Investigation of Automated Variance Reduction Techniques for Monte Carlo Shielding Problems 22.106 Project Report

1. Introduction. The Monte Carlo method is widely believed to be the most accurate method for solving problems in radiation transport. Unfortunately, due to its very nature—following individual particle histories—certain classes of problems are particularly challenging for the method. One such class of problems consist of so-called deep penetration shielding problems. Because the purpose of a shield is to attenuate a particle population by several orders of magnitude, to use the Monte Carlo method requires a sufficient number of histories to ensure that the population, once attenuated, can still provide adequate statistics. For deep penetration problems, the level of attenuation makes Monte Carlo prohibitively expensive. To circumvent this issue, several approaches for variance reduction have been developed over the years. Variance reduction techniques aim to modify (i.e. bias) in some manner the underlying physics in such a way that an unbiased solution with lower statistical error is found than an unbiased simulation using the same computational resources. Haghighat and Wagner [1] classify variance reduction techniques in three ways: modified sampling methods (e.g. source biasing, implicit capture), population control methods (e.g. geometry splitting/roulette, weight windows), and semi-analytical methods (e.g. point detectors and DXTRAN). In this paper, we only analyze methods falling in the first two categories, namely (automated) approaches for source biasing, geometry splitting/roulette, and weight windows. The rest of this paper is organized as follows. In Section 2, we describe several approaches to variance reduction that use the adjoint or forward fluxes computed via the discrete ordinates method to select parameters for source biasing, geometry splitting, or weight windows. We apply those techniques in Section 3 to some simple slab problems and summarize the impact each technique has on various problem types. Section 4 provides several concluding remarks. Finally, Appendix A provides a sample input file based on the problems analyzed in this project, and Appendix B provides a complete listing of the Matlab code.