求解单调变分不等式的两类迭代算法

给出了求解单调变分不等式的两类迭代算法．通过解强单调变分不等式子问题,产生两个迭代点列,都弱收敛到变分不等式的解．最后,给出了这两类新算法的收敛性分析．     

箱式约束变分不等式的一类新光滑gap函数

针对箱式约束变分不等式问题,利用一类积分型全局最优性条件,提出了一个新光滑gap函数．该光滑gap函数形式简单且具有较好的性质．利用该gap函数,箱式约束变分不等式可转化为等价光滑优化问题进行求解．进一步地,讨论了可保证等价光滑优化问题的任意聚点为箱式约束变分不等式问题解的条件．以一个简单的摩擦接触问题为例阐释了该方法的应用．最后,利用标准的变分不等式考题验证了方法的有效性．     

单调变分不等式可行与非可行点组合的连续算法

本文给出了单调变分不等式问题一个新的连续型求解方法，方法的实现依赖于一系列含有四个参数的摄动单调变分不等式的求解．其中摄动参数要求的条件较为温和，这使得本文方法成为可行点与非可行点算法的有机组合和统一．在适当的假设条件下，我们分析和证明了摄动变分不等式问题解的存在性，唯一性和算法的强收敛性．     

向量变分不等式的严格可行性问题

本篇文章首先定义了向量变分不等式的严格可行点概念,其次在假设了映射是强(D)-伪单调的情况下,证明了向量变分不等式解集非空有界与其严格可行点存在的等价性问题,推广了在数量变分不等式上得到的相应结果．     

广义集值强非线性混合似变分不等式的辅助原理和三步迭代算法

使用辅助原理技巧研究了一类广义集值强非线性混合变分不等式．证明了此类集值强非线性混合变分不等式辅助问题解的存在性和唯一性;构建了一个新的三步迭代算法,通过辅助原理技巧,构建并计算此类非线性混合变分不等式的近似解,进一步证明非线性混合变分不等式解的存在性以及由算法产生的三个序列的收敛性．所得结论推广了近年来许多混合变分不等式和准变分不等式以及他们的有关结果．     

解拟变分不等式的超平面投影算法

拟变分不等式问题是变分不等式问题的一种推广,超平面投影算法是解变分不等式的一种重要方法.通过构造严格分离当前点与拟变分不等式解集的超平面,建立了解拟变分不等式的超平面投影算法.在一定的条件下,证明了该算法的全局收敛性.     

不等式组与变分不等式的极大熵函数方法

利用极大熵函数方法将不等式组及变分不等式的求解问题转化为近似可微优化问题，给出了不等式组及变分不等式问题近似解的可微优化方法，得到了不等式组和变分不等式问题的解集合的示性函数．     

一般多值混合隐拟变分不等式的解的存在性与算法

引入了实Hilbert空间中一类新的一般多值混合隐拟变分不等式.它概括了丁协平教授引入与研究过的熟知的广义混合隐拟变分不等式类成特例.运用辅助变分原理技巧来解这类一般多值混合隐拟变分不等式.首先,定义了具真凸下半连续的二元泛函的新的辅助变分不等式,并选取了一适当的泛函,使得其唯一的最小值点等价于此辅助变分不等式的解.其次,利用此辅助变分不等式,构造了用于计算一般多值混合隐拟变分不等式逼近解的新的迭代算法.在此,等价性保证了算法能够生成一列逼近解.最后,证明了一般多值混合隐拟变分不等式解的存在性与逼近解的收敛性.而且,给算法提供了新的收敛判据.因此,结果对M.A.Noor提出的公开问题给出了一个肯定答案,并推广和改进了关于各种变分不等式与补问题的早期与最近的结果,包括最近文献中涉及单值与集值映象的有关混合变分不等式、混合拟变不等式与拟补问题的相应结果.     

具有双障碍的退缩抛物变分不等式解的存在性和唯一性(英)

本文研究具有双障碍的退缩抛物变分不等式．我们利用罚技巧，有限逼近和先验估计方法，得到一类退缩抛物变分不等式弱解的存在性，并在一定条件之下，建立了弱解的唯一性．本文结论对广泛的一类抛物型变分不等式成立．     

关于求解变分不等式问题的2-次梯度外梯度算法收敛性的一个补注

Yair Censor,Aviv Gibali和Simeon Reich为求解变分不等式问题提出了2-次梯度外梯度算法。关于此算法的收敛性,作者给出了部分证明,有一个问题：由算法产生的迭代点列能否收敛到变分不等式问题的一个解上,没有得到解决。此问题作为一个公开问题在文章“Extensions of Korpelevich's extragradient method for the variational inequalityproblem in Euclidean space”(Optimization,61(9)：1119-1132,2012)中被提出。在这篇简短的补注性文章中,对所提出的问题给出了答案：由算法产生的迭代点列能收敛到变分不等式问题的一个解上。给出2-次梯度外梯度算法的全局收敛性的一个完整证明,证明了从任意起始点开始,由算法产生的迭代点列都能收敛到变分不等式问题的一个解上。     

A variational inequality formulation for unconfined seepage problems in porous media

In the existing variational inequality formulations for the unconfined seepage problem in porous media, the seepage point, namely the exit point of the free surface, is a singular point and how to locate the seepage point exactly has been an open issue. By generalizing Darcy’s law applied solely to the saturated zone in an earth dam to the entire dam including the no-flow zone, a new variational inequality formulation is presented. The new formulation imposes a boundary condition of Signorini’s type on the potential seepage boundary and the seepage point turns out to be such a point that makes both inequalities in Signorini’s complementary condition become equalities. Singularity of the seepage point is accordingly eliminated. A strategy is developed for overcoming the mesh-dependency in the finite element implementation.     

非稳态渗流的自由边界问题

随着经济建设的高速发展,各种地下工程大量增加,如水坝和高层建筑的基础、地铁和隧道、水井和油井等。那里,介质中的渗流现象往往是工程单位需要考虑的重要问题。佘颖禾等在《应用数学和力学》第17卷6期中曾经给出了具有自由边界的稳态渗流的变分不等式模式及有限元解。本文中,以抽水井为例,进一步研究了非稳态渗流问题的变分不等式模式及其有限元解法。结果表明,对于非稳态的渗流问题,这种方法同样能避免传统的自由边界的迭代过程,为简单而快速地进行数值分析提供方便。     

Numerical simulation of tridimensional electromagnetic shaping of liquid metals

Summary We describe a numerical method to compute free surfaces in electromagnetic shaping and levitation of liquid metals. We use an energetic variational formulation and optimization techniques to compute, a critical point. The surfaces are represented by piecewise linear finite elements. Each step of the algorithm requires solving an elliptic boundary value problem in the exterior of the intermediate surfaces. This is done by using an integral representation on these surfaces.     

The construction of dual variational principles of the theory of the seepage of an incompressible fluid in some deformable complex media

A method of constructing dual variational principles for problems of the seepage of an incompressible fluid in deformable media with complex rheology is presented. Dual variational principles of the fluid seepage in a typical deformable media with complex rheology [1, 2], namely, a Maxwell element, connected in parallel with a visco-plastic or elastic element, are constructed. The variational principles are derived from variational problems compiled for the constitutive relations for fluid and solid phases.     

Projection iterative methods for solving some systems of general nonconvex variational inequalities

In this article, we introduce and consider a new system of general nonconvex variational inequalities involving four different operators. We use the projection operator technique to establish the equivalence between the system of general nonconvex variational inequalities and the fixed points problem. This alternative equivalent formulation is used to suggest and analyse some new explicit iterative methods for this system of nonconvex variational inequalities. We also study the convergence analysis of the new iterative method under certain mild conditions. Since this new system includes the system of nonconvex variational inequalities, variational inequalities and related optimization problems as special cases, results obtained in this article continue to hold for these problems. Our results can be viewed as a refinement and an improvement of the previously known results for variational inequalities.     

Minimal surfaces with prescribed topological type on a Schwarzian chain inMc3

Similar to the investigations of unstable polygonal minimal surfaces by Courant [1] we introduce here a variational principle for the free boundary problem with prescribed topological type which produces minimal surfaces in Riemannian manifolds with constant curvature. For special boundary configurations the surfaces have no branch points. The approach can be applied to numerical algorithms since it is constructive.     

Some iterative schemes for general mixed variational inequalities

In this paper, we consider a class of variational inequalities which is called the general mixed variational inequality. It is known that the general mixed variational inequalities are equivalent to the fixed point problems. This equivalent formulation is used to suggest and analyze some three-step iterative schemes for finding the common element of the set of fixed points of a nonexpansive mappings and the set of solutions of the mixed variational inequalities. We also study the convergence criteria of three-step iterative method under some mild conditions. Our results include the previous results as special cases and may be considered as an improvement and refinement of the previously known results.     

Algorithms for solutions of extended general mixed variational inequalities and fixed points

In this paper, we introduce and study a new class of extended general nonlinear mixed variational inequalities and a new class of extended general resolvent equations and establish the equivalence between the extended general nonlinear mixed variational inequalities and implicit fixed point problems as well as the extended general resolvent equations. Then by using this equivalent formulation, we discuss the existence and uniqueness of solution of the problem of extended general nonlinear mixed variational inequalities. Applying the aforesaid equivalent alternative formulation and a nearly uniformly Lipschitzian mapping S, we construct some new resolvent iterative algorithms for finding an element of set of the fixed points of nearly uniformly Lipschitzian mapping S which is the unique solution of the problem of extended general nonlinear mixed variational inequalities. We study convergence analysis of the suggested iterative schemes under some suitable conditions. We also suggest and analyze a class of extended general resolvent dynamical systems associated with the extended general nonlinear mixed variational inequalities and show that the trajectory of the solution of the extended general resolvent dynamical system converges globally exponentially to the unique solution of the extended general nonlinear mixed variational inequalities. The results presented in this paper extend and improve some known results in the literature.     

一类单边约束问题的数值方法

本文分别基于原始变分形式与对偶混合变分形式，对一类单边约束问题进行了数值求解，提出了求解离散对偶混合变分问题的Uzawa型算法，并用数值例子验证了算法的有效性．     

Solution of plane seepage problems for a multivalued seepage law when there is a point source

The steady seepage of an incompressible fluid in a uniform porous medium, occupying an arbitrary bounded two-dimensional region, when there is a point source present is considered. Part of the boundary of the region is free, while the remaining part is impermeable for the fluid. It is assumed that the function defining the seepage law is multivalued and has a linear increase at infinity. A generalized formulation of the problem is proposed in the form of a variational inequality of the second kind. An approximate solution of the problem is obtained by an iterative splitting method, which enables approximate values of both the solution itself (the pressure) and its gradient to be found. Analytic expressions describing the boundaries of the region where the modulus of the pressure gradient takes a constant value are obtained for model problems of a line of bore holes. Numerical experiments are carried out for model problems, which confirm the effectiveness of the proposed method. Good agreement is observed between the results of calculations obtained analytically and by approximate methods.