【作者】 何朗；

【导师】 黄樟灿；

【作者基本信息】 武汉理工大学 ， 计算数学， 2008， 硕士

【摘要】 本文主要分三个部分,第一部分为研究背景的介绍,给出了偏微分方程求解问题的描述,主要概括了求解偏微分方程的方法—变分法、有限元法和无网格法。第二部分是对基于解析逼近的并行算法的研究:简要介绍了演化算法和基因表达式编程基因表达式编程(Gene Expression Programming,GEP),在此基础上,提出了一种基于解析逼近求解偏微分方程的并行算法,并对该算法的可行性做了一定的理论分析。第三部分是并行算法的4个检验实验:将所设计的算法分别应用到常微分方程、抛物线方程、椭圆方程和双曲线方程求解的典型实例中,结果显示本文设计的算法表现出良好的性能。本文主要的工作和创新点:1、定义了运算函数和运算符号库,以基本初等函数为基础,通过函数四则运算和复合运算来构造函数表达式。2、引入∞范数作为算法适应度函数,以迭代解与精确解之间的距离作为解优劣性的评价准则。3、将问题转化为一个双目标(边界条件和方程的解)变分问题。针对传统数值解方法要求在满足边界条件下寻找近似解,从而使得解的空间非常狭窄,本文提出个体在迭代过程中不尽要向精确解逼近,而且需要向边界条件逼近。4、利用具有内在并行性的GEP技术使迭代向精确解逼近,将一个待求解的任务分成一个主任务(主进程)和一些从任务子进程。通过子进程间的划分、通信、组合、映射来完成并行计算,并最终向主进程发回各自的计算结果。5、从智能性和并行性的角度分析了算法的可行性,通过典型实例验证了算法的可行性。由于方法不依托于方程的形式,其计算过程适合于大规模并行,它适合求解大多数偏微分方程。更多还原

【Abstract】 This dissertation is composed of three parts. In the first part we introduce the background of our research, describe the partial differential equation (PDE) problem and give general solutions such as variation principle, finite element method (FEM) and Meshless method. In the second part, we do search on parallel algorithm. A brief analysis is made on evolutionary algorithm and genetic expression programming (GEP). Based on those techniques, we proposed a new parallel algorithm to solve PDE based on analytical approximation. In addition, we made some theory analysis on the algorithm. In the third part, we applied the algorithm into 4 test problems from ordinary differential, parabolic equation, elliptic equation and hyperbolic equation respectively. The experimental results of the new algorithm show the advantageous performance. The main research work and algorithm show innovative points as follows:1、Define the function warehouse and operation warehouse; it is through the Compound Calculations and Four Fundamental Operations to produce shape functions.2、Introduce oo norm as the algorithm’s fitness function, make the distance between analytical and accurate solutions as the criterion of solution’s level.3、Turn the function solution to a double-object (boundary problem and function solution) optimization problem. The traditional method is to search approximated results of PDE under their satisfying with the boundary constraints; in this case, the result space is very simple. In this paper, we proposed two demands that analytical solutions should approach accurate one as well as boundary constraints, therefore, the results space can be enlarged to a broadened one.4、Make analytical solutions to approach accurate outputs by using the GEP which is strong in paralleled computation. We divided a main task into several sub processes, it is through operations of Partitioning、Communication、Agglomeration and Mapping among processes to parallel computation.5、Analysis the availability of the new algorithm based on intelligence and parallelism. And it is proved to be useful by 4 tests. Due to the independency on function formats, this method is suited to parallel computing, and also deal with most PDE.更多还原