Master of Science - Science / Relationships between symmetries and conservation laws of perturbed partial differential equations are reviewed. Potential symmetries and their applications to perturbed partial differential equations and conservation laws are presented in detail. An example of a perturbed wave equation for an inhomogeneous medium is solved in detail. Proofs of some of the lesser-known theorems are outlined. A wide range of examples is given to further explain these concepts.

The solution of initial value problems is a fundamental component of many scientific simulations of physical phenomena. In many cases these initial value problems arise from a method of lines approach to solving partial differential equations, resulting in very large systems of equations that require the use of numerical time integration methods to solve. Many problems of scientific interest exhibit stiff behavior for which implicit methods are favorable, however standard implicit methods are computationally expensive. They require the solution of one or more large nonlinear systems at each timestep, which can be impractical to solve exactly and can behave poorly when solved approximately. The recently introduced ``lightly-implicit'' K-methods seek to avoid this issue by directly coupling the time integration methods with a Krylov based approximation of linear system solutions, treating a portion of the problem implicitly and the remainder explicitly. This work seeks to further two primary objectives: evaluation of these K-methods in large-scale parallel applications, and development of new linearly implicit methods for contexts where improvements can be made. To this end, Rosenbrock-Krylov methods, the first K-methods, are examined in a scalability study, and two new families of time integration methods are introduced: biorthogonal Rosenbrock-Krylov methods, and linearly implicit multistep methods. For the scalability evaluation of Rosenbrock-Krylov methods, two parallel contexts are considered: a GPU accelerated model and a distributed MPI parallel model. In both cases, the most significant performance bottleneck is the need for many vector dot products, which require costly parallel reduce operations. Biorthogonal Rosenbrock-Krylov methods are an extension of the original Rosenbrock-Krylov methods which replace the Arnoldi iteration used to produce the Krylov approximation with Lanczos biorthogonalization, which requires fewer vector dot products, leading to lower overall cost for stiff problems. Linearly implicit multistep methods are a new family of implicit multistep methods that require only a single linear solve per timestep; the family includes W- and K-method variants, which admit arbitrary or Krylov based approximations of the problem Jacobian while maintaining the order of accuracy. This property allows for a wide range of implementation optimizations. Finally, all the new methods proposed herein are implemented efficiently in the MATLODE package, a Matlab ODE solver and sensitivity analysis toolbox, to make them available to the community at large. / Doctor of Philosophy / Differential equations are a fundamental building block of the mathematical description of many physical phenomena. Thus, solving problems involving complex differential equations is necessary for construction of scientific models of these phenomena, which can then be used to make useful predictions, such as weather forecasts. Aside from some simplified cases, complex differential equations cannot be solved exactly. Time integration methods are a class of numerical algorithms used to compute approximate solutions to differential equations, by stepping a given initial solution forward in time, producing a new solution at each timestep. Time integration methods are generally categorized as either explicit or implicit methods. Explicit methods are simpler, but have significant restrictions on the size of timesteps for challenging differential equations. Implicit methods relax this timestep restriction, but are much more expensive to compute. The recently introduced ``lightly-implicit'' K-methods provide a way to fuse the advantages of both implicit and explicit methods, by effectively treating a portion of the problem implicitly and the remainder explicitly. This work seeks to further two primary objectives: evaluation of these K-methods on very large problems, and development of new time integration methods. To this end, Rosenbrock--Krylov methods, the first K-methods, are applied to a large-scale problem and examined in a scalability study, and two new families of time integration methods are introduced: biorthogonal Rosenbrock--Krylov methods, and linearly implicit multistep methods. Ultimately, the goal is to develop new methods which allow for the creation of larger, more detailed, and more accurate scientific models, in order to get better and faster predictions.

A fundamental result in the theory of elliptic PDEs shows that the hessian of solutions of uniformly elliptic PDEs belong to the Sobolev space ��^2,ε. New results show that for the right choice of c, the optimal hessain integrability exponent ε* is given by ε* = ������ ����(1−������) / ����(1−��), �� ∈ (0,1) Through the techniques of asymptotic analysis, the behavior and properties of this function are better understood to establish improved quantitative estimates for the optimal integrability exponent in the ��^2,ε-regularity theory.

Partial differential equations (PDEs) are used to describe multi-dimensional physical phenomena. However, some of these phenomena are described by a more general class of systems called fractional systems. Indeed, fractional calculus has emerged as a new tool for modeling complex phenomena thanks to the memory and hereditary properties of fraction derivatives. In this thesis, we explore a class of controllers and estimators that respond to some control and estimation challenges for both PDE and FPDE. We first propose a backstepping controller for the flow control of a first-order hyperbolic PDE modeling the heat transfer in parabolic solar collectors. While backstepping is a well-established method for boundary controlled PDEs, the process is less straightforward for in-domain controllers. One of the main contributions of this thesis is the development of a new integral transformation-based control algorithm for the study of reference tracking problems and observer designs for fractional PDEs using the extended backstepping approach. The main challenge consists of the proof of stability of the fractional target system, which utilizes either an alternative Lyapunov method for time FPDE or a fundamental solution for the error system for reference tracking, and observer design of space FPDE. Examples of applications involving reference tracking of FPDEs are gas production in fractured media and solute transport in porous media. The designed controllers, require knowledge of some system’s parameters or the state. However, these quantities may be not measurable, especially, for space-evolving PDEs. Therefore, we propose a non-asymptotic and robust estimation algorithm based on the so-called modulating functions. Unlike the observers-based methods, the proposed algorithm has the advantage that it converges in a finite time. This algorithm is extended for the state estimation of linear and non-linear PDEs with general non-linearity. This algorithm is also used for the estimation of parameters and disturbances for FPDEs. This thesis aims to design an integral transformation-based algorithm for the control and estimation of PDEs and FDEs. This transformation is defined through a suitably designed function that transforms the identification problem into an algebraic system for non-asymptotic estimation purposes. It also maps unstable systems to stable systems to achieve control goals.

In recent years, technological advances in animal tracking have renewed interests in collective animal behavior, and in particular, locust swarms. These swarms pose a major threat to agriculture in northern Africa, the Middle East, and other regions. In their early life stages, locusts move in hopper bands, which are huge aggregations traveling on the ground. Our main goal is to understand the underlying mechanisms for the emergence and organization of these bands. We construct an agent-based model that tracks individual locusts and a continuum model that tracks the evolution of locust density. Both these models are motivated by experimental observations of individuals’ behavior. The macroscopic emergent behavior of the group is studied through numerical simulation of these models.

Dermal wound healing involves a myriad of highly regulated and sophisticated mechanisms, which are coordinated and carried out via several specialized cell types. The dominant players involved in this process include platelets, neutrophils, macrophages and fibroblasts. These cells play a vital role in the repair of the wound by orchestrating tasks such as forming a fibrin clot to stanch blood flow, removing foreign organisms and cellular debris, depositing new collagen matrix and establishing the contractile forces which eventually bridge the void caused by the initial infraction.\\[5pt] \indent Our current understanding of these mechanisms has been primarily based upon animal models. Unfortunately, these models lack insight into pathologic conditions, which plague human beings, such as keloid scar or chronic ulcer formation. Consequently, investigators have proposed a number of {\it in vivo} techniques to study wound repair in humans in order to overcome this barrier. One approach, which has been devised to increase our level of understanding of these chronic conditions, involves the cutaneous placement of a small cylindrical structure within the appendage of a human test subject.\\[5pt] \indent Researches have designed a variety of these implantable structures to examine different aspects of wound healing in both healthy subjects and individuals that experience some trauma related condition. In each case, several implants are surgically positioned at multiple locations under sterile conditions. These structures are later removed at distinct time intervals at which point they are histologically analyzed and biochemically assayed to deduce the presence of biological markers involved in the repair process. Implantable structures used in this way are often referred to as Human Implantable Models or Systems.\\[5pt] \indent Clinical studies with implantable models open up tremendous opportunities in fields such as biomathematics because they provide an experimentally controlled setting that aids in the development and validation of mathematical models. Furthermore, experiments carried out with implants greatly simplify the mathematics required to describe the repair process because they minimize the modeling of complex features associated with healing such as wound geometry and the evolution of contractile forces.\\[5pt] \indent In this work, we present a notional mathematical model, which accounts for two fundamental processes involved in the repair of an acute dermal wound. These processes include the inflammatory response and fibroplasia. Our system describes each of these events through the time evolution of four primary species or variables. These include the density of initial damage, inflammatory cells, fibroblasts and deposition of new collagen matrix. Since it is difficult to populate the equations of our model with coefficients that have been empirically derived, we fit these constants by carrying out a large number of simulations until there is reasonable agreement between the time response of the variables of our system and those reported by the literature for normal healing. Once a suitable choice of parameters has been made, we then compare simulation results with data obtained from clinical investigations. While more data is desired, we have a promising first step toward describing the primary events of wound repair within the confines of an implantable system.

The market for mortgage backed securities (MBS) was active and fast growing from the issuance of the first MBS in 1981. This enabled financial firms to transform risky individual mortgages into liquid and tradable market instruments. The subprime mortgage crisis of 2007 shows the need for a better understanding and development of mathematical models for these securities. The aim of this thesis is to develop a model for MBS that is flexible enough to capture both regular and subprime MBS. The thesis considers two models, one for a single mortgage in an intensity based framework and the second for mortgage backed securities using a stochastic partial differential equation approach. In the model for a single mortgage, we capture the prepayment and default incentives of the borrower using intensity processes. Using the minimum of the two intensity processes, we develop a nonlinear equation for the mortgage rate and solve it numerically and present some case studies. In modelling of an MBS in a structural framework using stochastic PDEs (SPDEs), we consider a large number of individuals in a mortgage pool and assume that the wealth of each individual follows a stochastic process, driven by two Brownian mo- tions, one capturing the idiosyncratic noise of each individual and the second a common market factor. By defining the empirical measure of a large pool of these individuals we study the evolution of the limit empirical measure and derive an SPDE for the evolution of the density of the limit empirical measure. We numerically solve the SPDE to demonstrate its flexibility in different market environments. The calibration of the model to financial data is the focus of the final part of thesis. We discuss the different parameters and demonstrate how many can be fitted to observed data. Finally, for the key model parameters, we present a strategy to estimate them given observations of the loss function and use this to determine implied model parameters of ABX.HE.

We consider the application of multilevel Monte Carlo methods to elliptic partial differential equations with random coefficients. Such equations arise, for example, in stochastic groundwater ow modelling. Models for random coefficients frequently used in these applications, such as log-normal random fields with exponential covariance, lack uniform coercivity and boundedness with respect to the random parameter and have only limited spatial regularity. To give a rigorous bound on the cost of the multilevel Monte Carlo estimator to reach a desired accuracy, one needs to quantify the bias of the estimator. The bias, in this case, is the spatial discretisation error in the numerical solution of the partial differential equation. This thesis is concerned with establishing bounds on this discretisation error in the practically relevant and technically demanding case of coefficients which are not uniformly coercive or bounded with respect to the random parameter. Under mild assumptions on the regularity of the coefficient, we establish new results on the regularity of the solution for a variety of model problems. The most general case is that of a coefficient which is piecewise Hölder continuous with respect to a random partitioning of the domain. The established regularity of the solution is then combined with tools from classical discretisation error analysis to provide a full convergence analysis of the bias of the multilevel estimator for finite element and finite volume spatial discretisations. Our analysis covers as quantities of interest several spatial norms of the solution, as well as point evaluations of the solution and its gradient and any continuously Fréchet differentiable functional. Lastly, we extend the idea of multilevel Monte Carlo estimators to the framework of Markov chain Monte Carlo simulations. We develop a new multilevel version of a Metropolis Hastings algorithm, and provide a full convergence analysis.

Asymptotic properties of solutions of the linear dispersion equation ut = uxxx in R × R+, and its (2k + 1)th-order generalisations are studied. General Hermitian spectral theory and asymptotic behaviour of its kernel, for the rescaled operator B = D3 + 1 3 yDy + 1 3 I, is developed, where a complete set of bi-orthonormal pair of eigenfunctions, {ψβ}, {ψ∗β }, are found. The results apply to the construction of VSS (very singular solutions) of the semilinear equation with absorption ut = uxxx − |u|p−1u in R × R+, where p > 1, which serves as a basic model for various applications, including the classic KdV area. Finally, the nonlinear dispersion equations such as ut = (|u|nu)xxx in R × R+, and ut = (|u|nu)xxx − |u|p−1u in R × R+, where n > 0, are studied and their “nonlinear eigenfunctions” are constructed. The basic tools include numerical methods and “homotopy-deformation” approaches, where the limits n → 0 and n → +∞ turn out to be fruitful. Local existence and uniqueness is proved and some bounds on the highly oscillatory tail are found. These odd-order models were not treated in existing mathematical literature, from the proposed point of view.

Cette thèse a pour but d'étudier quelques applications des fonctions à variation bornée et des ensembles de périmètre fini. Nous nous intéressons en particulier à des applications en traitement d'images et en géométrie de dimension finie et infinie. Nous étudions tout d'abord une méthode dite Primale-Duale proposée par Appleton et Talbot pour la résolution de nombreux problèmes en traitement d'images. Nous réinterprétons cette méthode sous un oeil nouveau, ce qui aide à mieux la comprendre mathématiquement. Ceci permet par exemple de démontrer sa convergence et d'établir de nouvelles estimations a posteriori qui sont d'une grande importance pratique. Nous considérons ensuite le problème de courbure moyenne prescrite en milieu périodique. A l'aide de la théorie des ensembles de périmètre fini, nous démontrons l'existence de solutions approchées compactes de ce problème. Nous étudions également le comportement asymptotique de ces solutions lorsque leur volume tend vers l'infini. Les deux dernières parties de la thèse sont consacrées à l'étude de problèmes géométriques dans les espaces de Wiener. Nous étudions d'une part les liens entre symétrisations, semi-continuité et inégalités isopérimétriques ce qui permet d'obtenir un résultat d'approximation et de relaxation pour le périmètre dans ces espaces de dimension infinie. Nous démontrons d'autre part la convexité des solutions de certains problèmes variationnels dans ces espaces, en développant au passage l'étude de la semi-continuité et de la relaxation dans ce contexte.