Errors of regularisation under range inclusions using variable Hilbert scales

TL;DR

This paper develops new, simpler error bounds for regularisation methods in inverse problems using variable Hilbert scales, with applications demonstrated in image processing and spectral enhancement.

Contribution

It introduces novel, simplified formulae for error bounds and modulus of continuity in the context of regularisation under range inclusions, improving upon previous methods.

Findings

New error bounds derived using variable Hilbert scales

Simpler formulae for modulus of continuity of inverse operators

Applications demonstrated in image processing and spectral enhancement

Abstract

Based on the variable Hilbert scale interpolation inequality bounds for the error of regularisation methods are derived under range inclusions. In this context, new formulae for the modulus of continuity of the inverse of bounded operators with non-closed range are given. Even if one can show the equivalence of this approach to the version used previously in the literature, the new formulae and corresponding conditions are simpler than the former ones. Several examples from image processing and spectral enhancement illustrate how the new error bounds can be applied.