Reconstruction in the partial data Calder\'on problem on admissible manifolds

TL;DR

This paper develops methods to reconstruct a potential on admissible manifolds from partial boundary data, extending previous results and providing constructive procedures for local and global inversion of geodesic ray transforms.

Contribution

It introduces new reconstruction algorithms for the Calderón problem on admissible manifolds using partial data, including constructive inversion of geodesic ray transforms.

Findings

Reconstruction of local attenuated geodesic ray transform of Fourier transform of potential.

Global reconstruction under certain geometric conditions.

Extension and improvement of previous Euclidean reconstruction methods.

Abstract

We consider the problem of developing a method to reconstruct a potential $q$ from the partial data Dirichlet-to-Neumann map for the Schr\"odinger equation $(-\Delta_g+q)u=0$ on a fixed admissible manifold $(M,g)$. If the part of the boundary that is inaccessible for measurements satisfies a flatness condition in one direction, then we reconstruct the local attenuated geodesic ray transform of the one-dimensional Fourier transform of the potential $q$. This allows us to reconstruct $q$ locally, if the local (unattenuated) geodesic ray transform is constructively invertible. We also reconstruct $q$ globally, if $M$ satisfies certain concavity condition and if the global geodesic ray transform can be inverted constructively. These are reconstruction procedures for the corresponding uniqueness results given by Kenig and Salo. Moreover, the global reconstruction extends and improves the constructive proof of Nachman and Street in Euclidean setting. We derive a certain boundary integral equation which involves the given partial data and describes the traces of complex geometrical optics solutions. For construction of complex geometrical optics solutions, following Nachman and Street and improving their arguments, we use a new family of Green's functions for the Laplace-Beltrami operator and the corresponding single layer potentials. The constructive inversion problem for local or global geodesic ray transforms is one of the major topics of interest in integral geometry.