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Among the most formidable challenges facing our world is the need for safe, clean, affordable energy sources. Growing concerns over global warming induced climate change and the rising costs of fossil fuels threaten conventional means of electricity production and are driving the current nuclear renaissance. One concept at the forefront of international development efforts is the High Temperature Gas-Cooled Reactor (HTGR). With numerous passive safety features and a meltdown-proof design capable of attaining high thermodynamic efficiencies for electricity generation as well as high temperatures useful for the burgeoning hydrogen economy, the HTGR is an extremely promising technology. Unfortunately, the fundamental understanding of neutron behavior within HTGR fuels lags far behind that of more conventional watercooled reactors. HTGRs utilize a unique heterogeneous fuel element design consisting of thousands of tiny fissile fuel kernels randomly mixed with a non-fissile graphite matrix. Monte Carlo neutron transport simulations of the HTGR fuel element geometry in its full complexity are infeasible and this has motivated the development of more approximate computational techniques. A series of MATLAB codes was written to perform Monte Carlo simulations within HTGR fuel pebbles to establish a comprehensive understanding of the parameters under which the accuracy of the approximate techniques diminishes. This research identified the accuracy of the chord length sampling method to be a function of the matrix scattering optical thickness, the kernel optical thickness, and the kernel packing density. Two new Monte Carlo methods designed to focus the computational effort upon the parameter conditions shown to contribute most strongly to the overall computational error were implemented and evaluated. An extended memory chord length sampling routine that recalls a neutron’s prior material traversals was demonstrated to be effective in fixed source calculations containing densely packed, optically thick kernels. A hybrid continuous energy Monte Carlo algorithm that combines homogeneous and explicit geometry models according to the energy dependent optical thickness was also developed. This resonance switch approach exhibited a remarkably high degree of accuracy in performing criticality calculations. The versatility of this hybrid modeling approach makes it an attractive acceleration strategy for a vast array of Monte Carlo radiation transport applications. / text

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published_or_final_version / Physics / Master / Master of Philosophy

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This thesis contains the author’s work in and contributions to the field of Monte Carlo sampling for undirected graphical models, a class of statistical model commonly used in machine learning, computer vision, and spatial statistics; the aim is to be able to use the methodology and resultant samples to estimate integrals of functions of the variables in the model. Over the course of the study, three different but related methods were proposed and have appeared as research papers. The thesis consists of an introductory chapter discussing the models considered, the problems involved, and a general outline of Monte Carlo methods. The three subsequent chapters contain versions of the published work. The second chapter, which has appeared in (Hamze and de Freitas 2004), is a presentation of new MCMC algorithms for computing the posterior distributions and expectations of the unknown variables in undirected graphical models with regular structure. For demonstration purposes, we focus on Markov Random Fields (MRFs). By partitioning the MRFs into non-overlapping trees, it is possible to compute the posterior distribution of a particular tree exactly by conditioning on the remaining tree. These exact solutions allow us to construct efficient blocked and Rao-Blackwellised MCMC algorithms. We show empirically that tree sampling is considerably more efficient than other partitioned sampling schemes and the naive Gibbs sampler, even in cases where loopy belief propagation fails to converge. We prove that tree sampling exhibits lower variance than the naive Gibbs sampler and other naive partitioning schemes using the theoretical measure of maximal correlation. We also construct new information theory tools for comparing different MCMC schemes and show that, under these, tree sampling is more efficient. Although the work discussed in Chapter 2 exhibited promise on the class of graphs to which it was suited, there are many cases where limiting the topology is quite a handicap. The work in Chapter 3 was an exploration in an alternative methodology for approximating functions of variables representable as undirected graphical models of arbitrary connectivity with pairwise potentials, as well as for estimating the notoriously difficult partition function of the graph. The algorithm, published in (Hamze and de Freitas 2005), fits into the framework of sequential Monte Carlo methods rather than the more widely used MCMC, and relies on constructing a sequence of intermediate distributions which get closer to the desired one. While the idea of using “tempered” proposals is known, we construct a novel sequence of target distributions where, rather than dropping a global temperature parameter, we sequentially couple individual pairs of variables that are, initially, sampled exactly from a spanning treeof the variables. We present experimental results on inference and estimation of the partition function for sparse and densely-connected graphs. The final contribution of this thesis, presented in Chapter 4 and also in (Hamze and de Freitas 2007), emerged from some empirical observations that were made while trying to optimize the sequence of edges to add to a graph so as to guide the population of samples to the high-probability regions of the model. Most important among these observations was that while several heuristic approaches, discussed in Chapter 1, certainly yielded improvements over edge sequences consisting of random choices, strategies based on forcing the particles to take large, biased random walks in the state-space resulted in a more efficient exploration, particularly at low temperatures. This motivated a new Monte Carlo approach to treating complex discrete distributions. The algorithm is motivated by the N-Fold Way, which is an ingenious event-driven MCMC sampler that avoids rejection moves at any specific state. The N-Fold Way can however get “trapped” in cycles. We surmount this problem by modifying the sampling process to result in biased state-space paths of randomly chosen length. This alteration does introduce bias, but the bias is subsequently corrected with a carefully engineered importance sampler.

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