The contribution of K.-H. Elster to generalized conjugation theory and nonconvex duality

This article surveys the main contributions of K.-H. Elster to the theory of generalized conjugate functions and its applications to duality in nonconvex optimization.     

Comparison of multigrid algorithms for high‐order continuous finite element discretizations

We present a comparison of different multigrid approaches for the solution of systems arising from high‐order continuous finite element discretizations of elliptic partial differential equations on complex geometries. We consider the pointwise Jacobi, the Chebyshev‐accelerated Jacobi, and the symmetric successive over‐relaxation smoothers, as well as elementwise block Jacobi smoothing. Three approaches for the multigrid hierarchy are compared: (1) high‐order h‐multigrid, which uses high‐order interpolation and restriction between geometrically coarsened meshes; (2) p‐multigrid, in which the polynomial order is reduced while the mesh remains unchanged, and the interpolation and restriction incorporate the different‐order basis functions; and (3) a first‐order approximation multigrid preconditioner constructed using the nodes of the high‐order discretization. This latter approach is often combined with algebraic multigrid for the low‐order operator and is attractive for high‐order discretizations on unstructured meshes, where geometric coarsening is difficult. Based on a simple performance model, we compare the computational cost of the different approaches. Using scalar test problems in two and three dimensions with constant and varying coefficients, we compare the performance of the different multigrid approaches for polynomial orders up to 16. Overall, both h‐multigrid and p‐multigrid work well; the first‐order approximation is less efficient. For constant coefficients, all smoothers work well. For variable coefficients, Chebyshev and symmetric successive over‐relaxation smoothing outperform Jacobi smoothing. While all of the tested methods converge in a mesh‐independent number of iterations, none of them behaves completely independent of the polynomial order. When multigrid is used as a preconditioner in a Krylov method, the iteration number decreases significantly compared with using multigrid as a solver. Copyright © 2015 John Wiley & Sons, Ltd.     

The construction and approximation of feedforward neural network with hyp erb olic tangent function

In this paper, we discuss some analytic properties of hyperbolic tangent function and estimate some approximation errors of neural network operators with the hyperbolic tangent activation functionFirstly, an equation of partitions of unity for the hyperbolic tangent function is givenThen, two kinds of quasi-interpolation type neural network operators are constructed to approximate univariate and bivariate functions, respectivelyAlso, the errors of the approximation are estimated by means of the modulus of continuity of functionMoreover, for approximated functions with high order derivatives, the approximation errors of the constructed operators are estimated.     

使用压缩感知的遥感图像振荡畸变几何校正方法

卫星平台振动和反射镜震颤会引起遥感图像中的振荡畸变。这类畸变难以通过常用的几何校正方法消除。对此,提出了一种使用压缩感知的几何校正方法。该方法基于有理函数模型(RFM)进行几何校正。在校正过程中,利用初始的RFM计算出地面控制点(GCPs)在图像中的投影坐标与实际成像坐标之间的偏差(称为投影偏差),以地面控制点处的投影偏差作为采样值,使用压缩感知技术重构出所有像元处的投影偏差,并据此对RFM进行像方补偿;利用经过补偿的RFM进行遥感图像纠正。通过补偿,消除了振荡畸变引起的RFM模型误差,进而提高校正性能。利用实测数据验证了该方法的有效性,并通过仿真数据分析了地标点的数量与分布对该几何校正方法性能的影响。     

悬链曲面上的点粒子动力学及扩展空间约束系统量子化

扩张型正则量子化方案的核心内容是位置、动量以及哈密顿量同时量子化. 通过分析悬链面上粒子的扩张型正则量子化方案, 并且与薛定谔理论进行比较, 发现内禀几何中二维悬链面给不出与薛定谔理论相一致的结果, 而考虑将二维悬链面嵌入在三维欧氏空间之后, 还需要将正则量子化方案进行扩张, 可以得到体系的几何势能和几何动量, 并与薛定谔理论相一致.     

几何画板模拟阿贝折射仪测折射率

几何画板模拟阿贝折射仪测量液体折射率的过程,相对传统教学手段,更加科学直观并动态演示。通过修改棱镜折射率和待测液体折射率的参数,通过直观几何光学分析,得到为何阿贝折射仪必须满足棱镜折射率大于待测折射率。利用程序易修改特点研究,棱镜不同顶角和考虑色散现象的模拟结果,定量研究差别。     

基于频率调制激光驱动的囚禁离子几何相位逻辑门

为了避免激光相位的起伏对几何相位逻辑门保真度的影响, 提出一种基于囚禁离子的量子几何相位逻辑门的新方案。该机制是利用一束频率调制的行波激光场作用于两个囚禁离子上实现的。它的优点有：操作简单,仅需一步就能实现。不灵敏于激光场的相位也不需要对囚禁离子进行个别寻址。     

自然弯扭梁的非线性分析

研究了两端边界均为完全约束的自然弯扭梁在小应变，大位移和大转动情况下的非线性性质，并考虑了横向剪切变形和扭转翘曲变形的影响，分析中还包括了拉伸，弯曲和扭转的各种弹性耦合。由最小势能原理可以导出所给问题的平衡方程。这里欧拉角可以用来表示任意大的转动。该方法还可方便地推广到其他各种不完全约束边界的情况。此外，利用上述结果还可以得到该梁在小位移理论中的基本方程和有关公式。     

横向磁场激励下铁磁梁式板的混沌运动分析

在磁体力分布的磁弹性理论模型和磁场准静态假定模式基础上，对于处在周期时变磁场 中的不可移简支铁磁梁式板非线性磁弹性动力特性进行定性与定量分析．首先利用磁场的摄 动技术和结构变形的模态法，导出了关于模态坐标的非线性动力方程；然后利用Ｍｅｌｎｉｋｏｖ方 法，从理论上给出这一磁弹性动力系统可能出现混沌运动的必要条件及参数范围；最后采用变 步长Ｒｕｎｇｅ－Ｋｕｔｔａ数值积分方法对其磁弹性相互作用的混沌现象进行了定量搜索与模拟，并 利用其轨迹的Ｐｏｉｎｃａｒｅ截面图与Ｌｉａｐｕｎｏｖ指数加以判断．结果表明磁弹性简支梁式板在横 向周期时变磁场中存在混沌吸引子，且在机械阻尼很小时其混沌吸引子表现出稠的特性．     

桁架结构稳定分析的几何非线性欧拉稳定理论

目前,桁架结构稳定分析有两个理论:经典的特征值理论和几何非线性临界点理论。前者发展较早,在许多结构力学书中都有叙述。但是经过工程应用检验,它过高地估计了结构抗稳定的能力。后者是20年前提出的,提出者认为适用于所有的扁桁架。经过几年的研究,作者连续提出三个稳定理论:线性与非线性欧拉理论;非线性临界点-欧拉理论和两个计算方法,进行桁架结构的线性和非线性截面优化设计。经过理论研究和对前人文献中数值例题的计算和比较,发现特征值理论不符合实际,临界点理论只适用于大扁度的桁架,而不适用于一切扁桁架。研究结果提供了若干有用的结论,给出了各种稳定理论的适用范围,指出了国际文献中若干例题的正确和错误,补充了桁架结构的稳定分析理论,使之更为完整和实用了。