DISCRETIZATION INDEPENDENT CONVERGENCE RATES FOR NOISE LEVEL-FREE PARAMETER CHOICE RULES FOR THE REGULARIZATION OF ILL-CONDITIONED PROBLEMS

We develop a convergence theory for noise level-free parameter choice rules for Tikhonov regularization of finite-dimensional, linear, ill-conditioned problems. In particular, we derive convergence rates with bounds that do not depend on troublesome parameters such as the small singular values of the system matrix. The convergence analysis is based on specific qualitative assumptions on the noise, the noise conditions, and on certain regularity conditions. Furthermore, we derive several sufficient noise conditions both in the discrete and infinite-dimensional cases. This leads to important conclusions for the actual implementation of such rules in practice. For instance, we show that for the case of random noise, the regularization parameter can be found by minimizing a parameter choice functional over a subinterval of the spectrum (whose size depends on the smoothing properties of the forward operator), yielding discretization independent convergence rate estimates, which are of optimal order under regularity assumptions for the exact solution.