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

这个项目的目标是应用不精确的牛顿 框架，以解决大型非线性问题， 来自物理科学的各个学科。 在每个 在这种情况下，基本的迭代算法将需要解 一个重要的子问题。 为指导 不精确的牛顿理论，每个子问题将只解决， 确保快速收敛所需的精度 迭代法 将考虑以下问题： 1)大约束优化问题，由 Skyrme介子模型有限元离散化 领域 2)一种区域分解算法的实现 椭圆型边值问题 3)求解时间的一种有效方法 非独立Navier-Stokes方程 4)求解连续非线性边界的牛顿法 价值问题。 在初步研究活动期间，P.I.将尝试 （a）确定解决问题的准确性要求 塔皮亚的拉格朗日乘子公式中的两个线性系统， B）完成增广拉格朗日量的实现 区域分解问题的算法准备 对于不精确的迭代修改，c）将应用 预条件共轭梯度算法 正定线性系统，d）将尝试一个简单的 不精确牛顿法的有限元法。

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.