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

这个项目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。其方法是将稀疏的非线性方程组变换成一些特殊的结构，这样计算就可以分解为并行处理。研究将集中在两类重要的自然结构系统：1)大规模块边界非线性方程组，它是通过对一般的非线性系统进行置换而得到的。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、奇异或近似奇异块边界方程组的摄动方法、割线方法的应用以及这些方法在不同类型的并行计算机上的实现。研究还将扩展到求解块三角非线性方程组的不同的直接方法和迭代方法，包括收敛分析，拟牛顿方法和异步并行方法在求解这类问题中的应用。该方法的设计是基于并行计算机的使用，其实现和测试将在并行计算机上进行。然而，我们的目标是发现在并行计算机上以结构化方法高效地求解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.