Peterson, Jacob Ross (2014-05). Exponentially-convergent Monte Carlo for the One-dimensional Transport Equation. Master's Thesis.
An exponentially-convergent Monte Carlo (ECMC) method is analyzed using the one-group, one-dimension, slab-geometry transport equation. The method is based upon the use of a linear discontinuous finite-element trial space in position and direction to represent the transport solution. A space-angle h-adaptive algorithm is employed to maintain exponential convergence after stagnation occurs due to in- adequate trial-space resolution. In addition, a biased sampling algorithm is used to adequately converge singular problems. Computational results are presented demonstrating the efficacy of the new approach. We tested our ECMC algorithm against standard Monte Carlo and found the ECMC method to be generally much more efficient. For a manufacture solution the ECMC algorithm was roughly 200 times more effective than the standard Monte Carlo. When considering a highly singular pure attenuation problem, the ECMC method was roughly 4000 times more effective.