This talk is about multilevel techniques for solving a nonlinear problem f(u) = 0 ! in a Hilbert space H, arising from a least squares formulation for a system of nonlinear partial differential equations of first order. After discretizing the space H, this is a minimization problem that may be solved using a linear multilevel method for successive Gauss-Newton problems or using a nonlinear multilevel method like FAS.
Although these approaches both yield nice numerical results, there are some difficulties in the convergence theory. Since the discretized problem is not consistent, the core problem turns from finding a zero of f(u) to finding a zero of df/du which requires second order information about f for the theory. One more problem for describing theory for the linear multilevel case is the lack of a direct relation between the error due to the discretization and the algebraical error due to the iterative method in order to check on overall convergence. For the FAS-case, the Gauss-Newton method does not fit into existing nonlinear multilevel theory with respect to smoothing properties.
These problems disappear if the (inexact) Gauss-Newton solver is applied directly to the infinite dimensional problem in H and the emerging linear problems are discretized and solved using linear multilevel after that. The problem remains consistent and the above-mentioned errors both contribute to the inaccuracy of the Gauss-Newton method. Consequently, the number of linear multilevel sweeps may be monitored by easily computable error bounds via the multiplicative multilevel norm ensuring convergence of the overall method.
In this talk, these results will be presented in more detail. Numerical examples show the competitiveness of this method.