Models, numerical methods, and uncertainty quantification for radiation therapy

The evolution of particles traveling through a background medium can be described by the linear Boltzmann transport equation. For this equation, we study several numerical methods with the focus on radiation therapy. The main challenges to solve the linear transport equation numerically are the large dimensionality of the problem, the integro-differential form of the equation, and the highly varying material parameters in the scattering operator. In this work, we consider four main topics. First, we introduce a numerical method for electron transport based on the continuous slowing down approximation. We adapt a second order staggered grid method for spherical harmonic moment equations of radiative transfer to our application. Secondly, we investigate the filtered spherical harmonic (FPN ) equations, which have the potential to reduce the Gibbs phenomena around discontinuities. In particular, we prove global L2 convergence properties of the FPN equations and show how the convergence rates depend on the smoothness of the solution and the order of the filter. Thirdly, we extend a one-dimensional staggered grid method to two spatial dimensions. Due to the staggering, the method requires less unknowns than one might have expected and leads to a compact stencil in the discrete diffusion limit. In addition, we rigorously analyze the underlying one-dimensional scheme and show that the scheme is asymptotic preserving. Lastly, we derive a twolevel sampling strategy based on a model hierarchy to improve a stochastic collocation sparse grid method for uncertainty quantification. The method combines a reduced model for the hyperbolic relaxation system with a correction in a two-level framework to balance the different error terms and to minimize the computational cost. For all major topics, we present several numerical experiments to confirm the theoretical predictions.