odd lattices avoid the notorious spectral-doubling problem.

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Kirincich, Anthony. "Improved Detection of the First-Order Region for Direction-Finding HF Radars Using Image Processing Techniques." Journal of Atmospheric and Oceanic Technology 34, no. 8 (August 2017): 1679–91. http://dx.doi.org/10.1175/jtech-d-16-0162.1.

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AbstractFor direction-finding high-frequency (HF) radar systems, the correct separation of backscattered spectral energy due to Bragg resonant waves from that due to more complex double-scattering represents a critical first step toward attaining accurate estimates of surface currents from the range-dependent radar backscatter. Existing methods to identify this “first order” region of the spectra, generally sufficient for lower-frequency radars and low-velocity or low-surface gravity wave conditions, are more likely to fail in higher-frequency systems or locations with more variable current, wave, or noise regimes, leading to elevated velocity errors. An alternative methodology is presented that uses a single and globally relevant smoothing length scale, careful pretreatment of the spectra, and marker-controlled watershed segmentation, an image processing technique, to separate areas of spectral energy due to surface currents from areas of spectral energy due to more complex scattering by the wave field or background noise present. Applied to a number of HF radar datasets with a range of operating frequencies and characteristic issues, the new methodology attains a higher percentage of successful first-order identification, particularly during complex current and wave conditions. As operational radar systems continue to expand to more systematically cover areas of high marine traffic, close approaches to ports and harbors, or offshore energy installations, use of this type of updated methodology will become increasingly important to attain accurate current estimates that serve both research and operational interests.

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Gheorghiu, Călin-Ioan. "Accurate Spectral Collocation Solutions to 2nd-Order Sturm–Liouville Problems." Symmetry 13, no. 3 (February 27, 2021): 385. http://dx.doi.org/10.3390/sym13030385.

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This work is about the use of some classical spectral collocation methods as well as with the new software system Chebfun in order to compute the eigenpairs of some high order Sturm–Liouville eigenproblems. The analysis is divided into two distinct directions. For problems with clamped boundary conditions, we use the preconditioning of the spectral collocation differentiation matrices and for hinged end boundary conditions the equation is transformed into a second order system and then the conventional ChC is applied. A challenging set of “hard” benchmark problems, for which usual numerical methods (FD, FE, shooting, etc.) encounter difficulties or even fail, are analyzed in order to evaluate the qualities and drawbacks of spectral methods. In order to separate “good” and “bad” (spurious) eigenvalues, we estimate the drift of the set of eigenvalues of interest with respect to the order of approximation N. This drift gives us a very precise indication of the accuracy with which the eigenvalues are computed, i.e., an automatic estimation and error control of the eigenvalue error. Two MATLAB codes models for spectral collocation (ChC and SiC) and another for Chebfun are provided. They outperform the old codes used so far and can be easily modified to solve other problems.

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Childs, Andrew M., Jin-Peng Liu, and Aaron Ostrander. "High-precision quantum algorithms for partial differential equations." Quantum 5 (November 10, 2021): 574. http://dx.doi.org/10.22331/q-2021-11-10-574.

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Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for linear ordinary differential equations are well established, the best previous quantum algorithms for linear partial differential equations (PDEs) have complexity poly(1/ϵ), where ϵ is the error tolerance. By developing quantum algorithms based on adaptive-order finite difference methods and spectral methods, we improve the complexity of quantum algorithms for linear PDEs to be poly(d,log⁡(1/ϵ)), where d is the spatial dimension. Our algorithms apply high-precision quantum linear system algorithms to systems whose condition numbers and approximation errors we bound. We develop a finite difference algorithm for the Poisson equation and a spectral algorithm for more general second-order elliptic equations.

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Qiu, Lin, Linhai Jing, Baoxin Hu, Hui Li, and Yunwei Tang. "A New Individual Tree Crown Delineation Method for High Resolution Multispectral Imagery." Remote Sensing 12, no. 3 (February 10, 2020): 585. http://dx.doi.org/10.3390/rs12030585.

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In current individual tree crown (ITC) delineation methods for high-resolution multispectral imagery, either a spectral band or a brightness component of the multispectral image is employed in delineation with reference to edges or shapes of crowns, whereas spectra of tree crowns are seldom taken into account. Such methods normally perform well in coniferous forests with obvious between-crown shadows, but fail in dense deciduous or mixed forests, in which tree crowns are close to each other, between-crown shadows and boundaries are unobvious, whereas adjacent tree crowns may be of distinguishable spectra. In order to effectively delineate crowns in dense deciduous or mixed forests, a new ITC delineation method using both brightness and spectra of the image is proposed in this study. In this method, a morphological gradient map of the image is first generated, treetops of multi-scale crowns are extracted from the gradient map and refined regarding the spectral differences between neighboring crowns, the gradient map is segmented using a watershed approach with treetops as markers, and the resulting segmentation map is refined to yield a crown map. Evaluated on images of a rainforest and a deciduous forest, the proposed method more accurately delineated adjacent broad-leaved tree crowns with similar brightness but different spectra than the other two typical ITC delineation algorithms, achieving a delineation accuracy of up to 76% in the rainforest and 63% in the deciduous forest.

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Abd-Elhameed, Waleed Mohamed, Mohamed Salem Al-Harbi, Amr Kamel Amin, and Hany M. M. Ahmed. "Spectral Treatment of High-Order Emden–Fowler Equations Based on Modified Chebyshev Polynomials." Axioms 12, no. 2 (January 17, 2023): 99. http://dx.doi.org/10.3390/axioms12020099.

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This paper is devoted to proposing numerical algorithms based on the use of the tau and collocation procedures, two widely used spectral approaches for the numerical treatment of the initial high-order linear and non-linear equations of the singular type, especially those of the high-order Emden–Fowler type. The class of modified Chebyshev polynomials of the third-kind is constructed. This class of polynomials generalizes the class of the third-kind Chebyshev polynomials. A new formula that expresses the first-order derivative of the modified Chebyshev polynomials in terms of their original modified polynomials is established. The establishment of this essential formula is based on reducing a certain terminating hypergeometric function of the type 5F4(1). The development of our suggested numerical algorithms begins with the extraction of a new operational derivative matrix from this derivative formula. Expansion’s convergence study is performed in detail. Some illustrative examples of linear and non-linear Emden–Flower-type equations of different orders are displayed. Our proposed algorithms are compared with some other methods in the literature. This confirms the accuracy and high efficiency of our presented algorithms.

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ZHU, JUN, NING ZHAO, and HUASHENG ZHENG. "HIGH ORDER LOCALIZED ENO SCHEMES ON UNSTRUCTURED MESHES FOR CONSERVATION LAWS." Modern Physics Letters B 19, no. 28n29 (December 20, 2005): 1563–66. http://dx.doi.org/10.1142/s0217984905009912.

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We construct a localized finite volume method by applying ENO (essentially non-oscillatory) reconstruction to solve hyperbolic conservation laws following the partitions of the spectral volume methods. The main idea is as follows: Firstly, separate the calculating domain into intervals, named main-cells, then divide the intervals into subintervals, named sub-cells. Secondly, use ENO methodology to reconstruct conservative variables in the main-cells by using the cell averages of proper sub-cells. After that, use the TVD Runge-Kutta time discrete method to obtain fully discrete scheme. Several classic numerical tests show that this scheme has capabilities to capture discontinuities in high resolution and robustness.

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Hambli, Walid, James Slaughter, Filipe Fabian Buscariolo, and Spencer Sherwin. "Extension of Spectral/hp Element Methods towards Robust Large-Eddy Simulation of Industrial Automotive Geometries." Fluids 7, no. 3 (March 14, 2022): 106. http://dx.doi.org/10.3390/fluids7030106.

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A spectral/hp element methodology is utilised to investigate the SAE Notchback geometry with 20∘ backlight and 3∘ diffuser at Re=2.3×106. The study presented here considered two different mesh approaches: one focusing on classical h-type refinement with standard solution polynomial order (HFP3) and a second case considering relatively coarse mesh combined with high solution polynomial order (HCP5). For the same targeted number of degrees of freedom in both meshes, the results show significant differences in vorticity, flow structures and surface pressure. The first guidelines for hp refinement strategy are deduced for complex industrial cases. Further work on investigating the requirements for these hybrid techniques is required in order to maximize the benefits of the solution and mesh refinements in spectral/hp element method simulations.

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Han, Yanling, Cong Wei, Ruyan Zhou, Zhonghua Hong, Yun Zhang, and Shuhu Yang. "Combining 3D-CNN and Squeeze-and-Excitation Networks for Remote Sensing Sea Ice Image Classification." Mathematical Problems in Engineering 2020 (April 7, 2020): 1–15. http://dx.doi.org/10.1155/2020/8065396.

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Sea ice is one of the most prominent marine disasters in high latitudes. Remote sensing technology provides an effective means for sea ice detection. Remote sensing sea ice images contain rich spectral and spatial information. However, most traditional methods only focus on spectral information or spatial information, and do not excavate the feature of spectral and spatial simultaneously in remote sensing sea ice images classification. At the same time, the complex correlation characteristics among spectra and small sample problem in sea ice classification also limit the improvement of sea ice classification accuracy. For this issue, this paper proposes a new remote sensing sea ice image classification method based on squeeze-and-excitation (SE) network, convolutional neural network (CNN), and support vector machines (SVMs). The proposed method designs 3D-CNN deep network so as to fully exploit the spatial-spectrum features of remote sensing sea ice images and integrates SE-Block into 3D-CNN in-depth network in order to distinguish the contributions of different spectra to sea ice classification. According to the different contributions of spectral features, the weight of each spectral feature is optimized by fusing SE-Block in order to further enhance the sample quality. Finally, information-rich and representative samples are chosen by combining the idea of active learning and input into SVM classifier, and this achieves superior classification accuracy of remote sensing sea ice images with small samples. In order to verify the effectiveness of the proposed method, we conducted experiments on three different data from Baffin Bay, Bohai Bay, and Liaodong Bay. The experimental results show that compared with other classical classification methods, the proposed method comprehensively considers the correlation among spectral features and the small samples problems and deeply excavates the spatial-spectrum characteristics of sea ice and achieves better classification performance, which can be effectively applied to remote sensing sea ice image classification.

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