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The linear Boltzmann equation governs the absorption and scattering of a population of particles in a medium with an ambient field, represented by a Riemannian metric, where particles follow geodesics. In this paper, we study the possible issues of uniqueness and stability in recovering the absorption and scattering coefficients from the boundary knowledge of the albedo operator. The albedo operator takes the incoming flux to the outgoing flux at the boundary. For simple compact Riemannian manifolds of dimension $n \geq 2$, we study the stability of the absorption coefficient from the albedo operator up to a gauge transformation. We derive that when the absorption coefficient is isotropic then the albedo operator determines uniquely the absorption coefficient and we establish a stability estimate. We also give an identification result for the reconstruction of the scattering parameter. The approach in this work is based on the construction of suitable geometric optics solutions and the use of the invertibility of the geodesic ray transform., Comment: 32 pages

This paper deals with an inverse problem for a non-self-adjoint Schr\"odinger equation on a compact Riemannian manifold. Our goal is to stably determine a real vector field from the dynamical Dirichlet-to Neumann map. We establish in dimension n greater than 2, an H\"older type stability estimate for the inverse problem under study. The proof is mainly based on the reduction to an equivalent problem for an electro-magnetic Schr\"odinger equation and the use of a Carleman estimate designed for elliptic operators.

In this paper we consider the inverse problem of determining on a compact Riemannian manifold the electric potential and the absorption coefficient in the wave equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n \geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uni, Comment: arXiv admin note: text overlap with arXiv:1510.04247