In fiber optic communication systems, the main linear phenomenon that causes optical pulse broadening is called dispersion, which limits the transmission data rate and distance. The principle nonlinear effect, called self-phase modulation, can also limit the system performance by causing spectral broadening. Hence, to achieve the optimal system performance, high data rate and low bandwidth occupancy, those effects must be overcome or compensated. In a nonlinear dispersive fiber, properties of a transmitting pulse: width, chirp, and spectra, are changed along the way and are complicated to predict. Although there is a well-known differential equation, called the Nonlinear Schrodinger Equation, which describes the complex envelope of the optical pulse subject to the nonlinear and dispersion effects, the equation cannot generally be solved in closed form. Although, the split-step Fourier method can be used to numerically determine pulse properties from this nonlinear equation, numerical results are time consuming to obtain and provide limited insight into functional relationships and how to design input pulses. One technique, called the Variational Method, is an approximate but accurate way to solve the nonlinear Schrodinger equation in closed form. This method is exploited throughout this thesis to study the pulse properties in a nonlinear dispersive fiber, and to explore ways to compensate dispersion for both single link and concatenated link systems. In a single link system, dispersion compensation can be achieved by appropriately pre-chirping the input pulse. In this thesis, the variational method is then used to calculate the optimal values of pre-chirping, in which: (i) the initial pulse and spectral width are restored at the output, (ii) output pulse width is minimized, (iii) the output pulse is transform limited, and (iv) the output time-bandwidth product is minimized. For a concatenated link system, the variational calculation is used to (i) show the symmetry of pulse width around the chirp-free point in the plot of pulse width versus distance, (ii) find the optimal dispersion constant of the dispersion compensation fiber in the nonlinear dispersive regime, and (iii) suggest the dispersion maps for two and four link systems in which initial conditions (or parameters) are restored at the output end. The accuracy of the variational approximation is confirmed by split-step Fourier simulation throughout this thesis. In addition, the comparisons show that the accuracy of the variational method improves as the nonlinear effects become small. / Master of Science

The primary objective of this thesis is to develop and implement a global optimization algorithm to solve a class of nonconvex programming problems, and to test it using a collection of engineering design problem applications.The class of problems we consider involves the optimization of a general nonconvex factorable objective function over a feasible region that is restricted by a set of constraints, each of which is defined in terms of nonconvex factorable functions. Such problems find widespread applications in production planning, location and allocation, chemical process design and control, VLSI chip design, and numerous engineering design problems. This thesis offers a first comprehensive methodological development and implementation for determining a global optimal solution to such factorable programming problems. To solve this class of problems, we propose a branch-and-bound approach based on linear programming (LP) relaxations generated through various approximation schemes that utilize, for example, the Mean-Value Theorem and Chebyshev interpolation polynomials, coordinated with a {em Reformulation-Linearization Technique} (RLT). The initial stage of the lower bounding step generates a tight, nonconvex polynomial programming relaxation for the given problem. Subsequently, an LP relaxation is constructed for the resulting polynomial program via a suitable RLT procedure. The underlying motivation for these two steps is to generate a tight outer approximation of the convex envelope of the objective function over the convex hull of the feasible region. The bounding step is thenintegrated into a general branch-and-bound framework. The construction of the bounding polynomials and the node partitioning schemes are specially designed so that the gaps resulting from these two levels of approximations approach zero in the limit, thereby ensuring convergence to a global optimum. Various implementation issues regarding the formulation of such tight bounding problems using both polynomial approximations and RLT constructs are discussed. Different practical strategies and guidelines relating to the design of the algorithm are presented within a general theoretical framework so that users can customize a suitable approach that takes advantage of any inherent special structures that their problems might possess. The algorithm is implemented in C++, an object-oriented programming language. The class modules developed for the software perform various functions that are useful not only for the proposed algorithm, but that can be readily extended and incorporated into other RLT based applications as well. Computational results are reported on a set of fifteen engineering process control and design test problems from various sources in the literature. It is shown that, for all the test problems, a very competitive computational performance is obtained. In most cases, the LP solution obtained for the initial node itself provides a very tight lower bound. Furthermore, for nine of these fifteen problems, the application of a local search heuristic based on initializing the nonlinear programming solver MINOS at the node zero LP solution produced the actual global optimum. Moreover, in finding a global optimum, our algorithm discovered better solutions than the ones previously reported in the literature for two of these test instances. / Master of Science

This thesis involves restoration, reconstruction, and enhancement of a digitized library of hand-written documents. Imaging systems that perform this digitization often degrade the quality of the original documents. Many techniques exist for reconstructing, restoring, and enhancing digital images; however, many require <i> a priori </i> knowledge of the imaging system. In this study, only partial <i> a priori </i> knowledge is available, and therefore unknown parameters must be estimated before restoration, reconstruction, or enhancement is possible. The imaging system used to digitize the documents library has degraded the images in several ways. First, it has introduced a ringing that is apparent around each stroke. Second, the system has eliminated strokes of narrow widths. To restore these images, the imaging system is modeled by estimating the point spread function from sample impulse responses, and the image noise is estimated in an attempt to apply standard linear restoration techniques. The applicability of these techniques is investigated in the first part of this thesis. Then nonlinear filters, structural techniques, and enhancement techniques are applied to obtain substantial improvements in image quality. / Master of Science

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A complete study was conducted on optical limiting characterization in samples of carbon black microparticles in a mixture of deionized water and ethylene glyccol using nanosecodn and picosecond later pulses at 532 nm and 1064 nm.

Doctor of Philosophy / Department of Mathematics / Louis Crane / We exhibit a static, cylindrically symmetric, exact solution to the Euler-Heisenberg field equations (EHFE) and prove that its effective geometry contains (optical) black holes. It is conjectured that there are also soliton solutions to the EHFE which contain black hole geometries.

In this dissertation, we focus on solving Nonlinear Least Squares problems using a supervised approach. In particular, we developed a Supervised Descent Method (SDM), performed thorough theoretical analysis, and demonstrated its effectiveness on optimizing analytic functions, and four other real-world applications: Inverse Kinematics, Rigid Tracking, Face Alignment (frontal and multi-view), and 3D Object Pose Estimation. In Rigid Tracking, SDM was able to take advantage of more robust features, such as, HoG and SIFT. Those non-differentiable image features were out of consideration of previous work because they relied on gradient-based methods for optimization. In Inverse Kinematics where we minimize a non-convex function, SDM achieved significantly better convergence than gradient-based approaches. In Face Alignment, SDM achieved state-of-the-arts results. Moreover, it was extremely computationally efficient, which makes it applicable for many mobile applications. In addition, we provided a unified view of several popular methods including SDM on sequential prediction, and reformulated them as a sequence of function compositions. Finally, we suggested some future research directions on SDM and sequential prediction.

Thesis (MEng)--University of Stellenbosch, 2000. / ENGLISH ABSTRACT: The growth in intelligent control has primarily been a reaction to the realisation that nonlinear control theory has been unable to provide practical solutions to present day control challenges. Consequently the chemical industry may be cited for numerous instances of overdesign, which result as an attempt to avoiding operation near or within complex (often more economically viable) operating regimes. Within these complex operating regimes robust control system performance may prove difficult to achieve using conventional (algorithmic) control methodologies. Biological neuronal control mechanisms demonstrate a remarkable ability to make accurate generalisations from sparse environmental information. Neural networks, with their ability to learn and their inherent massive parallel processing ability, introduce numerous opportunities for developing superior control structures for complex nonlinear systems. To facilitate neural network learning, reinforcement learning techniques provide a framework which allows for learning from direct interactions with a dynamic environment. lts promise as a means of automating the knowledge acquisition process is beguiling, as it provides a means of developing control strategies from cause and effect (reward and punishment) interaction information, without needing to specify how the goal is to be achieved. This study aims to establish evolutionary reinforcement learning as a powerful tool for developing robust neurocontrollers for application in highly nonlinear process systems. A novel evolutionary algorithm; Symbiotic, Adaptive Neuro-Evolution (SANE), is utilised to facilitate neurocontroller development. This study also aims to introduce SANE as a means of integrating the process design and process control development functions, to obtain a single comprehensive calculation step for maximum economic benefit. This approach thus provides a tool with which to limit the occurrence of overdesign in the process industry. To investigate the feasibility of evolutionary reinforcement learning in achieving these aims, the SANE algorithm is implemented in an event-driven software environment (developed in Delphi 4.0), which may be applied for both simulation and real world control problems. Four highly nonlinear reactor arrangements are considered in simulation studies. As a real world application, a novel batch distillation pilot plant, a Multi-Effect Batch Distillation (MEBAD) column, was constructed and commissioned. The neurocontrollers developed using SANE in the complex simulation studies, were found to exhibit excellent robustness and generalisation capabilities. In comparison with model predictive control implementations, the neurocontrollers proved far less sensitive to model parameter uncertainties, removing the need for model mismatch compensation to eliminate steady state off-set. The SANE algorithm also proved highly effective in discovering the operating region of greatest economic return, while simultaneously developing a neurocontroller for this optimal operating point. SANE, however, demonstrated limited success in learning an effective control policy for the MEBAD pilot plant (poor generalisation), possibly due to limiting the algorithm's search to a too small region of the state space and the disruptive effects of sensor noise on the evaluation process. For industrial applications, starting the evolutionary process from a random initial genetic algorithm population may prove too costly in terms of time and financial considerations. Pretraining the genetic algorithm population on approximate simulation models of the real process, may result in an acceptable search duration for the optimal control policy. The application of this neurocontrol development approach from a plantwide perspective should also have significant benefits, as individual controller interactions are so doing implicitly eliminated. / AFRIKAANSE OPSOMMING: The huidige groei in intelligente beheerstelsels is primêr 'n reaksie op die besef dat nie-liniêre beheerstelsel teorie nie instaat is daartoe om praktiese oplossings te bied vir huidige beheer kwelkwessies nie. Gevolglik kan talle insidente van oorontwerp in die chemiese nywerhede aangevoer word, wat voortvloei uit 'n poging om bedryf in of naby komplekse bedryfsgebiede (dikwels meer ekonomies vatbaar) te vermy. Die ontwikkeling van robuuste beheerstelsels, met konvensionele (algoritmiese ) beheertegnieke, in die komplekse bedryfsgebiede mag problematies wees. Biologiese neurobeheer megamsmes vertoon 'n merkwaardige vermoë om te veralgemeen vanaf yl omgewingsdata. Neurale netwerke, met hulle vermoë om te leer en hulle inherente paralleie verwerkingsvermoë, bied talle geleenthede vir die ontwikkeling van meer doeltreffende beheerstelsels vir gebruik in komplekse nieliniêre sisteme. Versterkingsleer bied a raamwerk waarbinne 'n neurale netwerk leer deur direkte interaksie met 'n dinamiese omgewing. Versterkingsleer hou belofte in vir die inwin van kennis, deur die ontwikkeling van beheerstrategieë vanaf aksie en reaksie (loon en straf) interaksies - sonder om te spesifiseer hoe die taak voltooi moet word. Hierdie studie beaam om evolutionêre versterkingsleer as 'n kragtige strategie vir die ontwikkeling van robuuste neurobeheerders in nie-liniêre prosesomgewings, te vestig. 'n Nuwe evolutionêre algoritme; Simbiotiese, Aanpasbare, Neuro-Evolusie (SANE), word aangewend vir die onwikkeling van die neurobeheerders. Hierdie studie beoog ook die daarstelling van SANE as 'n weg om prosesontwerp en prosesbeheer ontwikkeling vir maksimale ekonomiese uitkering, te integreer. Hierdie benadering bied dus 'n strategie waardeur die insidente van oorontwerp beperk kan word. Om die haalbaarheid van hierdie doelwitte, deur die gebruik van evolusionêre versterkingsleer te ondersoek, is die SANE algoritme aangewend in 'n Windows omgewing (ontwikkel in Delphi 4.0). Die Delphi programmatuur geniet toepassing in beide die simulasie en werklike beheer probleme. Vier nie-liniêre reaktore ontwerpe is oorweeg in die simulasie studies. As 'n werklike beheer toepassing, is 'n nuwe enkelladingsdistillasie kolom, 'n Multi-Effek Enkelladingskolom (MEBAD) gebou en in bedryf gestel. Die neurobeheerders vir die komplekse simulasie studies, wat deur SANE ontwikkel is, het uitstekende robuustheid en veralgemeningsvermoë ten toon gestel. In vergelyking met model voorspellingsbeheer implementasies, is gevind dat die neurobeheerders heelwat minder sensitief is vir model parameter onsekerheid. Die noodsaak na modelonsekerheid kompensasie om gestadigde toestand afset te elimineer, word gevolglik verwyder. The SANE algoritme is ook hoogs effektief vir die soek na die mees ekonomies bedryfstoestand, terwyl 'n effektiewe neurobeheerder gelyktydig vir hierdie ekonomies optimumgebied ontwikkel word. SANE het egter beperkte sukses in die leer van 'n effektiewe beheerstrategie vanaf die MEBAD toetsaanleg getoon (swak veralgemening). Die swak veralgemening kan toegeskryf word aan 'n te klein bedryfsgebied waarin die algoritme moes soek en die negatiewe effek van sensor geraas op die evaluasie proses. Vir industriële applikasies blyk dit dat die uitvoer van die evolutionêre proses vanaf 'n wisselkeurige begintoestand nie koste effektief is in terme van tyd en finansies nie. Deur die genetiese algoritme populasie vooraf op 'n benaderde modelop te lei, kan die soek tydperk na 'n optimale beheerstrategie aansienlik verkort word. Die aanwending van die neurobeheer ontwikkelingstrategie vanuit 'n aanlegwye oogpunt mag aanleiding gee tot aansienlike voordele, aaangesien individuele beheerder interaksies sodoende implisiet uitgeskakel word.