Variable Hilbert Scales and Their Interpolation Inequalities with Applications to Tikhonov Regularization

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. 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. for compact operators only. Here, they are generalized to include unbounded operators. A detailed discussion of Tikhonov regularization by Nair et al. 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 diicult situations, however, the choice of regularizor cannot improve the convergence. 1. Introduction Practical inverse problems are ill-posed. The solution of ill-posed problems is particularly diicult as the information which is to be recovered can be severely corrupted by measurement errors, even if these errors are small. Unless special methods, such as Tikhonov regularization, are used, it is often impossible to get sensible results. Even when regularization is used, there is a need to understand how the regular-ized solution is improved as the accuracy of the errors is reened. It is often thought that a tenfold increase in measurement precision will result in a tenfold increase in the precision of the results. This is not true for ill-posed problems. For them, even with the best methods, the increase in precision of the results is limited. The interpolation inequalities of Hilbert scales 19] lead to bounds for actual and optimal errors for a large class of regularization methods. Some cases are not covered