Stable determination of X-ray transforms of time dependent potentials from the dynamical Dirichlet-to-Neumann map

TL;DR

This paper demonstrates the stable recovery of the X-ray transform of time-dependent potentials on Riemannian manifolds from boundary measurements, using Gaussian beams without full boundary data, advancing inverse boundary value problem techniques.

Contribution

It introduces a stable method to recover the X-ray transform of time-dependent potentials from partial boundary data on Riemannian manifolds, utilizing Gaussian beam techniques.

Findings

Established Hölder stability estimates for the X-ray transform of q(t,x).

Achieved recovery without measurements on the entire boundary.

Applied minimal geometric assumptions for Gaussian beam construction.

Abstract

We consider compact smooth Riemmanian manifolds with boundary of dimension greater than or equal to two. For the initial-boundary value problem for the wave equation with a lower order term $q(t,x)$, we can recover the X-ray transform of time dependent potentials $q(t,x)$ from the dynamical Dirichlet-to-Neumann map in a stable way. We derive H\"older stability estimates for the X-ray transform of $q(t,x)$ which do not require measurements on the entirety of the boundary. The essential technique involved is the Gaussian beam Ansatz, and the proofs are done with the minimal assumptions on the geometry for the Ansatz to be well-defined.