数值算法

Chapter 9: We report a large quantity of numerical experiments of 13 different algebraic multigrid algorithms for solving the Poisson equation, anisotropic equation, equation with cross-derivative terms, general matrix problems with large off-diagonal positive entries, biharmonic equation, Toeplity matrix, elasticity systems, finite element discretization of the Laplacian and even 3D problems. Particular attention is focused on asymptotic convergence factors and CPU-time consumed. Numerical results for many different types of practical problems demonstrate the efficiency and robustness of the proposed algebraic multigrid methods.

第九章：在各种代数多重网格算法的基础上，进行了大量的数值试验，具体给出了十三种不同的代数多重网格方法求解泊松方程，各向异性方程，带混合导数项的方程，带有大的非对角正元素的一般矩阵问题，重调和方程，托普利兹矩阵，弹性力学方程组，拉普拉斯算子的有限元离散，甚至三维问题的较为丰富的数值结果，重点关注它们的渐近收敛因子和所需的CPU时间，来源于不同类型问题的计算结果既为代数多重网格理论分析和算法的改进提供了很实用的资料，同时也证实了本文给出的代数多重网格算法的效绩和稳健性。

It shows that a remarkable advantage of the symplectic methods applied to the Schrodinger equation is the precise preservation of charge conservation law. In order to study the numerical stability, we also adopt the different time step sizes.

为了比较，我们还给出了同阶的非辛算法的数值模拟，从而得到辛摄动配置算法和其在理论上等价的辛龙格-库塔方法在数值上的等效性以及辛算法在数值计算中的优越性。

Furthermore, the first-order optimality condition and its equivalent reformulations for generalized semi-infinite max-min programming with a non-compact set are presented using the lower-Hadamard directional derivative and subdifferential.2. Chapter 3 studies the gradient-type methods for unconstrained optimization problems. Section 1 proposes a new class of three-term memory gradient methods. The global convergence property of the method is established. Furthermore, in order to improve the convergence property of the method, a new class of memory gradient projection methods is presented with the property that the whole sequence of iterates converges to a solution to the problem under the conditions such as pseudo-convexity and continuous differentiability of objective function. In section 2, two new classes of methods, called gradient-type method with perturbations and hybrid projection method with perturbations, are proposed. In these methods, non-monotone line search technique is employed, which makes them easily executed in computer.

第3章研究了无约束优化问题的梯度型算法，第1节提出了一类新的三项记忆梯度算法，讨论了算法的全局收敛性，进一步提出了一类新的具有更好收敛性质的记忆梯度投影算法，并证明了该算法在函数伪凸的情况下具有整体收敛性，第2节在非单调步长搜索下提出了带扰动项的梯度型算法及其混合投影算法，这两类算法的一个重要特征就是步长采用线搜索确定而不象许多文献中那样要求步长趋于零，这样更容易在计算机上实现，在较弱的条件下证明了这些算法的全局收敛性，数值算例表明了算法的有效性。

Based on forecasting model, five algorithms are presented, that is, 1 algorithm for free motion of the plant when the analytical solution is available, 2 numerical differentiation algorithm, 3 improved numerical differentiation algorithm, 4 sensitivity matrix algorithm and 5 reference integral curve method.

推导了具有预测模型的算法的形成过程，指出了这种算法的物理实质，给出了基于预测模型算法的五种具体算法：1、系统自由运动有解析解时的算法；2、数值微分算法；3、改进的数值微分算法；4、灵敏度矩阵算法；5、基准积分曲线方法。

That is to utilize general MPI programming environment, program on the NHBL parallel computational model, analyze and predicate the running cost of program by using the cost function of this model, and analyze the scalability of program by using the time-scale scalability metric. Finally, we analyze the numeric algorithm example---matrix multiplication and the nonnumeric algorithm example---PSRS sorting on Dawning-1000 MPP and NHPCC-cluster NOW system by using the integrative method.

提出了将并行计算模型、算法、编程进行一体化的研究的方法，即利用通用的标准的MPI编程环境，将编程建立在NHBL并行计算模型上，并且利用该并行计算模型的评价函数进行分析、预测程序的运行代价，再利用时间比例的可扩放性分析方法来分析程序的可扩放性性能；最后我们用PSRS排序算法和矩阵乘法这两个典型的数值和非数值的算法，在曙光-1000并行机和NHPCC-Cluster工作站集群上进行了一体化的分析。

The symplectic schemes have the ability to preserve the global symplectic structure of the phase space for a Hamiltonian system. They have substantial benefits in numerical computation for Hamiltonian system, especially in long-term simulations.

辛算法正是保持Hamilton系统内在性质的一种新型数值方法，该算法在长时间的数值计算中，具有常见数值方法无可比拟的计算优势。

Fist, comprehensive analysis on the current study on data processing is made, and characters of Newton methods about nonlinear surveying and mapping data processing are discussed, and then new solutions to parameters estimate with multi-sources, multi-types, multi-dimension, multi-precision bynonlinear least square are presented such as PSB algorithm, digital continuation and generalized digital continuation algorithm, cone model method, tensor analysis method, GCMA(mixed algorithm of gradient method and conjugate gradient method), combining algorithm based on Newton method and gradient method and confidence region and so on, and a new fast difference iterative algorithm is proposed towards parameters estimation containing random parameters in nonlinear models, and a new solutions to nonlinear least squares surveying and mapping adjustment by parameters estimation both considering the random and nonrandom parameters is presented after studying on nonlinear data processing in deformation monitoring, and at last primary analysis on error propagation of spatial data is made and approximate error propagation formula and error analysis formula to length and area are proposed.

首先比较全面分析了目前测量数据处理理论的研究现状，讨论了牛顿类方法在处理非线性模型参数估计方面的特点，研究并给出了求解多源、多类型、多维、多精度、非线性最小二乘测量平差参数估计的若干种新方法，这包括PSB算法、数值延拓及其广义数值延拓算法、锥模型法、张量分析法、基于最速下降法和牛顿法的组合算法、基于最速下降

The constructing process of the algorithms itself indicate that they are brevity and have definite geometric meaning, scientific iterative terminal criterions and no strict requirement to function. Finally, some digital experimentation is presented to indicate the algorithms application in solving the problem of maximum of function, transcendental equation,integer programming and so on. The results indicate that the algorithms have more global and more adaptability compared to traditional algorithms and have more calculating efficiency and more scientific iterative terminal criterions compared to some present popular artificial intelligence algorithms.

算法的构造过程本身表明了该算法简单明了、几何意义明确、迭代终止准则科学、对函数的要求较低；最后举了几个数值试验的例子以说明该类算法在求函数的最值，解超越方程以及整数规划等问题中的应用；数值试验的结果标明该算法比单纯由"微分"思想演化而来的传统算法有更好的全局性和对问题的更广泛的适应性，比现在流行的一些人工智能方法有更高的运算效率和更适当的迭代中止准则。

In this paper, we also discuss the theory of the parallel simplex method. Undej the 2-dimensional space case, we present that simplex seriary have some properties, but we adopt some repairing measures. We proof the convergence of the parallel algorithm. At last, We give some numerical examples to indicate that the new idea is sometimes useful and helpful.

本文也对算法做了一些理论分析，主要在二维情形下，讨论了算法形成的单纯形序列本身所具有的性质，并对其做了相应的补救措施，对算法的收敛性给予了证明，最后做了一些数值实验，由于软硬件条件的限制，并行算法未能在并行计算机上实现，鉴于这种情况，我们所做的数值实验均是在串行机上完成的。

We analyze theoretically the various algorithms of the fundamental matrix, and suggest that the input data should remove the errors in measurements through pre-processing for the algorithm stability. The robust algorithms, the cross-ratio based algorithm and the regularized pre-revised algorithm, are presented.

通过对目前常用算法分析及数值模拟指出，要提高算法的数值稳定性，最有效的办法是直接对输入数据作预处理去除噪声，并给出了较为鲁棒的基于交比预处理算法，和正则化预处理算法。

The split between the two groups can hardly be papered over.

这两个团体间的分歧难以掩饰。

This approach not only encourages a greater number of responses, but minimizes the likelihood of stale groupthink.

这种做法不仅鼓励了更多的反应，而且减少跟风的可能性。