Semi-iterative Regularization in Hilbert Scales

In this paper we investigate the regularizing properties of semi-iterative regularization methods in Hilbert scales for linear ill-posed problems and perturbed data. It is well known that standard Landweber iteration can be remarkably accelerated by polynomial acceleration methods leading to optimal speed of convergence, which can be obtained by several efficient two-step methods, e.g., the $\nu$-methods by Brakhage. It was observed earlier 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. We show that a combination of both ideas allows for a further acceleration, yielding optimal convergence rates with only the square root of iterations as compared to the $\nu$-methods or Landweber iteration in Hilbert scales. The theoretical results are illustrated by several examples and numerical tests, including a comparison to the method of conjugate gradients.

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