基于相对误差意义下的最小二乘法

最小二乘法一直广泛应用于科学技术各领域 ,但仅适用于等精度数据———因其评价依据是大致相同的绝对误差。而大多数科研及观测数据往往按被观测量的相对误差进行评价 ,即被观测量愈大 ,允许及实际测量误差也愈大。从这个意义上讲 ,若以相对拟合误差的平方和最小为条件 ,建立一套新的最小二乘法 ,那么当然可以期望新方法将具更广泛的用途和实际的意义。本文给出了新最小二乘法的实用结果     

The regularization method for an obstacle problem

Summary. We give a relatively complete analysis for the regularization method, which is usually used in solving non-differentiable minimization problems. The model problem considered in the paper is an obstacle problem. In addition to the usual convergence result and a-priori error estimates, we provide a-posteriori error estimates which are highly desired for practical implementation of the regularization method. Received March 22, 1993 / Revised version received October 11, 1993     

Some notes on split Newton iterative algorithm

In this study, a parameterized split Newton method is derived by using the accelerating technique. Convergence and error estimates of the method are obtained. In practical application, the proposed method can give a better result in view of computational CPU time. Numerical examples on several partial differential equations are shown to illustrate our findings. Copyright © 2015 John Wiley & Sons, Ltd.     

Localized sampling in the presence of noise

When the sampled values are corrupted by noise, error estimates for the localized sampling series for approximating a band-limited function are obtained. The result provides error bounds for practical cases including error caused by average sampling, jitter error and amplitude error.     

The error and guaranteed accuracy of cubature formulas in multidimensional periodic Sobolev spaces

We give an upper bound for the deviation of the norm of a perturbed error from the norm of the original error of a cubature formula in a multidimensional bounded domain. The deviation arises as a result of the joint influence on the computations of small variations of the weights of a cubature formula and rounding in the subsequent calculations of the cubature sum in the given standards (formats) of approximation to real numbers. We estimate the practical error of a cubature formula acting on an arbitrary function from the unit ball of a normed space of integrands. The resulting estimates are applied to studying the practical error of cubature formulas in the case of integrands in Sobolev spaces on a multidimensional cube. The norm of the error in the dual space of the Sobolev class is represented as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for cubature formulas constructed as the direct product of quadrature formulas of rectangles along the edges of the unit cube. The weights of this direct product are positive.     

Error estimates for shooting methods in two-point boundary value problems for second-order equations

This paper is concerned with a procedure for estimating the global discretization error arising when a boundary value problem for a system of second order differential equations is solved by the simple shooting method, without transforming the original problem in an equivalent first order problem. Expressions of the global discretization error are derived for both linear and nonlinear boundary value problems, which reduce the error estimation for a boundary value problem to that for an initial value problem of same dimension. The procedure extends to second order equations a technique for global error estimation given elsewhere for first order equations. As a practical result the accuracy of the estimates for a second order problem is increased compared with the estimates for the equivalent first order problem.     

Errors, condition numbers, and guaranteed accuracy of higher-dimensional spherical cubatures

We give upper bounds for the deviation of the norm of a perturbed error functional from the norm of the original error of a higher-dimensional spherical cubature formula. The deviation arises as a result of the combined influence on the computation of small variations of the weights of the cubature formula and rounding for the subsequent calculation of the cubature sum in the given standards of approximation to real numbers. We estimate the practical error of the cubature formula for its action on an arbitrary function in the unit ball of the normed space of integrands. The resulting estimates are applied to studying the practical error of spherical cubature formulas in the case of integrands in Sobolev-type spaces on the higher-dimensional unit sphere. We represent the norm of the error functional in the dual space of the Sobolev class as a positive definite quadratic form in the weights of the cubature formula. We estimate the practical error for spherical cubature formulas, each of which is constructed as the direct product of Gauss’s quadrature formula along the meridian of the sphere and of the rectangle quadrature formula along the equator. The weights of this direct product with 2m ² nodes are positive. The formula itself is exact at all spherical harmonics up to order 2m ? 1.     

An inverse-free Jarratt type approximation in a Banach space

In this study, we introduce a new iterative method for solving nonlinear operator equations in Banach spaces. We establish a sufficient as well as a local convergence theorem. We also provide the best possible practical error bound for this method.     

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.     

A practical relative error criterion for augmented Lagrangians

This paper develops a new error criterion for the approximate minimization of augmented Lagrangian subproblems. This criterion is practical since it is readily testable given only a gradient (or subgradient) of the augmented Lagrangian. It is also “relative” in the sense of relative error criteria for proximal point algorithms: in particular, it uses a single relative tolerance parameter, rather than a summable parameter sequence. Our analysis first describes an abstract version of the criterion within Rockafellar’s general parametric convex duality framework, and proves a global convergence result for the resulting algorithm. Specializing this algorithm to a standard formulation of convex programming produces a version of the classical augmented Lagrangian method with a novel inexact solution condition for the subproblems. Finally, we present computational results drawn from the CUTE test set—including many nonconvex problems—indicating that the approach works well in practice.     

SQCF系统的全局时滞混沌同步

考虑到在通信系统中接收端存在时滞问题,针对SQCF系统,以Lyapunov稳定性理论和矩阵论为基础,提出一种系统的全局混沌同步方案.通过选取适当的耦合参数,使误差系统全局渐近稳定,即使得响应系统在t+τ时刻的状态渐进地趋向于驱动系统在t时刻的状态.并用Mathematic软件给出了数值模拟,理论分析和数值仿真表明,该方法可以较快地实现混沌同步.对系统的第三变量z(t)进行扰动,数值仿真表明在扰动系统下仍能很好地保持同步,表明同步系统具有抗干扰性.     

A note on error expressions for reflected and averaged implicit Runge-Kutta methods

In addition to their usefulness in the numerical solution of initial value ODE's, the implicit Runge-Kutta (IRK) methods are also important for the solution of two-point boundary value problems. Recently, several classes of modified IRK methods which improve significantly on the efficiency of the standard IRK methods in this application have been presented. One such class is the Averaged IRK methods; a member of the class is obtained by applying an averaging operation to a non-symmetric IRK method and its reflection. In this paper we investigate the forms of the error expressions for reflected and averaged IRK methods. Our first result relates the expression for the local error of the reflected method to that of the original method. The main result of this paper relates the error expression of an averaged method to that of the method upon which it is based. We apply these results to show that for each member of the class of the averaged methods, there exists an embedded lower order method which can be used for error estimation, in a formula-pair fashion.This work was supported by the Natural Science and Engineering Research Council of Canada.     

一元向量有理插值的误差公式

Ｇｒａｖｅｓ－Ｍｏｒｒｉｓ于１９８３年利用向量的Ｓａｍｅｌｓｏｎ逆变换建立了一种实用的向量有理插值方法。本文得到了该向量有理插值的一个精确的误差公式。     

A posteriori error estimate for finite volume element method of the parabolic equations

In this article, residual‐type a posteriori error estimates are studied for finite volume element (FVE) method of parabolic equations. Residual‐type a posteriori error estimator is constructed and the reliable and efficient bounds for the error estimator are established. Residual‐type a posteriori error estimator can be used to assess the accuracy of the FVE solutions in practical applications. Some numerical examples are provided to confirm the theoretical results. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 259–275, 2017     

带有误差变量混合分布模型的估计

在许多实际同题中，存在一些不可直接观测的变量，对此统计学家们提出了反卷积和混合分布模型来解决这一变量的分布的估计问题。本文对这一问题采用bootstrap模拟方法得出分布函数的估计，并进一步建立该分布函数的非参bootstrap百分位区间，在数值试验中将我们的处理方式与传统的EM算法得到的分布估计和正态逼近区间作比较，数值结果表明用bootstrap模拟方法得到的准确度更好，数值效果更理想。     

铁磁性设备补偿磁场的数学模型及其应用

研究用永磁体对铁磁性设备进行磁场补偿的问题,建立了补偿磁场的数学模型.将设备划分成若干个小长方体后,基于磁矩量法建立了数学模型,并对补偿磁场进行拟合.在计算模型中的耦合系数矩阵时,用多个点的平均值作为耦合系数的有效值,提高了计算结果的可靠性和稳定性.并且,针对永磁体距离设备很近时,设备呈现出的非线性磁化特性,通过优化方法求解各个单元的等效磁化率,这种方法不需要知道铁磁材料的磁化曲线和设备结构,便于计算和实际应用.最后,通过实验设计与数值计算,得到了永磁体对设备进行补偿的磁场分布,模型计算结果与实际测量数据误差11%以内,这说明该模型能够满足工业要求,具有实际应用价值.     

Quantized feedback stabilization for a class of linear systems with nonlinear disturbances

The quantized feedback stabilization problem for a class of linear system with nonlinear disturbances is addressed, in which the system and controller are connected via a communication channel. In this case, the effect of quantization errors is studied. A practical quantized scheme is designed such that the transmission error decays to zero exponentially. Meanwhile, a sufficient condition in terms of linear matrix inequality for input-to-state stability (ISS) of the system is presented with regard to the transmission error. Therefore, a quantized stabilization of the system is guaranteed based on the ISS property. A simulation example for the single-link flexible joint robot is presented to demonstrate the effectiveness of the result.     

On quasi-consistent integration by Nordsieck methods

In this paper we study advantages of numerical integration by quasi-consistent Nordsieck formulas. All quasi-consistent numerical methods possess at least one important property for practical use, which has not attracted attention yet, i.e. the global error of a quasi-consistent method has the same order as its local error. This means that the usual local error control will produce a numerical solution for the prescribed accuracy requirement if the principal term of the local error dominates strongly over remaining terms. In other words, the global error control can be as cheap as the local error control in the methods under discussion.     

A Family of Linear Multistep Methods for the Solution of Stiff and Non-stiff ODEs

A multistep method is described which contains a free parameter.One value of this parameter will give an Adams formula whileanother gives a backward differentiation formula. Intermediatevalues of the parameter give a method with intermediate stabilityproperties and truncation error. The method is presented invariable step form to avoid the difficulties, due to interpolation,inherent in fixed step techniques. Some details useful for producinga practical implementation are also given.