Characterizations of Adjoint Sobolev Embedding Operators with Applications in Inverse Problems

TL;DR

This paper provides a comprehensive overview of the adjoint Sobolev embedding operator, exploring its properties, various characterizations, and applications in inverse problem regularization, serving as a valuable reference and guide.

Contribution

It offers a detailed, generalized collection of characterizations of the adjoint Sobolev embedding operator, filling a gap in existing literature with rigorous proofs and practical insights.

Findings

Multiple characterizations of $E_s^*$ including variational, boundary value, Fourier, wavelet, and frame representations.

Connections established between $E_s^*$ and regularization methods in inverse problems.

Provides a reference framework for numerical implementation and theoretical analysis.

Abstract

We consider the Sobolev embedding operator $E_s : H^s(\Omega) \to L_2(\Omega)$ and its role in the solution of inverse problems. In particular, we collect various properties and investigate different characterizations of its adjoint operator $E_s^*$, which is a common component in both iterative and variational regularization methods. These include variational representations and connections to boundary value problems, Fourier and wavelet representations, as well as connections to spatial filters. Moreover, we consider characterizations in terms of Fourier series, singular value decompositions and frame decompositions, as well as representations in finite dimensional settings. While many of these results are already known to researchers from different fields, a detailed and general overview or reference work containing rigorous mathematical proofs is still missing. Hence, in this paper we aim to fill this gap by collecting, introducing and generalizing a large number of characterizations of $E_s^*$ and discuss their use in regularization methods for solving inverse problems. The resulting compilation can serve both as a reference as well as a useful guide for its efficient numerical implementation in practice.