Characterizations of variational source conditions, converse results, and maxisets of spectral regularization methods

TL;DR

This paper provides a unified framework for verifying variational source conditions in inverse problems, linking them to solution smoothness, operator ill-posedness, and Besov spaces, with implications for convergence rates.

Contribution

It introduces a general strategy for verifying variational source conditions and characterizes them via Besov spaces for certain operators, advancing understanding of convergence in inverse problems.

Findings

Variational source conditions are necessary and sufficient for certain convergence rates.

A general criteria for verifying variational source conditions based on subspace families.

Characterization of variational source conditions via Besov spaces for Laplace-Beltrami operators.

Abstract

We describe a general strategy for the verification of variational source condition by formulating two sufficient criteria describing the smoothness of the solution and the degree of ill-posedness of the forward operator in terms of a family of subspaces. For linear deterministic inverse problems we show that variational source conditions are necessary and sufficient for convergence rates slower than the square root of the noise level. A similar result is shown for linear inverse problems with white noise. If the forward operator can be written in terms of the functional calculus of a Laplace-Beltrami operator, variational source conditions can be characterized by Besov spaces. This is discussed for a number of prominent inverse problems.