Multiplicative Operator Splittings in Nonlinear Diffusion: From Spatial Splitting to Multiple Timesteps

Operator splitting is a powerful concept used in many diversed fields of applied mathematics for the design of effective numerical schemes. Following the success of the additive operator splitting (AOS) in performing an efficient nonlinear diffusion filtering on digital images, we analyze the possibility of using multiplicative operator splittings to process images from different perspectives.We start by examining the potential of using fractional step methods to design a multiplicative operator splitting as an alternative to AOS schemes. By means of a Strang splitting, we attempt to use numerical schemes that are known to be more accurate in linear diffusion processes and apply them on images. Initially we implement the Crank-Nicolson and DuFort-Frankel schemes to diffuse noisy signals in one dimension and devise a simple extrapolation that enables the Crank-Nicolson to be used with high accuracy on these signals. We then combine the Crank-Nicolson in 1D with various multiplicative operator splittings to process images. Based on these ideas we obtain some interesting results. However, from the practical standpoint, due to the computational expenses associated with these schemes and the questionable benefits in applying them to perform nonlinear diffusion filtering when using long timesteps, we conclude that AOS schemes are simple and efficient compared to these alternatives.We then examine the potential utility of using multiple timestep methods combined with AOS schemes, as means to expedite the diffusion process. These methods were developed for molecular dynamics applications and are used efficiently in biomolecular simulations. The idea is to split the forces exerted on atoms into different classes according to their behavior in time, and assign longer timesteps to nonlocal, slowly-varying forces such as the Coulomb and van der Waals interactions, whereas the local forces like bond and angle are treated with smaller timesteps. Multiple timestep integrators can be derived from the Trotter factorization, a decomposition that bears a strong resemblance to a Strang splitting. Both formulations decompose the time propagator into trilateral products to construct multiplicative operator splittings which are second order in time, with the possibility of extending the factorization to higher order expansions. While a Strang splitting is a decomposition across spatial dimensions, where each dimension is subsequently treated with a fractional step, the multiple timestep method is a decomposition across scales. Thus, multiple timestep methods are a realization of the multiplicative operator splitting idea. For certain nonlinear diffusion coefficients with favorable properties, we show that a simple multiple timestep method can improve the diffusion process.     

钻井井架力学性能分析

钻井井架作为钻机系统设备中的关键部分,其力学性能直接关系到整套钻机系统的安全生产。文章推导了常用于分析钻井井架力学性能的理论,即线性屈曲、几何非线性和双重非线性理论。提出同时用这三种理论对钻井井架进行综合比较分析,从而合理地预见井架力学性能的方法。以A形钻井井架为例,进行了详细的分析,得到了井架的失稳形式、极限承载、破坏特征及薄弱部位,为钻井井架的设计、优化与评定提供了参考。     

船舶操纵非线性系统的鲁棒变结构控制

本文采用伪线性化变换将船舶操纵非线性系统近似地化为线性可控正则型，并对线性化系统设计了一种连续的变结构以提高整个闭环系统的鲁棒性。该方案用于限制水域中船舶的航向航迹纠编控制中，取得了预期的结果。     

基于神经网络非线性系统辨识和控制的研究

本文提出了由静态的前馈网络和稳定的滤波器构成的非线性系统的辨识模型，在神经网络固有的逼近误差存在的情况下，从理论上讨论了神经网络应用于辨识控制过程中系统的稳定性问题，最后研究了在非线性系统的轨迹跟踪过程中增加滑动控制来偿神经网络的逼近误差，从而提高系统跟踪性能。     

高层结构几何非线性分析的QR法

本文利用ＱＲ法分析高层建筑结构几何非线性问题，建立了新的计算格式。该法不仅未知量的数目少，精度高，而且输入简单，计算速度快，可以用微机分析复杂问题。     

采用ZnO灭磁时阀片均流/均能的探讨

从ZnO电阻的非线性特性出发,对采用ZnO作为灭磁电阻时阀片的均流/均能进行理论论述,指出阀片均能性能的好坏主要依赖于阀片自身的特性;分析比较了ZnO阀片的不同选择方法对均流/均能的影响,指出阀片的均流/均能很大程度上取决于可供选择的阀片的库存量.该研究为发电机励磁系统的安全稳定运行提供了有力的保证.     

Nonlinear filtering structure for image smoothing in mixed-noise environments

This paper introduces a new nonlinear filtering structure for filtering image data that have been corrupted by both impulsive and nonimpulsive additive noise. Like other nonlinear filters, the proposed filtering structure uses order-statistic operations to remove the effects of the impulsive noise. Unlike other filters, however, nonimpulsive noise is smoothed by using a maximum a posteriori estimation criterion. The prior model for the image is a novel Markov random-field model that models image edges so that they are accurately estimated while additive Gaussian noise is smoothed. The Markov random-field-based prior is chosen such that the filter has desirable analytical and computational properties. The estimate of the signal value is obtained at the unique minimum of the a posteriori log likelihood function. This function is convex so that the output of the filter can be easily computed by using either digital or analog computational methods. The effects of the various parameters of the model will be discussed, and the choice of the predetection order statistic filter will also be examined. Example outputs under various noise conditions will be given.     

共轭高分子三阶非线性光学材料

概述了ｎ－共轭高分子材料的三阶非线性光学性质，着重讨论了化学结构对其光学非线性的影响，说明了ｎ－共轭聚合物的三阶非线性极化率主要取决于共轭主链的结构，侧链取代基的变化对其非线性影响不大，文中展望了共轭高分子材料在集成光学中的应用前景。     

铁铝酸盐复合膨胀水泥化学预应力机理及其应用研究

本文阐述铁铝酸盐自应力混凝土的主要物理力学特征.研究了限制膨胀应变与配筋率之间的定量函数关系.通过正交试验和方差分析,确定了复合外加剂对其性能的影响.针对其在自应力管中的应用,提出多层中空圆柱理论计算模型,按弹性理论计算出自应力管初始应力分布规律.对比以前的线性理论计算误差,给出相应的修正系数.本文对类似的自应力混凝土,具有普遍的指导意义.     

激光光束质量因子的研究进展

本文详细分析了各种光场下光束质量M2因子。在不同的场中,A.E.Siegman当初所定义的M2因子,已经有了很大的变化,因而M2因子的特性也与当初不同,不仅M2因子可以大于等于1,而且可以小于1,甚至小于0。其反映的物理意义,也从经典场中的衍射极限倍数,变化为非经典场中的准直性量度以及对空间相干性的某些反映。此外还列举并分析了几种特殊的光束质量因子,最后讨论了M2因子应用的问题并指出了其局限性。