2024/02/21 by Simon St-Amant, St-Amant, Simon · 1 citation
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.2402.13854
openalex publication_date 2024/02/21 · openalex created_date 2024/02/23 · openalex updated_date 2026/07/28
We consider the inverse problem of recovering a connection on a complex vector bundle over a compact smooth Riemannian manifold with boundary from a Dirichlet-to-Neumann (DN) map at a high fixed frequency. We construct Gaussian beams using the language of jet bundles and show that their value at the boundary can be recovered from those DN maps. This allows us to show injectivity up to gauge on manifolds whose non-abelian X-ray transform is injective. We also study DN maps with a cubic nonlinearity and show how to recover the broken non-abelian X-ray transform from them. This transform maps a connection to its parallel transport along broken geodesics with endpoints on the boundary. We show that the broken non-abelian X-ray transform is always injective up to gauge equivalence, regardless of the manifold's geometry.