Philosophiae Doctor - PhD / Fitted Numerical Methods for Delay Di erential Equations Arising in Biology E.B.M. Bashier PhD thesis, Department of Mathematics and Applied Mathematics,Faculty of Natural Sciences, University of the Western Cape. This thesis deals with the design and analysis of tted numerical methods for some delay di erential models that arise in biology. Very often such di erential equations are very complex in nature and hence the well-known standard numerical methods seldom produce reliable numerical solutions to these problems. Ine ciencies of these methods are mostly accumulated due to their dependence on crude step sizes and unrealistic stability conditions.This usually happens because standard numerical methods are initially designed to solve a class of general problems without considering the structure of any individual problems. In this thesis, issues like these are resolved for a set of delay di erential equations. Though the developed approaches are very simplistic in nature, they could solve very complex problems as is shown in di erent chapters.The underlying idea behind the construction of most of the numerical methods in this thesis is to incorporate some of the qualitative features of the solution of the problems into the discrete models. Resulting methods are termed as tted numerical methods. These methods have high stability properties, acceptable (better in many cases) orders of convergence, less computational complexities and they provide reliable solutions with less CPU times as compared to most of the other conventional solvers. The results obtained by these methods are comparable to those found in the literature. The other salient feature of the proposed tted methods is that they are unconditionally stable for most of the problems under consideration.We have compared the performances of our tted numerical methods with well-known software packages, for example, the classical fourth-order Runge-Kutta method, standard nite di erence methods, dde23 (a MATLAB routine) and found that our methods perform much better. Finally, wherever appropriate, we have indicated possible extensions of our approaches to cater for other classes of problems. May 2009.

L’objectif principal de cette thèse est de décliner et d’analyser l’algorithme kernel-LMS à noyau Gaussien dans trois cadres différents: celui des noyaux uniques et multiples, à valeurs réelles et à valeurs complexes, dans un contexte d’apprentissage distributé et coopératif dans les réseaux de capteurs. Plus précisement, ce travail s’intéresse à l’analyse du comportement en moyenne et en erreur quadratique de cas différents types d’algorithmes LMS à noyau. Les modèles analytiques de convergence obtenus sont validés par des simulations numérique. Tout d’abord, nous introduisons l’algorithme LMS, les espaces de Hilbert à noyau reproduisants, ainsi que les algorithmes de filtrage adaptatif à noyau existants. Puis, nous étudions analytiquement le comportement de l’algorithme LMS à noyau Gaussien dans le cas où les statistiques des éléments du dictionnaire ne répondent que partiellement aux statistiques des données d’entrée. Nous introduisons ensuite un algorithme LMS modifié à noyau basé sur une approche proximale. La stabilité de l’algorithme est également discutée. Ensuite, nous introduisons deux types d’algorithmes LMS à noyaux multiples. Nous nous concentrons en particulier sur l’analyse de convergence de l’un d’eux. Plus généralement, les caractéristiques des deux algorithmes LMS à noyaux multiples sont analysées théoriquement et confirmées par les simulations. L’algorithme LMS à noyau complexe augmenté est présenté et ses performances analysées. Enfin, nous proposons des stratégies de diffusion fonctionnelles dans les espaces de Hilbert à noyau reproduisant. La stabilité́ de cas de l’algorithme est étudiée. / The main objective of this thesis is to derive and analyze the Gaussian kernel least-mean-square (LMS) algorithm within three frameworks involving single and multiple kernels, real-valued and complex-valued, non-cooperative and cooperative distributed learning over networks. This work focuses on the stochastic behavior analysis of these kernel LMS algorithms in the mean and mean-square error sense. All the analyses are validated by numerical simulations. First, we review the basic LMS algorithm, reproducing kernel Hilbert space (RKHS), framework and state-of-the-art kernel adaptive filtering algorithms. Then, we study the convergence behavior of the Gaussian kernel LMS in the case where the statistics of the elements of the so-called dictionary only partially match the statistics of the input data. We introduced a modified kernel LMS algorithm based on forward-backward splitting to deal with $\ell_1$-norm regularization. The stability of the proposed algorithm is then discussed. After a review of two families of multikernel LMS algorithms, we focus on the convergence behavior of the multiple-input multikernel LMS algorithm. More generally, the characteristics of multikernel LMS algorithms are analyzed theoretically and confirmed by simulation results. Next, the augmented complex kernel LMS algorithm is introduced based on the framework of complex multikernel adaptive filtering. Then, we analyze the convergence behavior of algorithm in the mean-square error sense. Finally, in order to cope with the distributed estimation problems over networks, we derive functional diffusion strategies in RKHS. The stability of the algorithm in the mean sense is analyzed.

Ce manuscrit de thèse porte sur l'analyse numérique de schémas volumes finis pour la discrétisation de deux systèmes particuliers d'équations. Dans un premier temps nous étudions l'équation de Cahn-Hilliard associée à des conditions aux limites dynamiques dont l'une des principales difficultés est que cette condition aux limites est une équation parabolique, non linéaire, posée sur le bord et couplée avec l'intérieur du domaine. Nous proposons une discrétisation de type volumes finis en espace qui permet de coupler naturellement l'équation dans le domaine et celle sur sa frontière par un terme de flux et qui s'adapte facilement à la géométrie courbe du domaine. Nous montrons l'existence et la convergence des solutions discrètes vers une solution faible du système. Dans un second temps nous étudions la stabilité Inf-Sup du problème de Stokes pour un schéma volumes finis de type dualité discrète (DDFV). Nous donnons une analyse complète de la stabilité Inf-Sup inconditionnelle dans certains cas et de la stabilité de codimension 1 dans le cas de maillages cartésiens. Nous mettons également en place une méthode numérique permettant de calculer la constante Inf-Sup associée à ce schéma pour un maillage donné. On peut ainsi observer le comportement stable ou instable selon les cas en fonction de la géométrie des maillages. Dans une dernière partie nous proposons un schéma DDFV pour un modèle couplé Cahn-Hilliard/Stokes ce qui nécessite l'introduction de nouveaux opérateurs discrets. Nous démontrons la décroissance de l'énergie au niveau discret ainsi que l'existence d'une solution au problème discret. L'ensemble de ces travaux est validé par de nombreux résultats numériques. / This manuscript is devoted to the numerical analysis of finite-volume schemes for the discretization of two particular equations. First, we study the Cahn-Hilliard equation with dynamic boundary conditions whose one of the main difficulties is that this boundary condition is a non-linear parabolic equation on the boundary coupled with the interior of the domain. We propose a spatial finite-volume discretization which is well adapted to the coupling of the dynamics in the domain and those on the boundary by the flux term. Moreover this kind of scheme accounts naturally for the non-flat geometry of the boundary. We prove the existence and the convergence of the discrete solutions towards a weak solution of the system. Second, we study the Inf-Sup stability of the discrete duality finite volume (DDFV) scheme for the Stokes problem. We give a complete analysis of the unconditional Inf-Sup stability in some cases and of codimension 1 Inf-Sup stability for Cartesian meshes. We also implement a numerical method which allows us to compute the Inf-Sup constant associated with this scheme for a given mesh. Thus, we can observe the stable or unstable behaviour that can occur depending on the geometry of the meshes. In a last part we propose a DDFV scheme for a Cahn-Hilliard/Stokes phase field model that required the introduction of new discrete operators. We prove the dissipation of the energy in the discrete case and the existence of a solution to the discrete problem. All these research results are validated by extensive numerical results.

A class of Lagrange-Newton-SQP methods is investigated for optimal control problems governed by semilinear parabolic initial- boundary value problems. Distributed and boundary controls are given, restricted by pointwise upper and lower bounds. The convergence of the method is discussed in appropriate Banach spaces. Based on a weak second order sufficient optimality condition for the reference solution, local quadratic convergence is proved. The proof is based on the theory of Newton methods for generalized equations in Banach spaces.

Philosophiae Doctor - PhD / Fitted operator finite difference methods (FOFDMs) for singularly perturbed problems have been explored for the last three decades. The construction of these numerical schemes is based on introducing a fitting factor along with the diffusion coefficient or by using principles of the non-standard finite difference methods. The FOFDMs based on the latter idea, are easy to construct and they are extendible to solve partial differential equations (PDEs) and their systems. Noting this flexible feature of the FOFDMs, this thesis deals with extension of these methods to solve interior layer problems, something that was still outstanding. The idea is then extended to solve singularly perturbed time-dependent PDEs whose solutions possess interior layers. The second aspect of this work is to improve accuracy of these approximation methods via methods like Richardson extrapolation. Having met these three objectives, we then extended our approach to solve singularly perturbed two-point boundary value problems with variable diffusion coefficients and analogous time-dependent PDEs. Careful analyses followed by extensive numerical simulations supporting theoretical findings are presented where necessary.

<p>The advent of advanced communication technologies has given rise to large-scale networks comprised of numerous interconnected agents, which need to cooperate to accomplish various tasks, such as distributed message routing, formation control, robust statistical inference, and spectrum access coordination. These tasks can be formulated as distributed optimization problems, which require agents to agree on a parameter minimizing the average of their local cost functions by communicating only with their neighbors. However, distributed optimization algorithms are typically susceptible to malicious (or "Byzantine") agents that do not follow the algorithm. This thesis offers analysis and algorithms for such scenarios. As the malicious agent's function can be modeled as an unknown function with some fundamental properties, we begin in the first two parts by analyzing the region containing the potential minimizers of a sum of functions. Specifically, we explicitly characterize the boundary of this region for the sum of two unknown functions with certain properties. In the third part, we develop resilient algorithms that allow correctly functioning agents to converge to a region containing the true minimizer under the assumption of convex functions of each regular agent. Finally, we present a general algorithmic framework that includes most state-of-the-art resilient algorithms. Under the strongly convex assumption, we derive a geometric rate of convergence of all regular agents to a ball around the optimal solution (whose size we characterize) for some algorithms within the framework.</p>

This thesis examines the convergence of neoliberal rhetoric across popular media, academic, and institutional discourses, and draws connections between contemporary women's travel literature and common scripts in study abroad promotion. Finding such narratives to be freighted with ethnocentric constructs and tacit endorsements of market-based globalization, I critique the mainstreaming of neoliberal attitudes that depict travel as a commodity primarily valuable for its role in increasing the worth of U.S. American personhood. I question both the prevailing definitions of "global citizenship" and the ubiquitous claims that study abroad prepares students for "success in the global economy" as ideological signifiers of a higher education system that is increasingly corporatized. Utilizing a postcolonial and transnational feminist theoretical framework, the thesis offers a literary analysis of contemporary women's travel memoirs, examining patterns of narcissism and "othering" in their depictions of cross-cultural encounter, and connects these neoliberal trends to consumerism in higher education, study abroad, and post-second wave feminism. Shared themes in the representation of privileged U.S./Western women abroad and the student-consumer model in higher education bespeak a movement toward individual international engagements that reinforce corporate motives for travel and endorse the commodification of global environments, cultures, and people. In hopes of contesting this paradigm, I argue for the reassertion of a social justice-oriented definition of global citizenship and for educational models that foster self-criticism and the decolonization of knowledge. / Graduation date: 2012

This thesis treats two essentially different subjects: V-cycle schemes are considered in Chapters 2-4, whereas the aggregation-based coarsening is analysed in Chapters 5-6. As a matter of paradox, these two multigrid ingredients, when combined together, can hardly lead to an optimal algorithm. Indeed, a V-cycle needs more accurate prolongations than the simple piecewise-constant one, associated to aggregation-based coarsening. On the other hand, aggregation-based approaches use almost exclusively piecewise constant prolongations, and therefore need more involved cycling strategies, K-cycle <a href=http://www3.interscience.wiley.com/journal/114286660/abstract?CRETRY=1&SRETRY=0>[Num.Lin.Alg.Appl. vol.15(2008), pp.473-487]</a> being an attractive alternative in this respect.<p><br><p><br><p>Chapter 2 considers more precisely the well-known V-cycle convergence theories: the approximation property based analyses by Hackbusch (see [Multi-Grid Methods and Applications, 1985, pp.164-167]) and by McCormick [SIAM J.Numer.Anal. vol.22(1985), pp.634-643] and the successive subspace correction theory, as presented in [SIAM Review, vol.34(1992), pp.581-613] by Xu and in [Acta Numerica, vol.2(1993), pp.285-326.] by Yserentant. Under the constraint that the resulting upper bound on the convergence rate must be expressed with respect to parameters involving two successive levels at a time, these theories are compared. Unlike [Acta Numerica, vol.2(1993), pp.285-326.], where the comparison is performed on the basis of underlying assumptions in a particular PDE context, we compare directly the upper bounds. We show that these analyses are equivalent from the qualitative point of view. From the quantitative point of view,<p>we show that the bound due to McCormick is always the best one.<p><br><p><br><p>When the upper bound on the V-cycle convergence factor involves only two successive levels at a time, it can further be compared with the two-level convergence factor. Such comparison is performed in Chapter 3, showing that a nice two-grid convergence (at every level) leads to an optimal McCormick's bound (the best bound from the previous chapter) if and only if a norm of a given projector is bounded on every level.<p><br><p><br><p>In Chapter 4 we consider the Fourier analysis setting for scalar PDEs and extend the comparison between two-grid and V-cycle multigrid methods to the smoothing factor. In particular, a two-sided bound involving the smoothing factor is obtained that defines an interval containing both the two-grid and V-cycle convergence rates. This interval is narrow when an additional parameter α is small enough, this latter being a simple function of Fourier components.<p><br><p><br><p>Chapter 5 provides a theoretical framework for coarsening by aggregation. An upper bound is presented that relates the two-grid convergence factor with local quantities, each being related to a particular aggregate. The bound is shown to be asymptotically sharp for a large class of elliptic boundary value problems, including problems with anisotropic and discontinuous coefficients.<p><br><p><br><p>In Chapter 6 we consider problems resulting from the discretization with edge finite elements of 3D curl-curl equation. The variables in such discretization are associated with edges. We investigate the performance of the Reitzinger and Schöberl algorithm [Num.Lin.Alg.Appl. vol.9(2002), pp.223-238], which uses aggregation techniques to construct the edge prolongation matrix. More precisely, we perform a Fourier analysis of the method in two-grid setting, showing its optimality. The analysis is supplemented with some numerical investigations. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

<p>The swift emergence and wide-ranging utilization of machine learning (ML) across various industries, including healthcare, transportation, and robotics, have underscored the escalating need for efficient, scalable, and privacy-preserving solutions. Recognizing this, we present an integrated examination of three novel frameworks, each addressing different aspects of distributed learning and privacy issues: Two Timescale Hybrid Federated Learning (TT-HF), Delay-Aware Federated Learning (DFL), and Differential Privacy Hierarchical Federated Learning (DP-HFL). TT-HF introduces a semi-decentralized architecture that combines device-to-server and device-to-device (D2D) communications. Devices execute multiple stochastic gradient descent iterations on their datasets and sporadically synchronize model parameters via D2D communications. A unique adaptive control algorithm optimizes step size, D2D communication rounds, and global aggregation period to minimize network resource utilization and achieve a sublinear convergence rate. TT-HF outperforms conventional FL approaches in terms of model accuracy, energy consumption, and resilience against outages. DFL focuses on enhancing distributed ML training efficiency by accounting for communication delays between edge and cloud. It also uses multiple stochastic gradient descent iterations and periodically consolidates model parameters via edge servers. The adaptive control algorithm for DFL mitigates energy consumption and edge-to-cloud latency, resulting in faster global model convergence, reduced resource consumption, and robustness against delays. Lastly, DP-HFL is introduced to combat privacy vulnerabilities in FL. Merging the benefits of FL and Hierarchical Differential Privacy (HDP), DP-HFL significantly reduces the need for differential privacy noise while maintaining model performance, exhibiting an optimal privacy-performance trade-off. Theoretical analysis under both convex and nonconvex loss functions confirms DP-HFL’s effectiveness regarding convergence speed, privacy performance trade-off, and potential performance enhancement with appropriate network configuration. In sum, the study thoroughly explores TT-HF, DFL, and DP-HFL, and their unique solutions to distributed learning challenges such as efficiency, latency, and privacy concerns. These advanced FL frameworks have considerable potential to further enable effective, efficient, and secure distributed learning.</p>