Iterative Substructuring Methods for Elliptic Problems and Related Algorithms

这个项目继续研究迭代子结构方法。这些是求解大型线性代数方程组的迭代方法，当椭圆型问题被有限单元或有限差分离散时，这些代数方程组就会出现。将问题的区域划分为子区域（子结构），并在子区域上重复求解问题，同时使用预条件共轭梯度算法来满足划分区域的曲线或曲面的连续性要求。目前的主要研究问题是将这些算法扩展到非一致性有限元、混合方法等，以及新算法的开发，特别是在三维空间。该计划将理论工作与系统的数值实验结合起来，使用模型问题和工业有限元代码。这些算法中的最佳算法通常代表了对单处理器上先前已知方法的改进，并且它们对多处理器系统具有相当大的希望。此外，针对某些排队网络问题的新算法也将被开发出来。生成Kolmogorov平衡方程，目标是根据相应的非常大的、稀疏的、非对称矩阵的零元计算稳态概率分布。其基本思想是使用适当选择的附近可分解问题作为前置条件，这在许多情况下已被证明是相当成功的。这些算法将得到进一步发展，理论将得到完善，以试图解释迭代矩阵奇异值的有趣分布，并尝试将方法的适用性扩展到新的问题类别。这项研究解决了求解大型线性方程组的新方法和更快的方法，这是许多科学和工程问题的基础计算任务。

This project continues to work on iterative substructuring methods. These are iterative methods for large linear systems of algebraic equations which arise when elliptic problems are discretized by finite elements or finite differences. The region of the problem is divided into subregions (substructures) and the problems on the subregions are solved repeatedly while using a preconditioned conjugate gradient algorithm to satisfy the continuity requirements across the curves or surfaces that divide the region. Main current research issues are the extension of these algorithms to nonconforming finite elements, mixed methods, etc., and the development of new algorithms, particularly in three dimensions. This program will combine theoretical work with systematic numerical experiments using both model problems and industrial finite element codes. The best of these algorithms often represent an improvement over previously known methods on the uni processors and they hold considerable promise for multi processor systems. In addition, new algorithms for certain queuing network problems will be developed. Kolmogorov's balance equations are generated and the goal is to compute the steady state probability distribution in terms of the null element of the corresponding very large, sparse, nonsymmetric matrix. The basic idea, which has proven quite successful in a number of cases, is to use suitably chosen nearby decomposable problems as preconditioners. These algorithms will be further developed, the theory will be refined to try to explain the interesting distribution of the singular values of the iteration matrices and an attempt will be made to extend the applicability of methods to new classes of problems. This research addresses new and faster methods for solving large systems of linear equations, a computational task that is fundamental to many scientific and engineering problems.