FROM LINEAR TO NONLINEAR LARGE SCALE SYSTEMS

The solution of large scale nonlinear problems can be tackled by solving successive small scale nonlinear problems on subspaces of the original space. We propose here a natural method to construct small dimensional subspaces of the original space, that is an extension to the nonlinear case of Krylov spaces. In the linear case, we show that the Krylov spaces are the subspaces generated by the successive derivatives at the origin of a particular mapping. An extension of this description leads to the definition of algorithms that generalize to the nonlinear case the relaxation methods and GMRES. These methods are also generalizations to a higher degree of Newton's method and Newton's method with line search, and a local fast convergence rate is obtained. Numerical simulations are conducted in two test cases from the CUTEr test set.