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Explicit gaussian quadrature Rules for C1 Cubic Splines with Non-uniform Knot Sequences

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Communications in Mathematics and Statistics 2021年第3期9卷 331-345页

作者： Peng Chen Xin Li School of Mathematical Sciences University of Science and Technology of ChinaHefei 230026People’s Republic of China

This paper provides the explicit and optimal quadrature rules for the cubic C1 spline space,which is the extension of the results in Ait-Haddou et al.(J Comput Appl Math 290:543–552,2015)for less restricted non-uniform knot values.The rules are optimal in the sense that there exist no other quadrature rules with fewer quadrature points to exactly integrate the functions in the given spline space.The explicit means that the quadrature nodes and weights are derived via an explicit recursive formula.Numerical experiments and the error estimations of the quadrature rules are also presented in the end.

关键词： gaussian quadrature Non-uniform Isogeometric analysis Cubic splines

Reliability-Optimal Pilot-Assisted Transmission for URLLC over Non-Reciprocal MISO Channels: TDD or FDD?

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China Communications 2022年第4期19卷 14-27页

作者： Yuncong Xie Pinyi Ren School of Information and Communications Engineering Xi’an Jiaotong UniversityXi’an 710049China Shaanxi Smart Networks and Ubiquitous Access Research Center Xi’an 710049China

In this paper, we focus on the pilot-assisted transmission design for downlink URLLC over nonreciprocal channels, in which the multi-antenna controller sends mission-critical data signals to a singleantenna actuator. In this system, the prior knowledge of downlink channel state information(CSI) is a prerequisite for reliable data transmission. Generally, the acquisition of downlink CSI is completed either via the uplink pilot measurement exploiting channel reciprocity and time-division duplex(TDD) operation, or via the downlink pilot measurement with quantized feedback and frequency division duplex(FDD) operation. Inspired by this, we aim to investigate how the degree of channel non-reciprocity impacts the transmission reliability of our URLLC system, and the superiority between the TDD mode and FDD mode in terms of transmission reliability maximization. To describe the degree of reliability loss, we derive the closed-form approximations on the transmission error probability of URLLC in TDD and FDD modes, via leveraging the Gauss-Hermite and Gauss-Chebyshev quadrature rules. Following by the theoretical approximations, we demonstrate how to determine the optimal training pilot length and quantized feedback duration that maximize the transmission reliability under given latency constraint. Through numerical results,we validate the accuracy of theoretical approximations derived in this paper, and obtain some meaningful conclusions.

关键词： URLLC channel non-reciprocity channel training quantized feedback gaussian quadrature

EMT simulation and effect of TTI anisotropic media in EMT signal

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Petroleum Science 2021年第1期18卷 106-122页

作者： Qing-Yun Di Olalekan Fayemi Qi-Hui Zhen Tian Fei Key Laboratory of Shale Gas and Geoengineering Institute of Geology and GeophysicsChinese Academy of SciencesBeijing 100029China Frontier Technology and Equipment Development Center for Deep Resources Exploration Institute of Geology and GeophysicsChinese Academy of SciencesBeijing 100029China Innovation Academy for Earth Science Chinese Academy of SciencesBeijing 100029China College of Earth and Planetary Sciences University of Chinese Academy of SciencesBejing 100049China

An axisymmetric finite difference method is employed for the simulations of electromagnetic telemetry in the homogeneous and layered underground formation.In this method,we defined the anisotropy property using extensive 2D conductivity tensor and solved it in the transverse magnetic mode.Significant simplification arises in the decoupling of the anisotropic parameter.The developed method is cost-efficient,more straightforward in modeling anisotropic media,and easy to be implemented.In addition,we solved the integral operation in the estimation of measured surface voltage using gaussian quadrature technique.We performed a series of numerical modeling of EM telemetry signals in both isotropic and anisotropic models.Experiment with 2D tilt transverse isotropic media characterized by the tilt axis and anisotropy parameters shows an increase in the EMT signal with an increase in the angle of tilt of the principal axis for a moderate coefficient of anisotropy.We show that the effect of the tilt of the subsurface medium can be observed with sufficient accuracy and that it is an order of magnitude of 5 over the tilt of 90 degrees.Lastly,consistent results with existing field data were obtained by employing the gaussian quadrature rule for the computation of surface measured signal.

关键词： Electromagnetic telemetry Finite difference method Anisotropy gaussian quadrature

Comparison study of several numerical integration schemes used in calculations of density functional theory

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Chinese Journal of Chemistry 1996年第4期14卷 289-296页

作者： Hong, GY Li, LM BEIJING UNIV COLL CHEM & MOL ENGNSTATE KEY LAB RARE EARTH MAT CHEM & APPLICATBEIJING 100871PEOPLES R CHINA

Several numerical integration schemes for the evaluation of matrix elements in density functional theory calculations have been studied and compared by computational practice. The best scheme was found to be the combination of the atomic partition [function proposed by Becke with the scaled generalized Gauss‐Laguerre quadrature formula for radial integration suggested by Yang, which achieve the highest convergence rate to the numerical integration. With the same number of integration points, the accuracy of the calculated results by this scheme is higher by 1 to 2 orders of magnitudes than that by other schemes. The reason for achieving higher accuracy by this scheme has been proposed preliminarily.

关键词： density functional theory partition function numerical integration gaussian quadrature numerical error

A Spectral Time-Domain Method for Computational Electrodynamics

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Advances in Applied Mathematics and Mechanics 2009年第6期1卷 781-798页

作者： James V.Lambers Department of Mathematics University of Southern MississippiHattiesburgMS 39406-0001 USA

Ever since its introduction by Kane Yee over forty years ago,the finitedifference time-domain(FDTD)method has been a widely-used technique for solving the time-dependent Maxwell’s equations that has also inspired many other methods.This paper presents an alternative approach to these equations in the case of spatially-varying electric permittivity and/or magnetic permeability,based on Krylov subspace spectral(KSS)methods.These methods have previously been applied to the variable-coefficient heat equation and wave equation,and have demonstrated high-order accuracy,as well as stability characteristic of implicit timestepping schemes,even though KSS methods are explicit.KSS methods for scalar equations compute each Fourier coefficient of the solution using techniques developed by Golub and Meurant for approximating elements of functions of matrices by gaussian quadrature in the spectral,rather than physical,domain.We show how they can be generalized to coupled systems of equations,such as Maxwell’s equations,by choosing appropriate basis functions that,while induced by this coupling,still allow efficient and robust computation of the Fourier coefficients of each spatial component of the electric and magnetic fields.We also discuss the application of block KSS methods to problems involving non-self-adjoint spatial differential operators,which requires a generalization of the block Lanczos algorithm of Golub and Underwood to unsymmetric matrices.

关键词： Spectral methods gaussian quadrature block Lanczos method Maxwell’s equations

Some Relations Between Unscented and Cubature Kalman Filters

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Some Relations Between Unscented and Cubature Kalman Filters

A Numerical Tank through Linear and Quadratic Finite Element Methods

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作者： Yongfang Nie Tao Zhang Department of Automation Tsinghua University Department of Strategic Missile and Underwater Weapon Naval Submarine Academy

The unscented Kalman filter（UKF） and the cubature Kalman filter（CKF） are the popular nonlinear gaussian sigma point filters. UKF propagates the sigma points by function evaluations using the unscented transformation（UT） and it is at first glance very different from CKF which is based on the spherical-radial cubature rule. In this paper, some relations between UKF and CKF are analyzed from the perspective of Mahalanobis distance ellipsoid. Two examples are given to demonstrate why UKF and CKF can achieve nearly the same accuracy.

关键词： unscented Kalman filter cubature Kalman filter gaussian quadrature Mahalanobis distance ellipsoid

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A Numerical Tank through Linear and Quadratic Finite Element...