The study of fluvial geomorphology is one of the critical sciences in the 21st Century. The previous century witnessed a virtual disregard of the hydro and morphodynamic processes occurring in rivers when it came to design of transportation, flood control, and water resources infrastructure. This disregard, along with urbanization, industrialization, and other land uses has imperiled many waterways. New technologies including geospatially referenced data collection, laser-based measurement tools, and increasing computational powers by personal computers are significantly improving our ability to represent these complex and diverse systems. We can accomplish this through both the building of more sophisticated models and our ability to calibrate those models with more detailed data sets. The effort put forth in this dissertation is to first introduce the accomplishments and challenges in fluvial geomorphology and then to illustrate two specific efforts to add to the growing body of knowledge in this exciting field.First, we explore a dramatic phenomenon occurring in the Middle Rio Grande River. The San Marcial Reach of the Rio Grande River has experienced four events that completely filled the main channel with sediment over the past 20 years. This sediment plug has cost the nation millions of dollars in both costs to dredge and rebuild main channels and levees, along with detailed studies by engineering consultants. Previous efforts focused on empirical relations developed with historical data and very simple one dimensional representation of river hydrodynamics. This effort uses the state-of-the-art three-dimensional hydro and morphodynamic model Delft3D. We were able to use this model8to test those hypotheses put forth in previous empirical studies. We were also able to use this model to test theories associated with channel avulsion. Testing found that channel avulsions thresholds do exist and can be predicted based on channel bathymetric changes.The second effort included is a simple yet sophisticated model of river meander evolution. Prediction of river meandering planform evolution has proven to be one of the most difficult problems in all of geosciences. The limitations of using detailed three dimensional hydro and morphodynamic models is that the computational intensity precludes the modeling of large spatial or temporal scale phenomenon. Therefore, analytical solutions to the standard Navier-Stokes equations with simplifications made for hydrostatic pressure among others, along with sediment transport functions still have a place in our toolbox to understand and predict this phenomenon. One of the most widely used models of meander propagation is the Linear Bend Model that employs a bank erosion coefficient. Due to the various simplifications required to find analytical solutions to these sets of equations, efforts to build the stochasticity seen in nature into the models have proven useful and successful. This effort builds upon this commonly used meander propogation model by introducing stochasticity to the known variability in outer bank erodibility, resulting in a more realistic representation of model results.

This thesis describes a stochastic ship routing problem with inventory management. The problem involves finding a set of least costs routes for a fleet of ships transporting a single commodity when the demand for the commodity is uncertain. Storage at consumption and supply ports is limited and inventory levels are monitored in the model. Consumer demands are at a constant rate within each time period in the deterministic problem, and in the stochastic problem, the demand rate for a period is not known until the beginning of that period. The demand situation in each time period can be described by a scenario tree with corresponding probabilities. Several possible solution approaches for solving the problem are studied in the thesis. This problem can be formulated as a mixed integer programming (MIP) model. However solving the problem this way is very time consuming even for a deterministic problem with small problem size. In order to solve the stochastic problem, we develop a decomposition formulation and solve it using a Branch and Price framework. A master problem (set partitioning with extra inventory constraints) is built, and the subproblems, one for each ship, involve solving stochastic dynamic programming problems to generate columns for the master problem. Each column corresponds to one possible tree of schedules for one ship giving the schedule for the ship for all demand scenarios. In each branch-and-bound node, the node problem is solved by iterating between the master problem and the subproblems. Dual variables can be obtained solving the master problem and are used in the subproblems to generate the most promising columns for the master problem. Computational results are given showing that medium sized problems can be solved successfully. Several extensions to the original model are developed, including a variable speed model, a diverting model, and a model which allows ships to do extra tasks in return for a bonus. Possible solution approaches for solving the variable speed and the diverting model are presented and computational results are given.

The Black-Scholes model and corresponding option pricing formula has led to a wide and extensive industry, used by financial institutions and investors to speculate on market trends or to control their level of risk from other investments. From the formation of the Chicago Board Options Exchange in 1973, the nature of options contracts available today has grown dramatically from the single-date contracts considered by Black and Scholes (1973) to a wider and more exotic range of derivatives. These include American options, which can be exercised at any time up to maturity, as well as options based on the weighted sums of assets, such as the Asian and basket options which we consider. Moreover, the underlying models considered have also grown in number and in this work we are primarily motivated by the increasing interest in past-dependent asset pricing models, shown in recent years by market practitioners and prominent authors. These models provide a natural framework that considers past history and behaviour, as well as present information, in the determination of the future evolution of an underlying process. In our studies, we explore option pricing techniques for arithmetic Asian and basket options under a Stochastic Delay Differential Equation (SDDE) approach. We obtain explicit closed-form expressions for a number of lower and upper bounds before giving a practical, numerical analysis of our result. In addition, we also consider the properties of the approximate numerical integration methods used and state the conditions for which numerical stability and convergence can be achieved.

In the last decades, the explosion of data from quantitative techniques has revolutionised our understanding of biological processes. In this scenario, advanced statistical methods and algorithms are becoming fundamental to decipher the dynamics of biochemical mechanisms such those involved in the regulation of gene expression. Here we develop mechanistic models and approximate inference techniques to reverse engineer the dynamics of gene regulation, from mRNA and/or protein time series data. We start from an existent variational framework for statistical inference in transcriptional networks. The framework is based on a continuous-time description of the mRNA dynamics in terms of stochastic differential equations, which are governed by latent switching variables representing the on/off activity of regulating transcription factors. The main contributions of this work are the following. We speeded-up the variational inference algorithm by developing a method to compute a posterior approximate distribution over the latent variables using a constrained optimisation algorithm. In addition to computational benefits, this method enabled the extension to statistical inference in networks with a combinatorial model of regulation. A limitation of this framework is the fact that inference is possible only in transcriptional networks with a single-layer architecture (where a single or couples of transcription factors regulate directly an arbitrary number of target genes). The second main contribution in this work is the extension of the inference framework to hierarchical structures, such as feed-forward loop. In the last contribution we define a general structure for transcription-translation networks. This work is important since it provides a general statistical framework to model complex dynamics in gene regulatory networks. The framework is modular and scalable to realistically large systems with general architecture, thus representing a valuable alternative to traditional differential equation models. All models are embedded in a Bayesian framework; inference is performed using a variational approach and compared to exact inference where possible. We apply the models to the study of different biological systems, from the metabolism in E. coli to the circadian clock in the picoalga O. tauri.

In this dissertation we obtain an e cient hybrid numerical method for the solution of stochastic di erential equations (SDEs). Speci cally, our method chooses between two numerical methods (Euler and Milstein) over a particular discretization interval depending on the value of the simulated Brownian increment driving the stochastic process. This is thus a new1 adaptive method in the numerical analysis of stochastic di erential equation. Mauthner (1998) and Hofmann et al (2000) have developed a general framework for adaptive schemes for the numerical solution to SDEs, [30, 21]. The former presents a Runge-Kutta-type method based on stepsize control while the latter considered a one-step adaptive scheme where the method is also adapted based on step size control. Lamba, Mattingly and Stuart, [28] considered an adaptive Euler scheme based on controlling the drift component of the time-step method. Here we seek to develop a hybrid algorithm that switches between euler and milstein schemes at each time step over the entire discretization interval, depending on the outcome of the simulated Brownian motion increment. The bias of the hybrid scheme as well as its order of convergence is studied. We also do a comparative analysis of the performance of the hybrid scheme relative to the basic numerical schemes of Euler and Milstein. / Mathematical Sciences / M.Sc. (Applied Mathematics)

We develop and study the high-density limit of various new models in mathematical pop- ulation genetics. These models extend the Λ-Fleming–Viot process when there are two genetic types at the locus of study. Given a finite sample from a population undergoing these dynamics, a key tool for understanding the corresponding genealogy is the method of duality. We introduce the reproduction-linked mutation mechanism and consider how this affects the process of relative allelic frequencies and the genealogy. The second generalization incorporates two forms of natural selection – differential killing and differential birth. We contrast the structure of their genealogies. Several properties of the block size spectra of the Kingman and Beta coalescents are also investigated, including their behaviour as they come down from infinity.

We propose a new approach to optimize operations of hydro storage systems with multiple connected reservoirs whose operators participate in wholesale electricity markets. Our formulation integrates short-term intraday with long-term interday decisions. The intraday problem considers bidding decisions as well as storage operation during the day and is formulated as a stochastic program. The interday problem is modeled as a Markov decision process of managing storage operation over time, for which we propose integrating stochastic dual dynamic programming with approximate dynamic programming. We show that the approximate solution converges towards an upper bound of the optimal solution. To demonstrate the efficiency of the solution approach, we fit an econometric model to actual price and in inflow data and apply the approach to a case study of an existing hydro storage system. Our results indicate that the approach is tractable for a real-world application and that the gap between theoretical upper and a simulated lower bound decreases sufficiently fast. (authors' abstract)

In this thesis, we develop multiscale models for particle simulations in population dynamics. These models are characterised by prescribing particle motion on two spatial scales: microscopic and macroscopic.At the microscopic level, each particle has its own mass, position and velocity, while at the macroscopic level the particles are interpolated to a continuum quantity whose evolution is governed by a system of transport equations.This way, one can prescribe various types of interactions on a global scale, whilst still maintaining high simulation speed for a large number of particles. In addition, the interplay between particle motion and interaction is well tuned in both regions of low and high densities. We analyse links between models on these two scales and prove that under certain conditions, a system of interacting particles converges to a nonlinear coupled system of transport equations.We use this as a motivation to derive a model defined on both modelling scales and prescribe the intercommunication between them. Simulation takes place in inhomogeneous domains with arbitrary conditions at inflow and outflow boundaries. We realise this by modelling obstacles, sources and sinks.Integrating these aspects into the simulation requires a route planning algorithm for the particles. Several algorithms are considered and evaluated on accuracy, robustness and efficiency. All aspects mentioned above are combined in a novel open source prototyping simulation framework called Mercurial. This computational framework allows the design of geometries and is built for high performance when large numbers of particles are involved. Mercurial supports various types of inhomogeneities and global systems of equations. We apply our framework to simulate scenarios in crowd dynamics.We compare our results with test cases from literature to assess the quality of the simulations. / <p>Master Thesis in Industrial and Applied Mathematics</p>

This thesis consists of two parts. The first part (Chapters 2-4) focuses on the problem of inverting the signature of a path of bounded variation, and we present three results here. First, we give an explicit inversion formula for any axis path in terms of its signature. Second, we show that for relatively smooth paths, the derivative at the end point can be approximated arbitrarily closely by its signature sequence, and we provide explicit error estimates. As an application, we give an effective inversion procedure for piecewise linear paths. Finally, we prove a uniform estimate for the signatures of paths of bounded variations, and obtain a reconstruction theorem via that uniform estimate. Although this general reconstruction theorem is not computationally efficient, the techniques involved in deriving the uniform estimate are useful in other situations, and we also give an application in the case of expected signatures for Brownian motion. The second part (Chapter 5) deals with rough paths. After introducing proper backgrounds, we extend the uniform estimate above to the context of rough paths, and show how it can lead to simple proofs of distance bounds for Gaussian iterated integrals.

Binocular rivalry is an interesting phenomenon where perception oscillates between different images presented to the two eyes. This thesis is primarily concerned with modelling travelling waves of visual perception during transitions between these perceptual states. In order to model this effect in such a way that we retain as much analytical insight into the mechanisms as possible we employed neural field theory. That is, rather than modelling individual neurons in a neural network we treat the cortical surface as a continuous medium and establish integro-differential equations for the activity of a neural population. Our basic model which has been used by many previous authors both within and outside of neural field theory is to consider a one dimensional network of neurons for each eye. It is assumed that each network responds maximally to a particular feature of the underlying image, such as orientation. Recurrent connections within each network are taken to be excitatory and connections between the networks are taken to be inhibitory. In order for such a topology to exhibit the oscillations found in binocular rivalry there needs to be some form of slow adaptation which weakens the cross-connections under continued firing. By first considering a deterministic version of this model, we will show that, in fact, this slow adaptation also serves as a necessary "symmetry breaking" mechanism. Using this knowledge to make some mild assumptions we are then able to derive an expression for the shape of a travelling wave and its wave speed. We then go on to show that these predictions of our model are consistent not only with numerical simulations but also experimental evidence. It will turn out that it is not acceptable to completely ignore noise as it is a fundamental part of the underlying biology. Since methods for analyzing stochastic neural fields did not exist before our work, we first adapt methods originally intended for reaction-diffusion PDE systems to a stochastic version of a simple neural field equation. By regarding the motion of a stochastic travelling wave as being made up of two distinct components, firstly, the drift-diffusion of its overall position, secondly, fast fluctuations in its shape around some average front shape, we are able to derive a stochastic differential equation for the front position with respect to time. It is found that the front position undergoes a drift-diffusion process with constant coefficients. We then go on to show that our analysis agrees with numerical simulation. The original problem of stochastic binocular rivalry is then re-visited with this new toolkit and we are able to predict that the first passage time of a perceptual wave hitting a fixed barrier should be an inverse Gaussian distribution, a result which could potentially be experimentally tested. We also consider the implications of our stochastic work on different types of neural field equation to those used for modelling binocular rivalry. In particular, for neural fields which support pulled fronts propagating into an unstable state, the stochastic version of such an equation has wave fronts which undergo subdiffusive motion as opposed to the standard diffusion in the binocular rivalry case.