阻尼Gauss-Newton方法解非线性不等式组

本文研究了非线性不等式组的求解问题.利用了阻尼Gauss-Newton方法求解非线性方程组,获得了该算法的全局收敛性,推广了Gauss-Newton法在解非线性方程组方面的应用.     

一类非线性方程组的Newton-PSS迭代法

正定反Hermite分裂(PSS)方法是求解大型稀疏非Hermite正定线性代数方程组的一类无条件收敛的迭代算法.将其作为不精确Newton方法的内迭代求解器,我们构造了一类用于求解大型稀疏且具有非Hermite正定Jacobi矩阵的非线性方程组的不精确Newton-PSS方法,并对方法的局部收敛性和半局部收敛性进行了详细的分析.数值结果验证了该方法的可行性与有效性.     

凸约束非线性方程组的一种无导数投影方法

推广LCG共轭梯度方法并建立一种求解凸约束非线性单调方程组问题的无导数投影方法.在适当的条件下,证明了方法的全局收敛性.方法不需要任何导数信息,而且继承了共轭梯度方法储存量小的特征,因此它特别适合求解大规模非光滑的非线性单调方程组问题.大量数值结果和比较表明方法是有效的和稳定的.     

带凸约束的非线性方程组的无导数记忆法

基于无导数线搜索技术和投影方法,本文提出了一种新的求解带凸约束的非线性方程组的无导数记忆法.该方法在每步迭代时不需要计算和贮存任何矩阵,因而适合求解大规模非线性方程组问题.在较弱条件下,该算法具有全局收敛性.数值试验结果及其相关的比较表明该算法是比较有效的.     

求解非线性互补问题的一类光滑Broyden-like方法

非线性互补问题可以等价地转换为光滑方程组来求解. 基于一种新的非单调线搜索准则, 提出了求解非线性互补问题等价光滑方程组的一类新的非单调光滑 Broyden-like 算法.在适当的假设条件下, 证明了该算法的全局收敛性与局部超线性收敛性. 数值实验表明所提出的算法是有效的.     

一个解界约束非线性方程组的无导数回溯线搜索仿射内点信赖域方法（英文）

文章给出了一个求解界约束非线性方程组的无导数回溯线搜索仿射内点信赖域方法.该方法利用非线性方程组的特点,对方程组中每一个函数建立插值模型.通过利用信赖域模型和回溯先搜索技术的结合,利用插值信赖域子问题子问题求解搜索方向,并利用回溯先搜索技术保证可行性.在合理的假设条件下,证明了算法的全局和快速局部收敛性.并且,通过数值实验表明该种无导数算法对求解界约束非线性方程组问题是有效的.     

等式约束非线性规划的一种线性代数方程组解法

本文提出了一种求解等式约束非线性规划的新方法－线性代数方程组求解方法。我们证明了该算法具有全局收敛性和局部二阶收敛性。     

非线性绝对值方程组的类SOR迭代方法

提出了非线性绝对值方程组(AVE)问题解的存在性和唯一性的一个充分条件,构建了数值求解方程组的类超松弛迭代方法,并证明其收敛性.数值算例表明该迭代方法是非常有效的.     

基于一个新的NCP函数的光滑牛顿法求解非线性互补问题

本文研究了非线性互补的光滑化问题.利用一个新的光滑NCP函数将非线性互补问题转化为等价的光滑方程组,并在此基础上建立了求解P0-函数非线性互补问题的一个完全光滑化牛顿法,获得了算法的全局收敛性和局部二次收敛性的结果.并给出数值实验验证了理论分析的正确性.     

非线性单调方程组的三项无导数投影算法

本文讨论了一种求解非线性单调方程组问题的三项无导数投影算法,并在适当的条件下证明了算法的全局收敛性和R-线性收敛速度.由于无需利用任何导数信息,该算法适合求解大规模的非线性单调方程组问题.数值比较表明该算法是有效的.     

计算部分奇异值分解的隐式重新启动的双对角化Lanczos方法和精化的双对角化Lanczos方法

The singular value decomposition problem is mathematically equivalent to the eigenproblem of an argumented matrix. Golub et al. give a bidiagonalization Lanczos method for computing a number of largest or smallest singular values and corresponding singular vertors, but the method may encounter some convergence problems. In this paper we analyse the convergence of the method and show why it may fail to converge. To correct this possible nonconvergence, we propose a refined bidiagonalization Lanczos method and apply the implicitly restarting technique to it, and we then present an implicitly restarted bidiagonalization Lanczos algorithm(IRBL) and an implicitly restarted refined bidiagonalization Lanczos algorithm (IRRBL). A new implicitly restarting scheme and a reliable and efficient algorithm for computing refined shifts are developed for this special structure eigenproblem.Theoretical analysis and numerical experiments show that IRRBL performs much better than IRBL.     

The solution methods for the largest eigenvalue (singular value) of nonnegative tensors and convergence analysis

In this paper we study two solution methods for finding the largest eigenvalue (singular value) of general square (rectangular) nonnegative tensors. For a positive tensor, one can find the largest eigenvalue (singular value) based on the properties of the positive tensor and the power-type method. While for a general nonnegative tensor, we use a series of decreasing positive perturbations of the original tensor and repeatedly recall power-type method for finding the largest eigenvalue (singular value) of a positive tensor with an inexact strategy. We prove the convergence of the method for the general nonnegative tensor. Under a certain assumption, the computing complexity of the method is established. Motivated by the interior-point method for the convex optimization, we put forward a one-step inner iteration power-type method, whose convergence is also established under certain assumption. Additionally, by using embedding technique, we show the relationship between the singular values of the rectangular tensor and the eigenvalues of related square tensor, which suggests another way for finding the largest singular value of nonnegative rectangular tensor besides direct power-type method for this problem. Finally, numerical examples of our algorithms are reported, which demonstrate the convergence behaviors of our methods and show that the algorithms presented are promising.     

An efficient Levenberg–Marquardt method with a new LM parameter for systems of nonlinear equations

A modified Levenberg–Marquardt method for solving singular systems of nonlinear equations was proposed by Fan [J Comput Appl Math. 2003;21;625–636]. Using trust region techniques, the global and quadratic convergence of the method were proved. In this paper, to improve this method, we decide to introduce a new Levenberg–Marquardt parameter while also incorporate a new nonmonotone technique to this method. The global and quadratic convergence of the new method is proved under the local error bound condition. Numerical results show the new algorithm is efficient and promising.     

有奇异解的无约束最优化问题的一种混合法

本文研究求解含有奇异解的无约束最优化问题算法 .该类问题的一个重要特性是目标函数的Hessian阵可能处处奇异 .我们提出求解该类问题的一种梯度 -正则化牛顿型混合算法 .并在一定的条件下得到了算法的全局收敛性 .而且 ,经一定迭代步后 ,算法还原为正则化 Newton法 .因而 ,算法具有局部二次收敛性 .     

Approximation Spaces in the Numerical Analysis of Operator Equations

We show that generalized approximation spaces can be used to prove stability and convergence of projection methods for certain types of operator equations in which unbounded operators occur. Besides the convergence, we also get orders of convergence by this approach, even in case of non-uniformly bounded projections. We give an example in which weighted uniform convergence of the collocation method for an easy Cauchy singular integral equation is studied.     

关于加权差分方法的一点注记

在这篇短文中,对奇异摄动问题引进了新的差分逼近并证明了一致收敛的必要条件。选择适当的权因子可得到用Ilin方法所得的相同的差分格式。     

Convergence and quotient convergence of iterative methods for solving singular linear equations with index one

Singular systems with index one arise in many applications, such as Markov chain modelling. In this paper, we use the group inverse to characterize the convergence and quotient convergence properties of stationary iterative schemes for solving consistent singular linear systems when the index of the coefficient matrix equals one. We give necessary and sufficient conditions for the convergence of stationary iterative methods for such problems. Next we show that for the stationary iterative method, the convergence and the quotient convergence are equivalent.     

A mean value algorithm for Toeplitz matrix completion

In this paper, we propose a new mean value algorithm for the Toeplitz matrix completion based on the singular value thresholding (SVT) algorithm. The completion matrices generated by the new algorithm keep a feasible Toeplitz structure. Meanwhile, we prove the convergence of the new algorithm under some reasonal conditions. Finally, we show the new algorithm is much more effective than the ALM (augmented Lagrange multiplier) algorithm through numerical experiments and image inpainting.     

Regularized Newton Methods for Convex Minimization Problems with Singular Solutions

This paper studies convergence properties of regularized Newton methods for minimizing a convex function whose Hessian matrix may be singular everywhere. We show that if the objective function is LC², then the methods possess local quadratic convergence under a local error bound condition without the requirement of isolated nonsingular solutions. By using a backtracking line search, we globalize an inexact regularized Newton method. We show that the unit stepsize is accepted eventually. Limited numerical experiments are presented, which show the practical advantage of the method.     

An iterative method based on equation decomposition for the fourth‐order singular perturbation problem

In this article, we propose an iterative method based on the equation decomposition technique ( 1 ) for the numerical solution of a singular perturbation problem of fourth‐order elliptic equation. At each step of the given method, we only need to solve a boundary value problem of second‐order elliptic equation and a second‐order singular perturbation problem. We prove that our approximate solution converges to the exact solution when the domain is a disc. Our numerical examples show the efficiency and accuracy of our method. Our iterative method works very well for singular perturbation problems, that is, the case of 0 < ε ? 1, and the convergence rate is very fast. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013