Variable hilbert scales and their interpolation inequalities with applications to tikhonov regularization

Abstract

Variable Hilbert scales are constructed using the spectral theory of self-adjoint operators in Hilbert spaces. An embedding and an interpolation theorem (based on Jenssen's inequality) are proved. They generalize known results about “ordinary” Hilbert scales derived by Natterer [Applic. Anal., 18 (1984), pp.29-37].

Bounds on best possible and actual errors for regularization methods are obtained by applying the interpolation inequality. These bounds extend the standard ones, and, in particular, include exponential and logarithmic error laws. Similar results were established earlier by Hegland [SIAM J. Numer. Anal., 29 (1992), pp. 1446-14611 for compact operators only. Here, they are generalized to include unbounded operators.

A detailed discussion of Tikhonov regularization by Nair et al. [Tech. Rep. MR8-94, CMA, Aust. Nat. Uni., 19941 indicates that parameter choice strategies, which were thought to be suboptimal, can give substantially higher convergence rates than the so-called optimal choices! This improvement depends on how the regularizor is chosen. In some particularly difficult situations, however, the choice of regularizor cannot improve the convergence.

Keywords:

• Hilbert scales
• Tlkhonov regularization
• interpolation inequalities