Stability, Range and Statistical Aspects of non-Abelian X-ray tomography

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Bohr, Jan 

The non-Abelian X-ray transform on a Riemannian manifold with boundary (M,g) is a nonlinear map A → CA that associates to a matrix-valued and possibly direction-dependent function A, referred to as attenuation, its scattering data CA. The scattering data records initial-to-end-value maps for the ordinary differential equation u ̇ + Au = 0 along complete geodesics in M and contains indirect information on the attenuation A, observable at the boundary ∂M. This thesis is concerned with several aspects related to the inverse problem of determining an unknown A from possibly noisy observations of CA. One focus lies on theoretical guarantees for the Bayesian approach to inverse problems, advocated for by Stuart (2010), where inference on A is made by means of suitable posterior measures. All results hold true and are novel in the case that M is a Euclidean ball, most of them extend to contractible Riemannian manifolds satisfying certain geometric conditions, such as simpleness (for dim M = 2) or the existence of a strictly convex function (for dim M ≥ 3). In Chapter 2, we prove stability estimates in dim M ≥ 3 for direction independent attenuations, written as A(x, v) = Φ(x). These estimates are of the form ∥Φ − Ψ∥L2 ≤ c∥CΦ − CΨ∥^μ_L2 , with c > 0 and μ ∈ (0,1) depending on a bound of the C^k-norms (k ≫ 1) of Φ and Ψ. There is also a partial data version, where Φ − Ψ is estimated on suitable subsets O ⊂ M in terms of scattering data along geodesics that do not leave O. This is a quantitative version of the injectivity proof by Paternain–Salo–Uhlmann-Zhou (2019) and extends an analogous stability result of Monard–Nickl–Paternain (2021a) from dim M = 2 to higher dimensions. The proof employs the method of Uhlmann– Vasy (2016), which is based on Melrose’s microlocal scattering calculus and a layer stripping procedure, and develops a more quantitative version of their arguments. As an application, we show that with high probability, Φ can be recovered from noisy measurements of CΦ as L2-limit of Bayesian posterior means. In Chapter 3, we prove a linear stability estimate on the Euclidean unit disk M ≡ D ⊂ R2, leading to a log-concave approximation result for the Bayesian posterior distribution. The stability estimate for the linearisation C ̇Φ takes the form ∥h∥_{H ̃^−1/2} ≤ c∥C ̇Φ[h]∥_{L2}, where c depends continuously on Φ in the C4-topology and H ̃^{−1/2} is a Sobolev type space, defined in terms of Zernike polynomials. The estimate is based on a related isomorphism property, established by Monard–Nickl–Paternain (2021b). We then go on to show that with high probability, and restricted to suitable D-dimensional function spaces, the Bayesian posterior distribution is log-concave in a neighbourhood of the true Φ. This allows us to follow the strategy of Nickl–Wang (2022) and approximate the posterior in Wasserstein distance with a globally log-concave surrogate posterior. This makes available fast gradient-based Markov chain Monte Carlo algorithms that allow to sample from the posterior – and thus compute posterior means as above – with convergence guarantees that scale polynomially in the relevant quantities, assuming a warm start. In Chapter 4, we obtain a range characterisation for the map A → CA in dim M = 2, which is reminiscent of the Ward correspondence for anti-self-dual Yang– Mills fields, but without solitonic degrees of freedom. The result can be viewed as a nonlinear version of a characterisation by Pestov-Uhlmann (2004) and relies on the existence of so-called matrix holomorphic integrating factors on simple surfaces, which had been an open problem. This problem is solved using the inverse function theorem of Nash and Moser, and the required tame estimates are obtained from re- cent results on the injectivity of attenuated X-ray transforms and microlocal analysis of the associated normal operators. Finally, we introduce a novel twistor correspondence in dim M = 2, under which (M,g) corresponds to a complex surface Z and attenuations A correspond to holomorphic vector bundles over Z. We show that the twistor space Z of a simple surface behaves in many ways like a contractible Stein surface. In particular, there is a transport version of the classical Oka–Grauert principle, which says that Z does not support any non-trivial holomorphic vector bundles, and which follows from the existence of matrix holomorphic integrating factors.

Paternain, Gabriel
mathematics, geometric inverse problems
Doctor of Philosophy (PhD)
Awarding Institution
University of Cambridge
EPSRC (2114530)
This research was supported by the EPSRC Centre for Doctoral Training and the Munro–Greaves Bursary of Queens’ College Cambridge.