Fast fully iterative Newton-type methods for inverse problems

We study nonlinear inverse problems of the form F(x) = y, and their stable solution via iterative regularization methods, in particular by Newton-type methods, which are well-known for their fast convergence for well-posed problems. A basic step of Newton’s method consists of calculating the update Δx_k via solution of a linearized equation , which will in general be ill-posed if the nonlinear problem is. Thus, for ill-posed problems, the linearized equations have to be solved by some regularization method. In particular for large scale problems, e.g., inverse problems in partial differential equations, where F(x) is only defined implicitly via the solution of a PDE, iterative methods have to be used for this purpose. In order to keep the overall effort, i.e., the overall number of iterations, as small as possible, appropriate preconditioning has to be applied. We propose and analyse a general preconditioning strategy in Hilbert scales, and show that the overall number of iterations can be reduced to about the square root by preconditioning. Moreover, in many examples differential operators can be used as preconditioners, and thus preconditioning is almost for free. The theoretical results are illustrated in numerical examples. A comparison with preconditioned Landweber iteration shows that the iteration numbers can be further reduced, if fast iterative methods, e.g., the v-methods, are used for the solution of the linearized problems.