Self-adaptive strategy for one-dimensional finite element method based on EEP method with optimal super-convergence order

Based on the newly-developed element energy projection （EEP） method with optimal super-convergence order for computation of super-convergent results, an improved self-adaptive strategy for one-dimensional finite element method （FEM） is proposed. In the strategy, a posteriori errors are estimated by comparing FEM solutions to EEP super-convergent solutions with optimal order of super-convergence, meshes are refined by using the error-averaging method. Quasi-FEM solutions are used to replace the true FEM solutions in the adaptive process. This strategy has been found to be simple, clear, efficient and reliable. For most problems, only one adaptive step is needed to produce the required FEM solutions which pointwise satisfy the user specified error tolerances in the max-norm. Taking the elliptical ordinary differential equation of the second order as the model problem, this paper describes the fundamental idea, implementation strategy and computational algorithm and representative numerical examples are given to show the effectiveness and reliability of the proposed approach.

Optimal design study of high order FIR digital filters based on neural network algorithm

An optimal design approach of high order FIR digital filter is developed based on the algorithm of neural networks with cosine basis function . The main idea is to minimize the sum of the square errors between the amplitude response of the desired FIR filter and that of the designed by training the weights of neural networks, then obtains the impulse response of FIR digital filter . The convergence theorem of the neural networks algorithm is presented and proved, and the optimal design method is introduced by designing four kinds of FIR digital filters , i.e., low-pass, high-pass, bandpass , and band-stop FIR digital filter. The results of the amplitude responses show that attenuation in stop-bands is more than 60 dB with no ripple and pulse existing in pass-bands, and cutoff frequency of passband and stop-band is easily controlled precisely .The presented optimal design approach of high order FIR digital filter is significantly effective.

一维优化问题的一种Q-2阶收敛混合算法

利用牛顿法和不精确牛顿法构造了1维代化问题的混合算法。用该算法连续2次迭代只计算1次2阶导数值。在一合理的假设下证明了该算法具有Q-2阶收敛速率。

2-循环相容次序阵的AOR迭代的收敛域

设A∈Cn×n是2-循环相容次序阵,其Jacobi阵J的非零特征值均为纯虚数.记α=ρ(J).本文证明了A的AOR迭代阵Lr,ω(约定ω>0,r≠0)收敛当且仅当参数ω,r满足条件0<ω<21+α2,ω+ωα-22<r<12ω+(2ω-αω2)2,r≠0,或等价地,r≥rb,0<ω<2+rα2-α1+αr22α2+4r-4;rb≥r>-α22,r≠0,0<ω<21++rαα22,其中rb=1+21+α2.这一结果纠正了薛秋芳文给出的相应结果,并指出了其中的3个问题.

强奇异积分方程小波Petrov-Galerkin快速算法

通过构造具有高阶消失矩、小支集和半双正交性质的分片多尺度小波基底,给出第2类强奇异积分方程的小波Petrov-Galerk in快速算法,并证明该算法收敛阶达到最佳,条件数有界,计算复杂性几乎最佳。

基于MO-RAMP插值模型的阻尼板GCMOC法拓扑减振优化

为了有效实现板件的抗振性动力学设计,研究约束阻尼板拓扑动力学优化方法。建立约束阻尼板有限元动力学分析模型,推导出模态损耗因子计算公式;建立了基于模态损耗因子最大化目标,以阻尼层单元相对密度为拓扑变量,以阻尼材料使用量及结构频率作为控制的阻尼板优化数学模型;利用序列凸规划理论而对传统优化准则法进行改进,采用改进准则法GCMOC(global extreme point converged by method of optimization criterion)解算优化模型以求取全域性优化解,推导出面向GCMOC的拓扑变量迭代式;考虑到多阶次RAMP(rational approxination of material properties)函数的形状具有较理想的可控下凹几何特征,提出在优化迭代中采用多阶次RAMP材料插值模型(MO-RAMP)对拓扑变量集合进行惩罚以实现其快速的0,1二值化,并尽量减少处于0.3~0.7的中间拓扑变量值出现;编制了面向约束阻尼板的拓扑动力学优化程序,实现了基于MO-RAMP的约束阻尼板GCMOC法变密度式减振拓扑动力学优化过程。算例分析表明,MO-RAMP与GCMOC复合的算法用于阻尼板拓扑迭代时,可将阻尼单元密度值快速地推向逼近0或1的值。它能得到清晰的阻尼单元优化密度云并有利于优化构型的实现;能在大幅减少阻尼材料用量条件下充分发挥其黏弹耗能效应,能在保证阻尼板动力学特性基本稳定的前提下使结构获得更好的减振效果。

具有最佳收敛性的积分方程全离散Petrov-Galerkin快速算法

对第二类奇异积分方程提出新的全离散Petrov-Galerkin快速算法,通过调整截断参数,使得算法收敛性达到最优的同时,计算复杂度仍然保持几乎最优,条件数有界.

弱奇性积分方程的最优阶多尺度Petrov-Galerkin快速算法

考虑求解第二类Fredholm弱奇性积分方程的多尺度Petrov-Galerkin压缩格式,给出压缩策略中截断参数的选取范围,证明了相应的压缩格式在保持稳定性、计算复杂度和系数矩阵条件数一致有界的基础上,收敛阶达到最优。并以数值算例验证了理论结果的正确性和有效性。

New Variants of Newton’s Method for Nonlinear Unconstrained Optimization Problems

In this paper, we propose new variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems. It is proved that the order of convergence of the proposed family is three. Numerical comparisons are made to show the performance of the presented methods. Furthermore, numerical experiments demonstrate that the logarithmic mean Newton’s method outperform the classical Newton’s and other variants of Newton’s method. MSC: 65H05.

一类八阶收敛的最优修正牛顿迭代法

通过对牛顿迭代法进行改进,利用两步平移加权得到一类八阶收敛的求解非线性方程最优修正迭代算法,该算法中每一步迭代只需计算四个函数值即可实现最优阶数,收敛性分析和数值实例验证该方法具有较高的效率指数和收敛精度.这一非线性方程求根算法的实现,在模式识别、电力系统、轨迹预测等工程领域和人工智能领域具有一定的理论和使用价值.

MIXED DISCONTINUOUS GALERKIN METHOD FOR QUASI-NEWTONIAN STOKES FLOWS

In this paper,we introduce and analyze an augmented mixed discontinuous Galerkin(MDG)method for a class of quasi-Newtonian Stokes flows.In the mixed formulation,the unknowns are strain rate,stress and velocity,which are approximated by a discontinuous piecewise polynomial triplet ■for k≥0.Here,the discontinuous piecewise polynomial function spaces for the field of strain rate and the stress field are designed to be symmetric.In addition,the pressure is easily recovered through simple postprocessing.For the benefit of the analysis,we enrich the MDG scheme with the constitutive equation relating the stress and the strain rate,so that the well-posedness of the augmented formulation is obtained by a nonlinear functional analysis.For k≥0,we get the optimal convergence order for the stress in broken ■(div)-norm and velocity in L^(2)-norm.Furthermore,the error estimates of the strain rate and the stress in-norm,and the pressure in L^(2)-norm are optimal under certain conditions.Finally,several numerical examples are given to show the performance of the augmented MDG method and verify the theoretical results.Numerical evidence is provided to show that the orders of convergence are sharp.

分布鲁棒二阶随机占优约束优化问题的稳定性

为规避真实分布的不确定性导致的风险,提出分布鲁棒二阶随机占优约束优化问题,矩信息和Wasserstein球相结合构造分布集合,采用离散近似方法处理该问题,并在适当的假设条件下,讨论近似问题可行集、最优值和最优解集的收敛性。

Quadrature Based Optimal Iterative Methods with Applications in High-Precision Computing

We present a simple yet effective and applicable scheme,based on quadrature,for constructing optimal iterative methods.According to the,still unproved,Kung-Traub conjecture an optimal iterative method based on n+1 evaluations could achieve a maximum convergence order of 2n.Through quadrature,we develop optimal iterative methods of orders four and eight.The scheme can further be applied to develop iterative methods of even higher orders.Computational results demonstrate that the developed methods are efficient as compared with many well known methods.

具有最佳超收敛阶的EEP法计算格式:Ⅰ算法公式

对一维C0问题的高次有限元后处理中超收敛计算的EEP(单元能量投影)法提出改进的最佳超收敛计算格式,即用m次单元对足够光滑问题的有限元解答,采用该格式计算的任意一点的位移和应力都可以达到h2m阶的最佳超收敛结果。整个工作分为3个部分,分别给出算法公式、数值算例和数学证明。该文是系列工作的第一部分,针对高次单元提出了凝聚形函数的概念,并证明了相关的逼近定理和等价定理,在此基础上给出了具体的算法公式。

基于最佳超收敛阶EEP法的一维有限元自适应求解

基于新近提出的具有最佳超收敛阶的单元能量投影(EEP)超收敛算法,提出用具有最佳超收敛阶的EEP超收敛解对有限元解进行误差估计,用均差法进行网格划分,用拟有限元解进行多次遍历而不反复求解有限元真解,形成一套新型的一维有限元自适应求解策略.该法理论上简明清晰,算法上高效可靠,对于大多数问题,一步自适应迭代便可给出按最大模度量逐点满足误差限的有限元解答.以二阶椭圆型常微分方程模型问题为例,介绍了该法的基本思想、实施策略及具体算法,并给出具有代表性的数值算例,以展示该法的优良性能和效果.

解约束多目标优化问题的一种鲁棒的进化算法

将约束条件与目标函数融合在一起 ,对有约束的多目标优化问题 (MOP)建立了一种新的偏序关系 ,引入了约束占优的定义 ,并证明了在新的偏序关系意义下的Pareto最优集就是满足约束条件的Pareto最优集 ,从而在对种群中的个体进行评估或排序时 ,并不需要特别去关心个体是否可行 ,避免了罚函数选择参数的困难 尝试应用有限Markov链的有关理论证明了此进化算法的收敛性 用较复杂的Benchmark函数进行了大量的数值实验 ,测试结果表明新算法在解集分布的均匀性。

具有最佳超收敛阶的EEP法计算格式:Ⅱ数值算例

对一维C0问题的高次有限元后处理中超收敛计算的EEP(单元能量投影)法提出改进的最佳超收敛计算格式,即用m次单元对足够光滑问题的有限元解答,采用该格式计算的任意一点的位移和应力都可以达到h2m阶的最佳超收敛结果。整个工作分为3个部分,分别给出算法公式、数值算例和数学证明。该文是系列工作的第二部分,给出实施算法和数值算例,用以验证理论公式的有效性和正确性。

二阶微粒群算法

为了提高标准微粒群算法的全局收敛性,提出了一种新的微粒群算法——二阶微粒群算法.首先,介绍了二阶微粒群算法的引入,分析了其收敛性,并且研究了其参数的选择范围.其次,在分析二阶微粒群算法的进化方程的基础上,引出了具有随机惯性权重的标准微粒群算法.再次,在二阶微粒群算法中加入振荡因子来调整微粒的速度变化率,更好地使二阶微粒群算法收敛于全局最优.最后,利用这几种改进方法对典型测试函数进行仿真,实验结果表明,这些方法能够有效克服早熟问题,在全局收敛性和收敛速度方面均优于标准微粒群算法.

一类求解非线性方程最优的8阶收敛迭代法

利用权函数方法得到一类求非线性方程单根的最优8阶收敛迭代法.该方法每步迭代需要计算3个函数值和1个一阶导数值,效率指数为1.682.数值试验结果表明,该方法具有较高的收敛阶数和计算精度.

改进的二阶振荡粒子群算法

针对粒子群算法易陷入局部最优等问题,分析了粒子群算法的进化方程,提出了一种改进的粒子群优化算法。算法在振荡环节采用互不相同的参数取值来调节粒子群算法的全局和局部搜索能力,并通过对测试函数和机器人路径规划问题仿真模拟,与标准PSO、标准二阶PSO、二阶振荡PSO算法的实验结果进行对比分析,验证了所提出算法的有效性和可行性。