Nous nous intéressons dans cette thèse à certains modèles mathématiques permettant une description de systèmes ouverts classiques et quantiques. Dans l'étude de ces systèmes en interaction avec un environnement, nous montrons que la dynamique induite par l'environnement sur le système donne lieu à l'apparition de bruits. Dans une première partie de la thèse, dédiée aux systèmes classiques, le modèle décrit est le schéma d'interactions répétées. Etant à la fois hamiltonien et markovien, ce modèle en temps discret permet d'implémenter facilement la dissipation dans des systèmes physiques. Nous expliquons comment le mettre en place pour des systèmes physiques avant d'en étudier la limite en temps continu. Nous montrons la convergence Lp et presque sûre de l'évolution de certains systèmes vers la solution d'une équation différentielle stochastique, à travers l'étude de la limite de la perturbation d'un schéma d'Euler stochastique. Dans une seconde partie de la thèse sur les systèmes quantiques, nous nous intéressons dans un premier temps aux actions d'environnements quantiques sur des systèmes quantiques aboutissant à des bruits classiques. A cette fin, nous introduisons certains opérateurs unitaires appelés « classiques », que nous caractérisons à l'aide de variables aléatoires dites obtuses. Nous mettons en valeur comment ces variables classiques apparaissent naturellement dans ce cadre quantique à travers des 3-tenseurs possédant des symétries particulières. Nous prouvons notamment que ces 3-tenseurs sont exactement ceux diagonalisables dans une base orthonormée. Dans un second temps, nous étudions la limite en temps continu d'une variante des interactions répétées quantiques dans le cas particulier d'un système biparti, c'est-à-dire composé de deux systèmes isolés sans interaction entre eux. Nous montrons qu'à la limite du temps continu, une interaction entre ces sous-systèmes apparaît explicitement sous forme d'un hamiltonien d'interaction; cette interaction résulte de l'action de l'environnement et de l'intrication qu'il crée / This dissertation is dedicated to some mathematical models describing classical and quantum open systems. In the study of these systems interacting with an environment, we particularly show that the dynamics induced by the environment leads to the appearance of noises. In a first part of this thesis, devoted to classical open systems, the repeated interaction scheme is developed. This discrete-time model, being Hamiltonian and Markovian at the same time, has the advantage to easily implement the dissipation in physical systems. We explain how to set this scheme up in some physical examples. Then, we investigate the continuous-time limit of these repeated interactions. We show the Lp and almost sure convergences of the evolution of the system to the solution of a stochastic differential equation, by studying the limit of a perturbed Stochastic Euler Scheme. In a second part of this dissertation on quantum systems, we characterize in a first work classical actions of a quantum environment on a quantum system. In this study, we introduce some “classical” unitary operators representing these actions and we highlight a strong link between them and some random variables, called obtuse random variables. We explain how these random variables are naturally connected to some 3-tensors having some particular symmetries. We particularly show that these 3 tensors are exactly the ones that are diagonalizable in some orthonormal basis. In a second work of this part, we study the continuous-time limit of a variant of the repeated interaction scheme in a case of a bipartite system, that is, a system made of two isolated systems not interaction together. We prove that an explicit Hamiltonian interaction between them appears at the limit. This interaction is due to the action of the environment and the entanglement between the two systems that it creates

This thesis focuses on the methodology of controlling dynamic systems in real time. It contents a review of the control theory basis and the elementary base of regulators construction. Then the list of matemathic formulaes follows as well as the math basis for the system simulations using a difeerential count and the problem of difeerential equations solving. Furthermore, there is a systematic approach to the design of general regulator enclosed, using modern simulation techniques. After the results confirmation in the Matlab system, the problematics of transport delay & quantization modelling follow.

The thesis deals with speedup and accuracy of numerical computation, especially when differential equations are solved. Algorithms, which are fulling these conditions are named semi-analytical. One posibility how to accelerate computation of differential equation is paralelization. Presented paralelization is based on transformation numerical solution into residue number system, which is extended to floating point computation. A new algorithm for modulo multiplication is also proposed. As application applications in solution of differential calculus are the main goal it is discussed numeric integration with modified Euler, Runge - Kutta and Taylor series method in residue number system. Next possibilities and extension for implemented residue number system are mentioned at the end.

Economic uncertainty may affect significantly people’s behavior and hence macroeconomic variables. It is thus important to understand how people behave in presence of different kinds of economic risk. The present dissertation focuses therefore on the impact of the uncertainty in capital and labor income on the individual saving behavior. The underlying uncertain variables are here modeled as stochastic processes that each obey a specific stochastic differential equation, where uncertainty stems either from Poisson or Lévy processes. The results on the optimal behavior are derived by maximizing the individual expected lifetime utility. The first chapter is concerned with the necessary mathematical tools, the change-of-variables formula and the Hamilton-Jacobi-Bellman equation under Poisson uncertainty. We extend their possible field of application in order make them appropriate for the analysis of the dynamic stochastic optimization problems occurring in the following chapters and elsewhere. The second chapter considers an optimum-saving problem with labor income, where capital risk stems from asset prices that follow geometric L´evy processes. Chapter 3, finally, studies the optimal saving behavior if agents face not only risk but also uncertain spells of unemployment. To this end, we turn back to Poisson processes, which here are used to model properly the separation and matching process.

Infimalfolgen nichtkonvexer Variationsprobleme haben aufgrund feiner Oszillationen häufig keinen starken Grenzwert in Sobolevräumen. Diese Oszillationen haben eine physikalische Bedeutung; Finite-Element-Approximationen können sie jedoch im Allgemeinen nicht auflösen. Relaxationsmethoden ersetzen die nichtkonvexe Energie durch ihre (semi)konvexe Hülle. Das entstehende makroskopische Modell ist degeneriert: es ist nicht strikt konvex und hat eventuell mehrere Minimalstellen. Die fehlende Kontrolle der primalen Variablen führt zu Schwierigkeiten bei der a priori und a posteriori Fehlerschätzung, wie der Zuverlässigkeits- Effizienz-Lücke und fehlender starker Konvergenz. Zur Überwindung dieser Schwierigkeiten erweitern Stabilisierungstechniken die relaxierte Energie um einen diskreten, positiv definiten Term. Bartels et al. (IFB, 2004) wenden Stabilisierung auf zweidimensionale Probleme an und beweisen dabei starke Konvergenz der Gradienten. Dieses Ergebnis ist auf glatte Lösungen und quasi-uniforme Netze beschränkt, was adaptive Netzverfeinerungen ausschließt. Die vorliegende Arbeit behandelt einen modifizierten Stabilisierungsterm und beweist auf unstrukturierten Netzen sowohl Konvergenz der Spannungstensoren, als auch starke Konvergenz der Gradienten für glatte Lösungen. Ferner wird der sogenannte Fluss-Fehlerschätzer hergeleitet und dessen Zuverlässigkeit und Effizienz gezeigt. Für Interface-Probleme mit stückweise glatter Lösung wird eine Verfeinerung des Fehlerschätzers entwickelt, die den Fehler der primalen Variablen und ihres Gradienten beschränkt und so starke Konvergenz der Gradienten sichert. Der verfeinerte Fehlerschätzer konvergiert schneller als der Fluss- Fehlerschätzer, und verringert so die Zuverlässigkeits-Effizienz-Lücke. Numerische Experimente mit fünf Benchmark-Tests der Mikrostruktursimulation und Topologieoptimierung ergänzen und bestätigen die theoretischen Ergebnisse. / Infimising sequences of nonconvex variational problems often do not converge strongly in Sobolev spaces due to fine oscillations. These oscillations are physically meaningful; finite element approximations, however, fail to resolve them in general. Relaxation methods replace the nonconvex energy with its (semi)convex hull. This leads to a macroscopic model which is degenerate in the sense that it is not strictly convex and possibly admits multiple minimisers. The lack of control on the primal variable leads to difficulties in the a priori and a posteriori finite element error analysis, such as the reliability-efficiency gap and no strong convergence. To overcome these difficulties, stabilisation techniques add a discrete positive definite term to the relaxed energy. Bartels et al. (IFB, 2004) apply stabilisation to two-dimensional problems and thereby prove strong convergence of gradients. This result is restricted to smooth solutions and quasi-uniform meshes, which prohibit adaptive mesh refinements. This thesis concerns a modified stabilisation term and proves convergence of the stress and, for smooth solutions, strong convergence of gradients, even on unstructured meshes. Furthermore, the thesis derives the so-called flux error estimator and proves its reliability and efficiency. For interface problems with piecewise smooth solutions, a refined version of this error estimator is developed, which provides control of the error of the primal variable and its gradient and thus yields strong convergence of gradients. The refined error estimator converges faster than the flux error estimator and therefore narrows the reliability-efficiency gap. Numerical experiments with five benchmark examples from computational microstructure and topology optimisation complement and confirm the theoretical results.

In dieser Arbeit betrachten wir das Strömungs-Diffusions-(Reaktions)-Problem für passive Markerteilchen, die in zweidimensionalen laminaren Strömungsmustern mit geringem thermischem Rauschen gelöst sind. Der deterministische Fluss umfasst Zellen in Form von Quadraten oder Katzenaugen. In ihnen tritt Rotationsbewegung auf. Einige der Strömungen bestehen aus wellenförmigen Bereichen mit gerader Vorwärtsbewegung. Alle Systeme sind entweder periodisch oder durch Wände begrenzt. Eine untersuchte Familie von Strömungen interpoliert kontinuierlich zwischen Reihen von Wirbeln und Scherflüssen. Wir analysieren zahlreiche numerische Simulationen, die bisherige theoretische Vorhersagen bestätigen und neue Phänomene offenbaren. Ohne Rauschen sind die Teilchen in einzelnen Bestandteilen des Flusses für immer gefangen. Durch Hinzufügen von schwachem thermischen Rauschen wird die normale Diffusion für lange Zeiten stark verstärkt und führt zu verschiedenen Diffusionsarten für mittlere Zeiten. Mit Continuous-Time-Random-Walk-Modellen leiten wir analytische Ausdrücke in Übereinstimmung mit den numerischen Ergebnissen her, die je nach Parametern, Anfangsbedingungen und Alterungszeiten von subdiffusiver bis superballistischer anomaler Diffusion für mittlere Zeiten reichen. Wir sehen deutlich, dass einige der früheren Vorhersagen nur für Teilchen gelten, die an der Separatrix des Flusses starten - der einzige Fall, der in der Vergangenheit ausführlich betrachtet wurde - und dass das System zu vollkommen anderem Verhalten in anderen Situationen führen kann, einschließlich einem Schwingenden beim Start im Zentrum einesWirbels nach einer gewissen Alterungszeit. Darüber hinaus enthüllen die Simulationen, dass Teilchenreaktionen dort häufiger auftreten, wo sich die Geschwindigkeit der Strömung stark ändert, was dazu führt, dass langsame Teilchen von schnelleren getroffen werden, die ihnen folgen. Die umfangreichen numerischen Simulationen, die für diese Arbeit durchgeführt wurden, mussten jetzt durchgeführt werden, da wir die Rechenleistung dafür besitzen. / In this thesis, we consider the advection-diffusion-(reaction) problem for passive tracer particles suspended in two-dimensional laminar flow patterns with small thermal noise. The deterministic flow comprises cells in the shape of either squares or cat’s eyes. Rotational motion occurs inside them. Some of the flows consist of sinusoidal regions of straight forward motion. All systems are either periodic or are bounded by walls. One examined family of flows continuously interpolates between arrays of eddies and shear flows. We analyse extensive numerical simulations, which confirm previous theoretical predictions as well as reveal new phenomena. Without noise, particles are trapped forever in single building blocks of the flow. Adding small thermal noise, leads to largely enhanced normal diffusion for long times and several kinds of diffusion for intermediate times. Using continuous time random walk models, we derive analytical expressions in accordance with numerical results, ranging from subdiffusive to superballistic anomalous diffusion for intermediate times depending on parameters, initial conditions and aging time. We clearly see, that some of the previous predictions are only true for particles starting at the separatrix of the flow - the only case considered in depth in the past - and that the system might show a vastly different behavior in other situations, including an oscillatory one, when starting in the center of an eddy after a certain aging time. Furthermore, simulations reveal that particle reactions occur more frequently at positions where the velocity of the flow changes the most, resulting in slow particles being hit by faster ones following them. The extensive numerical simulations performed for this thesis had to be done now that we have the computational means to do so. Machines are powerful tools in order to gain a deeper and more detailed insight into the dynamics of many complicated dynamical and stochastic systems.

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<div>The rise of artificial intelligence (AI) hinges on the efficient training of modern deep neural networks (DNNs) for non-convex optimization and uncertainty quantification, which boils down to a non-convex Bayesian learning problem. A standard tool to handle the problem is Langevin Monte Carlo, which proposes to approximate the posterior distribution with theoretical guarantees. However, non-convex Bayesian learning in real big data applications can be arbitrarily slow and often fails to capture the uncertainty or informative modes given a limited time. As a result, advanced techniques are still required.</div><div><br></div><div>In this thesis, we start with the replica exchange Langevin Monte Carlo (also known as parallel tempering), which is a Markov jump process that proposes appropriate swaps between exploration and exploitation to achieve accelerations. However, the na\"ive extension of swaps to big data problems leads to a large bias, and the bias-corrected swaps are required. Such a mechanism leads to few effective swaps and insignificant accelerations. To alleviate this issue, we first propose a control variates method to reduce the variance of noisy energy estimators and show a potential to accelerate the exponential convergence. We also present the population-chain replica exchange and propose a generalized deterministic even-odd scheme to track the non-reversibility and obtain an optimal round trip rate. Further approximations are conducted based on stochastic gradient descents, which yield a user-friendly nature for large-scale uncertainty approximation tasks without much tuning costs. </div><div><br></div><div>In the second part of the thesis, we study scalable dynamic importance sampling algorithms based on stochastic approximation. Traditional dynamic importance sampling algorithms have achieved successes in bioinformatics and statistical physics, however, the lack of scalability has greatly limited their extensions to big data applications. To handle this scalability issue, we resolve the vanishing gradient problem and propose two dynamic importance sampling algorithms based on stochastic gradient Langevin dynamics. Theoretically, we establish the stability condition for the underlying ordinary differential equation (ODE) system and guarantee the asymptotic convergence of the latent variable to the desired fixed point. Interestingly, such a result still holds given non-convex energy landscapes. In addition, we also propose a pleasingly parallel version of such algorithms with interacting latent variables. We show that the interacting algorithm can be theoretically more efficient than the single-chain alternative with an equivalent computational budget.</div>

The thesis deals with qualitative properties of solutions of singular initial value problems for ordinary differential and integrodifferential equations which occur in the theory of linear and nonlinear electrical circuits and the theory of therminionic currents. The research is concentrated especially on questions of existence and uniqueness of solutions, asymptotic estimates of solutions and modications of Adomian decomposition method for singular initial problems. Solution algoritms are derived for scalar differential equations of Lane-Emden type using Taylor series and modication of the Adomian decomposition method. For certain classes of nonlinear of integrodifferential equations asymptotic expansions of solutions are constructed in a neighbourhood of a singular point. By means of the combination of Wazewski's topological method and Schauder xed-point theorem there are proved asymptotic estimates of solutions in a region which is homeomorphic to a cone having vertex coinciding with the initial point. Using Banach xed-point theorem the uniqueness of a solution of the singular initial value problem is proved for systems of integrodifferential equations of Volterra and Fredholm type including implicit systems. Moreover, conditions of continuous dependence of a solution on a parameter are determined. Obtained results are presented in illustrative examples.