Well-defined forward operators in dynamic diffractive tensor tomography using viscosity solutions of transport equations

Abstract

We consider a general setting for dynamic tensor field tomography in an inhomogeneous refracting and absorbing medium as an inverse source problem for the associated transport equation. Following Fermat's principle, the Riemannian metric in the considered domain is generated by the refractive index of the medium. There is a wealth of results for the inverse problem of recovering a tensor field from its longitudinal ray transform in a static Euclidean setting, whereas there are only a few inversion formulas and algorithms existing for general Riemannian metrics and time-dependent tensor fields. It is a well-known fact that tensor field tomography is equivalent to an inverse source problem for a transport equation where the ray transform serves as given boundary data. We prove that this result extends to the dynamic case. Interpreting dynamic tensor tomography as an inverse source problem represents a holistic approach in this field. To guarantee that the forward mappings are well defined, it is necessary to prove existence and uniqueness for the underlying transport equations. Unfortunately, the bilinear forms of the associated weak formulations do not satisfy the coercivity condition. To this end we transfer to viscosity solutions and prove their unique existence in appropriate Sobolev (static case) and Sobolev-Bochner (dynamic case) spaces under a certain assumption that allows only small variations of the refractive index. Numerical evidence is given that the viscosity solution solves the original transport equation if the viscosity term turns to zero.