Time-dependent generalized polynomial chaos

Generalized polynomial chaos (gPC) has non-uniform convergence and tends to break down for long-time integration. The reason is that the probability density distribution (PDF) of the solution evolves as a function of time. The set of orthogonal polynomials associated with the initial distribution will therefore not be optimal at later times, thus causing the reduced efficiency of the method for long-time integration. Adaptation of the set of orthogonal polynomials with respect to the changing PDF removes the error with respect to long-time integration. In this method new stochastic variables and orthogonal polynomials are constructed as time progresses. In the new stochastic variable the solution can be represented exactly by linear functions. This allows the method to use only low order polynomial approximations with high accuracy. The method is illustrated with a simple decay model for which an analytic solution is available and subsequently applied to the three mode Kraichnan–Orszag problem with favorable results.     

Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws

We develop a new hierarchical reconstruction (HR) method  and  for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang [9]. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes.     

Gauss quadrature based finite temperature Lanczos method

The finite temperature Lanczos method (FTLM), which is an exact diagonalization method intensively used in quantum many-body calculations, is formulated in the framework of orthogonal polynomials and Gauss quadrature. The main idea is to reduce finite temperature static and dynamic quantities into weighted summations related to one- and two-dimensional Gauss quadratures. Then lower order Gauss quadrature, which is generated from Lanczos iteration, can be applied to approximate the initial weighted summation. This framework fills the conceptual gap between FTLM and kernel polynomial method, and makes it easy to apply orthogonal polynomial techniques in the FTLM calculation.     

On integrability of the Szekeres system. I

The Szekeres system is a four-dimensional system of ?rst-order ordinary differential equations with nonlinear but polynomial (quadratic) right-hand side. It can be derived as a special case of the Einstein equations, related to inhomogeneous and nonsymmetrical evolving spacetime. The paper shows how to solve it and ?nd its three global independent ?rst integrals via Darboux polynomials and Jacobi’s last multiplier method. Thus the Szekeres system is completely integrable. Its two-dimensional subsystem is also investigated: we present its solutions explicitly and discuss its behaviour at in?nity.     

中心遮拦干涉图的圆泽尼克拟合对计算赛德尔像差的影响分析

从波像差的幂级数和圆泽尼克多项式展开理论入手,介绍了圆泽尼克多项式和赛德尔多项式之间的联系,理论上分析了圆泽尼克多项式在环域的相关性,着重讨论了以中心遮拦干涉图的圆泽尼克多项式拟合系数计算赛德尔像差系数的影响。对理论分析进行了实验验证,其结果与理论分析具有良好的一致性,并提出了一种简单直观的误差容限设定方法。研究表明,随着遮拦比的增加,赛德尔系数误差增加,其变化规律和被测元件的像差类型和大小有关。当遮拦比达到某一特定的阈值时,误差曲线将产生较大的变化,为了获得较准确的赛德尔系数,圆泽尼克拟合时应选择适当的阶数;当遮拦比继续增加时,为了计算出准确的赛德尔系数,拟合时应选择环泽尼克多项式。此外,遮拦比对赛德尔系数中畸变、像散的影响较弱,对彗差、场曲、球差的影响较强。     

Calculation of binding energy from moments via Jacobi polynomials

Moments contain essential information for the density of states, and they can be readily computed within the tight-binding framework; therefore, it is desirable to obtain the bond energy from the moments. A linear-scaling moment-based potential via orthogonal polynomials (LMPO) was proposed recently by (Qin, 2010) [29]. Arbitrary orthogonal polynomials can be implemented within the LMPO. This paper will concentrate on Jacobi polynomials, and these polynomials have finite support intervals. In order to match these support intervals with the band edges of the density of states, the band edges need to be obtained first, ideally from the moments. Algorithms obtaining the band edges from the moments are analysed, and a more convergent algorithm is proposed. Then the combination of this band edge algorithm and the LMPO is examined on an s-band system and a d-band tungsten system. As demonstrated from the results, even though the band edges are not fully converged within only a limited number of moments, this combined method is able to converge to the tight-binding results in terms of binding energy. Consequently, this can significantly reduce the computation time.     

基于三维元胞模型的刻蚀工艺表面演化方法

为了更好地理解和认识刻蚀机理, 并为制造工艺提供优化指导, 采用三维元胞模型研究了刻蚀工艺的表面演化过程, 并着重探讨了离子对表面演化过程的影响.针对刻蚀离子入射角度的求解问题, 提出了一种降维分量拟合方法, 将一个三维曲面拟合问题转化为两个二维曲线拟合进行求解, 对入射点的表面法向量计算实现了快速求解, 与采用最小二乘多项式曲面拟合求解离子入射角度相比, 其计算精度和效率都有较大的提高; 对用于拟合计算的表面元胞的选取方法进行了改进, 提高了拟合的准确度.将这种方法应用到硅刻蚀工艺三维仿真中, 其模拟结果与相关实验结果对比, 验证了该方法对刻蚀工艺描述的有效性. 关键词： 刻蚀 三维元胞模型 表面演化算法 降维分量拟合     

基于DDE接口技术的计算机辅助装调方法

根据波前拟合和计算机辅助装调的算法原理,基于动态数据交换（DDE）接口技术和泽尼克多项式拟合技术,采用阻尼最小二乘法,运用MATLAB和ZEMAX联合计算实现了计算机辅助装调。利用DDE接口在ZEMAX和MATLAB之间进行通讯,并用MATLAB编写了能够生成各阶泽尼克圆多项式和环多项式的基底矩阵函数的可视化的计算机辅助装调程序,实现了程序的通用性及易用性。使用该程序对大遮拦比光学系统进行模拟装调,验证了程序的正确性。为了确认泽尼克圆多项式在环域上的相关性对计算机辅助装调结果的影响,分别采用泽尼克圆多项式和环多项式进行模拟装调。模拟结果表明：计算机辅助装调使用这两种泽尼克多项式均可行。     

嵌入随机多项式的抛物方程不确定声场快速算法

为了得到准确且高效计算起伏海洋介质中随机声场的算法,本文将随机多项式展开嵌入到宽角抛物方程声场计算模型(简称RAM模型)中,发展了一种不确定声场的快速算法.其计算结果比使用嵌入随机多项式的窄角抛物方程准确,计算时间小于作为参考的蒙特卡洛方法.在仿真算例中,随机多项式展开法对声强均值、方差、概率密度的计算准确;在一定的随...     

Polynomial approximation method for tomographic reconstruction of three-dimensional refractive index fields with limited data

We propose a polynomial approximation method (PAM) for reconstruction of three-dimensional refractive index fields by interferometric tomography using limited data. Based on the assumption that the fields to be reconstructed are usually smooth and can be decomposed into a finite order of (orthogonal) polynomials, a set of linear equations can be constructed using both the measured projection data and the Radon transform of the basis functions. By solving these equations, the least-squares solutions of expansion coefficients can be obtained and then substituted back to yield the desired fields. Numerical results have demonstrated that the proposed method is fast, robust to noise and can achieve satisfactory results for refractive index fields with limited projection views and large opaque objects.     

复合随机Duffing系统可靠性分析

基于参数α=2的Gegenbauer正交多项式展开方法,研究了大变异系数情况下复合随机强Duffing体系的可靠性分析问题.应用随机空间的正交多项式展开方法,Edgeworth级数逼近技术求取了强非线性随机振动系统响应的前四阶矩以及概率密度函数.基于首次超越模型,讨论了复合随机强Duffing体系的可靠性分析问题.提出了系统动态可靠度与系统平均可靠度的区别、联系以及各自特点,发展了可靠度数值计算公式.分析计算结果与Monte-Carlo模拟结果较好符合,表明该方法的正确与有效. 关键词： 随机Duffing方程 动态可靠度 平均可靠度 Gegenbauer正交多项式     

Orthogonal harmonic polynomials onU(2)

Electric charges and free electromagnetic waves are supposed to be described locally with the same wave differential equation. It is only the topology that is considered to be different. The calculated nonlocalU(2) individuals are characterized by polynomials that belong neither to the classical nor to the Szegö polynomials. The construction of the polynomial solution in component form, their orthogonality over singular measures, the relationships to the Jacobi polynomials, Rodriguez formulas, product decomposition, asymptotic formulas, and completeness are presented in some detail. The possibility is discussed of whether this highly nonlocal model for electric charges can have a physical significance. This work is intended to be a first step for the realization of an old idea of Einstein's (and also commented on by Dirac) to start with the electric charge, not with the Planck constant, as the primary concept for quantum theory.     

Application of group representation theory to derive Hermite interpolation polynomials on a triangle

Current methods used to devise sets of Hermite interpolation polynomials of minimal order that ensure C⁽ⁿ⁾ continuity across triangular element boundaries in two dimensions are not readily extensible to higher dimensions. The extension of such methods is especially difficult when the number of degrees of freedom afforded by data at points is different from the number of degrees of freedom determined by the coefficients of a complete polynomial basis to a particular order. This work introduces a formalism based on group representation theory that can accomplish this task in general. The method is introduced through the derivation of C⁽¹⁾ continuous Hermite polynomials that interpolate data at the three vertices of an equilateral triangular element. These interpolation polynomials are reported here for the first time. The polynomials derived here are compared to the standard polynomials defined in a right triangle by using the two sets of polynomials to solve the Laplace equation over finite elements. The methodology presented here is of use in higher dimensional elements when the complete polynomial degrees of freedom exceed the total C⁽ⁿ⁾ degrees of freedom at the nodes.     

一个新的求解点堆中子动力学方程组的数值方法

使用中子密度一阶泰劳多项式分段近似技术,给出一个新的求解点堆中子动力学方程组的数值方法并采用全隐格式以克服方程组的刚性,同时确保解的必要精度。数值结果表明:在隐式一阶多项式近似下,对合适的反应性输入能够取得足够精确的结果。当反应性给定时,对于求解反应堆动力学问题,能给出一个简法的计算过程。     

基于Wassermann-Wolf方程的共形光学系统设计研究

提出了基于多项式拟合Wassermann-Wolf曲面设计共形光学系统的设计方法，并给出了完整的设计结果.共形光学系统要求导弹整流罩具有流线型几何外观以减少空气阻力，其次考虑导引头的光学系统设计，所以共形整流罩引入的像差通常高达几十甚至一百个波长量级，为导引头光学系统的设计带来了极大的困难.通过多项式拟合Wassermann-Wolf曲面提供共形光学系统初始结构，建立Zernike多项式特殊优化函数取代传统的光学系统评价函数，克服了用传统光学设计方法设计共形光学系统时系统评价函数收敛缓慢的问题，实现了共形光学系统的设计.设计结果表明，系统的调制传递函数在整个目标视场范围内达到了衍射极限.     

Topological charge inversion in polynomial astigmatic Gaussian beams

The evolution of the global topological charge in a general polynomial astigmatic Gaussian beam is investigated. The leading order terms of the polynomial prefactor determines the global topological charge and can be expressed as a product of first order polynomials, each representing an optical vortex function. We show that the global topological charge is bounded by the order of the polynomial and change during propagation in steps of 2 every time one of the optical vortices undergo topological charge inversion. We investigate the locations of the flip planes where charge inversions occur and provide expressions for a number of special cases. Numerical results are provided for an example of such a polynomial astigmatic Gaussian beam.     

On invariant and covariant Schwartz distributions in the case of a compact linear group

A proof is given for the representations of invariant and covariant (Schwartz) distributions onR ⁿ , which are often used in theoretical physics. We express invariant distributions as distributions of standard polynomial invariants and decompose covariant distributions in standard polynomial covariants. Our consideration is restricted to compact groups acting linearly onR ⁿ . The representation for invariant distributions is obtained provided the standard invariants form an algebraically independent generating set in the ring of invariant polynomials. As for the standard covariants we assume that in the class of covariant polynomials they provide a unique decomposition into a sum of the standard covariants multiplied with invariant polynomials.     

空间遥感器中窗口的热光学特性研究

光学窗口是空间光学遥感器应用系统中的重要组成部分 ,其热光学特性是其热控设计的指南。拟定某种温度边界条件后利用有限元法计算其稳态温度场分布 ,再将温度载荷和其它有关广义力载荷作为载荷 ,计算光学玻璃的变形。将变形值拟合为 Zemike多项式 ,输入到光学设计程序 Zmax TM中 ,计算出光学玻璃的变形所引入的波差。最终根据光学遥感器许用的系统波像差来给定光学玻璃的许用温度梯度     

Comparison of simultaneous signals obtained from a dual-field-of-view lidar and its application to noise reduction based on empirical mode decomposition

Although the empirical mode decomposition (EMD) method is an effective tool for noise reduction in lidar signals, evaluating the effectiveness of the denoising method is difficult. A dual-field-of-view lidar for observing atmospheric aerosols is described. The backscattering signals obtained from two channels have different signal-to-noise ratios (SNRs). The performance of noise reduction can be investigated by comparing the high SNR signal and the denoised low SNR signal without a simulation experiment. With this approach, the signal and noise are extracted to one intrinsic mode function (IMF) by the EMD- based denoising; thus, the threshold method is applied to the IMFs. Experimental results show that the improved threshold method can effectively perform noise reduction while preserving useful sudden-change information.     

On non-linear oscillations with slowly varying system parameters

The paper deals with an approximate analysis of non-linear oscillation problems with slowly varying system parameters. From the differential equations for amplitude and phase, set up by the method of variation of parameters, the approximate solutions are obtained by using the generalized averaging method of Sinha and Srinivasan based on ultraspherical polynomial expansions. The Bogoliubov-Mitropolsky results are given by a particular set of these polynomials. Problems of a single degree of freedom system as well as monofrequency oscillations in systems with multiple degrees of freedom are considered. The approach has been illustrated by an example and the results are compared with the numerical solutions. A close agreement is found.