Statistical Inverse Problems in Hilbert Scales

Abstract

In this paper, we study the Tikhonov regularization scheme in Hilbert scales for the nonlinear statistical inverse problem with a general noise. The regularizing norm in this scheme is stronger than the norm in Hilbert space. We focus on developing a theoretical analysis for this scheme based on the conditional stability estimates. We utilize the concept of the distance function to establish the high probability estimates of the direct and reconstruction error in Reproducing kernel Hilbert space setting. Further, the explicit rates of convergence in terms of sample size are established for the oversmoothing case and the regular case over the regularity class defined through appropriate source condition. Our results improve and generalize previous results obtained in related settings.

References in corpus (6)

• Convergence Analysis of (Statistical) Inverse Problems under Conditional Stability Estimates
• Convergence analysis of Tikhonov regularization for non-linear statistical inverse learning problems
• Bayesian linear inverse problems in regularity scales
• Tikhonov regularization with oversmoothing penalty for nonlinear statistical inverse problems
• Inverse learning in Hilbert scales
• Designing truncated priors for direct and inverse Bayesian problems