Volume 59, pp. 116-144, 2023.

Characterizations of adjoint Sobolev embedding operators with applications in inverse problems

Simon Hubmer, Ekaterina Sherina, and Ronny Ramlau

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.

Full Text (PDF) [877 KB], BibTeX

Key words

Sobolev spaces, embedding operators, inverse problems

AMS subject classifications

46N40, 46E35, 47A52, 65J20

Links to the cited ETNA articles

[36]	Vol. 30 (2008), pp. 54-74 Ronny Ramlau: Regularization properties of Tikhonov regularization with sparsity constraints

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