Inverse learning in Hilbert scales

TL;DR

This paper investigates the use of regularization schemes in Hilbert scales to solve linear ill-posed inverse problems with noisy data, providing convergence rates and explicit error bounds based on source conditions.

Contribution

It introduces a framework for analyzing inverse problems in Hilbert scales with noisy data, deriving convergence rates and error bounds under specific prior assumptions.

Findings

Derived explicit convergence rates for regularized solutions.

Established error bounds based on source conditions.

Analyzed the impact of prior assumptions on reconstruction accuracy.

Abstract

We study the linear ill-posed inverse problem with noisy data in the statistical learning setting. Approximate reconstructions from random noisy data are sought with general regularization schemes in Hilbert scale. We discuss the rates of convergence for the regularized solution under the prior assumptions and a certain link condition. We express the error in terms of certain distance functions. For regression functions with smoothness given in terms of source conditions the error bound can then be explicitly established.