Stable Determination of X-Ray Transforms of Time Dependent Potentials from Partial Boundary Data

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

Keywords:

• Inverse Problems
• Partial Data
• Radon Transforms
• Wave Equations

Mathematics Subject Classification:

• 35R01
• 35R30
• 35L20
• 58J45
• 35A22

Acknowledgments

The author would like to thank James Ralston for his ongoing encouragement to pursue mathematics. She would like to thank Mikko Salo for his suggestion to investigate this problem in the context of his work.