Abstract
This thesis studies numerical solutions of large-scale nonlinear systems using unconstrained optimization techniques. We focus on Quasi-Newton methods of order 1 and 2. We describe the methods, the corresponding algorithms, and their costs and convergence rates in order to allow a motivated methods choice. We limit ourselves to nonlinear linear systems resulting from finite elements discretization of boundary value problems. The first chapter is dedicated to a general introduction and preliminaries on the subject. The second chapter is dedicated to an optimization problem review. Problems and basic methods are the subjects of the third chapter. We study Broyden methods and rank-1 updating in the fourth chapter, then the construction of Quasi-Newton methods of rank-2 for unconstrained optimization problem in the fifth chapter. The sixth and seventh chapters have the goal of presenting BFGS algorithm and its algebraic properties, links with Conjugate Gradient method with fixed metric H0, then the implementation of BFGS. Some convergence theorems are the subject of the eighth chapter. And, the next chapter presents some comments on preconditioning and large scale sparse case. We compare numerically all presented methods in the tenth chapter. Then, we summarize and present some conclusions and future work in the last chapter. All MATLAB codes are presented in the last pages of the thesis
