非线性方程组迭代方法特征研究及并行计算

项目名称： 非线性方程组迭代方法特征研究及并行计算

项目编号： No.10871178

项目类型： 面上项目

立项/批准年度： 2009

项目学科： 金属学与金属工艺

项目作者： 吴庆标

作者单位： 浙江大学

项目金额： 26万元

中文摘要： 研究非线性方程组迭代方法本质特征及并行计算，对非线性方程组著名的迭代方法，如Newton法、不精确Newton法等方法及一些免导数计算的高阶方法如变形Halley法等在导数或差商Lipschitz连续、Holder连续等条件下，研究这些迭代方法的局部收敛半径和半局部收敛半径问题。将非线性方程组迭代的Newton法、不精确Newton法等方法、线性迭代的Krylov子空间法、预处理技术中预条件的构造方式进行综合考虑，提出求解大型非线性方程组迭代方法的高效算法。在理论和算法研究的基础上，进行非线性方程组迭代方法的并行计算，针对来自于非线性结构有限元计算中的非线性方程组应用本质特征的研究成果进行具体的并行算法设计和并行计算，开展交叉性学科的应用研究。将解决非线性方程组迭代方法的本质特征结果与实际工程并行计算、应用之间脱节的问题，为理论研究、并行计算和应用建立一种本质的联系。

中文关键词： 非线性方程组；迭代方法；本质特征；并行计算；有限元

英文摘要： The essential character of iterative methods for solving nonlinear system of equations and its applications in parallel computation are studied. Many famous iterative methods: Newton's method,inexact Newton's method,and some high order methods free from derivative such as the deformed Halley's method, are considered. Under the condition of Lipschitz continuity or Holder continuity of derivatives or divided difference, the local or semilocal radii of the convergence balls of these iterative methods will be given. By the comprehensive investigation of Newton's method, inexact Newton's method, Krylov-subspace method and many preconditioning techniques, the effective iterative methods for solving nonlinear system of equations will be presented. On the bases of the theoretic and algorithmic researches, the parallel computation of iterative methods for solving nonlinear system of equations will be carried out. The research results of the essential character of iterative methods for solving nonlinear system of equations will be applied in the nonlinear system of equations arising from the computations of the nonlinear structure finite element, and the application researches among some interdiscipline subjects will be carried out. The gap between the theoretic researches of the essential character of iterative methods for solving nonlinear system of equations and the parallel computations in many engineering problems will be filled up. The essential relations will be established among the theoretic researches, the parallel computations and the applications.

英文关键词： Nonlinear equations system;Iterative methods;Essential character;Parallel computation;Finite element