# Mathematical modeling and numerical methods for non-classical transport in correlated media

• Mathematische Modellierung und numerische Methoden für nichtklassischen Transport in korrelierten Medien

In this work, we investigate a new and non-classical linear transport equation for the transport of particles in correlated background media. We derive a time-dependent non-classical transport equation that is capable of reproducing arbitrary path length distributions, in contrast to the classical theory. This equation governs the distribution function of a microscopic particle game in the Boltzmann-Grad limit. This rigorous mathematical derivation is based on recent results on the periodic Lorentz gas, and it relies on an analytic expression for the distribution of free path lengths in the limit. The resulting equation has the distance s to the next collision as an additional independent variable, which makes it non-classical''. A Monte-Carlo study of path length distributions in homogeneous but correlated media suggests that this equation is valid for a wider class of spatially correlated media. We discuss generalizations of the resulting equation, as well as the connection to another recently proposed non-classical steady state transport equation.In the following, we develop numerical methods for this non-classical steady state transport equation. Therefore, we investigate a variation of a standard finite volume HLL method for moment models of this equation. In a detailed numerical analysis we show that these schemes preserve the analytic asymptotic limit in a diffusive scaling. Furthermore, the schemes preserve the convex set of admissible and realizable sets. A coupling of the initial value and the full solution strongly suggests to use an iterative solution method. The naive source iteration method is shown to become arbitrarily slow in the scattering dominated regime, therefore we adapt a Diffusion Synthetic Acceleration method for the classical equation based on the diffusion approximation for the non-classical equation. We investigate the contraction rates via a von Neumann analysis of the full equation and the corresponding moment models, which shows that they are significantly decreased by the acceleration method.Equipped with accurate and efficient numerical schemes, we present a method for the solution of inverse problems based on the non-classical steady state transport equation. We formulate the parameter estimation problem as an optimal control problem, and derive a first order optimality system using a Lagrange formalism. This formalism is based on adjoint calculus, therefore we formally derive the adjoint equation of the non-classical steady state transport equation. Then we adapt the numerical schemes to discretize this first order optimality system. We also show that these schemes are consistent with the underlying discrete optimality system. Numerical solutions of the optimality system are obtained using gradient based optimization methods and show that parameters can be reconstructed accurately at a reasonable computational cost.