变系数偏微分方程的Galerkin KPOD模型降阶方法

研究了变系数偏微分方程的 Galerkin KPOD (Krylov Enhanced Proper Orthogonal Decomposition)模型降阶方法.首先基于Galerkin有限元理论建立变系数偏微分方程的空间离散格式,得到具有时变系数矩阵的常微分方程组,并对该常微分方程组应用KPOD模型降阶方法进行降阶并求解.其次,对降阶投影算子进行了分析,给出了 Galerkin有限元解与Galerkin KPOD降阶解之间的误差界.最后用数值算例验证了变系数偏微分方程的Galerkin KPOD模型降阶求解方法的可行性,通过有限元离散解与Galerkin KPOD降阶解、Galerkin POD降阶解之间的误差比较,体现Galerkin KPOD降阶方法的优势.     

Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials

We extend the definition of the classical Jacobi polynomials withindexes α, β>−1 to allow α and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials leads to much simplified analysis, more precise error estimates and well conditioned algorithms.Mathematics subject classification 1991. 65N35, 65N22, 65F05, 35J05     

Discrete least-norm approximation by nonnegative (trigonometric) polynomials and rational functions

Polynomials, trigonometric polynomials, and rational functions are widely used for the discrete approximation of functions or simulation models. Often, it is known beforehand that the underlying unknown function has certain properties, e.g., nonnegative or increasing on a certain region. However, the approximation may not inherit these properties automatically. We present some methodology (using semidefinite programming and results from real algebraic geometry) for least-norm approximation by polynomials, trigonometric polynomials, and rational functions that preserve nonnegativity.     

Derivation of operational matrices of differentiation for orthogonal polynomials

A direct method for deriving an operational matrix of differentiation for Legendre polynomials is presented. The general case is also considered. The presented formulas are formally proved. Results are compared with the other known method. The new method is accurate for all matrix dimensions, while the other method may fail for higher dimensions.     

Chebyshev collocation method for solving singular integral equation with cosecant kernel

A collocation method based on Chebyshev polynomials is proposed for solving cosecant-type singular integral equations (SIE). For solving SIE, difficulties lie in its singular term. In order to remove singular term, we introduce Gauss–Legendre integration and integral properties of the cosecant kernel. An advantage of this method is to approximate the best uniform approximation by the best square approximation to obtain the unknown coefficients in the method. On the other hand, the convergence is fast and the accuracy is high, which is verified by the final numerical experiments compared with the existing references.     

A Rational Approximation and Its Applications to Differential Equations on the Half Line

An orthogonal system of rational functions is introduced. Some results on rational approximations based on various orthogonal projections and interpolations are established. These results form the mathematical foundation of the related spectral method and pseudospectral method for solving differential equations on the half line. The error estimates of the rational spectral method and rational pseudospectral method for two model problems are established. The numerical results agree well with the theoretical estimates and demonstrate the effectiveness of this approach.     

A highly accurate adaptive finite difference solver for the Black–Scholes equation

In this paper, we develop a highly accurate adaptive finite difference (FD) discretization for the Black–Scholes equation. The final condition is discontinuous in the first derivative yielding that the effective rate of convergence in space is two, both for low-order and high-order standard FD schemes. To obtain a method that gives higher accuracy, we use an extra grid in a limited space- and time-domain. This new method is called FD6G2. The FD6G2 method is combined with space- and time-adaptivity to further enhance the method. To obtain solutions of high accuracy, the adaptive FD6G2 method is superior to both a standard and an adaptive second-order FD method.     

线性时变时滞系统分析与参数辨识的PMCP方法

本文将[1]定义的按段多重chebyshev多项式系应用于线性时变时滞系统分析与参数辨识,提出了系统分析和参数辨识算法。数值例子表明本文提出的算法既简单又有效,大大优于由移位chebyshev多项式系所导出来的相应算法。     

高次幂函数逼近的阴影图反走样算法

针对传统阴影图算法由于阴影贴图分辨率的约束而在阴影的边缘会出现走样现象的问题, 提出一种基于高次幂函数逼近的预滤波阴影图反走样算法.该算法使用高次幂函数来逼近传统阴影图中的深度比较函数, 并对高次幂函数进行泰勒展开, 在展开的各项上分别进行预滤波;在绘制过程中, 利用泰勒展开式计算得到的数值代替传统阴影图中的深度比较函数得到的二值结果, 从而实现了对阴影边缘的反走样.实验结果表明, 该算法在现有图形硬件中能够达到很好的效果.     

线性时变时滞系统分析与参数辨识的PMCP方法(英文)

本文将文[1]定义的按段多重chebyshev多项式系应用于线性时变时滞系统分析与参数辨识,提出了系统分析和参数辨识算法。数值例子表明本文提出的算法既简单又有效,大大优于由移位chebyshev多项式系所导出的相应算法。     

Reconstruction of Piecewise Smooth Functions from Non-uniform Grid Point Data

Spectral series expansions of piecewise smooth functions are known to yield poor results, with spurious oscillations forming near the jump discontinuities and reduced convergence throughout the interval of approximation. The spectral reprojection method, most notably the Gegenbauer reconstruction method, can restore exponential convergence to piecewise smooth function approximations from their (pseudo-)spectral coefficients. Difficulties may arise due to numerical robustness and ill-conditioning of the reprojection basis polynomials, however. This paper considers non-classical orthogonal polynomials as reprojection bases for a general order (finite or spectral) reconstruction of piecewise smooth functions. Furthermore, when the given data are discrete grid point values, the reprojection polynomials are constructed to be orthogonal in the discrete sense, rather than by the usual continuous inner product. No calculation of optimal quadrature points is therefore needed. This adaptation suggests a method to approximate piecewise smooth functions from discrete non-uniform data, and results in a one-dimensional approximation that is accurate and numerically robust.