A Dissipation-Preserving Integrator for Damped Oscillatory Hamiltonian Systems

In this paper, based on discrete gradient, a dissipation-preserving integrator for weakly dissipative perturbations of oscillatory Hamiltonian system is established. The solution of this system is a damped nonlinear oscillator. Basically, lots of nonlinear oscillatory mechanical systems including frictional forces lend themselves to this approach. The new integrator gives a discrete analogue of the dissipation property of the original system. Meanwhile, since the integrator is based on the variation-of-constants formula for oscillatory systems, it preserves the oscillatory structure of the system. Some properties of the new integrator are derived. The convergence is analyzed for the implicit iterations based on the discrete gradient integrator, and it turns out that the convergence of the implicit iterations based on the new integrator is independent of $\|M\|$, where $M$ governs the main oscillation of the system and usually $\|M\|\gg1$. This significant property shows that a larger stepsize can be chosen for the new schemes than that for the traditional discrete gradient integrators when applied to the oscillatory Hamiltonian system. Numerical experiments are carried out to show the effectiveness and efficiency of the new integrator in comparison with the traditional discrete gradient methods in the scientific literature.     

Ekeland变分原理的一个注记

给出了一般形式的Ekeland变分原理,并根据新得到的结论讨论了泛函强制性条件与一般性弱PS条件之间的关系.     

A Variational Complex for Difference Equations

An analogue of the Poincaré lemma for exact forms on a lattice is stated and proved. Using this result as a starting-point, a variational complex for difference equations is constructed and is proved to be locally exact. The proof uses homotopy maps, which enable one to calculate Lagrangians for discrete Euler–Lagrange systems. Furthermore, such maps lead to a systematic technique for deriving conservation laws of a given system of difference equations (whether or not it is an Euler–Lagrange system).     

Ekeland变分原理的一种推广

研究完备度量空间中一类拟均衡问题的可解性,由此导出名的Ekeland变分原理。     

一类非强制的变分不等式

讨论了一类非强制的变分不等式解之存在性的一个必要条件和一个充分条件,改进了文[1]中的相应结果。     

A Comparison of Two Variational Methods

A discussion is given of the relation between the maximum functionaltheorem and Schwinger's variational principle, and it is shownthat the bound given by Schwinger's principle is better thanor the same as that given by the maximum functional theorem.     

一类Ekeland变分原理的推广

本文给出一个向量值形式的Ekeland变分原理,其目标函数是从完备的度量空间映到锥序拓扑线性空间.此结果是对钟承奎建立的Ekeland变分原理的推广.     

建立广义变分原理的一种方法

本文建议了一种根据问题的力学意义来建立广义变分原理的方法,本方法对于那些尚未建立起与之相应的变分原理的问题建立其相应的变分原理是有用的.文中不从最小势能原理的推广出发而从力学意义出发导出了弹性力学中的Hu-Washizu广义变分原理和胡海昌广义余能原理,给出了这两个广义变分原理的正确证明.本文并证明了,如果根据Hu-Washizu广义变分原理及胡海昌广义余能原理中含有σ_ij,e_ij和u_i三类变量,就认为这三类变量相互独立,就会导致错误.文中并阐明了这两个广义变分原理正确运用的条件.     

A Linearization Method for Nondegenerate Variational Conditions

We employ recent results about constraint nondegeneracy in variational conditions to design and justify a linearization algorithm for solving such problems. The algorithm solves a sequence of affine variational inequalities, but the variational condition itself need not be a variational inequality: that is, its underlying set need not be convex. However, that set must be given by systems of differentiable nonlinear equations with additional polyhedral constraints. We show that if the variational condition has a solution satisfying nondegeneracy and a standard regularity condition, and if the linearization algorithm is started sufficiently close to that solution, the algorithm will produce a well defined sequence that converges Q-superlinearly to the solution.     

A Logarithmic-Quadratic Proximal Method for Variational Inequalities

We present a new method for solving variational inequalities on polyhedra. The method is proximal based, but uses a very special logarithmic-quadratic proximal term which replaces the usual quadratic, and leads to an interior proximal type algorithm. We allow for computing the iterates approximately and prove that the resulting method is globally convergent under the sole assumption that the optimal set of the variational inequality is nonempty.     

A Variational Problem Pertaining to Biharmonic Maps

The second derivative of a map into a Riemannian manifold is given by a nonlinear differential operator. We study minimizers and critical points of the L ²-norm of this second derivative. We show existence of minimizers with the direct method and we prove a partial regularity result.     

A Projection Algorithm for Non-Monotone Variational Inequalities

Set-Valued and Variational Analysis - We introduce a projection-type algorithm for solving the variational inequality problem for point-to-set operators, and establish its convergence properties....     

A Range Condition for Polyconvex Variational Regularization

In the context of convex variational regularization, it is a known result that, under suitable differentiability assumptions, source conditions in the form of variational inequalities imply range conditions, while the converse implication only holds under an additional restriction on the operator. In this article, we prove the analogous result for polyconvex regularization. More precisely, we show that the variational inequality derived by the authors in 2017 implies that the derivative of the regularization functional must lie in the range of the dual-adjoint of the derivative of the operator. In addition, we show how to adapt the restriction on the operator in order to obtain the converse implication.     

一类非线性拟变分不等式

本文利用伪单调算子理论研究如下变分不等式问题：求ｘ∈Ｍ，使得〈Ａｘ，ｙ－ｘ〉＋〈Ｇｘ，ｙ－ｘ〉≥〈ｆ，ｙ－ｘ〉，?ｙ∈Ｍ．并将所得结果应用于拟线性椭圆型边值问题的求解．     

A Variational Principle in Krein Space II

Let A be a self-adjoint operator in a Krein space Under certain natural assumptions, it is shown precisely which real eigenvalues of A can be given a max-inf characterization generalizing the usual one in Hilbert space. This result unifies several approaches in the recent literature.     

A One Perturbation Variational Principle and Applications

Set-Valued and Variational Analysis - We study a variational principle in which there is one common perturbation function ? for every proper lower semicontinuous extended real-valued function...