Regularization by projection with a posteriori discretization level choice for linear and nonlinear ill-posed problems

In this paper we study the regularization of linear and nonlinear ill-posed operator equations by projection onto finite-dimensional spaces with a posteriori chosen space dimension. We show that this results in a regularization method, i.e. a stable solution method for the ill-posed problem, that converges to an exact solution as the data noise level goes to zero, with optimal rates under additional regularity conditions.