For centuries, mathematicians have attempted to predict and emulate real-life physical phenomena through the use of mathematical models, often in the form of PDEs. More recently, understanding the effects of uncertainty on these models has led to an active area of research known as "Uncertainty Quantification". One method in Uncertainty Quantification, that avoids the so-called "curse of dimensionality", is the Multilevel Monte Carlo (MLMC) method.

The underlying theory of MLMC contains three assumptions on the numerical scheme: the (rate of) convergence; the (rate of) variance reduction; and the (rate of) cost growth. It can be then be shown that the cost of the overall MLMC method depends on each of these three rates. However, finding such rates is very much a non-trivial problem.

This analysis is well documented for the Darcy "fruit-fly" problem, an elliptic PDE. However for our application, the Radiative Transport equation, a hyperbolic integro-differential equation, the theory is non-existent. We will prove error estimates for a simplification of the (full) Radiative Transport Equation, where there is random spatial variation in the coefficients, with potentially low smoothness and discontinuities. Such error estimates can then be used to prove two of the MLMC assumptions.

This work is in collaboration with Ivan Graham and Rob Scheichl.