This article considers the attenuated transport equation on Riemannian surfaces in the light of a novel twistor correspondence under which matrix attenuations correspond to holomorphic vector bundles on a complex surface. The main result is a transport version of the classical Oka-Grauert principle and states that the twistor space of a simple surface supports no nontrivial holomorphic vector bundles. This solves an open problem on the existence of matrix holomorphic integrating factors on simple surfaces and is applied to give a range characterisation for the non-Abelian X-ray transform. The main theorem is proved using the inverse function theorem of Nash and Moser and the required tame estimates are obtained from recent results on the injectivity of attenuated X-ray transforms and microlocal analysis of the associated normal operators.Résumé (Le principe de transport d'Oka-Grauert pour les surfaces simples)Cet article étudie l'équation de transport atténuée sur les surfaces riemanniennes à la lumière d'une nouvelle correspondance de twisteurs dans laquelle les atténuations de matrice correspondent à des fibrés vectoriels holomorphes sur une surface complexe. Le résultat principal est une version de transport du principe classique d'Oka-Grauert et stipule que l'espace des twisteurs d'une surface simple ne supporte aucun fibré vectoriel holomorphe non trivial. Ceci résout un problème ouvert sur l'existence de facteurs intégrants holomorphes matriciels sur des surfaces simples et est appliqué pour donner une caractérisation du domaine pour la transformation en rayons X non abélienne. Le théorème principal est démontré en utilisant le théorème d'inversion locale de Nash et Moser, et les estimations nécessaires sont obtenues à partir de résultats récents sur l'injectivité des transformées en rayons X atténuées et l'analyse microlocale des opérateurs normaux associés.

The problem of efficiently generating random samples from high-dimensional and non-log-concave posterior measures arising from nonlinear regression problems is considered. Extending investigations from [45], local and global stability properties of the model are identified under which such posterior distributions can be approximated in Wasserstein distance by suitable log-concave measures. This allows the use of fast gradient based sampling algorithms, for which convergence guarantees are established that scale polynomially in all relevant quantities (assuming 'warm' initialisation). The scope of the general theory is illustrated in a non-linear inverse problem from integral geometry for which new stability results are derived.

This article considers the attenuated transport equation on Riemannian surfaces in the light of a novel twistor correspondence under which matrix attenuations correspond to holomorphic vector bundles on a complex surface. The main result is a transport version of the classical Oka-Grauert principle and states that the twistor space of a simple surface supports no nontrivial holomorphic vector bundles. This solves an open problem on the existence of matrix holomorphic integrating factors on simple surfaces and is applied to give a range characterisation for the non-Abelian X-ray transform.The main theorem is proved using the inverse function theorem of Nash and Moser and the required tame estimates are obtained from recent results on the injectivity of attenuated X-ray transforms and microlocal analysis of the associated normal operators.

Non-abelian X-ray tomography seeks to recover a matrix potential $$\Phi :M\rightarrow {\mathbb {C}}^{m\times m}$$ Φ : M → C m × m in a domain M from measurements of its so-called scattering data $$C_\Phi $$ C Φ at $$\partial M$$ ∂ M . For $$\dim M\ge 3$$ dim M ≥ 3 (and under appropriate convexity and regularity conditions), injectivity of the forward map $$\Phi \mapsto C_\Phi $$ Φ ↦ C Φ was established in (Paternain et al. in Am J Math 141(6):1707–1750, 2019). The present article extends this result by proving a Hölder-type stability estimate. As an application, a statistical consistency result for $$\dim M =2$$ dim M = 2 (Monard et al. in Commun Pure Appl Math, 2019) is generalised to higher dimensions. The injectivity proof in (Paternain et al. in Am J Math 141(6):1707–1750, 2019) relies on a novel method by Uhlmann and Vasy (Invent Math 205(1):83–120, 2016), which first establishes injectivity in a shallow layer below $$\partial M$$ ∂ 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.

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.