Statistical Inverse Problems in Hilbert Scales

TL;DR

This paper analyzes Tikhonov regularization in Hilbert scales for nonlinear statistical inverse problems, providing theoretical error estimates and convergence rates that extend previous results in the field.

Contribution

It develops a comprehensive theoretical framework for Tikhonov regularization in Hilbert scales, including high probability error bounds and explicit convergence rates.

Findings

High probability estimates for direct and reconstruction errors.

Explicit convergence rates in terms of sample size.

Generalization of previous results in related inverse problem settings.

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.