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.
