Stability of the non-abelian $X$-ray transform in dimension $\ge 3$

Abstract

Non-abelian X-ray tomography seeks to recover a matrix potential Φ : M → C m×m in a domain M from measurements of its so called scattering data C Φ at ∂M . For dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map Φ → C Φ was established in [20]. In this article we extend [20] by proving a Hölder-type stability estimate. As an application we generalise a statistical consistency result for dim M = 2 [14] to higher dimensions. The injectivity proof in [20] relies on a novel method by Uhlmann-Vasy [27], which first establishes injectivity in a shallow layer below ∂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.

On log-concave approximations of high-dimensional posterior measures and stability properties in non-linear inverse problems

“…Here C k (S D) (k ∈ N 0 ) denotes the Banach space of functions on SD which are k-times continuosly differentiable up to the boundary. In the compactly supported case, the previous two displays are stated as equation (5.17) in [38] and Proposition 2.2 in [6] and are both consequences of Lemma 5.2 of the first cited paper; one easily observes that the support condition can be dropped for k = 0. (for Φ,Ψ assumed to be continuous and of full support), also obeys…”