Inexact Iterative Methods for Large-Scale Nonlinear Problems(Preliminary Planning)

大规模非线性问题的不精确迭代方法（初步规划）

• 批准号: 8917691
• 负责人: Edward Dean
• 金额: $ 1.2万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1990
• 资助国家: 美国
• 起止时间: 1990-03-15 至 1991-02-28
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8917691&HistoricalAwards=false
• 关键词: Inexact; Iterative; Methods; Large; Scale

The goal of this project is to apply the inexact Newton framework to the solution of large nonlinear problems arising from various disciplines in the physical sciences. In each case, the basic iterative algorithm will require the solution of a substantial subproblem at each iteration. Guided by the inexact Newton theory, each subproblem will be solved only to the accuracy needed to ensure the rapid convergence of the iterative method. The following problems will be considered: 1) The large constrained optimization problem arising from the finite element discretization of Skyrme's model for meson field. 2) The implementation of a domain decomposition algorithm for elliptic boundary value problems. 3) An effective method for the solution of the time dependent Navier-Stokes equation. 4) Newton's method for solving continuous nonlinear boundary value problems. During the preliminary research activities the P.I. will try to; a) determine how the accuracy requirements for solving the two linear systems in Tapia's Lagrange multiplier formula, b) complete the implementation of the augmented Lagrangian algorithm for the domain decomposition problem in preparation for the inexact iteration modification, c) will apply a preconditioned conjugate gradient algorithm to a symmetric positive definite linear system, and d) will try a simple finite element method for inexact Newton's method.

该项目的目标是将不精确牛顿框架应用于解决物理科学中各个学科产生的大型非线性问题。在每种情况下，基本迭代算法都需要在每次迭代中解决一个重要的子问题。在不精确牛顿理论的指导下，每个子问题只求解到保证迭代方法快速收敛所需的精度。将考虑以下问题：1)Skyrme介子场模型的有限元离散所引起的大型约束优化问题。2)椭圆型边值问题的区域分解算法的实现。3)求解时变Navier-Stokes方程的一种有效方法。求解连续非线性边值问题的牛顿方法。在初步研究活动中，私家侦探将尝试；a)确定Tapia拉格朗日乘子公式中求解两个线性系统的精度要求如何；b)完成对区域分解问题的增广拉格朗日算法的实现，为不精确迭代修正做准备；c)将对对称正定线性系统应用预条件共轭梯度算法，d)将对不精确牛顿法尝试简单的有限元方法。