Semi-iterative Regularization in Hilbert Scales

In this paper we investigate the regularization properties of semiiterative regularization methods in Hilbert scales for linear ill-posed problems and perturbed data. It is well known that Landweber iteration can be remarkably accelerated by polynomial acceleration methods leading to the notion of optimal speed of convergence, which can be obtained by several efficient two-step methods, e.g., the ν−methods by Brakhage. It was observed in [3] that a similar speed of convergence, i.e., similar iteration numbers yielding optimal convergence rates, can be obtained if Landweber iteration is performed in Hilbert scales. Combining both ideas, we show that semiiterative methods can be further accelerated yielding optimal convergence rates with only the square root of iterations compared to semiiterative regularization methods or Landweber iteration in Hilbert scales. We conclude with several examples and numerical tests confirming the theoretical results, including a comparison to the method of conjugate gradients.