论文标题
伴随Sobolev嵌入操作员的表征与应用程序中的应用
Characterizations of Adjoint Sobolev Embedding Operators with Applications in Inverse Problems
论文作者
论文摘要
我们考虑sobolev嵌入操作员$ e_s：h^s（ω）\ to l_2（ω）$及其在反问题解决方案中的作用。特别是，我们收集了各种属性，并研究其伴随运算符$ e_s^*$的不同特征，这是迭代和变异正则化方法中的常见组件。这些包括变异表示和与边界价值问题，傅立叶和小波表示的连接以及与空间过滤器的连接。此外，我们考虑了傅立叶级数，奇异值分解和框架分解以及有限维度设置中的表示形式。尽管来自不同领域的研究人员已经知道其中的许多结果，但仍然缺少包含严格数学证明的详细概述或参考工作。因此，在本文中，我们旨在通过收集，引入和推广$ e_s^*$的大量特征来填补这一空白，并讨论它们在解决反问题的正则化方法中的用途。由此产生的汇编既可以用作参考，也可以作为其在实践中有效的数值实现的有用指南。
We consider the Sobolev embedding operator $E_s : H^s(Ω) \to L_2(Ω)$ 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.
