基于GMRES(m)法的双连通区域数值保角变换的计算法

本文研究了基于模拟电荷法的双连通区域的数值保角变换问题.利用限制Krylov子空间最大维数的算法–GMRES(m)算法,求解基于模拟电荷法的双连通区域数值保角变换中的约束方程,获得了模拟电荷和变换半径,构造了近似保角变换函数.数值实验表明了本文算法的有效性.     

计算机求解渐开线齿轮齿廓的保角映射函数

平面弹性理论的复变函数保角映射解法可以求得齿轮的应力和位移的精确解.而相应于各种不同参数的轮齿齿廓的保角映射函数的求得却是比较困难的.以往均采用试算法,这是费时且昂贵的.作者编制了求解保角映射函数的计算机程序,并通过大量的计算证明这一程序是成功的,所取得的映射函数是精确的.从而解决了保角映射法求解渐开线齿轮应力和位移应用于实际工程计算的主要障碍.     

A Fast Space-decomposition Scheme for Nonconvex Eigenvalue Optimization

In this paper, a nonsmooth bundle algorithm to minimize the maximum eigenvalue function of a nonconvex smooth function is presented. The bundle method uses an oracle to compute separately the function and subgradient information for a convex function, and the function and derivative values for the smooth mapping. Using this information, in each iteration, we replace the smooth inner mapping by its Taylor-series linearization around the current serious step. To solve the convex approximate eigenvalue problem with affine mapping faster, we adopt the second-order bundle method based on ????-decomposition theory. Through the backtracking test, we can make a better approximation for the objective function. Quadratic convergence of our special bundle method is given, under some additional assumptions. Then we apply our method to some particular instance of nonconvex eigenvalue optimization, specifically: bilinear matrix inequality problems.     

Moreau–Yosida regularization of Lagrangian-dual functions for a class of convex optimization problems

In this paper, we consider the Lagrangian dual problem of a class of convex optimization problems, which originates from multi-stage stochastic convex nonlinear programs. We study the Moreau–Yosida regularization of the Lagrangian-dual function and prove that the regularized function η is piecewise C ², in addition to the known smoothness property. This property is then used to investigate the semismoothness of the gradient mapping of the regularized function. Finally, we show that the Clarke generalized Jacobian of the gradient mapping is BD-regular under some conditions.      

Composite proximal bundle method

We consider minimization of nonsmooth functions which can be represented as the composition of a positively homogeneous convex function and a smooth mapping. This is a sufficiently rich class that includes max-functions, largest eigenvalue functions, and norm-1 regularized functions. The bundle method uses an oracle that is able to compute separately the function and subgradient information for the convex function, and the function and derivatives for the smooth mapping. With this information, it is possible to solve approximately certain proximal linearized subproblems in which the smooth mapping is replaced by its Taylor-series linearization around the current serious step. Our numerical results show the good performance of the Composite Bundle method for a large class of problems.     

On approximately microperiodic mappings

We show that, under some weak regularity conditions, an approximately microperiodic function, mapping a semitopological group into the set of reals, is close to a constant function.      

ON A NEW SINGULAR DIRECTION OF QUASIMEROMORPHIC MAPPINGS

By applying Ahlfors＇ theory of covering surface, we establish a fundamental inequality for quasimeromorphic mapping in an angular domain. As an application, we prove the existence of a new singular direction for quasimeromorphic mapping f, namely, a precise S direction, for which the spherical characteristic function S（r, f） is used as a comparison function.     

Newton’s method for generalized equations: a sequential implicit function theorem

In an extension of Newton’s method to generalized equations, we carry further the implicit function theorem paradigm and place it in the framework of a mapping acting from the parameter and the starting point to the set of all associated sequences of Newton’s iterates as elements of a sequence space. An inverse function version of this result shows that the strong regularity of the mapping associated with the Newton sequences is equivalent to the strong regularity of the generalized equation mapping.     

Two-level iterative method for non-stationary mixed variational inequalities

We consider a mixed variational inequality problem involving a set-valued nonmonotone mapping and a general convex function, where only approximation sequences are known instead of exact values of the cost mapping and function, and feasible set. We suggest to apply a two-level approach with inexact solutions of each particular problem with a descent method and partial penalization and evaluation of accuracy with the help of a gap function. Its convergence is attained without concordance of penalty, accuracy, and approximation parameters under coercivity type conditions.     

A nonlinear descent method for a variational inequality on a nonconvex set

In this paper, we consider a generalized variational inequality problem which involves the integrable cost mapping and a nonsmooth mapping with convex components. We propose a new gradient-type method which determines a stepsize by using the smooth part of the cost function. Thus, the method does not utilize analogs of derivatives of nonsmooth functions. We show that its convergence does not require additional assumptions.     

A generalization of D. A. Grave's method for plane boundary-value problems in harmonic potential theory

The article describes and proves D. A. Grave's method that solves classical plane boundary-value problems for the Green's function of the Laplace equation in regions whose boundaries are smooth analytical curves defined by finite-order irreducible polynomials. The proposed method has certain advantages compared with the method that constructs the Green's function by conformally mapping the original region onto the unit disk. A class of regions are identified for which Grave's methods produces an explicit analytical solution in convergent-series form. This is a natural generalization of the conformal mapping method for simplest regions. Translated from Prikladnaya Matematika i Informatika, No. 1, pp. 5–19, 1999.     

非线性优化修正Frisch函数方法的乘子映射

对于同时含有等式与不等式约束的非线性优化问题的修正Frisch函数方法,给出其乘子映射和解映射的导数的估计.将得到的估计用于建立修正Frisch函数方法的线性收敛速率.在线性无关的约束规范,严格互补条件和二阶充分性条件成立的前提下,证得该收敛率与1/c成正比.本文的收敛性分析依赖于矩阵的奇异值分解,其方法可以用来分析其他的修正Lagrange方法.     

一类非线性演化方程的周期解

通过直接积分和映射方法,得到了一类非线性演化方程的一系列Jacobi椭圆函数周期解.在极限情况下,得到三角函数解.     

A new application of Lagrange-Bürmann expansions. I. General principle

A new scheme is developed for improving the convergence of slowly convergent series solutions. The method is based on a transformation of variables of similarity form whereby the resulting composite function is constructed by its Lagrange-Bürmann expansion. It is the improved convergence of the new expansion that we take most advantage of in this method. The convergence of the Lagrange-Bürmann expansion as well as its inversion scheme is proved for analytic (object) functions. The inversion is required to recover from the Lagrange-Bürmann expansion the object function which is imbedded in the mapping functions. Several numerical examples demonstrate the improved convergence of the new method. The improvement owes much to the invariance properties of the mapping function under a group and the “built-in” feature of analytic continuation of the method. These features are elucidated in detail.     

Descent method for nonsmooth variational inequalities

A descent method with a gap function is proposed for a finite-dimensional variational inequality with nonintegrable and nonsmooth mapping. The convergence of the method with line search is established under strong monotonicity conditions on the underlying mapping. Published in Russian in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 7, pp. 1251–1257. This article was translated by the author.     

Self-adaptive projection method for co-coercive variational inequalities

In some real-world problems, the mapping of the variational inequalities does not have any explicit forms and only the function value can be evaluated or observed for given variables. In this case, if the mapping is co-coercive, the basic projection method is applicable. However, in order to determine the step size, the existing basic projection method needs to know the co-coercive modulus in advance. In practice, usually even if the mapping can be characterized co-coercive, it is difficult to evaluate the modulus, and a conservative estimation will lead an extremely slow convergence. In view of this point, this paper presents a self-adaptive projection method without knowing the co-coercive modulus. We also give a real-life example to demonstrate the practicability of the proposed method.     

Extragradient Methods and Linesearch Algorithms for Solving Ky Fan Inequalities and Fixed Point Problems

In this paper, we introduce some new iterative methods for finding a common element of the set of points satisfying a Ky Fan inequality, and the set of fixed points of a contraction mapping in a Hilbert space. The strong convergence of the iterates generated by each method is obtained thanks to a hybrid projection method, under the assumptions that the fixed-point mapping is a ??-strict pseudocontraction, and the function associated with the Ky Fan inequality is pseudomonotone and weakly continuous. A?Lipschitz-type condition is assumed to hold on this function when the basic iteration comes from the extragradient method. This assumption is unnecessary when an Armijo backtracking linesearch is incorporated in the extragradient method. The particular case of variational inequality problems is examined in a last section.     

Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we construct new explicit exact solutions for the coupled the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation (KD equation) by using a improved mapping approach and variable separation method. By means of the method, new types of variable-separation solutions (including solitary wave solutions, periodic wave solutions and rational function solutions) for the KD system are successfully obtained. The improved mapping approach and variable separation method can be applied to other higher-dimensional coupled nonlinear evolution equations.     

A backtracking method for constructing perfect hash functions from a set of mapping functions

This paper presents a backtracking method for constructing perfect hash functions from a given set of mapping functions. A hash indicator table is employed in the composition. By the nature of backtracking, the method can always find a perfect hash function when such a function does exist according to the composing scheme. Simulation results show that the probability of getting a perfect hash function by the backtracking method is much higher than by the single-pass and multipass methods previously proposed.     

一类非线性双曲型发展方程的孤子解

研究了一类非线性发展方程.首先在无扰动情形下,利用待定函数和泛函同伦映射方法得到了非扰动发展方程的孤子精确解和扰动方程的任意次近似行波孤子解.接着引入一个同伦映射,并选取初始近似函数,再用同伦映射理论,依次求出非线性双曲型发展扰动方程孤子解的各次近似解析解.再利用摄动理论举例说明了用该方法得到的近似解析解的有效性和各次近似解的近似度.最后,简述了用同伦映射方法得到的近似解的意义,指出了用上述方法得到的各次近似解具有便于求解、精度高等优点.