A Posteriori Error Estimation for a Finite Volume Discretization on Anisotropic Meshes

A singularly perturbed reaction-diffusion problem is considered. The small diffusion coefficient generically leads to solutions with boundary layers. The problem is discretized by a vertex-centered finite volume method. The anisotropy of the solution is reflected by using anisotropic meshes which can improve the accuracy of the discretization considerably. The main focus is on a posteriori error estimation. A residual type error estimator is proposed and rigorously analysed. It is shown to be robust with respect to the small perturbation parameter. The estimator is also robust with respect to the mesh anisotropy as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution (which is almost always the case for sensible discretizations). Altogether, reliable and efficient a posteriori error estimation is achieved for the finite volume method on anisotropic meshes. Numerical experiments in 2D underline the applicability of the theoretical results in adaptive computations.     

Exact Discretization of a Matrix Differential Riccati Equation With Constant Coefficients

An exact method is presented for discretizing a constant-coefficient, non-square, matrix differential Riccati equation, whose solution is assumed to exist. The resulting discrete-time equation gives the values that have no error at discrete-time instants for any discrete-time interval. The method is based on a matrix fractional transformation, which is more general than existing ones, for linearizing the differential Riccati equation. A numerical example is presented to compare the proposed method with that based on gage invariance and bilinearization, which has better performances than the conventional forward-difference method.      

A Spectral Mortar Element Discretization of the Poisson Equation with Mixed Boundary Conditions

In this paper, we study a spectral mortar element discretization of the Poisson equation on a square subject to mixed boundary conditions of Dirichlet and Neumann type. We carry out the numerical analysis of the method and derive error estimates. An efficient algorithm for the solution of the problem is proposed and numerical tests confirming the theoretical results are presented.     

信息表离散格的进一步研究

在文献[1]的基础上,得到了离散格的表示定理.进一步证明了在离散格到划分格的映射下交运算可以保持运算,而并运算不能保持运算,因此该映射不是同态映射.根据离散化后得到的对象域划分定义了离散化方案之间的等价关系,证明了随着离散化等价类[DR]的加粗,离散化方案对应的正区域下降,而条件信息熵上升.最后分析了另外两种离散格搜索算法.     

二维线性对流扩散问题的NURBS等几何分析

基于NURBS的等几何分析法有机地结合了CAGD和有限元分析.为将该方法应用于线性对流扩散问题的求解,提出将SUPG(streamline upwind Petrov-Galerkin)法与等几何法相结合的稳定化离散方案.首先对空间域进行等几何离散,然后用θ加权法离散时间域建立了完全离散的等几何求解格式;同时引入罚函数法处理NURBS基函数的非插值性所造成的本质边界处理误差.最后通过数值算例验证了文中方法的有效性.     

广义代数Riccati方程正解的研究

研究一类广义代数Riccati方程,该方程在随机控制领域有重要的应用,因而被许多学者所研究.通过引进能稳性的概念,给出了该方程有唯一正解的一个充分必要条件,所得结果不仅改进了以往的结论,也推广了确定性代数Riccati方程的相应性质.     

A Posteriori Error Estimators for a Class of Variational Inequalities

In this paper, we present an a posteriori error analysis for the finite element approximation of a variational inequality. We derive a posteriori error estimators of residual type, which are shown to provide upper bounds on the discretization error for a class of variational inequalities provided the solutions are sufficiently regular. Furthermore we derive sharp a posteriori error estimators with both lower and upper error bounds for a subclass of the obstacle problem which are frequently met in many physical models. For sufficiently regular solutions, these estimates are shown to be equivalent to the discretization error in an energy type norm. Our numerical tests show that these sharp error estimators are both reliable and efficient in guiding mesh adaptivity for computing the free boundaries.     

Stability at Nonconforming Grid Interfaces for a High Order Discretization of the Schr?dinger Equation

In this paper we extend the results from our earlier work on stable boundary closures for the Schr?dinger equation using the summation-by-parts-simultaneous approximation term (SBP?CSAT) method to include stability and accuracy at nonconforming grid interfaces. Stability at the grid interface is shown by the energy method, and the estimates are generalized to multiple dimensions. The accuracy of the grid interface coupling is investigated using normal mode analysis for operators of 2nd and 4th order formal interior accuracy. We show that full accuracy is retained for the 2nd and 4th order operators. The accuracy results are extended to 6th and 8th order operators by numerical simulations, in which case two orders of accuracy is gained with respect to the lower order approximation close to the interface.     

Accuracy and Stability of the Petrov–Galerkin Method for Solving the Stationary Convection-Diffusion Equation

The accuracy and stability of numerical solution of the stationary convection-diffusion equation by the finite element Petrov–Galerkin method are analyzed with the use of weight functions with different stabilization parameters as test functions, and estimates are obtained for the accuracy of the method depending on the choice of a collection of stabilization parameters. The convergence of the method is shown.     

Discretization of a Continuous Curve

We consider the problem of approximating a finite-length continuous curve by a piecewise linear one whose segments are assumed to be connected by 2 DOF joints. We solve the problem under the assumption that the endpoints of the line segments lie on the continuous curve. Analytical expressions for the relative orientations of each pair of line segments as a function of a single rotational DOF are found. This angle can be chosen arbitrarily or used to optimize a secondary task. The motivating application for this paper is the control of a snake-like robot using gaits designed from shape primitives.     

Discretization of the Wave Equation Using Continuous Elements in Time and a Hybridizable Discontinuous Galerkin Method in Space

We provide an error analysis of two methods for time stepping the wave equation. These are based on the Hybridizable Discontinuous Galerkin (HDG) method to discretize in space, and the continuous Galerkin method to discretize in time. Two variants of HDG are proposed: a dissipative method based on the standard numerical flux used for elliptic problems, and a non-dissipative method based on a new choice of the flux involving time derivatives. The analysis of the fully discrete problem is based on simplified arguments using projections rather than explicit interpolants used in previous work. Some numerical results are shown that illuminate the theory.     

一种基于不变量的工作流协同模型分解方法

现代企业计算的业务过程越来越复杂,有很多分散且相对独立的组织机构,为了协同来自不同组织的业务过程,文中提出一种IOPN模型(面向交互的Petri网)用于描述跨组织的工作流协同,该模型包含组织内的过程模型和组织间的交互关系.为了确保IOPN模型能够被正确地执行,文中提出IOPN模型的弱合理性(relaxed soundness)作为IOPN模型的正确性标准之一.IOPN模型是一种复合模型,其规模一般较大,采用基于状态空间的分析方法,容易产生状态空间爆炸问题,为此文中提出基于不变量的分解方法,能够将一个弱合理的无回路IOPN模型分解为一组顺序图,并提出相关定理:一个无回路IOPN模型是弱合理的当且仅当其可以被分解为一组合法的顺序图.     

A Posteriori Error Control for a Weakly Over-Penalized Symmetric Interior Penalty Method

A reliable and efficient residual based a posteriori error estimator is constructed for a weakly over-penalized symmetric interior penalty method for second order elliptic problems. Numerical results that demonstrate the performance of the error estimator are presented.     

A Posteriori Error Estimates and an Adaptive Finite Element Method for the Allen–Cahn Equation and the Mean Curvature Flow

This paper develops an a posteriori error estimate of residual type for finite element approximations of the Allen–Cahn equation u_t − Δu+ ε⁻² f(u)=0. It is shown that the error depends on ε⁻¹ only in some low polynomial order, instead of exponential order. Based on the proposed a posteriori error estimator, we construct an adaptive algorithm for computing the Allen–Cahn equation and its sharp interface limit, the mean curvature flow. Numerical experiments are also presented to show the robustness and effectiveness of the proposed error estimator and the adaptive algorithm.     

基于PCA与最大后验概率分类的人脸识别方法

在运用主成分分析进行人脸识别的过程中,由于实际图像可能符合某种概率密度分布,并且实际用到的图像可能受到不同程度的噪声污染,简单的距离分类已不再适用。基于核函数的最大后验概率分类是将概率密度函数估计中的参数估计、核函数以及贝叶斯理论结合起来,能很好地考虑到概率分布情况,用多元高斯分布下的基于核函数的最大后验概率分类取代距离分类,对于含有不同参数值的高斯噪声图像有较好的识别率。用ORL标准人脸库进行验证,实验结果表明了可行性。     

循环不变式开发技术研究

高可靠性软件是当今软件开发的热点问题。确保算法程序逻辑结构正确最理想的途径是算法程序的形式化推导和证明,而循环不变式是算法程序形式推导和证明的关键。循环不变式的开发一直是算法程序设计领域中最具挑战性、最富有创造性、也是最困难的问题之一。本文研究了众多现有循环不变式开发方法中较为典型的几种方法,指出了它们的基本原理、技术难点、特点及效果,旨在探寻循环不变式本质特征,从而为研究更简单、有效的生成方法提出指导。     

周期黎卡提微分方程正定解的存在性

讨论了标准的周期黎卡提微分方程,给出了其存在埃尔米特周期正定（HPPD）解的一个完整的充分必要条件。准确地说,在经过一个适当的状态空间基底变换后该条件通过能稳性和能检测性概念表述。结果表明,当HP－PD解存在时,它或者是唯一的,或者有无限多个。这一结果可以看作是Richardson和Kwong的结果对周期时变情况的扩展。