几类改进的新的两步六阶Chebyshev-Halley方法

利用权函数法,给出非线性方程求根的Chebyshev-Halley方法的几类改进方法,证明方法六阶收敛到单根.Chebyshev-Halley方法的效率指数为1.442,改进后的两步方法的效率指数为1.565.最后给出数值试验,且与牛顿法,Chebyshev-Halley 方法及其它已知的方程求根方法做了比较.结果表明方法具有一定的优越性.     

Convergence results for general linear methods on singular perturbation problems

Many numerical methods used to solve Ordinary Differential Equations, or Differential Algebraic Equations can be written as general linear methods. The B-convergence results for general linear methods are for algebraically stable methods, and therefore useless for nearly A-stable methods. The purpose of this paper is to show convergence for singular perturbation problems for the class of general linear methods without assuming A-stability.     

Direct prediction methods in Hilbert space with applications to control problems

In this paper, the convergence of variable-metric methods without line searches (direct prediction methods) applied to quadratic functionals on a Hilbert space is established. The methods are then applied to certain control problems with both free endpoints and fixed endpoints. Computational results are reported and compared with earlier results. The methods discussed here are found to compare favorably with earlier methods involving line searches and with other direct prediction quasi-Newton methods.     

高阶优化算法分析简介

高阶优化算法是利用目标函数的高阶导数信息进行优化的算法,是最优化领域中的一个新兴的研究方向.高阶算法具有更低的迭代复杂度,但是需要求解一个更难的子问题.主要介绍三种高阶算法,分别为求解凸问题的高阶加速张量算法和A-HPE框架下的最优张量算法,以及求解非凸问题的ARp算法.同时也介绍了怎样求解高阶算法的子问题.希望通过对高阶算法的介绍,引起更多学者的关注与重视.     

Fourth-Order Splitting Methods for Time-Dependant Differential Equations

This study was suggested by previous work on the simulation of evolution equations with scale-dependent processes,e.g.,wave-propagation or heat-transfer,that are modeled by wave equations or heat equations.Here,we study both parabolic and hyperbolic equations.We focus on ADI (alternating direction implicit) methods and LOD (locally one-dimensional) methods,which are standard splitting methods of lower order,e.g.second-order.Our aim is to develop higher-order ADI methods,which are performed by Richardson extrapolation,Crank-Nicolson methods and higher-order LOD methods,based on locally higher-order methods.We discuss the new theoretical results of the stability and consistency of the ADI methods.The main idea is to apply a higher- order time discretization and combine it with the ADI methods.We also discuss the dis- cretization and splitting methods for first-order and second-order evolution equations. The stability analysis is given for the ADI method for first-order time derivatives and for the LOD (locally one-dimensional) methods for second-order time derivatives.The higher-order methods are unconditionally stable.Some numerical experiments verify our results.     

The composite Milstein methods for the numerical solution of Stratonovich stochastic differential equations

In this paper, we present two composite Milstein methods for the strong solution of Stratonovich stochastic differential equations driven by d-dimensional Wiener processes. The composite Milstein methods are a combination of semi-implicit and implicit Milstein methods. The criterion for choosing either the implicit or the semi-implicit method at each step of the numerical solution is given. The stability and convergence properties of the proposed methods are analyzed for the linear test equation. It is shown that the proposed methods converge to the exact solution in Stratonovich sense. In addition, the stability properties of our methods are found to be superior to those of the Milstein and the composite Euler methods. The convergence properties for the nonlinear case are shown numerically to be the same as the linear case. Hence, the proposed methods are a good candidate for the solution of stiff SDEs.     

Iterative operator-splitting methods with higher-order time integration methods and applications for parabolic partial differential equations

In this paper we design higher-order time integrators for systems of stiff ordinary differential equations. We combine implicit Runge–Kutta and BDF methods with iterative operator-splitting methods to obtain higher-order methods. The idea of decoupling each complicated operator in simpler operators with an adapted time scale allows to solve the problems more efficiently. We compare our new methods with the higher-order fractional-stepping Runge–Kutta methods, developed for stiff ordinary differential equations. The benefit is the individual handling of each operator with adapted standard higher-order time integrators. The methods are applied to equations for convection–diffusion reactions and we obtain higher-order results. Finally we discuss the applications of the iterative operator-splitting methods to multi-dimensional and multi-physical problems.     

A new two-parameter family of nonlinear conjugate gradient methods

Conjugate gradient methods are an important class of methods for unconstrained optimization, especially for large-scale problems. Recently, they have been much studied. In this paper, we propose a new two-parameter family of conjugate gradient methods for unconstrained optimization. The two-parameter family of methods not only includes the already existing three practical nonlinear conjugate gradient methods, but has other family of conjugate gradient methods as subfamily. The two-parameter family of methods with the Wolfe line search is shown to ensure the descent property of each search direction. Some general convergence results are also established for the two-parameter family of methods. The numerical results show that this method is efficient for the given test problems. In addition, the methods related to this family are uniformly discussed.     

Mehar’s methods for fuzzy assignment problems with restrictions

In this paper, limitations of existing methods [5, 11] for solving fuzzy assignment problems (FAPs) are pointed out. In order to overcome the limitations of existing methods, two new methods named Mehar’s methods are proposed. To show the advantages of Mehar’s methods over existing methods, some FAPs are solved. The Mehar’s methods can solve the problems solved by existing methods as well as those which cannot be solved by existing methods.     

A family of Numerov-type exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation

A family of predictor-corrector exponential Numerov-type methods is developed for the numerical integration of the one-dimensional Schrödinger equation. The formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions. The new methods are very simple and integrate more exponential functions than both the well-known fourth-order Numerov-type exponentially fitted methods and the sixth algebraic order Runge-Kutta-type methods. Numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above.     

On the choice of correctors for semi-implicit Picard deferred correction methods

The goal of this study is to assess the implications of the choice of correctors for semi-implicit Picard integral deferred correction (SIPIDC) methods. The SIPIDC methods previously developed compute a high-order approximation by first computing a low-order provisional solution using a semi-implicit method and then using a first-order semi-implicit method to solve a series of correction equations, each of which raises the order of accuracy of the solution by one. In this study, we examine the efficiency of SIPIDC methods that instead use standard second-order semi-implicit methods to solve the correction equations. The accuracy, efficiency, and stability of the resulting methods are compared to previously developed methods, in the context of both nonstiff and stiff problems.     

A family of hybrid exponentially fitted predictor-corrector methods for the numerical integration of the radial Schrödinger equation

A family of hybrid, exponentially fitted, predictor-corrector methods is developed for the numerical integration of the one-dimensional Schrödinger equation. The formula considered contains certain free parameters which allow it to be fitted automatically to exponential functions. The new methods are of algebraic order six, they are very simple and integrate more exponential functions than both the well-known fourth-order Numerov-type exponentially fitted methods and the Runge-Kutta-type methods of algebraic order six. Numerical results also indicate that the new methods are much more accurate than the other exponentially fitted methods mentioned above.     

Expanding stability regions of explicit advective-diffusive finite difference methods by Jacobi preconditioning

Explicit time differencing methods for solving differential equations are advantageous in that they are easy to implement on a computer and are intrinsically very parallel. The disadvantage of explicit methods is the severe restrictions that are placed on stable time-step intervals. Stability bounds for explicit time differencing methods on advective–diffusive problems are generally determined by the diffusive part of the problem. These bounds are very small and implicit methods are used instead. The linear systems arising from these implicit methods are generally solved by iterative methods. In this article we develop a methodology for increasing the stability bounds of standard explicit finite differencing methods by combining explicit methods, implicit methods, and iterative methods in a novel way to generate new time-difference schemes, called preconditioned time-difference methods. A Jacobi preconditioned time differencing method is defined and analyzed for both diffusion and advection–diffusion equations. Several computational examples of both linear and nonlinear advective-diffusive problems are solved to demonstrate the accuracy and improved stability limits. © 1995 John Wiley & Sons, Inc.     

电力系统黑启动决策方法比较研究

黑启动作为电力体系安全防御和事故后快速恢复的措施之一,其路径的合理选择对电力系统快速恢复供电具有重要意义。近年来,学者们从不同角度提出了多种黑启动方案决策方法,然而并没有实现各决策方法间的优劣比较。本文引入平均绝对偏差公式,设计了一种黑启动决策方法比较策略,实现了黑启动决策方法的量化比较。在所提比较策略基础上,对常用的黑启动权重确定方法和排序方法进行了实验分析,广东电网上的实验结果表明基于标准差权重和TOPSIS排序的黑启动决策方法具有最高的准确性。本文的价值在于：(1)提出了一种新的比较策略,使黑启动决策方法的量化比较成为可能;(2)通过大量实验确定了一种优化的黑启动决策方法,为后续黑启动决策研究提供了比较基准。     

An Experimental Evaluation of Some Classification Methods

The classification problem is of major importance to a plethora of research fields. The outgrowth in the development of classification methods has led to the development of several techniques. The objective of this research is to provide some insight on the relative performance of some well-known classification methods, through an experimental analysis covering data sets with different characteristics. The methods used in the analysis include statistical techniques, machine learning methods and multicriteria decision aid. The results of the study can be used to support the design of classification systems and the identification of the proper methods that could be used given the data characteristics.     

Two-parameter families of predictor-corrector methods for the solution of ordinary differential equations

Two-parameter families of predictor-corrector methods based upon a combination of Adams- and Nyström formulae have been developed. The combinations use correctors of order one higher than that of the predictors. The methods are chosen to give optimal stability properties with respect to a requirement on the form and size of the regions of absolute stability. The optimal methods are listed and their regions of absolute stability are presented. The efficiency of the methods is compared to that of the corresponding Adams methods through numerical results from a variable order, variable stepsize program package.     

Supermemory descent methods for unconstrained minimization

The supermemory gradient method of Cragg and Levy (Ref. 1) and the quasi-Newton methods with memory considered by Wolfe (Ref. 4) are shown to be special cases of a more general class of methods for unconstrained minimization which will be called supermemory descent methods. A subclass of the supermemory descent methods is the class of supermemory quasi-Newton methods. To illustrate the numerical effectiveness of supermemory quasi-Newton methods, some numerical experience with one such method is reported.The authors are indebted to Dr. H. Y. Huang for his helpful criticism of this paper.     

特征选择方法在信用评估指标选取中的应用

在信用评分模型中所运用的指标变量对模型的表现有重要的影响,指标选取方法的科学化规范化水平有待于进一步提高。本文研究了机器学习领域的特征选择方法在定量确定信用评分模型指标体系上的应用。以实际信用评估问题为例,对四种特征选择方法(ReliefF方法、基于相关性的方法、基于一致性的方法和包裹性)进行了比较试验,验证了特征选择方法可以在精简性、速度和准确率三个方面提高信用评分模型的表现。其中基于一致性的方法和包裹法表现优于Reli-efF方法和基于相关性的方法。     

Waveform Relaxation Methods for Lie-Group Equations

In this paper, we derive and analyse waveform relaxation (WR) methods for solving differential equations evolving on a Lie-group. We present both continuous-time and discrete-time WR methods and study their convergence properties. In the discrete-time case, the novel methods are constructed by combining WR methods with Runge-Kutta-Munthe-Kaas (RK-MK) methods. The obtained methods have both advantages of WR methods and RK-MK methods, which simplify the computation by decoupling strategy and preserve the numerical solution of Lie-group equations on a manifold. Three numerical experiments are given to illustrate the feasibility of the new WR methods.     

A note on some improvements of the simultaneous methods for determination of polynomial zeros

Applying Gauss-Seidel approach to the improvements of two simultaneous methods for finding polynomial zeros, presented in [9], two iterative methods with faster convergence are obtained. The lower bounds of the R-order of convergence for the accelerated methods are given. The improved methods and their accelerated modifications are discussed in view of the convergence order and the number of numerical operations. The considered methods are illustrated numerically in the example of an algebraic equation.