| This thesis develops and investigates iterative tensor methods for solving large-scale systems of nonlinear equations. Standard tensor methods for nonlinear equations have performed especially well on small, dense problems where the Jacobian matrix at the solution is singular or ill-conditioned, something that occurs on many classes of large-scale problems, such as identifying or approaching turning points. The success of tensor methods originates from a special, restricted form of the second-order term included in the local tensor model that provides information lacking in a (nearly) singular Jacobian. This research has three areas of emphasis. First, we study the performance of tensor methods on ill-conditioned problems. The results show that direct tensor methods and large-scale iterative tensor methods have clear computational advantages over Newton-based methods on ill-conditioned problems, similar to their performance on singular problems. Second, a new curvilinear linesearch globalization scheme is developed for tensor methods that smoothly combines the Newton and tensor directions. The results show that the curvilinear linesearch is more robust and efficient than previous linesearch implementations. Finally, this research extends direct tensor methods to large-scale problems by developing three tensor-Krylov methods that base each iteration upon a linear model augmented with a limited second-order term. The methods are implemented in an object-oriented nonlinear software package called NOX that is being developed at Sandia National Laboratories. The advantage of the new tensor-Krylov methods over existing large-scale tensor methods is their ability to solve the local tensor model to a specified accuracy, which produces a more accurate tensor step. The performance of these methods in comparison to Newton-GMRES and tensor-GMRES is explored on several problems, including three Navier-Stokes fluid flow problems. The numerical results provide evidence that tensor-Krylov methods are generally more robust and more efficient than Newton-GMRES on some important and difficult problems, including ill-conditioned problems. In addition, the results show that the new tensor-Krylov methods and tensor-GMRES each perform better in certain situations. |
