【作者】 陈良生；

【导师】 朱伟勇；

【作者基本信息】 东北大学 ， 计算机软件与理论， 2005， 博士

【摘要】 最优设计理论及应用是一门迅速发展起来的统计学分支．它对于科学试验中数学模型的建立、最佳工艺条件的获得等问题，是一种极其有效的统计学工具．它的应用遍及采矿、金属材料、钢铁冶金、自动控制、机械制造、化工等领域，并取得了较好的实际效果。为此，本文选择了最优设计的计算机构造及软件设计作为研究课题，将混沌分形理论与最优设计理论相结合，开辟了一个崭新的研究领域． 本文详细地研究了D-最优设计的数值构造法以及对称算法理论，提出了D-最优设计的单纯形构造法及软件设计．运用这种新方法构造了多分量对数项混料模型及高阶对数项混料模型的D-最优设计，并运用单点交换法构造了多分量对数项混料模型的Dn-最优确切设计．将新得到的最优设计方案应用到焊接工艺中，给出了混料模型的回归方程及方差分析，并对回归方程及试验结果进行Lyapunov指数等非线性特征的量化分析，得到其混沌特征的综合描述． 本文将单纯形法进一步应用到复映射的M-J集的构造之中，与逃逸时间算法相结合，给出了构造M-J集分形图的单纯形旋转对称逃逸时间构造法，简称单纯形分形图构造法，并绘制了大量的M-J集分形图． 经过对广义M-J集分形图的细致分析，发现了其内在的几何特征，证实了曼德勃罗特的“Mandelbrot集是Julia集的微缩字典”的精辟论述，并给出了从M-J集寻找其几何特征的新方法． 本文主要内容归纳为以下几个方面： (1) 概述了最优设计理论、最优设计数值算法、混料回归设计的发生、发展和目前的状况．详细地介绍了D-最优设计的数值算法中著名的Fedorov算法和Dn-最优确切设计的数值算法中的Wynn-Mitchell单点交换法． (2) 系统地研究了D-最优设计的对称性，给出了对称区域、对称剖分的概念，介绍了D最优设计的对称构造法． (3) 应用Fibonacci技术，对Evans方法进行改进，并通过选取合适的初始单纯形，提出了局部寻优的新方法——单纯形法． (4) 提出了构造D-最优设计及Dn-最优确切设计的新方法——单纯形最优设更多还原

【Abstract】 The theory and application of optimal design is a statistic branch which grew rapidly in recent years. It is an effective statistics tool for the establishing of mathematical model and the obtaining of optimum processing condition of scientific experiment. It is widely used in many fields such as mining, metallic material, ferrous metallurgy, automatic control, mechanism manufacture, chemical industry, etc, and has obtained interesting result. So the author choices the computer constructing and software design of optimal design as research subject, combines ihe theory of chaotic fractal with the theory of optimal design, which creates a new researching area.The modified simplex algorithm and software design of constructing D-optimal design are proposed by studying the numerical constructing algorithm and symmetric algorithm of D-optimal design. The D-optimal designs of multi-component mixture regression models and high-order mixture regression models with logarithmic terms have been constructed by the new algorithm. At the same time, the Dn-optimal exact designs of multi-component mixture regression models with logarithmic terms have been constructed by the means of Wynn’s exchange algorithm. The new constructed optimal designs are used in the welding process and the regression analysis is given. The chaotic characteristic is described in detail by analyzing the Lyapunov exponent for the regression equation and test result.The modified simplex method has been further used in the constructing of M-J sets, and the simplex rotational symmetry escape time algorithm is proposed that combines the simplex algorithm with the classic escape time algorithm, which is called simplex fractal constructing algorithm for short, and a great deal of M-J set fractal images are drawn out by the new algorithm.By analyzing the general M-J set intensively, the inner geometry characteristic is found. The express of "Mandelbrot set is compact dictionary of Julia set" has been proved, and the method of finding the geometry characteristic from M-J sets is given.更多还原