In the talk we investigate the problem of reconstructing a tensor field from its known longitudinal ray transform. Our setting considers the general situation of a refractive medium with absorption. In this case the rays are geodesic curves of a Riemannian metric which is connected with the index of refraction n(x). Whereas for n=1, i.e., the absence of refraction and straight lines as rays, inversion formulas and a well-established theory are known, this holds no longer true in case of non-Euclidean geometries. It is known that
the tensor field problem can be equivalently formulated as an inverse source problem for a transport equation where the unknown field appears as source term and the given ray transform serves as boundary data and can be interpreted as outflow through the boundary. We discuss the forward operator and investigate existence and uniqueness of the boundary value problem. It shows that the transport equation lacks of the important coercivity property. This is why we turn over to define viscosity solutions and present a condition on n which yields existence and uniqueness of such solutions. The result even generalizes to time-dependent tensor fields and might be seen as a starting point for numerical regularization methods to solve the inverse problem in a holistic approach.
(This is joint work with Lukas Vierus and Alfred Louis from Saarland University Saarbrücken in cooperation with the Sobolev Institute for Mathematics in Novosibirsk, Russia)