The broken non-abelian X-ray transform and inverse problems for connections at high fixed frequency

This thesis is concerned with the recovery of a u(n) -valued connection A on a Hermitian vector bundle E of rank n over a smooth Riemannian or Lorentzian manifold with boundary (M, g) from different geometric inverse problems. The connection is to be recovered up to a gauge that is the identity on the boundary. The main inverse problem we study is the broken non-abelian X-ray transform. Given a connection A, its broken non-abelian X-ray transform is given by its parallel transports over a set of broken geodesics that start and end on the boundary. Here, a broken geodesic is a piecewise smooth curve formed by the composition of two geodesics that intersect transversally at a point. In Chapter 2, we consider the case where M = D\Ω is a causal diamond D in Minkowski space to which a small neighbourhood Ω of the observer's worldline has been removed, g is the induced Lorentzian metric, and the set of broken geodesics are the broken light rays that start and end in Ω. This transform was introduced by Chen-Lassas-Oksanen-Paternain (2022) to study inverse problems related to the Yang-Mills-Higgs equations. It is also known as the broken light ray transform. We show a stability estimate that takes the gauge into account, and we use it to show that, under a suitable fixing of the gauge, one can consistently recover a connection from noisy measurements of its broken non-abelian X-ray transform through Bayesian inversion. The broken non-abelian X-ray transform on an arbitrary Riemannian or Lorentzian manifold is studied in Chapter 3. We show that regardless of the geometry of M, the transform is injective up to gauge if we consider all possible broken geodesics with endpoints on the boundary. We also give a criterion to determine when the broken non-abelian X-ray transform is injective up to gauge over a fixed set of broken geodesics. In the subsequent chapters, we consider two inverse problems of Calderón type on a compact Riemannian manifold with boundary. The first asks whether one can recover a connection A from the Dirichlet-to-Neumann map DN_λ^A associated with the linear equation (Δ_A - λ^2)u = 0 as λ gets large. Here, Δ_A is the connection Laplacian. The second problem is analogous to the first but concerns the Dirichlet-to-Neumann map Λ_λ^A associated with the nonlinear equation (Δ_A - λ^2)u + |u|^2 u = 0 instead. We study both problems by using special quasimodes called Gaussian beams. Gaussian beams are approximate solutions of (Δ_A - λ^2)u = 0 that are supported on a small neighbourhood of a geodesic. We develop the theory of Gaussian beams on a vector bundle in Chapter 4. Our approach is novel because we view the solutions globally as sections of a jet bundle associated with the geodesic, we take care in prescribing their values on the boundary and our estimates are uniform with respect to certain families of geodesics. In Chapter 5, we show that given two connections A_1 and A_2, either the boundary values of their respective Gaussian beams agree to all order or there exists λ_0 depending on A_1 and A_2 such that DN_λ^{A_1} ≠ DN_λ^{A_2} for almost all λ > λ_0. Importantly, if the boundary values of the Gaussian beam agree to order 1, then the non-abelian X-ray transforms of the connections agree. Depending on the geometry of the underlying manifold, this can be enough to show that A_1 and A_2 are gauge equivalent. The proof relies on establishing a solvability result with vanishing boundary terms, computing lower-order terms in the asymptotics of an integral with a stationary phase and seeing the k-th power of the Laplacian as an inner product on the space of homogeneous polynomials of order k. In Chapter 6, we show that given A_1 and A_2 as well as a family of broken geodesics, either their broken non-abelian X-ray transforms agree on that family or there exists λ_0 such that Λ_λ^{A_1} ≠ Λ_λ ^{A_2} for almost all λ > λ_0. By the results of Chapter 3, if the family of broken geodesics is large enough, then the broken non-abelian X-ray transforms agree if and only if the connections are gauge equivalent. The proof relies on an identity derived from the third-order linearisation of Λ_λ^A. Finally, in Chapter 7, we relate the order of conjugacy of a manifold with boundary to the existence of pairs of nontrapped geodesics that intersect exactly once.

Name dc:type.qualificationname

Doctor of Philosophy (PhD)

Level dc:type.qualificationlevel

Doctoral

Grantor dc:publisher.institution

University of Cambridge

Year dc:date.issued

2024

Author dc:creator

• St-Amant, Simon

Advisor dc:contributor.advisor

• Paternain, Gabriel

• Mathematics
• Geometric inverse problems
• Gaussian beams
• Differential geometry
• Analysis of PDEs
• Dirichlet-to-Neumann map
• High frequency
• Non-abelian X-ray transform

Licence

• https://www.repository.cam.ac.uk/bitstreams/52cd3163-e493-437e-9bb4-482cecf7ae92/download
• https://www.rioxx.net/licenses/all-rights-reserved/

Language dc:language

eng

DOI dc:identifier.doi

https://doi.org/10.17863/CAM.112085

OAI identifier oai:identifier

oai:www.repository.cam.ac.uk:1810/373804