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

这个项目的目标是将不精确的牛顿框架应用于解决物理科学中不同学科中出现的大型非线性问题。在每一种情况下，基本迭代算法都需要在每一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，每个子问题的求解将只达到保证迭代法快速收敛所需的精度。将考虑以下问题：1)介子场Skyrme模型有限元离散所产生的大型约束优化问题。2)椭圆边值问题的区域分解算法的实现。3)求解含时N-S方程的一种有效方法。4)求解连续非线性边值问题的牛顿方法。在初步研究活动中，P.I.将尝试；A)确定如何在Tapia的拉格朗日乘子公式中确定求解两个线性系统的精度要求，b)完成区域分解问题的增广拉格朗日算法的实现，为不精确迭代修改做准备，c)将预条件共轭梯度算法应用于对称正定线性系统，和d)将尝试一种简单的有限元方法来处理不精确的牛顿方法。

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.