Mode coupling due to scattering by weak random inhomogeneities leads to the loss of coherence in the wave field measured a long distances of propagation. This in turn leads to the deterioration of coherent source localization methods such as matched-field. In this dissertation, we study with analysis and numerical simulations how such deterioration occurs and we introduce an original incoherent source localization approach for random waveguides. This approach is based on a special form of transport theory for the incoherent fluctuations of the wave field. The statistical stability of the method is analyzed and its performance is illustrated with numerical simulations. In addition, this method is used to estimate the correlation function of the random fluctuations of the wave speed.

Contrary to the Black-Scholes paradigm, an option-pricing model which incorporates the possibility of jumps more accurately reflects the evolution of stocks in the real world. However, hedging a contingent claim in such a model is a non-trivial issue: in many cases, an infinite number of hedging instruments are required to eliminate the risk of an option position. This thesis develops practical techniques for hedging contingent claims in markets with jumps. Both regime-switching and jump-diffusion models are considered.

Interest in stochastic simulations of chemical systems is growing. One of the aspects of simulation of chemical systems that has been the prime focus over the past few years is accelerated simulation methods applicable when there is a separation of time scale. With so many new methods being developed we have decided to look at four methods that we consider to be the main foundation for this research area. The four methods that will be the focus of this thesis are: the slow scale stochastic simulation algorithm, the quasi steady state assumption applied to the stochastic simulation algorithm, the nested stochastic simulation algorithm and the implicit tau leaping method. These four methods are designed to deal with stiff chemical systems so that the computational time is decreased from that of the "gold standard" Gillespie algorithm, the stochastic simulation algorithm. These approximation methods will be tested against a variety of sti examples such as: a fast reversible dimerization, a network of isomerizations, a fast species acting as a catalyst, an oscillatory system and a bistable system. Also, these methods will be tested against examples that are marginally stiff, where the time scale separation is not that distinct. From the results of testing stiff examples, the slow scale SSA was typically the best approximation method to use. The slow scale SSA was highly accurate and extremely fast in comparison with the other methods. We also found for certain cases, where the time scale separation was not as distinct, that the nested SSA was the best approximation method to use.

Image segmentation is a branch of computer vision that has received a considerable amount of interest in recent years. Segmentation describes a process that divides or partitions the pixels of a digital image into groups that correspond to the entities represented in the image. One such segmentation method is the Segmentation by Weighted Aggregation algorithm (SWA). Inspired by Algebraic Multigrid (AMG), the SWA algorithm provides a fast multilevel method for image segmentation. The SWA algorithm takes a graph-based approach to the segmentation problem. Given an image Ω, the weighted undirected graph A = (N,E) is constructed with each pixel corresponding to a node in N and each weighted edge connecting neighbouring nodes in E. The edge weight between nodes is calculated as a function of the difference in intensity between connected pixels. To determine whether a group of pixels should be declared as a segment in the SWA algorithm, a new scale-invariant measure to calculate the saliency of the group of pixels is introduced. This new measure determines the saliency of a potential segment by taking the ratio of the average similarity to its neighbours and its internal similarity. For complex images, intensity alone is not sufficient in providing a suitable segmentation. The SWA algorithm provides a way to improve the segmentation by incorporating other vision cues such as texture, shape and colour. The SWA algorithm with the new scale-invariant saliency measure was implemented and its performance was tested on simple test images and more complex aerial-view images.

Under certain conditions, flow down an inclined plane destabilizes and a persistent series of interfacial waves develop. An interest in determining under what conditions a flow becomes unstable and how the interface develops has motivated researchers to derive several models for analyzing this problem. The first part of this thesis compares three models for flow down a wavy, inclined plane with the goal of determining which best predicts features of the flow. These models are the shallow-water model (SWM), the integral-boundary-layer (IBL) model, and the weighted residual model (WRM). The model predictions for the critical Reynolds number for flow over an even bottom are compared to the theoretical value, and the WRM is found to match the theoretical value exactly. The neutral stability curves predicted by the three models are compared to two sets of experimental data, and again the WRM most closely matches the experimental data. Numerical solutions of the IBL model and the WRM are compared to numerical solutions of the full Navier-Stokes equations; both models compare well, although the WRM matches slightly better. Finally, the critical Reynolds numbers for the IBL model and the WRM for flow over a wavy incline are compared to experimental data. Both models give results close to the data and perform equally well. These comparisons indicate that the WRM most accurately models the flow. In the second part of the thesis, the WRM is extended to include the effects of bottom heating and permeability. The model is used to predict the effect of heating and permeability on the stability of the flow, and the results are compared to theoretical predictions from the Benney equation and to a perturbation solution of the Orr-Sommerfeld equation from the literature. The results indicate that the model does faithfully predict the theoretical critical Reynolds number with heating and permeability, and both effects destabilize the flow. Finally, numerical simulations of the model equations are compared to full numerical solutions of the Navier-Stokes equations for the case with bottom permeability. The results are found to agree, which indicates that the WRM remains appropriate when permeability is included.

This work seeks, through numerical simulations as well as analysis, to derive from relatively simple models an intuitive understanding of the dynamics and behaviour of flow in lakes near the bottom boundary. The main body is divided into two equally important sections, the analysis and simulation of the effects of nonlinear (quadratic) bottom drag on the flow, and the simulation of the effects of topography on lake dynamics as it relates to the redistribution of sediment from the lakebed. The simulations all follow a structured scheme, beginning with relatively simple one-dimensional models to build intuition and proceeding to full two-dimensional simulations using the weakly nonhydrostatic shallow water equations. Thus this work seeks to build an understanding of the behaviour of the modified shallow water equations (a good representation of lake behaviour) and to analyze the effects of nonlinear drag and bottom topography on these systems. The nonlinear drag chapters demonstrate that the addition of a nonlinear friction term, while very efficient at removing energy from the system, also causes interesting new behaviour. In the pendulum (a good one dimensional analogy to the shallow water equations) the presence of nonlinear drag alters the parameter space enough to induce or destroy chaotic behaviour. A phenomenon worth considering in relation the shallow water equations. Additionally, the presence of drag causes as a cascade in spectral space, similar to the classical turbulent cascade. This work considers this effect and seeks to differentiate it from the turbulent cascade wherever and whenever possible. The final section of the thesis deals with the presence and effects of bottom topography (namely protrusions from the lake bed) on wave velocities due to a basin-scale seiche. This section examines both the dynamics of the system, through deflection about topography and the modification of the wave due to nonlinearity and bathymetry, as well as the relationship between lake dynamics and sediment redistribution. Finally, possible future directions are suggested as natural extensions to the work already done, as well as more sophisticated numerical models which could provide further insight into the problems discussed herein.

This paper analyzes the dynamics of non-Newtonian fluids, those whose viscosity is not constant. First, the Navier-Stokes equations are modified by introducing a new parameter with units of viscosity. Then, the energy equation and micro-scale of the model are derived. This allows the value of the parameter to be determined in order to make the micro-scale the order of the mesh width. Finally, the Finite Element Method with Backward Euler discretization is programmed using FreeFEM++ to simulate the model; a problem with known exact solution is used to test convergence of the method, and the step problem is also discussed.

We have used a combination of theory and computation to investigate collective aspects of discrete mechanical systems. The analysis involves considerations from geometry, elasticity and hydrodynamics. We have developed continuum theories to describe these systems, in the spirit of compressing information by mathematical abstraction from the discrete description. / Engineering and Applied Sciences

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