离散问题

The discrete variable action space enables the complex decision-making problem to fall into several simple and independent sub problems, and each of them can be resolved via existing theories.

离散变量动作空间能够将复杂的决策问题分解为多个独立的子问题，且每个子问题都有相应的理论来解决。

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。

Three solution algorithms for classical transportation continuous/discrete/ mixed network design problems are designed based on particle swarm optimization approach, in which the continuous NDP is studied in detail, discrete and mixed NDP are only described the frameworks of the solution algorithms.2. Generally, road capacity constraints must be added to the lower level problem of classical discrete NDPs bilevel programming model in order to avoid traffic assignment on candidate roads.

本论文研究工作主要有如下几个方面：(1)基于粒子群优化思想设计了求解传统的城市道路交通网络设计问题的算法，其中详细研究了连续网络设计问题，并用两个网络例子进行了数值实验；并给出了离散网络设计问题和混合网络设计问题的求解算法思路和主要步骤。

Based on the Saint-venant equations describing the channel flow movement, the nonlinear algebraic equations derived by the use of Preissmann weighted implicit four-point scheme are solved with the Netwon-Raphson method. A combined Gauss main-element elimination method with compress-storage technique is proposed. The relaxation iterative method for solving branch river networks is extended, which can turn the problem of any complex river networks into the problem of a series of single rivers. The coupling of hydraulic model with water quality model is built up. The combined model was applied to a large complicated river networks with numerical hydraulic structure which influenced by strong tide, storm and heave rain. Visualization of computational results is realized by means of MapInfo, which displays the computational results in fast, convenient and accurate way.

水力模型以描述河道洪水波运动的Saint-Venant方程组为基础，用Newton-Raphson方法直接求解按Preissmann加权四点格式进行离散所得的非线性代数方程组，提出了压缩存贮形式的Gauss列主元消去法并对数值解的收敛性进行了讨论；将支流流量的松弛迭代方法从树状河系推广到了环状河网，使任意复杂河网水力计算的问题都可化为一系列的单一河道的水力计算问题；建立了水力模型与水质模型的联接，初步应用于既有洪水、暴雨、强潮等综合影响，又存在多种水工建筑物及水力调度问题的大型复杂河网中的非恒定流水力、水质计算问题；利用MapInfo对计算结果的可视化进行了研究，有利于决策者快速、方便、准确地作出决策。

Note that regularized problems are usually defined in an infinite setting and have to be discTetized for an implementation and there have been many comparatively adult methods of calculating singular system. Therefore, there is no need to avoid the calculation of singular system in inverse problems calculating of numeric value. Here TSVD Regularization Method is very simple and quite effective one.

考虑到反问题的数值计算需要将问题离散化，化为有限维的问题来进行处理，而对于有限维算子的奇异系的计算已经有了相当成熟的各种算法，因此在反问题的数值计算中没有必要避开奇异系的计算，此时TSVD正则化方法是十分简单并相当有效的正则化方法。

The paper gives a special discussion of regularization method in inverse problem. On this basis, an ideal solution for ill-posed problem is built-the best disturbed iteration.

论述了反问题的高度非线性及求解的不适定性，分析了反问题求解的各种方法，并对在反问题求解过程中经常遇到的一类离散不适定问题进行了详细地分析和研究，并给出了具体的算法。

The discretized frictional contact problems can be reduced as a linear complementarity problem using parametric variational principle and parametric quadratic programming method. Then we can apply the algorithms proposed in this thesis to the solution of the contact problems.

利用参变量变分原理和参数二次规划法，离散后的接触问题可化为线性互补问题，之后，将第二、三章提出的互补问题算法应用到接触问题的数值求解中。

The first kind of the digital signature schemes is based on the discrete logarithm problem and second kind of the digital signature scheme is based on discrete logarithm and factorization problem, the third kind of the digital signature schemes is based on quadratic residue problem.

本文第一章对数字签名方案进行分类，根据数字签名方案所基于的数学问题，将签名方案分为：基于离散对数问题的数字签名方案，基于素因子分解问题的数字签名方案和基于二次剩余问题的签名方案。

As the shape optimization design of structural element sections belong to structural optimization,which has its own traits such as discreteness, nondifferentiable and non-convexity,while the traditional optimal methods are not applicable.

结构截面形状优化设计属于工程结构优化问题，由于工程结构优化问题具有离散性、不可微性和非凸性等特点，而传统的优化算法对所优化的问题往往要求连续、可微、非凹等前提条件，所以已不再适合用于工程结构优化问题的求解。

Otherwise, the algorithm restores the domain of variables that were eliminated due to the instantiation of current variable, and backtracks sequently. This thesis gives design for "Ming Yue "[1.1], and briefly states impleme -nt of problem solving system on discrete domains combined with examples. It also gives some experiment results of typical problems, such as car sequencing, timetabling problem, job shop scheduling problem and so on.

本文给出了&明月&的设计，并且结合实例简述了离散论域上约束求解系统的实现，给出了一些典型问题的实验结果，如汽车排序、时间表调度问题、车间作业调度问题等，实验证明该系统对于某些类问题的确是行之有效的。

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。