Le couplage important entre les différentes échelles d’un écoulement est une des caractéristiques prin-cipales des turbulences. Cela est exprimé mathématiquement par les termes non linéaires présents dans les équations d’équilibre de l’écoulement, dominants en dynamique turbulente. En magnétohy-drodynamique (MHD), la force de Lorentz influe sur l’équation de conservation de l’impulsion et le nombre de termes non linéaires passe à quatre au lieu d’un seul pour un fluide non conducteur. L’objectif principal de cette thèse est d’analyser le transport d’énergie inter-échelles en utilisant une simulation numérique directe d’un écoulement turbulent MHD. Les propriétés de localité du transport de l’énergie entre les échelles pour un écoulement anisotropique ou isotropique, généré par la présence d’un champ magnétique constant, sont renforcées. Un objectif secondaire est d’établir un cadre de travail pour l’étude du transport de particules test chargées dans un champ électromagnétique turbu-lent, i.e. généré par le mouvement d’un fluide conducteur, qui possède des structures à plusieurs ordres de grandeur. La structure de la thèse est présentée ci-dessous. Dans la première partie, composée des deux premiers chapitres, l’auteur présente les notions de turbu-lences, aussi bien hydrodynamiques que MHD. Ces deux chapitres sont des synthèses. La deuxième partie est la principale source de nouveaux résultats. Le chapitre 3 présente les méthodes numériques pour la résolution des équations, les méthodes pseudo-spectrales. Un nouveau type de force est introduit, imposant un niveau de dissipation pour tous les invariants. Dans le chapitre 4, il est effectué une analyse du transfert d'énergie entre ordres de grandeur pour les turbulences MHD. Pour explorer ces transferts d'énergie, le domaine spectral est décomposé en une série de coques de même nombre d'onde. Le transfert moyen d'énergie entre ces coques est analysé. Les transferts d'énergie s'avèrent être surtout locaux en ordre de grandeur, alors qu'une contribution non locale existe due à la force. En présence d'un champ magnétique, l'écoulement développe une direction préférentielle, une anisotropie, où une idée nouvelle de décomposition de l'espace spectral en structures annulaires est présentée. Utilisant cette décomposition annulaire on trouve que le transfert entre anneaux est local, surtout dans les anneaux de direction perpendiculaire au champ magnétique. Pour les turbulences isotropiques, dans le chapitre 5, la localité des flux d'énergie est explorée par le biais de fonctions de localité. Dans le cas de la turbulence MHD, nous avons un comportement non local plus prononcé. La dernière partie, les chapitres 6 et 7, présente le formalisme de suivi des trajectoires de particules chargées évoluant dans un champ électromagnétique turbulent. L'influence de la méthode d'interpola-tion du solveur de particules est étudiée avant la présentation des concepts liés au transport de particu-les et aux régimes de diffusion. L'adiabatisme du mouvement des particules chargées est discuté et le transport de particules chargées dans un champ magnétique turbulent est montré en exemple.

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The effect of finite Reynolds numbers on chaotic advection is investigated for two dimensional lid-driven cavity flows that exhibit topological chaos in the creeping flow regime. The emphasis in this endeavor is to study how the inertial effects present due to small, but non-zero, Reynolds number influence the efficacy of mixing. A spectral method code based on the Fourier-Chebyshev method for two-dimensional flows is developed to solve the Navier-Stokes and species transport equations. The high sensitivity to initial conditions and the exponentional growth of errors in chaotic flows necessitate an accurate solution of the flow variables, which is provided by the exponentially convergent spectral methods. Using the spectral coefficients of the basis functions as solved through the conservation equations, exponentially accurate values of velocity everywhere in the flow domain are obtained as required for the Lagrangian particle tracking. Techniques such as Poincare maps, the stirring index based on the box counting method, and the tracking of passive scalars in the flow are used to analyze the topological chaos and quantify the mixing efficiency.

The linear receptivity and stability of plane idealized detonation with one-step Arrhenius type reaction kinetics is explored in the case of three-dimensional perturbations to a Zel'dovich-von Neumann-Doering base flow. This is explored in both overdriven and explicitly Chapman-Jouguet detonation. Additionally, the use of a multi-domain spectral collocation method for solving the conventional stability problem is explored within the context of normal-mode detonation. An extension of the stability analysis to confined detonations in a slightly porous walled tube is also carried out. Finally, an asymptotic analysis of a detonation with two-step reaction kinetics in the limit of large activation energy and for general overdrive and reaction order is performed yielding a nonlinear evolution equation for perturbations that produce stable limit cycle solutions.

As the availability of unstructured data on the web continues to increase, it is becoming increasingly necessary to develop machine learning methods that rely less on human annotated training data. In this thesis, we present methods for learning from weakly labeled data. We present a unifying framework to understand weakly labeled data in terms of bias and noise and identify methods that are well suited to learning from certain types of weak labels. To compensate for the tremendous sizes of weakly labeled datasets, we leverage computationally efficient and statistically consistent spectral methods. Using these methods, we present results from four diverse, real-world applications coupled with a unifying simulation environment. This allows us to make general observations that would not be apparent when examining any one application on its own. These contributions allow us to significantly improve prediction when labeled data is available, and they also make learning tractable when the cost of acquiring annotated data is prohibitively high.

Probabilistic modeling with latent variables is a powerful paradigm that has led to key advances in many applications such natural language processing, text mining, and computational biology. Unfortunately, while introducing latent variables substantially increases representation power, learning and modeling can become considerably more complicated. Most existing solutions largely ignore non-identifiability issues in modeling and formulate learning as a nonconvex optimization problem, where convergence to the optimal solution is not guaranteed due to local minima. In this thesis, we propose to tackle these problems through the lens of linear/multi-linear algebra. Viewing latent variable models from this perspective allows us to approach key problems such as structure learning and parameter learning using tools such as matrix/tensor decompositions, inversion, and additive metrics. These new tools enable us to develop novel solutions to learning in latent variable models with theoretical and practical advantages. For example, our spectral parameter learning methods for latent trees and junction trees are provably consistent, local-optima-free, and 1-2 orders of magnitude faster thanEMfor large sample sizes. In addition, we focus on applications in Natural Language Processing, using our insights to not only devise new algorithms, but also to propose new models. Our method for unsupervised parsing is the first algorithm that has both theoretical guarantees and is also practical, performing favorably to theCCMmethod of Klein and Manning. We also developed power low rank ensembles, a framework for language modeling that generalizes existing n-gram techniques to non-integer n. It consistently outperforms state-of-the-art Kneser Ney baselines and can train on billion-word datasets in a few hours.

Spectral methods have been widely used in various fields of engineering and applied mathematics. In the field of computer arithmetic: data compression, polynomial multiplication and the spectral integer multiplication of Sch&uml / onhage and Strassen are among the most important successful utilization. Recent advancements in technology report the spectral methods may also be beneficial for modular operations heavily used in public key cryptosystems. In this study, we evaluate the use of spectral methods in modular multiplication. We carefully compare their timing performances with respect to the full return algorithms. Based on our evaluation, we introduce new approaches for spectral modular multiplication for polynomials and exhibit standard reduction versions of the spectral modular multiplication algorithm for polynomials eliminating the overhead of Montgomery&rsquo / s method. Moreover, merging the bipartite method and standard approach, we introduce the bipartite spectral modular multiplication to improve the hardware performance of spectral modular multiplication for polynomials. Finally, we introduce Karatsuba combined bipartite method for polynomials and its spectral version.

In this work, we construct numerical methods to solve a wide range of problems in finance. This includes the valuation under affine jump diffusion processes, chaotic and hyperchaotic systems, and pricing fractional cryptocurrency models. These problems are of extreme importance in the area of finance. With today’s rapid economic growth one has to get a reliable method to solve chaotic problems which are found in economic systems while allowing synchronization. Moreover, the internet of things is changing the appearance of money. In the last decade, a new form of financial assets known as cryptocurrencies or cryptoassets have emerged. These assets rely on a decentralized distributed ledger called the blockchain where transactions are settled in real time. Their transparency and simplicity have attracted the main stream economy players, i.e, banks, financial institutions and governments to name these only. Therefore it is very important to propose new mathematical models that help to understand their dynamics. In this thesis we propose a model based on fractional differential equations. Modeling these problems in most cases leads to solving systems of nonlinear ordinary or fractional differential equations. These equations are known for their stiffness, i.e., very sensitive to initial conditions generating chaos and of multiple fractional order. For these reason we design numerical methods involving Chebyshev polynomials. The work is done from the frequency space rather than the physical space as most spectral methods do. The method is tested for valuing assets under jump diffusion processes, chaotic and hyperchaotic finance systems, and also adapted for asset price valuation under fraction Cryptocurrency. In all cases the methods prove to be very accurate, reliable and practically easy for the financial manager. / Thesis (PhD)--University of Pretoria, 2021. / Mathematics and Applied Mathematics / PhD / Unrestricted

This thesis focuses on the improvement of the Hermite-Fourier spectral method for solving kinetic plasma problems. In the first part of the thesis a novel dynamically adaptive techniques for changing the number of Hermite modes and Hermite basis are presented. Preconditioning of the problem and use of high-end scientific toolkit PETSc are discussed. The technique of changing the number of modes and preconditioning are believed to reduce computational time, while a change of basis improves robustness and numerical convergence of the method. The second part of the thesis focuses on paralellization strategies and performance analysis of the code implemented in the Fortran programming language.

Although spectral methods have been in use for decades, there is still room for innovation, refinement and improvement of the methods in terms of efficiency and accuracy, for generalized homogeneous turbulent flows, and especially for specialized applications like the computation of atmospheric flows and numerical weather prediction. In this thesis, two such innovations are presented. First, inspired by the adaptive mesh refinement (AMR) technique, which was developed for the computation of fluid flows in physical space, an algorithm is presented for accelerating direct numerical simulation (DNS) of isotropic homogeneous turbulence in spectral space. In the adaptive spectral resolution (ASR) technique developed here the spectral resolution in spectral space is dynamically refined based on refinement criteria suited to the special features of isotropic homogeneous turbulence in two, and three dimensions. Applying ASR to computations of two- and three-dimensional turbulence allows significant savings in the computational time with little to no compromise in the accuracy of the solutions. In the second part of this thesis the effect of explicit filtering on large eddy simulation (LES) of atmospheric flows in spectral space is studied. Apply an explicit filter in addition to the implicit filter due to the computational grid and discretization schemes in LES of turbulent flows allows for better control of the numerical error and improvement in the accuracy of the results. Explicit filtering has been extensively applied in LES of turbulent flows in physical space while few studies have been done on explicitly filtered LES of turbulent flows in spectral space because of perceived limitations of the approach, which are shown here to be incorrect. Here, explicit filtering in LES of the turbulent barotropic vorticity equation (BVE) as a first model of the Earth's atmosphere in spectral space is studied. It is shown that explicit filtering increases the accuracy of the results over implicit filtering, particularly where the location of coherent structures is concerned. / Ph. D.