利用改进的同伦算法求解特征值问题

矩阵特征值问题是机器学习、数据处理以及工程分析和计算中经常需要解决的问题之一.同伦算法是求解矩阵特征值的经典方法;自动微分可以有效、快速地计算出大规模问题相关函数的导数项,并且可以达到机器精度.充分利用自动微分的优点,设计自动微分技术与同伦算法相结合的方法求解矩阵特征值问题.数值实验验证了该算法的有效性.     

Schrdinger特征值问题的Legendre-Galerkin-Chebyshev配置法

提出了求解多维的Schrodinger特征值问题的Legendre-Galerkin-Chebyshev配置法(LGCC).LGCC方法是Legendre-Galerkin方法和Chebyshev配置法的耦合,便于处理变系数和非线性项且保持了Legendre-Galerkin方法的稳定性和高精度.对于拟线性SchrodingerPoisson特征值问题,建立了基于同伦连续法的LGCC方法.数值结果显示了该方法的有效性.     

基于Bregman距离函数的可靠性分析

针对概率结构可靠性问题,引入Bregman距离函数,建立了基于同伦算法(HM)的可靠性分析模型.利用极限状态方程,将可靠性指标求解转化为一个非线性约束优化问题.结合同伦思想的基本理论和Bregman距离函数,构造同伦方程组,采用路径跟踪算法对该方程组进行求解.通过相应的数值算例探讨了不同函数形式以及不同程度非线性问题的可靠性计算,并与其他方法计算结果进行了对比,分析结果表明该模型能够有效求解概率结构可靠性问题.     

求解混合三角多项式方程组的直接GBQ方法

许多科学与工程领域,我们经常需要求混合三角多项式方程组的全部解.一般来说,混合三角多项式方程组可以通过变量替换及增加二次多项式转化为多项式方程组,进而利用数值方法进行求解,但这种转化会增大问题的规模从而增加计算量.在本文中,我们不将问题转化,考虑利用直接同伦方法求解,并给出基于GBQ方法构造的初始方程组及同伦定理的证明.数值实验结果表明我们构造的直接同伦方法较已有的直接同伦方法更加有效.     

同伦方法求解无界域上非凸规划问题的收敛性定理

利用同伦方法求解非凸规划时,一般只能得到问题的K-K-T点.本文得到无界域上同伦方法求解非凸规划的几个收敛性定理,证明在一定条件下,通过构造合适的同伦方程,同伦算法收敛到问题的局部最优解.     

信号处理中一类非线性方程组的快速求解

在声纳和雷达信号处理中,需要求解一类维数可变的非线性方程组,这类方程组具有混合三角多项式方程组形式.由于该问题有很多解,且其对应的最小二乘问题有很多局部极小点,用牛顿法等传统的迭代法很难找到有物理意义的解.若把它化为多项式方程组,再用解多项式方程组的符号计算方法或现有的同伦方法求解,由于该问题规模太大而不能在规定的时间内求解,而当考虑的问题维数较大时,利用已有的方法甚至根本无法求解.综合利用我们提出的解混合三角多项式方程组的混合同伦方法和保对称的系数参数同伦方法,我们给出该类问题一种有效的求解方法.利用这种方法,可以达到实时求解的目的,满足实际问题的需要.     

解约束非凸规划问题的同伦方法的收敛性定理

本文在利用组合内点同伦方法求解约束非凸规划问题时,得到了一些新的收敛性定理.证明了同伦映射为正则映射的条件下,选取合适的同伦方程,用此同伦方法得到的K-K-T点一定是问题局部最优解.     

多目标规划的光滑同伦方法

本文给出了求解多目标规划的一种连续同伦方法 .首先 ,运用光滑熵函数将多目标多约束的问题化为单目标单约束的问题 ,然后构造了求解单目标问题的同伦方法 ,并证明了其大范围收敛性 .     

解非凸规划问题动边界组合同伦方法

本文给出了一个新的求解非凸规划问题的同伦方法,称为动边界同伦方程,并在较弱的条件下,证明了同伦路径的存在性和大范围收敛性．与已有的拟法锥条件、伪锥条件下的修正组合同伦方法相比,同伦构造更容易,并且不要求初始点是可行集的内点,因此动边界组合同伦方法比修正组合同伦方法及弱法锥条件下的组合同伦内点法和凝聚约束同伦方法更便于应用．     

求解多目标规划最小弱有效解的同伦内点方法

本文利用非线性规划中的组合同伦方法;给出了求解目标规划问题最小弱有效解的同伦内点方法,并证明了该方法是整体收敛的。     

To solving inverse eigenvalue problems for parametric matrices

Some statements of inverse eigenvalue problems for one-parameter and multiparameter regular polynomial matrices with linear and nonlinear dependences on spectral parameters are considered. Methods for solving inverse eigenvalue problems based on rank factorization, exhaustion, and reduction to nonlinear equations are proposed. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 334, 2006, pp. 174–192.     

Kinematics of an offset 3-UPU translational parallel manipulator by the homotopy continuation method

For most parallel manipulators, the inverse kinematics is straightforward, while the direct kinematics is challenging. The latter requires the solution of a system of nonlinear equations. In this paper we use the homotopy continuation method to solve the forward and inverse kinematic problems of an offset 3-UPU translational parallel manipulator. The homotopy continuation method is a novel method which alleviates drawbacks of the traditional numerical techniques, namely; the acquirement of good initial guess values, the problem of convergence and computing time. The direct kinematics problem of the manipulator leads to 16 real solutions.     

Adomian polynomials: A powerful tool for iterative methods of series solution of nonlinear equations

In this article, we illustrate how the Adomian polynomials can be utilized with different types of iterative series solution methods for nonlinear equations. Two methods are considered here: the differential transform method that transforms a problem into a recurrence algebraic equation and the homotopy analysis method as a generalization of the methods that use inverse integral operator. The advantage of the proposed techniques is that equations with any analytic nonlinearity can be solved with less computational work due to the properties and available algorithms of the Adomian polynomials. Numerical examples of initial and boundary value problems for differential and integro-differential equations with different types of nonlinearities show good results.     

He’s homotopy perturbation method for systems of integro-differential equations

In this article, the homotopy perturbation method [He JH. Homotopy perturbation technique. Comput Meth Appl Mech Eng 1999;178:257–62; He JH. A coupling method of homotopy technique and perturbation technique for nonlinear problems. Int J Non-Linear Mech 2000;35(1):37–43; He JH. Comparison of homotopy perturbation method and homotopy analysis method. Appl Math Comput 2004;156:527–39; He JH. Homotopy perturbation method: a new nonlinear analytical technique. Appl Math Comput 2003;135:73–79; He JH. The homotopy perturbation method for nonlinear oscillators with discontinuities. Appl Math Comput 2004;151:287–92; He JH. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005;26:695–700] is applied to solve linear and nonlinear systems of integro-differential equations. Some nonlinear examples are presented to illustrate the ability of the method for such system. Examples for linear system are so easy that has been ignored.     

Laplace transform for the solution of higher order deformation equations arising in the homotopy analysis method

A new analytic approach for solving nonlinear ordinary differential equations with initial conditions is proposed. First, the homotopy analysis method is used to transform a nonlinear differential equation into a system of linear differential equations; then, the Laplace transform method is applied to solve the resulting linear initial value problems; finally, the solutions to the linear initial value problems are employed to form a convergent series solution to the given problem. The main advantage of the new approach is that it provides an effective way to solve the higher order deformation equations arising in the homotopy analysis method.     

Closure of the squared Zakharov-Shabat eigenstates

By solving the inverse scattering problem for a third-order (degenerate) eigenvalue problem, we can find the closure of the squared eigenfunctions of the Zakharov-Shabat equations. The question of the completeness of squared eigenstates occurs in many aspects of “inverse scattering transforms” (solving nonlinear evolution equations exactly by inverse scattering techniques) as well as in various aspects of the inverse scattering problem. The method we use is quite suggestive as to how one might find the closure of the squared eigenfunctions of other eigenvalue equations, and we point the strong analogy between our results and the problem of finding the closure of the eigenvectors of a nonself-adjoint matrix.     

求解参数识别反问题的同伦正则化方法

提出了求解参数识别反问题的同伦正则化方法,给出了相应的收敛性定理.数值结果表明该方法是一种快速的大范围收敛方法.     

Probability-one homotopies in computational science

Probability-one homotopy algorithms are a class of methods for solving nonlinear systems of equations that, under mild assumptions, are globally convergent for a wide range of problems in science and engineering. Convergence theory, robust numerical algorithms, and production quality mathematical software exist for general nonlinear systems of equations, and special cases such as Brouwer fixed point problems, polynomial systems, and nonlinear constrained optimization. Using a sample of challenging scientific problems as motivation, some pertinent homotopy theory and algorithms are presented. The problems considered are analog circuit simulation (for nonlinear systems), reconfigurable space trusses (for polynomial systems), and fuel-optimal orbital rendezvous (for nonlinear constrained optimization). The mathematical software packages HOMPACK90 and POLSYS_PLP are also briefly described.     

THE SUFFICIENT CONDITIONS FOR SOLVABILITY OF ALGEBRAIC INVERSE EIGENVALUE PROBLEMS

Applying constructed homotopy and its properties,we gel some sufficient conditions for the solvability of algebraic inverse eigenvalue problems,which are better than that of the paper [4] in some cases. Inverse eigenvalue problems,solvability,sufficient conditions.     

Solving generalized equations via homotopies

Global Newton methods for computing solutions of nonlinear systems of equations have recently received a great deal of attention. By using the theory of generalized equations, a homotopy method is proposed to solve problems arising in complementarity and mathematical programming, as well as in variational inequalities. We introduce the concepts of generalized homotopies and regular values, characterize the solution sets of such generalized homotopies and prove, under boundary conditions similar to Smale’s [10], the existence of a homotopy path which contains an odd number of solutions to the problem. We related our homotopy path to the Newton method for generalized equations developed by Josephy [3]. An interpretation of our results for the nonlinear programming problem will be given.