Evolutionary game theory has recently emerged as a key paradigm in various behavioral science disciplines. In particular it provides powerful tools and a conceptual framework for the analysis of the time evolution of strategic interdependence among players and its consequences, especially when the players are spatially distributed and linked in a complex social network. We develop various evolutionary game models, analyze these models using appropriate techniques, and study their applications to complex phenomena. In the second chapter, we derive integro-differential equations as deterministic approximations of the microscopic updating stochastic processes. These generalize the known mean-field ordinary differential equations and provide powerful tools to investigate the spatial effects on the time evolutions of the agents' strategy choices. The deterministic equations allow us to identify many interesting features of the evolution of strategy profiles in a population, such as standing and traveling waves, and pattern formation, especially in replicator-type evolutions. We introduce several methods of decomposition of two player normal form games in the third chapter. Viewing the set of all games as a vector space, we exhibit explicit orthonormal bases for the subspaces of potential games, zero-sum games, and their orthogonal complements which we call anti-potential games and anti-zero-sum games, respectively. Perhaps surprisingly, every anti-potential game comes either from Rock-paper-scissors type games (in the case of symmetric games) or from Matching Pennies type games (in the case of asymmetric games). Using these decompositions, we prove old (and some new) cycle criteria for potential and zero-sum games (as orthogonality relations between subspaces). We illustrate the usefulness of our decompositions by (a) analyzing the generalized Rock-Paper-Scissors game, (b) completely characterizing the set of all null-stable games, (c) providing a large class of strict stable games, (d) relating the game decomposition to the Hodge decomposition of vector fields for the replicator equations, (e) constructing Lyapunov functions for some replicator dynamics, (f) constructing Zeeman games -games with an interior asymptotically stable Nash equilibrium and a pure strategy ESS. The hierarchical modeling of evolutionary games provides flexibility in addressing the complex nature of social interactions as well as systematic frameworks in which one can keep track of the interplay of within-group dynamics and between-group competitions. For example, it can model husbands and wives' interactions, playing an asymmetric game with each other, while engaging coordination problems with the likes in other families. In the fourth chapter, we provide hierarchical stochastic models of evolutionary games and approximations of these processes, and study their applications

Since augmented Lagrangian methods were introduced by Powell and Hestenes, this class of methods has been investigated very intensively. While the finite dimensional case has been treated in a satisfactory manner, the infinite dimensional case is studied much less. The general approach to solve an infinite dimensional optimization problem subject to equality constraints is as follows: First one proves convergence for a basic algorithm in the Hilbert space setting. Then one discretizes the given spaces and operators in order to make numerical computations possible. Finally, one constructs a discretized version of the infinite dimensional method and tries to transfer the convergence results to the finite dimensional version of the basic algorithm. In this thesis we discuss a globally convergent augmented Lagrangian algorithm and discretize it in terms of functional analytic restriction operators. Given this setting, we prove global convergence of the discretized version of this algorithm to a stationary point of the infinite dimensional optimization problem. The proposed algorithm includes an explicit rule of how to update the discretization level and the penalty parameter from one iteration to the next one - questions that had been unanswered so far. In particular the latter update rule guarantees that the penalty parameters stay bounded away from zero which prevents the Hessian of the discretized augmented Lagrangian functional from becoming more and more ill conditioned. / Master of Science

The detrimental effects rapid power fluctuation has on wireless narrowband communication channels has long been a concern of the mobile radio community as appropriate channel models seek to gauge link quality. Furthermore, advances in signal processing capabilities and the desire for spectrally efficient and low power radio systems have rekindled the interest for adaptive transmission schemes, hence some method of quickly probing the link quality and/or predicting channel conditions is required. Mathematical distributions for modeling the channel profile seek to estimate fading parameters from a finite number of discrete time samples of signal amplitude. While the statistical inference of such estimators has proven to be robust to rapidly shifting channel conditions, the benefits are quickly realized at the expense of processing complexity. Furthermore, computations of the best-known estimation techniques are often iterative, tedious, and complex. This thesis takes a renewed look at estimating fading parameters for the Nakagami-m, Rice-K, and Weibull distributions, specifically by showing that the need to solve transcendental equations in the estimators can be circumvented through use of polynomial approximation in the least-squared error sense or via asymptotic series expansion which often lead to closed-form and simplified expressions. These new estimators are compared to existing ones, the performances of which are comparable while preserving a lower computational complexity. In addition, the thesis also investigates the impact knowledge of the fading profile has on systems employing adaptive switching modulation schemes by characterizing performance in terms of average bit error rates (BER) and spectral efficiency. A channel undergoing Rice-$K$ fading on top of log-normal shadowing is simulated by correlating samples of received signal amplitude according to the user's doppler speed, carrier frequency, etc. The channel's throughput and BER performances are analyzed using the above estimation techniques and compared to non-estimation assumptions. Further discussion on narrowband fading parameter estimation and its applicability to wireless communication channels is provided. / Master of Science

The study of stochastic queueing networks is quite important due to the many applications including transportation, telecommunication, and manufacturing industries. Since there is often no explicit solution to these types of control problems, numerical methods are needed. Following the method of Boué-Dupuis, we use a Dynamic Programming approach of optimization on a controlled Markov Chain that simulates the behavior of a fluid limit of the original process. The search for an optimal control in this case involves a Skorokhod problem to describe the dynamics on the boundary of closed, convex domain. Using relaxed stochastic controls we show that the approximating numerical solution converges to the actual solution as the size of the mesh in the discretized state space goes to zero, and illustrate with an example. / Ph. D.

2025 / This dissertation investigates how early career teachers' orientations of students and their mathematical activity, as surfaced their experience in a simulation, can reveal when their well-intentioned decisions might negatively impact traditionally underserved students. Three early career teachers engaged in two online simulations where they could speak, write, or select a response to a classroom scenario described with text and images, where their choices impact what part of the scenario they engage with next. The simulations were designed to elicit a tension between the teachers’ intended positive orientations about students and their decisions made in the simulations. A narrative analysis (Slocum-Bradley, 2010; Wortham, 2001) of the participants’ choices in the simulations, and their rationale behind those choices, revealed multiple instances where the teachers aligned themselves with the same orientations but sometimes positioned their role as teachers in conflict with that orientation. The narratives that emerged regarding these specific moments, the teachers' interpretation of that moment, and their decisions in the simulation showed multiple ways the rationalization of decisions can conflict with their intention. This dissertation informs how making sense of the narratives early career teachers make when engaging with simulations could help teachers' interpretations and actions better align with their intentions. Therefore, teacher learning could shift from focusing on content knowledge, pedagogical content knowledge, and pedagogical choices to the continual attention and decision-making around the interpretation that supported that decision.

L'imagerie par résonance magnétique fonctionnelle (IRMf) est une technique non invasive permettant l'étude de l'activité cérébrale au travers des changements hémodynamiques associés. Récemment, une technique de détection-estimation conjointe (DEC) a été développée permettant d'alterner (1) la détection de l'activité cérébrale induite par une stimulation ainsi que (2) l'estimation de la fonction de réponse hémodynamique caractérisant la dynamique vasculaire; deux problèmes qui sont généralement traités indépendamment. Cette approche considère une parcellisation a priori du cerveau en zones fonctionnellement homogènes et alterne (1) et (2) sur chacune d'entre elles séparément. De manière standard, l'analyse DEC suppose que le cerveau entier peut être activé par tous les types de stimuli (visuel, auditif, etc.). Cependant la spécialisation fonctionnelle des régions cérébrales montre que l'activité d'une région n'est due qu'à certains types de stimuli. La prise en compte de stimuli non pertinents dans l'analyse, peut dégrader les résultats. La sous-famille des types de stimuli pertinents n'étant pas la même à travers le cerveau une procédure de sélection de modèles serait très coûteuse en temps de calcul. De plus, une telle sélection a priori n'est pas toujours possible surtout dans les cas pathologiques. Ce travail de thèse propose une extension de l'approche DEC permettant la sélection automatique des conditions (types de stimuli) pertinentes selon l'activité cérébrale qu'elles suscitent, cela simultanément à l'analyse et adaptativement à travers les régions cérébrales. Des exemples d'analyses sur des jeux de données simulés et réels, illustrent la capacité de l'approche DEC parcimonieuse proposée à sélectionner les conditions pertinentes ainsi que son intérêt par rapport à l'approche DEC standard. / Functional magnetic resonance imaging (fMRI) is a noninvasive technique allowing the study of brain activity via the measurement of hemodynamic changes. Recently, a joint detection-estimation (JDE) framework was developed and relies on both (1) the brain activity detection and (2) the hemodynamic response function estimation, two steps that are generally addressed in a separate way. The JDE approach is a parcel-based model that alternates (1) and (2) on each parcel successively. The JDE analysis assumes that all delivered stimuli (e.g. visual, auditory, etc.) possibly generate a response everywhere in the brain although activation is likely to be induced by only some of them in specific brain areas. Inclusion of irrelevant events may degrade the results. Since the relevant conditions or stimulus types can change between different brain areas, a model selection procedure will be computationally expensive. Furthermore, criteria are not always available to select the relevant conditions prior to activation detection, especially in pathological cases. The goal of this work is to develop a JDE extension allowing an automatic selection of the relevant conditions according to the brain activity they elicit. This condition selection is done simultaneously to the analysis and adaptively through the different brain areas. Analysis on simulated and real datasets illustrate the ability of our model to select the relevant conditions and its interest compare to the standard JDE analysis.

Nous considérons la résolution de très grands systèmes linéaires creux à l'aide d'une méthode de factorisation directe appelée méthode multifrontale. Bien que numériquement robustes et faciles à utiliser (elles ne nécessitent que des informations algébriques : la matrice d'entrée A et le second membre b, même si elles peuvent exploiter des stratégies de prétraitement basées sur des informations géométriques), les méthodes directes sont très coûteuses en termes de mémoire et d'opérations, ce qui limite leur applicabilité à des problèmes de taille raisonnable (quelques millions d'équations). Cette étude se concentre sur l'exploitation des approximations de rang-faible dans la méthode multifrontale, pour réduire sa consommation mémoire et son volume d'opérations, dans des environnements séquentiel et à mémoire distribuée, sur une large classe de problèmes. D'abord, nous examinons les formats rang-faible qui ont déjà été développé pour représenter efficacement les matrices denses et qui ont été utilisées pour concevoir des solveurs rapides pour les équations aux dérivées partielles, les équations intégrales et les problèmes aux valeurs propres. Ces formats sont hiérarchiques (les formats H et HSS sont les plus répandus) et il a été prouvé, en théorie et en pratique, qu'ils permettent de réduire substantiellement les besoins en mémoire et opération des calculs d'algèbre linéaire. Cependant, de nombreuses contraintes structurelles sont imposées sur les problèmes visés, ce qui peut limiter leur efficacité et leur applicabilité aux solveurs multifrontaux généraux. Nous proposons un format plat appelé Block Rang-Faible (BRF) basé sur un découpage naturel de la matrice en blocs et expliquons pourquoi il fournit toute la flexibilité nécéssaire à son utilisation dans un solveur multifrontal général, en terme de pivotage numérique et de parallélisme. Nous comparons le format BRF avec les autres et montrons que le format BRF ne compromet que peu les améliorations en mémoire et opération obtenues grâce aux approximations rang-faible. Une étude de stabilité montre que les approximations sont bien contrôlées par un paramètre numérique explicite appelé le seuil rang-faible, ce qui est critique dans l'optique de résoudre des systèmes linéaires creux avec précision. Ensuite, nous expliquons comment les factorisations exploitant le format BRF peuvent être efficacement implémentées dans les solveurs multifrontaux. Nous proposons plusieurs algorithmes de factorisation BRF, ce qui permet d'atteindre différents objectifs. Les algorithmes proposés ont été implémentés dans le solveur multifrontal MUMPS. Nous présentons tout d'abord des expériences effectuées avec des équations aux dérivées partielles standardes pour analyser les principales propriétés des algorithmes BRF et montrer le potentiel et la flexibilité de l'approche ; une comparaison avec un code basé sur le format HSS est également fournie. Ensuite, nous expérimentons le format BRF sur des problèmes variés et de grande taille (jusqu'à une centaine de millions d'inconnues), provenant de nombreuses applications industrielles. Pour finir, nous illustrons l'utilisation de notre approche en tant que préconditionneur pour la méthode du Gradient Conjugué. / We consider the solution of large sparse linear systems by means of direct factorization based on a multifrontal approach. Although numerically robust and easy to use (it only needs algebraic information: the input matrix A and a right-hand side b, even if it can also digest preprocessing strategies based on geometric information), direct factorization methods are computationally intensive both in terms of memory and operations, which limits their scope on very large problems (matrices with up to few hundred millions of equations). This work focuses on exploiting low-rank approximations on multifrontal based direct methods to reduce both the memory footprints and the operation count, in sequential and distributed-memory environments, on a wide class of problems. We first survey the low-rank formats which have been previously developed to efficiently represent dense matrices and have been widely used to design fast solutions of partial differential equations, integral equations and eigenvalue problems. These formats are hierarchical (H and Hierarchically Semiseparable matrices are the most common ones) and have been (both theoretically and practically) shown to substantially decrease the memory and operation requirements for linear algebra computations. However, they impose many structural constraints which can limit their scope and efficiency, especially in the context of general purpose multifrontal solvers. We propose a flat format called Block Low-Rank (BLR) based on a natural blocking of the matrices and explain why it provides all the flexibility needed by a general purpose multifrontal solver in terms of numerical pivoting for stability and parallelism. We compare BLR format with other formats and show that BLR does not compromise much the memory and operation improvements achieved through low-rank approximations. A stability study shows that the approximations are well controlled by an explicit numerical parameter called low-rank threshold, which is critical in order to solve the sparse linear system accurately. Details on how Block Low-Rank factorizations can be efficiently implemented within multifrontal solvers are then given. We propose several Block Low-Rank factorization algorithms which allow for different types of gains. The proposed algorithms have been implemented within the MUMPS (MUltifrontal Massively Parallel Solver) solver. We first report experiments on standard partial differential equations based problems to analyse the main features of our BLR algorithms and to show the potential and flexibility of the approach; a comparison with a Hierarchically SemiSeparable code is also given. Then, Block Low-Rank formats are experimented on large (up to a hundred millions of unknowns) and various problems coming from several industrial applications. We finally illustrate the use of our approach as a preconditioning method for the Conjugate Gradient.

Les structures soumises à des vibrations sont rencontrées dans diverses applications. Dans denombreux cas, elles sont de nature linéaires, mais quand les amplitudes des oscillations deviennentimportantes, cela provoque un comportement non-linéaire. Par ailleurs, les oscillations desstructures dans un milieu fluide entrainent une interaction fluide-structure. Cette thèse porte surla modélisation du problème fluide-structure non-linéaire. Les cas de non-linéarités étudiés sont lanon-linéarité grands-déplacements caractéristique des structures minces, la non-linéarité localiséegéométrique décrivant une liaison non-linéaire entre deux structures et la non-linéarité acoustiqueparticularité des très hauts niveaux de pression.Pour la modélisation de ces problèmes, il se peut que le calcul en réponse demeure infaisable enraison du temps de calcul. D’une part, on est amené à résoudre des systèmes matriciels (symétriquesou non) de grandes tailles générés par la méthode des éléments finis et d’autre part, cetterésolution demande une évaluation de la force non-linéaire à chaque itération. Afin de diminuer lecoût de calcul, la réduction de modèle par des bases de réductions couplées avec un algorithmeparallélisant l’évaluation de la force non-linéaire, est une alternative à la résolution du systèmecomplet. La construction des bases de réduction doit s’adapter au mieux à chaque problème traité.La base modale du problème linéaire est une première approximation puis elle est enrichie par desinformations qui proviennent à la fois de la nature du couplage et du comportement non-linéaire / Structures subjected to vibrations are found in various applications. In many cases, they behave ina linear way, but when the amplitudes of the oscillations become important, it causes a nonlinearbehavior. Moreover, the oscillations of structures in a fluid field lead to a fluid-structureinteraction. This thesis focuses on the modeling of nonlinear fluid-structure problem. Differentkind of nonlinearities are studied in this work including the large-displacement nonlinearitycharacteristic of thin structures, the localized geometrical nonlinearity describing a nonlinear linkbetween two structures, and the acoustic nonlinearity characteristic of very high levels ofpressure.Modeling such problems are time and memory consuming, that may lead to a limitations of themodel. Therefore, it is necessary to solve a large matrix system (either symmetric or not)generated by the finite element method and the resolution needs an evaluation of the nonlinearforce at each iteration. In order to reduce the computational cost, model reduction with reducedbases combined with parallelization of the nonlinear force evolution is proposed as an alternative tothe resolution of complete systems. Building reduction bases must be adapted to each concernedproblem. The eigenmode of the linear problem is a first approximation and it is enriched withinformation coming from both coupling and nonlinear behaviors.

The behaviour of free-surface gravity waves on small Froude number fluid flow past some obstacle cannot be determined using ordinary asymptotic power series methods, as the amplitude of the waves is exponentially small. An exponential asymptotic method is used by Chapman and Vanden-Broeck (2006) to consider the problem of two-dimensional, steady flow past a submerged obstacle in the small Froude number limit, finding that a steady downstream wavetrainis switched on rapidly across a curve known as a Stokes line. Here, equivalent wavetrains on three-dimensional and unsteady flow configurations are considered, and Stokes switching causedby the interaction between exponentially small free-surface components is shown to play an important role in both cases. The behaviour of free-surface gravity waves is introduced by considering the problem of steady free-surface flow due to a line source. A steady wavetrain is shown to exist in the far field, and the behaviour of these waves is compared to existing numerical results. The problem of unsteady flow over a step is subsequently investigated, with the flow behaviour formulated in terms of Lagrangian coordinates so that the position of the free surface is fixed. Initially, the problem is linearized in the step-height, and the steady wavetrain is shown to spread downstream over time. The position of the wavefront is determined by considering the full Stokes structure present in the problem. The equivalent fully-nonlinear problem is then considered, with the position of the Stokes lines, and hence the wavefront, being determined numerically. Finally, linearized three-dimensional free-surface flow past an obstacle is considered in both the steady and unsteady case. The surface is shown to contain downstream longitudinal and transverse waves. These waves are shown to propagate downstream in the unsteady case, with the position of the wavefront again determined by considering the full Stokes structure of the problem.

In this Ph. D. thesis we discuss four different problems in analysis: (a) sharp inequalities related to the restriction phenomena for the Fourier transform, with emphasis on some Strichartz-type estimates; (b) extremal approximations of exponential type for the Gaussian and for a class of even functions, with applications to analytic number theory; (c) radial symmetrization approach to convolution-like inequalities for the Boltzmann collision operator; (d) regularity of maximal operators with respect to weak derivatives and weak continuity. / text