On Tikhonov regularization in Banach spaces - optimal convergence rates results

In the present article, we deal with convergence rates for a Tikhonov-like regularization approach for linear and non-linear ill-posed problems in Banach spaces. Under validity of a source condition, we derive convergence rates which are well known as optimal in a Hilbert space situation. Moreover, we show how this convergence rate depends on the convexity of the penalty functional and the smoothness of the image space. Additionally, we give an a posteriori choice of the regularization parameter leading to optimal convergence rates.