Les travaux exposés dans cette thèse sont consacrés à l’étude de méthodesprécises pour approcher des équations différentielles stochastiques ou deséquations aux dérivées partielles (EDP) déterministes. La première parties’inscrit dans le cadre du développement de méthodes visant à corriger le biaisdans les processus de diffusion paramétrique. Trois modèles sont étudiés enparticulier : Ornstein-Uhlenbeck, Auto-régressif et Moyenne mobile. A l’issuede ce travail, plusieurs approximations de biais ont été proposées suivant deuxapproches : la première consiste en un développement de Taylor del’estimateur obtenu alors que la seconde s'appuie sur une expansionstochastique de celui-ci.La deuxième partie de cette thèse porte sur l’approximation de l’équation de lachaleur obtenue après changement de variables à partir du modèle de Black etScholes. En général, on préfère utiliser des méthodes implicites pour résoudredes EDP paraboliques mais depuis quelques années, les méthodes dites deRunge-Kutta explicites stabilisées, sont de plus en plus utilisées. Nousmontrons que l’utilisation de ce type de méthodes explicites et notamment lesschémas ROCK donnent de très bons résultats même si les conditions initialessont peu régulières, ce qui est le cas dans les modèles financiers / The work presented in this thesis is devoted to the study of precise methods forapproximating stochastic differential equations (SDE) or deterministic partialdifferential equations (PDE). The first part is devoted to the development ofbias correction methods in parametric diffusion processes. Three models arestudied in particular : Ornstein-Uhlenbeck, auto-regressive and Movingaverage. At the end of this work, several approximations of bias have beenproposed following two approaches : the first consists in a Taylor developmentof the obtained estimator while the second one relies on a stochastic expansionof the latter.The second part of this thesis deals with the approximation of the heatequation obtained after changing variables from the Black-Scholes model. Likethe vast majority of PDE, this equation does not have an exact solution, sosolutions must be approached using explicit or implicit time schemes. Itis often customary to prefer the use of implicit methods to solve parabolic PDEsuch as the heat equation, but in the past few years, the stabilized explicitRunge-Kutta methods which have the largest possible domains of stabilityalong the negative real axis, are increasingly used. We show that the useof this type of explicit methods and in particular the ROCK (Runge-Orthogonal-Chebyshev-Kutta) schemes give very good results even if the initial conditionsare not very regular, which is the case in the financial models

Les réseaux de lois de bilan sont définis par l'interconnexion, via des conditions aux bords, de modules élémentaires individuellement caractérisés par la conservation de certaines quantités. Des applications industrielles se trouvent dans les réseaux de lignes de transmission électriques (réseaux HVDC), hydrauliques et pneumatiques (réseaux de distribution du gaz, de l'eau et du fuel). La thèse se concentre sur l'analyse modulaire et la commande au bord d'une ligne élémentaire représentée par un système de lois de bilan en dimension infinie, où la dynamique de la ligne est prise en considération au moyen d'équations aux dérivées partielles hyperboliques linéaires du premier ordre et couplées deux à deux. Cette dynamique permet de modéliser d'une manière rigoureuse les phénomènes de transport et les vitesses finies de propagation, aspects normalement négligés dans le régime transitoire. Les développements de ces travaux sont des outils d'analyse qui testent la stabilité du système, et de commande au bord pour la stabilisation autour d'un point d'équilibre. Dans la partie analyse, nous considérons un système de lois de bilan avec des couplages statiques aux bords et anti-diagonaux à l’intérieur du domaine. Nous proposons des conditions suffisantes de stabilité, tant explicites en termes des coefficients du système, que numériques par la construction d'un algorithme. La méthode se base sur la reformulation du problème en une analyse, dans le domaine fréquentiel, d'un système à retard obtenu en appliquant une transformation backstepping au système de départ. Dans le travail de stabilisation, un couplage avec des dynamiques décrites par des équations différentielles ordinaires (EDO) aux deux bords des EDP est considéré. Nous développons une transformation backstepping (bornée et inversible) et une loi de commande qui, à la fois stabilise les EDP à l'intérieur du domaine et la dynamique des EDO, et élimine les couplages qui peuvent potentiellement mener à l’instabilité. L'efficacité de la loi de commande est illustrée par une simulation numérique. / Networks of balance laws are defined by the interconnection, via boundary conditions, of elementary modules individually characterized by the conservation of physical quantities. Industrial applications of such networks can be found in electric (HVDC networks), hydraulic and pneumatic (gas, water and oil distribution) transmission lines. The thesis is focused on modular analysis and boundary control of an elementary line represented by a system of balance laws in infinite dimension, where the dynamics of the line is taken into consideration by means of first order two by two coupled linear hyperbolic partial differential equations. This representation allows to rigorously model the transport phenomena and finite propagation speed, aspects usually neglected in transient regime. The developments of this work are analysis tools that test the stability, as well as boundary control for the stabilization around an equilibrium point. In the analysis section, we consider a system of balance laws with static boundary conditions and anti-diagonal in-domain couplings. We propose sufficient stability conditions, explicit in terms of the system coefficients, and numerical by constructing an algorithm. The method is based on reformulating the analysis problem as an analysis of a delay system in the frequency domain, obtained by applying a backstepping transform to the original system. In the stabilization work, couplings with dynamic boundary conditions, described by ordinary differential equations (ODE), at both boundaries of the PDEs are considered. We develop a backstepping (bounded and invertible) transform and a control law that at the same time, stabilizes the PDEs inside the domain and the ODE dynamics, and eliminates the couplings that are a potential source of instability. The effectiveness of the control law is illustrated by a numerical simulation.

This thesis deals with the concepts of numerical integrator using floating point arithmetic for solving partial differential equations. The integrator uses Euler method and Taylor series. Thesis shows parallel and serial approach to computing with exponents and significands. There is also a comparison between modern parallel systems and the proposed concepts.

Cette thèse porte sur l’étude de quelques questions liées à l’identifiabilité et l’identification d’un problème inverse non-linéaire de source. Il s’agit de l’identification d’une source ponctuelle dépendante du temps constituant le second membre d’une équation de type advection-dispersion-réaction à coefficients variables. Dans le cas monodimensionnel, la souplesse du modèle stationnaire nous a permis de développer des réponses théoriques concernant le nombre des capteurs nécessaires et leurs emplacements permettant d’identifier la source recherchée d’une façon unique. Ces résultats nous ont beaucoup aidés à définir la ligne de conduite à suivre afin d’apporter des réponses similaires pour le modèle transitoire. Quant au modèle bidimensionnel transitoire, en utilisant quelques résultats de nulle contrôlabilité frontière et des mesures de l’état sur la frontière sortie et de son flux sur la frontière entrée du domaine étudié, nous avons établi un théorème d’identifiabilité et une méthode d’identification permettant de localiser les deux coordonnées de la position de la source recherchée comme étant l’unique solution d’un système non-linéaire de deux équations, et de transformer l’identification de sa fonction de débit en la résolution d’un problème de déconvolution. La dernière partie de cette thèse discute la difficulté principale rencontrée dans ce genre de problèmes inverses à savoir la non identifiabilité d’une source dans sa forme abstraite, propose une alternative permettant de surmonter cette difficulté dans le cas particulier où le but est d’identifier le temps limite à partir duquel la source impliquée a cessé d’émettre, et donc ouvre la porte sur de nouveaux horizons. / The thesis deals with the two main issues identifiability and identification related to a nonlinear inverse source problem. This problem consists in the identification of a time-dependent point source occurring in the right hand-side of an advection-dispersion-reaction equation with spatially varying coefficients. Starting from the stationnary case in the one-dimensional model, we derived theoritical results defining the necessary number of sensors and their positions that enable to uniquely determine the sought source. Those results gave us a good visibility on how to proceed in order to obtain similar results for the time-dependent (evolution) case. As far as the two-dimensional evolution model is concerned, using some boundary null controllability results and the records of the generated state on the inflow boundary and its flux on the outflow boundary of the monitored domain, we established a constructive identifiability theorem as well as an identification method that localizes the two coordinates of the sought source position as the unique solution of a nonlinear system of two equations and transforms the identification of its time-dependent intensity function into solving a deconvolution problem. The last part of this thesis highlights the main difficulty encountred in such inverse problems namely the nonidentifiabilityof a source in its abstract form, proposes a method that enables to overcome this difficulty in the particular case where the aim is to identify the time active limit of the involved source. And thus, this last part opens doors on new horizons and prospects.

This study seeks to develop a method that generalises the use of Adams-Bashforth to solve or treat partial differential equations with local and non-local differentiation by deriving a two-step Adams-Bashforth numerical scheme in Laplace space. The resulting solution is then transformed back into the real space by using the inverse Laplace transform. This is a powerful numerical algorithm for fractional order derivative. The error analysis for the method is studied and presented. The numerical simulations of the method as applied to the four-dimensional model, Caputo-Lu-Chen model and the wave equation are presented. In the analysis, the bifurcation dynamics are discussed and the periodic doubling processes that eventually caused chaotic behaviour (butterfly attractor) are shown. The related graphical simulations that show the existence of fractal structure that is characterised by chaos and usually called strange attractors are provided. For the Caputo-Lu-Chen model, graphical simulations have been realised in both integer and fractional derivative orders. / Mathematical Sciences / M. Sc. (Applied Mathematics)

<div>Accurate, fast, large-scale, and circuit-intuitive electromagnetic analysis is of critical importance to the design of integrated circuits (IC) and systems. Existing methods for the analysis of integrated circuits and systems have not satisfactorily achieved these performance goals. In this work, rapid modeling and simulation methods are developed for large-scale and circuit-intuitive electromagnetic analysis of integrated circuits and systems. The derived model is correct from zero to high frequencies where Maxwell's equations are valid. In addition, in the proposed model, we are able to analytically decompose the layout response into static and full-wave components with neither numerical computation nor approximation. This decomposed yet rigorous model greatly helps circuit diagnoses since now designers are able to analyze each component one by one, and identify which component is the root cause for the design failure. Such a decomposition also facilitates efficient layout modeling and simulation, since if an IC is dominated by RC effects, then we do not have to compute the full-wave component; and vice versa. Meanwhile, it makes parallelization straightforward. In addition, we develop fast algorithms to obtain each component of the inverse rapidly. These algorithms are also applicable for solving general partial differential equations for fast electromagnetic analysis.</div><div><br></div><div>The fast algorithms developed in this work are as follows. First, an analytical method is developed for finding the nullspace of the curl-curl operator in an arbitrary mesh for an arbitrary order of curl-conforming vector basis function. This method has been applied successfully to both a finite-difference and a finite-element based analysis of general 3-D structures. It can be used to obtain the static component of the inverse efficiently. An analytical method for finding the complementary space of the nullspace is also developed. Second, using the analytically found nullspace and its complementary space, a rigorous method is developed to overcome the low-frequency breakdown problem in the full-wave analysis of general lossy problems, where both dielectrics and conductors can be lossy and arbitrarily inhomogeneous. The method is equally valid at high frequencies without any need for changing the formulation. Third, with the static component part solved, the full-wave component is also ready to obtain. There are two ways. In the first way, the full-wave component is efficiently represented by a small number of high-frequency modes, and a fast method is created to find these modes. These modes constitute a significantly reduced order model of the complementary space of the nullspace. The second way is to utilize the relationship between the curl-curl matrix and the Laplacian matrix. An analytical method to decompose the curl-curl operator to a gradient-divergence operator and a Laplacian operator is developed. The derived Laplacian matrix is nothing but the curl-curl matrix's Laplacian counterpart. They share the same set of non-zero eigenvalues and eigenvectors. Therefore, this Laplacian matrix can be used to replace the original curl-curl matrix when operating on the full-wave component without any computational cost, and an iterative solution can converge this modified problem much faster irrespective of the matrix size. The proposed work has been applied to large-scale layout extraction and analysis. Its performance in accuracy, efficiency, and capacity has been demonstrated.</div>

This study proposes novel optimization methodologies for the optimization of problems that reveal non-physical step discontinuities. More specifically, it is proposed to use gradient-only techniques that do not use any zeroth order information at all for step discontinuous problems. A step discontinuous problem of note is the shape optimization problem in the presence of remeshing strategies, since changes in mesh topologies may - and normally do - introduce non-physical step discontinuities. These discontinuities may in turn manifest themselves as non-physical local minima in which optimization algorithms may become trapped. Conventional optimization approaches for step discontinuous problems include evolutionary strategies, and design of experiment (DoE) techniques. These conventional approaches typically rely on the exclusive use of zeroth order information to overcome the discontinuities, but are characterized by two important shortcomings: Firstly, the computational demands of zero order methods may be very high, since many function values are in general required. Secondly, the use of zero order information only does not necessarily guarantee that the algorithms will not terminate in highly unfit local minima. In contrast, the methodologies proposed herein use only first order information, rather than only zeroth order information. The motivation for this approach is that associated gradient information in the presence of remeshing remains accurately and uniquely computable, notwithstanding the presence of discontinuities. From a computational effort point of view, a gradient-only approach is of course comparable to conventional gradient based techniques. In addition, the step discontinuities do not manifest themselves as local minima. / Thesis (PhD)--University of Pretoria, 2010. / Mechanical and Aeronautical Engineering / unrestricted

No description available.

This work entails studying unfitted finite element discretizations for convection-diffusion equations in domains that evolve in time. In particular, these partial differential equations model the evolution of the concentration of soluble surfactants in bulk-interface domains. The work in this thesis docuses on developing numerical methods which conserve the modeled physical quantities. In this work, we propose cut finite element discretizations based on the Discontinuous Galerkin framework which are both locally and globally conservative. Local conservation is achieved on so-called macro elements, and we investigate macro element partitioning of the mesh for both stationary and time-dependent domains. Additionally, we develop globally conservative methods for time-dependent problems. We analyze the proposed methods by studying the convergence of the L2-error with respect to mesh size, condition numbers of the associated linear system matrices, and the conservation error. In numerical experiments for time-dependent problems, we show that the proposed methods have optimal convergence and that the developed macro element stabilization for time-dependent problems leads to increased accuracy while retaining stable condition numbers. Moreover, the measured conservation errors verify the global conservation of the proposed methods. / Detta arbete undersöker diskretiseringar av partiella differentialekvationer i tidsberoende domäner där beräkningsnätet inte behöver anpassas till domänens rörelse. I synnerhet betraktar vi partiella differentalekvationer som modellerar koncentrationen av lösliga ytaktiva ämnen, och skurna finita elementmetoder baserade på den Diskontinuerliga Galerkinmetoden som bevarar de modellerade fysikaliska storheterna. I detta arbete föreslås diskretiseringar som är både lokalt och globalt konservativa. Lokal konservering uppnås i så kallade makroelement, och vi undersöker makroelementpartitionering för både stationära och tidsberoende domäner. Även globalt konservativa metoder utvecklas för tidsberoende problem. De föreslagna metoderna analyseras med hjälp av numeriska exempel. Vi studerar konvergensen av L2-felet med avseende på nätstorlek, konditionstalen för de linjära systemmatriserna samt konserveringsfelet. Metoderna uppvisar optimal konvergens och makroelementstabilisering som utvecklas för tidsberoende problem leder till ökad noggrannhet, samtidigt som konditionstalen förblir stabila. Dessutom veritifierar de uppmättta konserveringsfelen den globala konserveringen hos de föreslagna metoderna.

[pt] Esquemas semi-Lagrangeanos usados para a aproximação do princípio da programação dinâmica são baseados em uma discretização temporal reconstruída no espaço de estado. O uso de uma malha estruturada torna essa abordagem inviável para problemas de alta dimensão devido à maldição da dimensionalidade. Nesta tese, apresentamos uma nova abordagem para problemas de controle ótimo de horizonte infinito onde a função valor é calculada usando Funções de Base Radial (RBFs) pelo método de aproximação de mínimos quadrados móveis de Shepard em malhas irregulares. Propomos um novo método para gerar uma malha irregular guiada pela dinâmica e uma rotina de otimizada para selecionar o parâmetro responsável pelo formato nas RBFs. Esta malha ajudará a localizar o problema e aproximar o princípio da programação dinâmica em alta dimensão. As estimativas de erro para a função valor também são fornecidas. Testes numéricos para problemas de alta dimensão mostrarão a eficácia do método proposto. Além do controle ótimo de EDPs clássicas mostramos como o método também pode ser aplicado ao controle de equações não-locais. Também fornecemos um exemplo analisando a convergência numérica de uma equação não-local controlada para o modelo contínuo. / [en] Semi-Lagrangian schemes for the approximation of the dynamic programming principle are based on a time discretization projected on a state-space grid. The use of a structured grid makes this approach not feasible for highdimensional problems due to the curse of dimensionality. In this thesis, we present a new approach for infinite horizon optimal control problems where the value function is computed using Radial Basis Functions (RBF) by the Shepard s moving least squares approximation method on scattered grids. We propose a new method to generate a scattered mesh driven by the dynamics and an optimal routine to select the shape parameter in the RBF. This mesh will help to localize the problem and approximate the dynamic programming principle in high dimension. Error estimates for the value function are also provided. Numerical tests for high dimensional problems will show the effectiveness of the proposed method. In addition to the optimal control of classical PDEs, we show how the method can also be applied to the control of nonlocal equations. We also provide an example analyzing the numerical convergence of a nonlocal controlled equation towards the continuous model.