论文标题
非 - 亚伯利亚$ x $ ray transform in Dimension $ \ ge 3 $的稳定性
Stability of the non-abelian $X$-ray transform in dimension $\ge 3$
论文作者
论文摘要
非阿布莱恩$ x $ -Ray断层扫描旨在从域$ m $中恢复矩阵电位$φ：M \ rightArrow \ Mathbb {C}^{M \ times m} $，从所谓的散射数据$c_φ$ at $ \ partial m $中的测量中的测量值。对于$ \ dim m \ ge 3 $（并且在适当的凸度和规律性条件下），在[arxiv：1605.07894]中建立了向前地图$φ\ mapstoc_φ$的注射率。在本文中，我们通过证明Hölder-type稳定性估计值扩展了[ARXIV：1605.07894]。作为一个应用程序，我们将$ \ dim m = 2 $ [arxiv：1905.00860]的统计一致性结果推广到更高的维度。 [Arxiv：1605.07894]中的注射性证明依赖于Uhlmann-Vasy [Arxiv：1210.2084]的新方法，该方法首先在$ \ \ partial m $以下的浅层层中建立注射率，然后通过层剥离参数进行全球化。本文的主要技术贡献是这些论点的更定量版本，特别是在层深度和稳定性常数上证明了统一的界限。
Non-abelian $X$-ray tomography seeks to recover a matrix potential $Φ:M\rightarrow \mathbb{C}^{m\times m}$ in a domain $M$ from measurements of its so called scattering data $C_Φ$ at $\partial M$. For $\dim M\ge 3$ (and under appropriate convexity and regularity conditions), injectivity of the forward map $Φ\mapsto C_Φ$ was established in [arXiv:1605.07894]. In this article we extend [arXiv:1605.07894] by proving a Hölder-type stability estimate. As an application we generalise a statistical consistency result for $\dim M =2$ [arXiv:1905.00860] to higher dimensions. The injectivity proof in [arXiv:1605.07894] relies on a novel method by Uhlmann-Vasy [arXiv:1210.2084], which first establishes injectivity in a shallow layer below $\partial M$ and then globalises this by a layer stripping argument. The main technical contribution of this paper is a more quantitative version of these arguments, in particular proving uniform bounds on layer-depth and stability constants.
