Adaptive Monte Carlo Algorithms Applied to Heterogeneous Transport Problems

We apply three generations of geometrically convergent adaptive Monte Carlo algorithms to solve a model transport problem with severe heterogeneities in energy. In the first generation algorithms an arbitrarily precise solution of the transport equation is sought pointwise. In the second generation algorithms the solution is represented more economically as a vector of regionwise averages over a fixed uniform phase space decomposition. The economy of this representation provides geometric reduction in error to a precision limited by the granularity of the imposed phase space decomposition. With the third generation algorithms we address the question of how the second generation uniform phase space subdivision should be refined in order to achieve additional geometric learning. A refinement strategy is proposed based on an information density function that combines information from the transport equation and its dual.