对流扩散方程迎风有限元的自适应方法

本文对二维发展型对流扩散方程的迎风有限元格式给出了显式后验误差估计,证明了真实误差被后验误差估计器上下界定;并通过误差估计器建立了相应的自适应算法,数值例子表明了方法的有效性.     

直接误差估计的一个新方法

本文利用文[1]的方法进行误差估计产生了严重过估的事实,从而提出了一个估计误差的新方法。通过严格的分析树高定义,给出了绝对误差和相对误差二种形式的误差估计方法,并针对并行算法的特点,给出了向量运算算法的误差估计方法。利用该方法本文给出了几个典型问题的算法的误差估计,结果表明,它和已有的结论是一致的。     

频域有限元计算的扩展面向目标误差估计

研究了针对频域有限元直接动态分析的面向目标误差估计以及误差范围估计计算方法．面向目标的误差估计方法就是专门针对如何准确和经济地估算特定值误差的一种方法,利用原问题的共轭偶问题进行计算．频域有限元的直接动态分析是模拟频域扫描实验的一种计算方法,专门针对谐振激励的线性动态响应问题,利用将原自由度分解为实部和虚部描述频率的变化,从而计算变形体的动态响应．利用扩展针对有限元的面向目标误差估计的自由度,将该方法应用到直接动态分析中进行误差估计．通过建立同时包含实部和虚部自由度的能量弱形式及偶问题,并将其数值实现,估算频域直接动态分析有限元解的误差及误差范围,并通过悬臂梁的激振算例进行了验证．     

定常Navier-Stokes方程流函数形式两重网格算法的残量型后验误差估计

运用七种两重网格协调元方法得出了不可压Navier-Stokes方程流函数形式的残量型后验误差估计．对比标准有限元方法的后验误差估计,两重网格算法的后验误差估计多了一些额外项(三线性项)．说明了这些额外项在误差估计中对研究离散解渐近性的重要性,推出了对于最优网格尺寸,这些额外项的收敛阶不高于标准离散解的收敛阶．     

RAS法真实误差的实证研究

RAS法为更新投入产出表的主要非调查方法之一,作为非调查方法,更新误差的大小决定了方法的应用价值,但传统的误差估计方法由于在设计上的缺陷,其估计结果在本质上是RAS法误差的下限.本文利用1992、1997、2002年中国投入产出表及1997、2002年常规统计数据首次对实际应用中RAS法的真实误差进行了实证研究.结果表明:传统的误差估计方法显示使用RAS法将使投入产出系数的平均误差下降17.84%;而在实际应用中,使用RAS法只能使投入产出系数的平均误差下降5.92%.因此,传统的误差估计方法会非常显著低估RAS法的真实误差.从而夸大了RAS法的应用价值.     

CNC固有误差估计新方法

提出了一个估计数控机床固有误差的新方法.与已有利用动态最小二乘(MLS)的方法相比,采用径向基函数(RBF)直接对已有数据进行拟合估计,大大提高了计算效率.还观察到,RBF方法在某个采样半径下误差估计精度总是优于MLS方法,而大于这个采样半径后则MLS方法较好.由此提出了穿越半径的概念与一种基于RBF方法与MLS方法的混合方法,以得到更好的误差估计.实验结果证实了新方法的有效性.     

Wilson元插值误差渐近估计

Wilson元是工程界常用的一种有限元计算方法,但在理论分析中插值误差估计的常数只知道存在,不知道具体值.本文给出了在L~2、H~1范数意义下Wilson元在参考单元和一般单元上插值误差渐近估计,导出了主要常数.这种精确的估计为有限元后验误差估计和自适应计算提供保障.     

数据缺失情形线性回归模型中误差方差的经验似然估计

在随机缺失(MAR)机制下利用经验似然方法构造了线性回归模型中误差方差的估计.并在一定条件下,证明了该估计的渐近正态性,由此得出当误差的分布不对称时,该估计的渐近方差比常用估计的渐近方差小.     

多元线性模型回归系数的有偏估计与均方误差的无偏估计

本文对多元线性模型回归系数的最小二乘估计的任一线性变换，给出了均方误差的一个无偏估计，并应用统一方法，即极小化均方误差的无偏估计的方法，对岭估计和广义岭估计给出了确定偏参数的公式。最后给出了一个实例。     

分子动力学模拟中非成键相互作用计算的误差估计

高效的非成键相互作用计算对于分子动力学模拟具有核心意义.本文在一个统一的框架下,综述短程相互作用的截断方法、长程静电相互作用的光滑粒子网格Ewald方法和交错网格Ewald方法的误差估计.与传统的误差估计假设体系均匀且无相关性不同,本文介绍的误差估计可以推广到非均匀和有相关性的体系.本文通过具体例子讨论非均匀性和相关性对误差的本质性影响,以及可能的修正方式,并说明误差估计对于提高非成键相互作用的计算精度和速度有重要作用.本文还展示一个针对光滑粒子网格Ewald方法的实用参数优化方法,使得在保证精度的同时选取计算效率近似最优的参数组合成为可能,改善了传统上参数全凭经验选取的局面.     

Asymptotic Mean-Square Stability of Two-Step Methods for Stochastic Ordinary Differential Equations

We deal with linear multi-step methods for SDEs and study when the numerical approximation shares asymptotic properties in the mean-square sense of the exact solution. As in deterministic numerical analysis we use a linear time-invariant test equation and perform a linear stability analysis. Standard approaches used either to analyse deterministic multi-step methods or stochastic one-step methods do not carry over to stochastic multi-step schemes. In order to obtain sufficient conditions for asymptotic mean-square stability of stochastic linear two-step-Maruyama methods we construct and apply Lyapunov-type functionals. In particular we study the asymptotic mean-square stability of stochastic counterparts of two-step Adams–Bashforth- and Adams–Moulton-methods, the Milne–Simpson method and the BDF method. AMS subject classification (2000) 60H35, 65C30, 65L06, 65L20     

数值流形方法的变分原理与应用

针对线弹性体静力问题,根据数值流形方法的特点及相应的位移模式,得到了面向物理覆盖的数值流形方法的变分原理,详细推导了基于变分原理的数值流形方法的理论计算公式,建立了数值流形方法的控制方程。作为实际应用,给出了相应的数值算例,结果表明,求解精度和效益令人满意。     

A linearization‐based approach of homotopy analysis method for non‐linear time‐fractional parabolic PDEs

In this paper, a novel approach, namely, the linearization‐based approach of homotopy analysis method, to analytically treat non‐linear time‐fractional PDEs is proposed. The presented approach suggests a new optimized structure of the homotopy series solution based on a linear approximation of the non‐linear problem. A comparative study between the proposed approach and standard homotopy analysis approach is illustrated by solving two examples involving non‐linear time‐fractional parabolic PDEs. The performed numerical simulations demonstrate that the linearization‐based approach reduces the computational complexity and improves the performance of the homotopy analysis method.     

A numerical method for solving linear integral equations of the second kind on the non-rectangular domains based on the meshless method

The main purpose of this article is to describe a numerical scheme for solving two-dimensional linear Fredholm integral equations of the second kind on a non-rectangular domain. The method approximates the solution by the discrete collocation method based on radial basis functions (RBFs) constructed on a set of disordered data. The proposed method does not require any background mesh or cell structures, so it is meshless and consequently independent of the geometry of domain. This approach reduces the solution of the two-dimensional integral equation to the solution of a linear system of algebraic equations. The error analysis of the method is provided. The proposed scheme is also extended to linear mixed Volterra–Fredholm integral equations. Finally, some numerical examples are presented to illustrate the efficiency and accuracy of the new technique.     

线性随机分数阶微分方程Euler方法的弱收敛性与弱稳定性

本文主要研究了线性随机分数阶微分方程Euler方法的弱收敛性与弱稳定性.首先构造了数值求解线性随机分数阶微分方程的Euler方法,然后证明该方法是弱稳定的和α阶弱收敛的,文末给出的数值算例验证了所获得的理论结果的正确性.     

On the numerical solution of stochastic quadratic integral equations via operational matrix method

In this paper, stochastic operational matrix of integration based on delta functions is applied to obtain the numerical solution of linear and nonlinear stochastic quadratic integral equations (SQIEs) that appear in modelling of many real problems. An important advantage of this method is that it dose not need any integration to compute the constant coefficients. Also, this method can be utilized to solve both linear and nonlinear problems. By using stochastic operational matrix of integration together collocation points, solving linear and nonlinear SQIEs converts to solve a nonlinear system of algebraic equations, which can be solved by using Newton's numerical method. Moreover, the error analysis is established by using some theorems. Also, it is proved that the rate of convergence of the suggested method is O(h²). Finally, this method is applied to solve some illustrative examples including linear and nonlinear SQIEs. Numerical experiments confirm the good accuracy and efficiency of the proposed method.     

线性互补问题模系多重网格方法的AMSOR光滑算子

为了改进求解大型稀疏线性互补问题模系多重网格方法的收敛速度和计算时间,本文采用加速模系超松弛(AMSOR)迭代方法作为光滑算子.局部傅里叶分析和数值结果表明此光滑算子能有效地改进模系多重网格方法的收敛因子、迭代次数和计算时间.     

On the convergence of nonstationary column-oriented version of algebraic iterative methods

Recently, Elfving, Hansen, and Nikazad introduced a successful nonstationary block-column iterative method for solving linear system of equations based on flagging idea (called BCI-F). Their numerical tests show that the column-action method provides a basis for saving computational work using flagging technique in BCI algorithm. However, they did not present a general convergence analysis. In this paper, we give a convergence analysis of BCI-F. Furthermore, we consider a fully flexible version of block-column iterative method (FBCI), in which the relaxation parameters and weight matrices can be updated in each iteration and the column partitioning of coefficient matrix is allowed to update in each cycle. We also provide the convergence analysis of algorithm FBCI under mild conditions.     

地下结构和岩土介质弹塑性耦合分析的摄动半解析法*

本文研究了一个用于物理非线性相互作用分析的有效的数值方法。结构和介质耦合分析的弹塑性问题可用摄动法转化为几个线性问题,然后对相应的线性问题分别用有限条和有限层法分析地下结构和岩土介质以达到简化计算的目的。这种方法用了两次半解析技术——摄动和半解析解函数——将三维非线性耦合问题化为一维的数值问题。此外,本法是半解析法结合解析的摄动法应用于非线性问题的新进展,同时也是近年来发展的摄动数值法的一个分支。