RUI: Parallel methods and software for solving sparse nonlinear systems of equations in structured approaches

本课题的目标是在并行计算机上设计、分析、实现和测试几组求解稀疏非线性方程组的并行方法。该方法是将稀疏非线性方程组转化为一些特殊的结构，以便将计算分解为并行处理。研究将集中在两类重要的自然结构系统上：1)通过置换一般非线性系统得到的非线性方程的大规模块边界系统。计划发展求解块边界非线性方程组的并行显式和隐式方法和迭代方法，包括收敛分析、奇异或近奇异块边界系统的摄动方法、割线方法的应用以及这些方法在不同类型并行计算机上的实现。研究还将扩展到求解块三角形非线性方程组的不同直接和迭代方法，包括收敛分析、准牛顿方法的应用和求解这类问题的异步并行方法。该方法的设计是由并行计算机的使用驱动的，并且将在并行计算机上实现和测试。然而，目标是在并行计算机上发现有效解决结构化方法中的稀疏非线性方程组的新方法。

The goal of this project is to design, analyze, implement and test several groups of parallel methods for solving sparse nonlinear systems of equations on parallel computers. The approach is to transform sparse nonlinear systems of equations into some special structures so that the computations can be decomposed for parallel processing. The research will concentrate on two important classes of naturally structured systems: 1) large-scale block bordered systems of nonlinear equations, obtained by permuting general nonlinear systems. It is planned to develop parallel explicit and implicit methods and iterative methods for solving the block bordered nonlinear systems of equations, including convergence analysis, perturbation methods for singular or nearly singular block bordered systems, applications of the secant method, and the implementations of the methods on different types of parallel computers. The investigation will also be extended to different direct and iterative methods for solving the block triangular nonlinear systems of equations including the convergence analysis, applications of quasi- Newton methods and asynchronous parallel methods for solving this type of problems. The method design is motivated by the use of parallel computers, and the implementation and testing will be on parallel computers. The goal is, however, to discover new methods that efficiently solve sparse nonlinear systems of equations in structured approaches on a parallel computer.