Full Error Analysis and Uncertainty Quantification for the Heterogeneous Transport Equation in Slab Geometry

We present an analysis of multilevel Monte Carlo techniques for the forward problem of uncertainty quantification for the radiative transport equation, when the coefficients ({\em cross-sections}) are heterogenous random fields. To do this, we first give a new error analysis for the combined spatial and angular discretisation in the deterministic case, with error estimates which are explicit in the coefficients (and allow for very low regularity and jumps). This detailed error analysis is done for the 1D space - 1D angle slab geometry case with classical diamond differencing. Under reasonable assumptions on the statistics of the coefficients, we then prove an error estimate for the random problem in a suitable Bochner space. Because the problem is not self-adjoint, stability can only be proved under a path-dependent mesh resolution condition. This means that, while the Bochner space error estimate is of order $\mathcal{O}(h^\eta)$ for some $\eta$, where $h$ is a (deterministically chosen) mesh diameter, smaller mesh sizes might be needed for some realisations. Under reasonable assumptions we show that the expected cost for computing a typical quantity of interest remains of the same order as for a single sample. This leads to rigorous complexity estimates for Monte Carlo and multilevel Monte Carlo: For particular linear solvers, the multilevel version gives up to two orders of magnitude improvement over Monte Carlo. We provide numerical results supporting the theory.