The solution of stiff systems of Ordinary Differential Equations (ODEs), that typically arise after spatial discretization of many important evolutionary Partial Differential Equations (PDEs), constitutes a topic of wide interest in numerical analysis. A prominent way to numerically integrate such systems involves using exponential integrators. In general, these kinds of schemes do not require the solution of (non)linear systems but rather the action of the matrix exponential and of some specific exponential-like functions (known in the literature as φ-functions). In this PhD thesis we aim at presenting efficient tensor-based tools to approximate such actions, both from a theoretical and from a practical point of view, when the problem has an underlying Kronecker sum structure. Moreover, we investigate the application of exponential integrators to compute numerical solutions of important equations in various fields, such as plasma physics, mean-field optimal control and computational chemistry. In any case, we provide several numerical examples and we perform extensive simulations, eventually exploiting modern hardware architectures such as multi-core Central Processing Units (CPUs) and Graphic Processing Units (GPUs). The results globally show the effectiveness and the superiority of the different approaches proposed.

No / The transient liquid phase bonding process has long been dealt with as a pure diffusion process at the joint interface, that is, as a mass phenomenon. In spite of the advances in the application of this technique to bond complex engineering alloys, the available models have failed to incorporate the effect of surface phenomena on the joining process. In this work, a new reaction–controlled diffusion formulation model is proposed, and the observation of experimental results of joining Al6061 alloy using thin single (50, 100 micron) and double Cu foils is recorded. This work directly unveils the unique role played by surface reaction–controlled diffusion rather than purely mass diffusion bonding process. Our experimental and modeling results reveal a conceptually new understanding that may well explain the joint formation in TLP bonding process.

Chemotaxis, the phenomenon through which cells respond to external chemical signals, is one of the most important and universally observable in nature. It has been the object of considerable modelling effort in the last decades. The models for chemotaxis available in the literature cannot reconcile the dynamics of external chemical signals and the intracellular signalling pathways leading to the response of the cells. The reason is that models used for cells do not contain the distinction between the extracellular and intracellular domains. The work presented in this dissertation intends to resolve this issue. We set up a numerical hybrid simulation framework containing such description and enabling the coupling of models for phenomena occurring at extracellular and intracellular levels. Mathematically, this is achieved by the use of the fictitious domain method for finite elements, allowing the simulation of partial differential equations on evolving domains. In order to make the modelling of the membrane binding of chemical signals possible, we derive a suitable fictitious domain method for Robin boundary elliptic problems. We also display ways to minimise the computational cost of such simulation by deriving a suitable preconditioner for the linear systems resulting from the Robin fictitious domain method, as well as an efficient algorithm to compute fictitious domain specific linear operators. Lastly, we discuss the use of a simpler cell model from the literature and match it with our own model. Our numerical experiments show the relevance of the matching, as well as the stability and accuracy of the numerical scheme presented in the thesis.

Dans cette thèse nous nous intéressons aux questions de stabilité et d'instabilité dans certains problèmes inverses classiques pour l'équation de Schrödinger et l'équation acoustique en dimension d>=2. Les problèmes considérés sont le problème inverse de Gel'fand de valeurs au bord et les problèmes inverses de diffusion en champ proche et en champ lointain. Les résultats de stabilité et d'instabilité présentés dans cette thèse se complètent mutuellement et contribuent à une meilleure compréhension de la nature des problèmes précités. En particulier, nous démontrons des nouvelles estimations de stabilité globale qui dépendent explicitement de la régularité du coefficient et de l'énergie. En outre, nous considérons le problème inverse de valeurs au bord pour l'équation de Schrödinger à l'énergie fixée avec des mesures frontières représentées comme l'opérateur frontière d'impédance (ou l'opérateur Robin-Robin). Nous démontrons des estimations de stabilité globale pour détermination du potentiel à partir de mesures frontières dans cette représentation d'impédance. De plus, des techniques similaires donnent aussi une procédure de reconstruction globale pour ce problème.

This thesis is devoted to the mathematical analysis of models describing the energy of defects in crystalline solids via variational methods. The first part of this work studies a discrete model describing the energy of a point defect in a one dimensional chain of atoms. We derive an expansion of the ground state energy using Gamma-convergence, following previous work on similar models [BDMG99,BC07,SSZ11]. The main novelty here is an explicit characterisation of the first order limit as the solution of a variational problem in an infinite lattice. Analysing this variational problem, we prove a regularity result for the perturbation caused by the defect, and demonstrate the order of the next term in the expansion. The second main topic is a discrete model describing screw dislocations in body centred cubic crystals. We formulate an anti plane lattice model which describes the energy difference between deformations and, using the framework defined in [AO05], provide a kinematic description of the Burgers vector, which is a key geometric quantity used to describe dislocations. Apart from the anti plane restriction, this model is invariant under all the natural symmetries of the lattice and in particular allows for the creation and annihilation of dislocations. The energy difference formulation enables us to provide a clear definition of what it means to be a stable deformation. The main results of the analysis of this model are then first, a proof that deformations with unit net Burgers vector exist as globally stable states in an infinite body, and second, that deformations containing multiple screw dislocations exist as locally stable states in both infinite bodies and finite convex bodies. To prove the former result, we establish coercivity with respect to the elastic strain, and exploit a concentration compactness principle. In the latter case, we use a form of the inverse function theorem, proving careful estimates on the residual and stability of an ansatz which combines continuum linear elasticity theory with an atomistic core correction.

Les premières étapes d’activation des lymphocytes T sont cruciales pour déterminer leur comportement, ainsi que leur prolifération. Ces étapes dépendent fortement des conditions initiales, particulièrement de l’avidité du récepteur du lymphocyte (TCR) pour le ligand spécifique provenant de l’antigène. La reconnaissance du virus entraine une séquence des réactions biochimiques mettant en œuvre de protéines membranaires et cellulaires. Le processus peut être mesuré par cytométrie en flux. On propose ici plusieurs modèles de différents niveaux de complexité. Ces modèles décrivent une relation entre la population de lymphocytes T et leurs composants intracellulaires et extracellulaires. Ils conduisent à des systèmes d’EDO et d’EDP dont la résolution permet d’étudier la dynamique de la densité de population des lymphocytes au cours du processus d'activation. En outre, différentes hypothèses sont proposées pour le processus d'activation des cellules filles après prolifération. Les équations de bilan de population (EBPs) sont résolues par une nouvelle méthode validée par une solution analytique quand elle existe, ou par comparaison à différentes méthodes numériques disponibles dans la littérature. L’avantage de cette nouvelle méthode est d’être utilisable dans certains cas où les méthodes classiques ne le sont pas. / The early steps of activation are crucial in deciding the fate of T-cells leading to the proliferation. These steps strongly depend on the initial conditions, especially the avidity of the T-cell receptor for the specific ligand and the concentration of this ligand. The recognition induces a rapid decrease of membrane TCR-CD3 complexes inside the T-cell, then the up-regulation of CD25 and then CD25–IL2 binding which down-regulates into the T-cell. This process can be monitored by flow cytometry technique. We propose several models based on the level of complexity by using population balance modeling technique to study the dynamics of T-cells population density during the activation process. These models provide us a relation between the population of T-cells with their intracellular and extracellular components. Moreover, the hypotheses are proposed for the activation process of daughter T-cells after proliferation. The corresponding population balance equations (PBEs) include reaction term (i.e. assimilated as growth term) and activation term (i.e. assimilated as nucleation term). Further the PBEs are solved by newly developed method that is validated against analytical method wherever possible and various approximate techniques available in the literature.

Uma família de esquemas upwind denominada FUS-RF (Family of Upwind Scheme via Rational Functions), que é derivada via funções racionais e dependentes de parâmetros, é proposta para o cálculo de soluções aproximadas de equações de conservação. A fim de ilustrar a capacidade dos novos esquemas, vários resultados computacionais para sistemas hiperbólicos de leis de conservação são apresentados. Esses testes mostram a inflluência dos parâmetros escolidos sobre a qualidade dos resultados numéricos. Fazendo o uso de alguns testes de padrões, comparação dos novos limitadores de fluxo correspondentes com o esquema bem estabelecido van Albada e esquema atual EPUS (Eight-degree Polynomial Upwind Scheme) é também realizada. Os testes numéricos realizados em transporte de escalares e problemas de dinâmica dos gases confirmam que alguns esquemas da família FUS-RF são não oscilatórios e fornecem resultados confiáveis quando perfis descontínuos são transportados. Um esquema particular dessa nova família de esquemas upwind é então selecionado e utilizado para resolver escoamentos complexos com superfícies livres móveis / A family of upwind schemes named as FUS-RF (Family of Upwind Scheme via Rational Functions), which is derived via rational functions and dependent of parameters, is proposed for computing approximated solutions of conservation equations. In order to illustrate the capability of the new schemes, several computational results for system of hyperbolic conservation laws are presented. These results clarify the influence of the chosen parameters on the quality of the numerical calculations. Using some standard test cases, comparison of the new corresponding limiters with the well established van Albada and the recently introduced EPUS (Eight-degree Polynomial Upwind Scheme) limiters is also done. Numerical tests on both scalar and gas dynamics problems confirm that some schemes of the FUS-RF family are non-oscillatory and yield sharp results when solving profiles with discontinuities. A particular upwind scheme of this new family is then slected and used for solving complex incompressible moving free surface flows

Les systèmes d'Equations aux Dérivées Partielles Hyperboliques Linéaires du Premier Ordre (EDPs HLPO) permettent de modéliser de nombreux systèmes de lois de conservation. Ils apparaissent, par exemple, lors de la modélisation de problèmes de trafic routier, d'échangeurs de chaleurs, ou de problèmes multiphasiques. Différentes approches ont été proposées pour stabiliser ou observer de tels systèmes. Parmi elles, la méthode de backstepping consiste à transformer le système originel en un système découplé pour lequel la synthèse de la loi de commande est plus simple. Les contrôleur obtenus par cette méthode sont explicites.Dans la première partie de cette thèse, nous présentons des résultats généraux de théorie des systèmes. Plus précisément, nous résolvons les problèmes de stabilisation en temps fini pour une classe générale d'EDPs HLPO. Le temps de convergence minimal atteignable dépend du nombre d'actionneurs disponibles. Les observateurs associés à ces contrôleurs (nécessaires pour envisager une utilisation industrielle de tels contrôleurs) sont obtenus via une approche duale. Un des avantages importants de l'approche considérée dans cette thèse est de montrer que l'espace généré par les solutions de l'EDPs HLPO considérée est isomorphe à l'espace généré par les solutions d'une système neutre à retards distribués.Dans la seconde partie de cette thèse, nous montrons la nécessité d'un changement de stratégie pour résoudre les problèmes de contrôle robuste. Ces questions surviennent nécessairement lorsque sont considérées des applications industrielles pour lesquelles les différents paramètres du système peuvent être mal connus, pour lesquelles des dynamiques peuvent avoir été négligées, de même que des retards agissant sur la commande ou sur la mesure, ou encore pour lesquels les mesures sont bruitées. Nous proposons ainsi quelques modifications sur les lois de commande précédemment développées en y incorporant plusieurs degrés de liberté permettant d'effectuer un compromis entre performance et robustesse. L'analyse de stabilité et de robustesse sous-jacente est rendue possible en utilisant l'isomorphisme précédemment introduit. / Linear First-Order Hyperbolic Partial Differential Equations (LFOH PDEs) represent systems of conservation and balance law and are predominant in modeling of traffic flow, heat exchanger, open channel flow or multiphase flow. Different control approaches have been tackled for the stabilization or observation of such systems. Among them, the backstepping method consists to map the original system to a simpler system for which the control design is easier. The resulting controllers are explicit.In the first part of this thesis, we develop some general results in control theory. More precisely, we solve the problem of finite-time stabilization of a general class of LFOH PDEs using the backstepping methodology. The minimum stabilization time reachable may change depending on the number of available actuators. The corresponding boundary observers (crucial to envision industrial applications) are obtained through a dual approach. An important by-product of the proposed approach is to derive an explicit mapping from the space generated by the solutions of the considered LFOH PDEs to the space generated by the solutions of a general class of neutral systems with distributed delays. This mapping opens new prospects in terms of stability analysis for LFOH PDEs, extending the stability analysis methods developed for neutral systems.In the second part of the thesis, we prove the necessity of a change of strategy for robust control while considering industrial applications, for which the major limitation is known to be the robustness of the resulting control law to uncertainties in the parameters, delays in the loop, neglected dynamics or disturbances and noise acting on the system. In some situations, one may have to renounce to finite-time stabilization to ensure the existence of robustness margins. We propose some adjustments in the previously designed control laws by means of several degrees of freedom enabling trade-offs between performance and robustness. The robustness analysis is fulfilled using the explicit mapping between LFOH PDEs and neutral systems previously introduced.

First, we provide new classes of initial data, that grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. The main novelty here is the establishment of certain continuity properties near the initial time, for weak Leray-Hopf solutions with initial data in supercritical Besov spaces. The techniques used here build upon related ideas of Calderón. Secondly, we prove local regularity up to the at part of the boundary, for certain classes of solutions to the Navier-Stokes equations, provided that the velocity field belongs to L_∞(-1; 0; L^{3, β}(B(1) ⋂ ℝ³ ₊)) with 3 ≤ β < ∞. What enables us to build upon the work of Escauriaza, Seregin and Šverák [27] and Seregin [100] is the establishment of new scale-invariant estimates, new estimates for the pressure near the boundary and a convenient new ϵ-regularity criterion. Third, we show that if a weak Leray-Hopf solution in ℝ³ ₊×]0,∞[ has a finite blow-up time T, then necessarily lim_t↑T||v(·, t)||_{L^3,β(ℝ³ ₊)} = ∞ with 3 < β < ∞. The proof hinges on a rescaling procedure from Seregin's work [106], a new stability result for singular points on the boundary, suitable a priori estimates and a Liouville type theorem for parabolic operators developed by Escauriaza, Seregin and Šverák [27]. Finally, we investigate a notion of global-in-time solutions to the Navier- Stokes equations in ℝ³, with solenoidal initial data in the critical Besov space ?^-1/4_4,∞(ℝ³), which has certain continuity properties with respect to weak* convergence of the initial data. Such properties are motivated by the strategy used by Seregin [106] to show that if a weak Leray-Hopf solution in ℝ³×]0,∞[ has a finite blow-up time T, then necessarily lim_t↑T ||v(·, t)||_{L₃(ℝ³)} = ∞. We prove new decomposition results for Besov spaces, which are key in the conception and existence theory of such solutions.

The central theme of this thesis is the design of optimal balanced black-box stopping criteria in iterative solvers of symmetric positive-definite, symmetric indefinite, and nonsymmetric linear systems arising from finite element approximation of stochastic (parametric) partial differential equations. For a given stochastic and spatial approximation, it is known that iteratively solving the corresponding linear(ized) system(s) of equations to too tight algebraic error tolerance results in a wastage of computational resources without decreasing the usually unknown approximation error. In order to stop optimally-by avoiding unnecessary computations and premature stopping-algebraic error and a posteriori approximation error estimate must be balanced at the optimal stopping iteration. Efficient and reliable a posteriori error estimators do exist for close estimation of the approximation error in a finite element setting. But the algebraic error is generally unknown since the exact algebraic solution is not usually available. Obtaining tractable upper and lower bounds on the algebraic error in terms of a readily computable and monotonically decreasing quantity (if any) of the chosen iterative solver is the distinctive feature of the designed optimal balanced stopping strategy. Moreover, this work states the exact constants, that is, there are no user-defined parameters in the optimal balanced stopping tests. Hence, an iterative solver incorporating the optimal balanced stopping methodology that is presented here will be a black-box iterative solver. Typically, employing such a stopping methodology would lead to huge computational savings and in any case would definitely rule out premature stopping. The constants in the devised optimal balanced black-box stopping tests in MINRES solver for solving symmetric positive-definite and symmetric indefinite linear systems can be estimated cheaply on-the- fly. The contribution of this thesis goes one step further for the nonsymmetric case in the sense that it not only provides an optimal balanced black-box stopping test in a memory-expensive Krylov solver like GMRES but it also presents an optimal balanced black-box stopping test in memory-inexpensive Krylov solvers such as BICGSTAB(L), TFQMR etc. Currently, little convergence theory exists for the memory-inexpensive Krylov solvers and hence devising stopping criteria for them is an active field of research. Also, an optimal balanced black-box stopping criterion is proposed for nonlinear (Picard or Newton) iterative method that is used for solving the finite dimensional Navier-Stokes equations. The optimal balanced black-box stopping methodology presented in this thesis can be generalized for any iterative solver of a linear(ized) system arising from numerical approximation of a partial differential equation. The only prerequisites for this purpose are the existence of a cheap and tight a posteriori error estimator for the approximation error along with cheap and tractable bounds on the algebraic error.