We present a new iterative approach to solving neutral-particle transport problems. The scheme divides the transport solution into its particular and homogeneous or “source-free” components. The particular problem is solved directly, while the homogeneous problem is found iteratively. To organize the iterative inversion of the homogeneous components, we exploit the structures of the so called Case-modes that compose it. The asymptotic Case-modes, those that vary slowly in space and angle, are assigned to a diffusion solver. The remaining transient Case-modes, those with large spatial gradients, are assigned to a transport solver. The scheme iterates on the contribution from each solver until the particular plus homogeneous solution converges. The iterative method is implemented successfully in slab geometry with isotropic scattering and one energy group. The convergence rate of the method is only weakly dependent on the scattering ratio of the problem. Instead, the rate of convergence depends strongly on the material thickness of the slab, with thick slabs converging in few iterations. The transient solution is obtained by applying a One Cell Inversion scheme instead of a Source Iteration based scheme. Thus, the transient unknowns are calculated with little coordination between them. This independence among unknowns makes our scheme ideally suited for transport calculations on parallel architectures. The slab geometry iterative scheme is adapted to XY geometry. Unfortunately, this attempt to extend the slab geometry iterative scheme to multiple dimensions has not been successful. The exact filtering scheme needed to discriminate asymptotic and transient modes has not been obtained and attempts to approximate this filtering process resulted in a divergent iterative scheme. However, the development of this iterative scheme yield valuable analysis tools to understand the Case-mode structure of any spatial discretization under arbitrary material properties.

Iterative solutions of the Boltzmann transport equation are computationally intensive. Spatial multigrid methods have led to efficient iterative algorithms for solving a variety of partial differential equations; thus, it is natural to explore their application to transport equations. Manteuffel et al. conducted such an exploration in one spatial dimension, using two-cell inversions as the relaxation or smoothing operation, and reported excellent results. In this dissertation we extensively test Manteuffel??s one-dimensional method and our modified versions thereof. We demonstrate that the performance of such spatial multigrid methods can degrade significantly given strong heterogeneities. We also extend Manteuffel??s basic approach to two-dimensional problems, employing four-cell inversions for the relaxation operation. We find that for uniform homogeneous problems the two-dimensional multigrid method is not as rapidly convergent as the one-dimensional method. For strongly heterogeneous problems the performance of the two-dimensional method is much like that of the one-dimensional method, which means it can be slow to converge. We conclude that this approach to spatial multigrid produces a method that converges rapidly for many problems but not for others. That is, this spatial multigrid method is not unconditionally rapidly convergent. However, our analysis of the distribution of eigenvalues of the iteration operators indicates that this spatial multigrid method may work very well as a preconditioner within a Krylov iteration algorithm, because its eigenvalues tend to be relatively well clustered. Further exploration of this promising result appears to be a fruitful area of further research.

Iterative decoding and codes on graphs were first devised by Gallager in 1960, and then rediscovered by Berrou, Glavieux and Thitimajshima in 1993. This technique plays an important role in modern communications, especially in coding theory and practice. In particular, low-density parity-check (LDPC) codes, introduced by Gallager in the 1960s, are the class of codes at the heart of iterative coding. Since these codes are quite general and exhibit good performance under message-passing decoding, they play an important role in communications research today. A thorough analysis of iterative decoding systems and the relationship between maximum a posteriori (MAP) and belief propagation (BP) decoding was initiated by Measson, Montanari, and Urbanke. This analysis is based on density evolution (DE), and extrinsic information transfer (EXIT) functions, introduced by ten Brink. Following their work, this thesis considers the MAP decoding thresholds of three iterative decoding systems. First, irregular repeat-accumulate (IRA) and accumulaterepeataccumulate (ARA) code ensembles are analyzed on the binary erasure channel (BEC). Next, the joint iterative decoding of LDPC codes is studied on the dicode erasure channel (DEC). The DEC is a two-state intersymbol-interference (ISI) channel with erasure noise, and it is the simplest example of an ISI channel with erasure noise. Then, we introduce a slight generalization of the EXIT area theorem and apply the MAP threshold bound for the joint decoder. Both the MAP and BP erasure thresholds are computed and compared with each other. The result quantities the loss due to iterative decoding Some open questions include the tightness of these bounds and the extensions to non-erasure channels.

This study proposed a numerical algorithm for estimating the angular misalignments between sensors of an ultra short baseline (USBL) positioning system. The algorithm is based on positioning a seabed transponder by moving a vessel along a predetermined straight-line path. Under the scheme of straight-line survey, mathematical representations of positioning error arising from heading, pitch, and roll misalignments were derived, respectively. The effect of each misalignment angle and how the differences can be used to calibrate each misalignment angle in turn are presented. A USBL calibration procedure that takes advantage of the geometry of position errors resulting from angular misalignments is then developed. During the USBL measurement, temporal and spatial variations of sound speed structure in water column are the major error sources. Therefore, this study used the sound speed profile together with a ray tracing method to correct observations of the USBL measurement. In addition, this study developed a method to compensate the effects of cross-track error on the estimation of alignment errors, and this makes the proposed algorithm applicable for using a vessel without dynamic positioning (DP) systems to collect USBL observations. The performance of the algorithm is evaluated through simulation and field experiment. The simulation and experimental results have demonstrated the effectiveness and robustness of the iterative scheme in finding alignment errors. The proposed algorithm yields a very rapid convergence of the solution series; usually the estimates obtained in the first iteration approximate to true values, and only a few iterations are necessary to achieve fairly accurate solutions.

In many multimedia applications, a little error in the output results is allowable. Therefore, this thesis presents an iterative floating-point multiplier with multiple precision to reduce the energy consumption of floating-point multiplication operations. The multiplier can provide the users with three kinds of modes. The distinction among the three modes is the accepted output error and the achievable energy saving through reducing the length of mantissa in the multiplication operation. In addition, to reduce the area of multiple precision floating-point multiplier we use the iterative structure to implement the mantissa multiplier in a floating point multiplier. Moreover the C++ language is adopted to evaluate the product error between each mode and the IEEE754 single precision multiplier. When the multimedia applications request high precision, the multiple precision floating-point multiplier will iteratively execute the 4-2 compression tree three times and the product error is around 10e-5%. The second-mode with the middle accuracy will iteratively execute the 4-2 compression tree two times and the product error is around 10e-3%. The third mode with the lowest accuracy will execute the 4-2 compression tree once and the product error is around 1%, it requires less execution cycle number. When compared with the tree-stage IEEE754 single-precision multiplier, the proposed iterative floating-point multiplier can save 42.54% area. For IDCT application, it can save 37.78% energy under 1% error constraint, For YUV to RGB application, it can save 31.36% energy under 1.1% error constraint. The experimental results demonstrate that the proposed multiple precision iterative floating-point multiplier can significantly reduce the energy consumption of multimedia applications that allow a little output distortion

Dissertation (Ph.D.)--Auburn University, 2008. / Abstract. Includes bibliographic references (p.73-80).

Thesis (M.S.)--Worcester Polytechnic Institute. / Keywords: Newton; globalized; inexact Newton. Includes bibliographical references (p. 61).

We study local well-posedness of the Cauchy problem for two geometric wave equations that can be derived from Anti-Self-Dual Yang Mills equations on R2+2. These are the space-time Monopole Equation and the Ward Wave Map. The equations can be formulated in different ways. For the formulations we use, we establish local well-posedness results, which are sharp using the iteration methods. / text

published_or_final_version / Mathematics / Master / Master of Philosophy

published_or_final_version / Computer Science / Doctoral / Doctor of Philosophy