This thesis is a survey of the mathematical theory of Generative Adversarial Networks (GANs). The relevant theories discussed are game theory, information theory and optimal transport theory.

Thesis (Ph. D.)--University of Washington, 1984. / Vita. Includes bibliographical references.

Calculation of the neutron flux in a nuclear reactor core is ideally performed by solving the neutron transport equation for a detailed-geometry model using several tens of energy groups. However, performing such detailed calculations for an entire core is prohibitively expensive from a computational perspective. Full-core neutronic calculations for CANDU reactors are therefore performed customarily using two-energy-group diffusion theory (no angular dependence) for a node-homogenized reactor model. The work presented here is concerned with reducing the loss in accuracy entailed when going from Transport to Diffusion. To this end a new method of calculating the diffusion coefficient was developed, based on equating the neutron balance equation expressed by the transport equation with the neutron balance equation expressed by the diffusion equation. The technique is tested on a simple twelve-node model and is shown to produce transport-like accuracy without the associated computational effort. / UOIT

Current advancements in nuclear reactor core design are pushing reactor cores towards greater heterogeneity in an attempt to make nuclear power more sustainable in terms of fuel utilization and long-term disposal needs. These new designs are now being limited by the accuracy of the core simulators/methods. Increasing attention has been given to full core transport as the flux module in future core simulators. However, the current transport methods, due to their significant memory and computational time requirements, are not practical for whole core calculations. While most researchers are working on developing new acceleration and phase space parallelization techniques for the current fine mesh transport methods, this dissertation focuses on the development of a practical heterogeneous coarse mesh transport method. In this thesis, a heterogeneous coarse mesh transport method is extended from two to three dimensions in Cartesian geometry and new techniques are developed to reduce the strain on computational resources. The high efficiency of the method is achieved by decoupling the problem into a series of fixed source calculations in smaller sub-volume elements (e.g. coarse meshes). This decoupling lead to shifting the computation time to a priori calculations of response functions in unique sub-volumes in the system. Therefore, the method is well suited for large problems with repeated geometry such as those found in nuclear reactor cores. Even though the response functions can be generated with any available existing fine-mesh (deterministic or stochastic) code, a stochastic method was selected in this dissertation. Previous work in two dimensions used discrete polynomial expansions that are better suited for treating discrete variables found in pure deterministic transport methods. The amount of data needed to represent very heterogeneous problems accurately became quite large making the three dimensional extension impractical. The deterministic method was thus replaced by a stochastic response function generator making the transition to continuous variables fairly simple. This choice also improves the geometry handling capability of the coarse mesh method.

Tomography refers to the cross-sectional imaging of an object from either transmission or reflection data collected by illuminating the object from many different directions. Classical tomography fails to reconstruct the optical properties of thick scattering objects because it does not adequately account for the scattering component of the neutron beam intensity exiting the sample. We proposed a new method of computed tomography which employs an inverse problem analysis of both the transmitted and scattered images generated from a beam passing through an optically thick object. This inverse problem makes use of a computationally efficient, two-dimensional forward problem based on neutron transport theory that effectively calculates the detector readings around the edges of an object. The forward problem solution uses a Step-Characteristic (SC) code with known uncollided source per cell, zero boundary flux condition and Sn discretization for the angular dependence. The calculation of the uncollided sources is performed by using an accurate discretization scheme given properties and position of the incoming beam and beam collimator. The detector predictions are obtained considering both the collided and uncollided components of the incoming radiation. The inverse problem is referred as an optimization problem. The function to be minimized, called an objective function, is calculated as the normalized-squared error between predicted and measured data. The predicted data are calculated by assuming a uniform distribution for the optical properties of the object. The objective function depends directly on the optical properties of the object; therefore, by minimizing it, the correct property distribution can be found. The minimization of this multidimensional function is performed with the Polack Ribiere conjugate-gradient technique that makes use of the gradient of the function with respect to the cross sections of the internal cells of the domain. The forward and inverse models have been successfully tested against numerical results obtained with MCNP (Monte Carlo Neutral Particles) showing excellent agreements. The reconstructions of several objects were successful. In the case of a single intrusion, TNTs (Tomography Neutron Transport using Scattering) was always able to detect the intrusion. In the case of the double body object, TNTs was able to reconstruct partially the optical distribution. The most important defect, in terms of gradient, was correctly located and reconstructed. Difficulties were discovered in the location and reconstruction of the second defect. Nevertheless, the results are exceptional considering they were obtained by lightening the object from only one side. The use of multiple beams around the object will significantly improve the capability of TNTs since it increases the number of constraints for the minimization problem.

The source and nature, as well as the history of ray-effects, is described. A benchmark code, using piecewise constant functions in angle and diamond differencing in space, is derived in order to analyze four sample problems. The results of this analysis are presented showing the ray effects and how increasing the resolution (number of angles) eliminates them. The theory of wavelets is introduced and the use of wavelets in multiresolution analysis is discussed. This multiresolution analysis is applied to the transport equation, and equations that can be solved to calculate the coefficients in the wavelet expansion for the angular flux are derived. The use of thresholding to eliminate wavelet coefficients that are not required to adequately solve a problem is then discussed. An iterative sweeping algorithm, called the SN-WN method, is derived to solve the wavelet-based equations. The convergence of the SN-WN method is discussed. An algorithm for solving the equations is derived, by solving a matrix within each cell directly for the expansion coefficients. This algorithm is called the CWWN method. The results of applying the CW-WN method to the benchmark problems are presented. These results show that more research is needed to improve the convergence of the SN-WN method, and that the CW-WN method is computationally too costly to be seriously considered.

Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (leaves 111-116).

Thesis (Ph. D.)--University of Hong Kong, 2001. / Includes bibliographical references (leaves 86-94).

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 64-67). Also available in print.

Thesis (M. Phil.)--University of Hong Kong, 2010. / Includes bibliographical references (leaves 64-67). Also available online.