Browsing by Subject "Newton's method"
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Item Application of a Constrained Optimization Technique to the Imaging of Heterogeneous Objects Using Diffusion Theory(2011-02-22) Sternat, Matthew RyanThe problem of inferring or reconstructing the material properties (cross sections) of a domain through noninvasive techniques, methods using only input and output at the domain boundary, is attempted using the governing laws of neutron diffusion theory as an optimization constraint. A standard Lagrangian was formed consisting of the objective function and the constraints to satisfy, which was minimized through optimization using a line search method. The chosen line search method was Newton's method with the Armijo algorithm applied for step length control. A Gaussian elimination procedure was applied to form the Schur complement of the system, which resulted in greater computational efficiency. In the one energy group and multi-group models, the limits of parameter reconstruction with respect to maximum reconstruction depth, resolution, and number of experiments were established. The maximum reconstruction depth for one-group absorption cross section or multi-group removal cross section were only approximately 6-7 characteristic lengths deep. After this reconstruction depth limit, features in the center of a domain begin to diminish independent of the number of experiments. When a small domain was considered and size held constant, the maximum reconstruction resolution for one group absorption or multi-group removal cross section is approximately one fourth of a characteristic length. When finer resolution then this is considered, there is simply not enough information to recover that many region's cross sections independent of number of experiments or flux to cross-section mesh refinement. When reconstructing fission cross sections, the one group case is identical to absorption so only the multi-group is considered, then the problem at hand becomes more ill-posed. A corresponding change in fission cross section from a change in boundary flux is much greater then change in removal cross section pushing convergence criteria to its limits. Due to a more ill-posed problem, the maximum reconstruction depth for multi-group fission cross sections is 5 characteristic lengths, which is significantly shorter than the removal limit. To better simulate actual detector readings, random signal noise and biased noise were added to the synthetic measured solutions produced by the forward models. The magnitude of this noise and biased noise is modified and a dependency of the maximum magnitude of this noise versus the size of a domain was established. As expected, the results showed that as a domain becomes larger its reconstruction ability is lowered which worsens upon the addition of noise and biased noise.Item Iteration as an avenue for mathematical exploration(2013-08) Joyoprayitno, Anne Christine; Luecke, John EdwinThis report explores several applications of iteration and the various connections that can be made to different areas of mathematics. The ties iteration has to the Wada Property, bifurcation diagram, root finding, and applications in geometry are all investigated. Finally, a rationale for incorporating iteration into secondary mathematics courses to support a more robust curriculum is discussed.Item Iterations of the Newton Map of tan(z)(2013-05) Bray, Kasey; Dwyer, Jerry F.; Barnard, Roger W.; Williams, BrockThe dynamical systems of trigonometric functions are explored, with a focus on t(z)=tan(z) and the fractal image created by iterating the Newton map, F_t (z), of t(z). As a point of reference we present Newton’s method applied to polynomials and the iterations of families of trigonometric functions. The basins of attraction created from iterating F_t (z) are analyzed and, in an effort to determine the fate of each seed value, bounds are placed within the primary basins of attraction. We further prove x and y-axis symmetry of the function, and explore the infinite nature of the fractal images. Lastly, Newton iterations of the family〖 z〗^k tan(z) are explored in comparison with F_t (z) and Householder’s methods are discussed.Item The role of interactive visualizations in the advancement of mathematics(2012-08) Alvarado, Alberto; Daniels, Mark L.; Odell, Edward W.This report explores the effect of interactive visualizations on the advancement of mathematics understanding. Not only do interactive visualizations aid mathematicians to expand the body of knowledge of mathematics but it also allows students an efficient way to process the information taught in schools. There are many concepts in mathematics that utilize interactive visualizations and examples of such concepts are illustrated within this report.