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This thesis addresses the sampling problem in a high-dimensional space, i.e., the computation of averages with respect to a defined probability density that is a function of many variables. Such sampling problems arise in many application areas, including molecular dynamics, multiscale models, and Bayesian sampling techniques used in emerging machine learning applications. Of particular interest are thermostat techniques, in the setting of a stochastic-dynamical system, that preserve the canonical Gibbs ensemble defined by an exponentiated energy function. In this thesis we explore theory, algorithms, and numerous applications in this setting. We begin by comparing numerical methods for particle-based models. The class of methods considered includes dissipative particle dynamics (DPD) as well as a newly proposed stochastic pairwise Nosé-Hoover-Langevin (PNHL) method. Splitting methods are developed and studied in terms of their thermodynamic accuracy, two-point correlation functions, and convergence. When computational efficiency is measured by the ratio of thermodynamic accuracy to CPU time, we report significant advantages in simulation for the PNHL method compared to popular alternative schemes in the low-friction regime, without degradation of convergence rate. We propose a pairwise adaptive Langevin (PAdL) thermostat that fully captures the dynamics of DPD and thus can be directly applied in the setting of momentum-conserving simulation. These methods are potentially valuable for nonequilibrium simulation of physical systems. We again report substantial improvements in both equilibrium and nonequilibrium simulations compared to popular schemes in the literature. We also discuss the proper treatment of the Lees-Edwards boundary conditions, an essential part of modelling shear flow. We also study numerical methods for sampling probability measures in high dimension where the underlying model is only approximately identified with a gradient system. These methods are important in multiscale modelling and in the design of new machine learning algorithms for inference and parameterization for large datasets, challenges which are increasingly important in "big data" applications. In addition to providing a more comprehensive discussion of the foundations of these methods, we propose a new numerical method for the adaptive Langevin/stochastic gradient Nosé-Hoover thermostat that achieves a dramatic improvement in numerical efficiency over the most popular stochastic gradient methods reported in the literature. We demonstrate that the newly established method inherits a superconvergence property (fourth order convergence to the invariant measure for configurational quantities) recently demonstrated in the setting of Langevin dynamics. Furthermore, we propose a covariance-controlled adaptive Langevin (CCAdL) thermostat that can effectively dissipate parameter-dependent noise while maintaining a desired target distribution. The proposed method achieves a substantial speedup over popular alternative schemes for large-scale machine learning applications.

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[pt] Suspensões correspondem a uma classe de materiais amplamente utilizada em uma grande variedade de aplicações e indústrias. Devido à sua extrema versatilidade, elas têm sido foco de inúmeros estudos nas últimas décadas. Suspensões também são muito flexíveis e podem apresentar diferentes propriedades reológicas e respostas macroscópicas dependendo da escolha dos parâmetros usados como entrada no sistema. Mais especificamente, a resposta reológica de suspensões está intimamente associada ao arranjo microestrutural das partículas que compõem o meio e a fatores externos, como o quão confinadas elas se encontram e a rigidez das partículas. No presente estudo, o efeito da rigidez, confinamento e vazão na microestrutura de suspensões altamente concentradas é avaliado usando Dinâmica Dissipativa de Partículas com Núcleo Modificado. Precedento este estudo principal, foram necessárias outras duas etapas para garantir um sistema de simulação confiável e representativo, que consistiu, essencialmente, na realização de estudos paramétricos para compreender e estimar os valores adequados para os parâmetros de interacção parede-partícula. O presente trabalho aborda estudos paramétricos realizados para auxiliar na escolha dos parâmetros de entrada para evitar a penetração de partículas em um sistema delimitado por paredes. Inicialmente um sistema mais simples, composto por solvente e paredes é construído e os parâmetros de interação e densidades de parede foram ajustados. Em seguida as interações são definidas para suspensões. Neste último caso, vários parâmetros desempenham um papel na penetração e a maneira tradicional de investigar esses efeitos seria exaustiva e demorada. Por isso, optamos por usar uma abordagem de Machine Learning para realizar este estudo. Uma vez ajustados os parâmetros, o estudo de confinamento pôde ser realizado. O objetivo principal deste estudo foi entender como a microestrutura de suspensões concentradas é afetada pela vazão, rigidez das partículas e confinamento. Verificou-se que partículas muito flexíveis sempre formam um aglomerado gigante independente da razão de confinamento; a diferença está em quão compactadas são as partículas. No caso de partículas rígidas, um confinamento mais forte leva à formação de aglomerados maiores. O estudo final aborda um estudo de aprendizado de máquina realizado para prever a reologia de suspensões não confinadas. Com este trabalho foi possível entender e ajustar parâmetros de simulação e desenvolver um domínio computacional que permite estudar sistematicamente efeitos do confinamento em suspensões. / [en] Suspensions correspond to a class of materials vastly used in a large set of applications and industries. Due to its extreme versatility, they have been the focus of numerous studies over the past decades. Suspensions are also very flexible and can display different rheological properties and macroscopic responses depending on the choice of parameters used as input in the system. More specifically, the rheological response of suspensions is intimately associated to the microstructural arrangement of the particles composing the medium and external factors, such as how strongly they are confined and particle rigidity. In the present study, the effect of particle rigidity, confinement and flow rate on the microstructure of highly concentrated suspensions is studied using CoreModified Dissipative Particle Dynamics. Preceding this main study, two other steps were necessary to guarantee a reliable and realistic simulation system, which consisted, essentially, on performing parametric studies to understand and estimate the appropriate values for wall-particle interaction parameters. The present work address parametric studies performed to assist the input parameters choice to prevent particle penetration in a wall-bounded system. Initially a simpler system, composed of solvent and walls, is built and the interaction parameters and wall densities were adjusted. Following, the interactions are set for suspensions. In the latter case multiple parameters play a role in penetration and the traditional way to investigate these effects would be exhaustive and time consuming. Hence, we choose to use a Machine Learning approach to perform this study. Once the parameters were adjusted, the study of confinement could be carried out. The main goal of this study was to understand how the microstructure of concentrated suspensions is affected by flow rate, particle rigidity and confinement. It was found that very soft particles always form a giant cluster regardless the confinement ratio; the difference being on how packed the particles are. In the rigid case, a stronger confinement leads the formation of larger clusters. The final study addresses a machine learning study carried out to predict the rheology of unconfined suspensions. The main contribution of this work is that it was possible to understand and adjust simulation parameters and develop a computational domain that enables to systematically study confinement effects on suspensions.

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La thèse est consacrée à la modélisation discrète et continue des écoulements sanguins et des phénomènes connexes tels que la coagulation du sang et l'athérosclérose. Ce travail comprend l'élaboration des modèles mathématiques et numériques de la coagulation du sang, des simulations numériques et l'analyse mathématique d'un modèle d'inflammation chronique au cours d'athérosclérose. Une partie importante de la thèse est liée à la programmation, la mise en œuvre et l'optimisation des codes numériques. La partie principale de la thèse concerne la modélisation de la coagulation du sang in vivo tenant compte des écoulements sanguins, les réactions biochimiques dans le plasma et l'agrégation de plaquettes. La nouveauté principale de ce travail est l'élaboration d'un modèle hybride (discret-continu) de la coagulation du sang et de la formation de caillot sanguin dans le flux. La partie théorique de la thèse est consacrée à l'analyse mathématique d'un modèle d'inflammation chronique liée à l'athérosclérose. Les simulations numériques réalisées dans le cadre de cette thèse impliquent l'élaboration des algorithmes numériques pour les modèles mathématiques et le d´développement des logiciels. Vu le fait que les simulations numériques ont été coûteuse en temps de calcul, des efforts considérables ont été consacrés à la parallélisation des logiciels et à leur optimisation / The thesis is devoted to discrete and continuous modelling of blood flows and related phenomena such as blood coagulation and atherosclerosis. It includes the development of mathematical and numerical models of blood coagulation, numerical simulations and the mathematical analysis of a model problem of chronic inflammation during atherosclerosis. The main part of the thesis concerns modelling of blood coagulation in vivo which takes into account blood flows, biochemical reactions in plasma and platelet aggregation. The main novelty of this work is the development of a hybrid (discrete-continuous) model of blood coagulation and clot formation in flow. The model is used to study several aspects of blood coagulation in flow : platelet aggregation and its interaction with coagulation pathways, influence of the flow speed on the clot development, a possible mechanism by which clot stops growing. The theoretical part of the thesis is devoted to the mathematical analysis of a model of chronic inflammation related to atherosclerosis. In this thesis we study a model problem which describes the propagation of a reaction-diffusion wave in the 2D case with non-linear boundary conditions. For that we use the Leray-Schauder method and a priori estimates of solutions in order to prove the existence of waves in the bistable case. Numerical simulations carried out in the framework of this thesis were based on the numerical implementation of the corresponding models and on the software development. Since the numerical simulations were computationally expensive, a substantial effort was directed to software parallelization and optimization

Un fondu de polymères est un liquide complexe constitué de chaînes macromoléculaires. Ces chaînes présentent la particularité d'offrir une distribution de temps caractéristiques extrêmement importante. L'ensemble de ces différentes échelles représente donc un défi pour la simulation numérique de polymères longs et sont bien au-delà des capacités des ordinateurs actuels. Un thème actuel de recherche porte donc sur le développement de modèles mésoscopiques (modèle gros grains). La construction d'un tel modèle consiste à éliminer les degrés de liberté rapide en regroupant un certain nombre d'atomes en un monomère. Ce monomère est représenté par une sphère molle évoluant dans un bain thermique créé par les degrés de liberté rapides éliminés au cours du processus de nivellement. La dynamique des particules créées est donc stochastique. La dynamique particulaire dissipative qui intègre ces idées est une combinaison de dynamique moléculaire, de Lattice Gas Automata ainsi que de dynamique Brownienne. Le champ de force DPD est constitué d'une interaction molle et d'un thermostat (force dissipative et bruit) et les paramètres de ce champ de force sont généralement calibrés sur des données expérimentales (compressibilité et diffusion). Cette approche est difficilement applicable aux mélanges de polymères. Pour surmonter cette difficulté, l'intégralité du champ de force DPD est construit à partir d'une dynamique moléculaire pour des corps purs ainsi que pour des mélanges. Nous montrons également que pour calculer correctement la force dissipative, la dynamique moléculaire doit être altérée en contraignant la position des monomères. Les coefficients de transport sont calculés par DPD et comparés à ceux obtenus par dynamique moléculaire. Ce travail s'achève par une étude de la transferabilité du champ de force du monomère vers toute une chaîne de polymères. / A molten polymer is a complex liquid consisting of macromolecular chains. These chains have many different time scales. All these scales present a real challenge to numerical simulations and exceed the computational capabilities of today's computers. A current topic of research therefore focuses on the development of mesoscopic models. The main idea behind coarse-graining is to eliminate fast degrees of freedom grouping atoms or molecules into clusters (or monomers). This monomer is represented by a soft sphere operating in a thermal bath generated by the fast degrees of freedom eliminated during the coarse-graining. Particle dynamics is therfore stochastic. Dissipative particle dyna-mics, which includes these ideas, is a combination of molecular dynamics, Lattice Gas Automata and Brownian dynamics. DPD force field consist of a soft interaction and a thermostat (dissipative and random force) and parameters of DPD interaction are generally optimized to match some macroscopic properties like compressibility or self-diffusion coefficient. This approach is difficult to apply to polymer melt. To overcome this problem, we apply an operational procedure available in the literature to the cons-truction of conservative and dissipative forces of DPD force field for pure substances and mixtures. We also show that in order to calculate the dissipative forces, the underlying molecular dynamics must be altered by constraining the position of the mo-nomers. Transport coefficients are calculated by DPD and compared with those obtained by molecular dynamics. This work concludes with a study of the transferability of the force field of the monomer to a chain of polymers.

Polymer networks are an important class of materials that are ubiquitously found in natural, biological, and man-made systems. The complex mesoscale structure of these soft materials has made it difficult for researchers to fully explore their properties. In this dissertation, we introduce a coarse-grained computational model for permanently cross-linked polymer networks than can properly capture common properties of these materials. We use this model to study several practical problems involving dry and solvated networks. Specifically, we analyze the permeability and diffusivity of polymer networks under mechanical deformations, we examine the release of encapsulated solutes from microgel capsules during volume transitions, and we explore the complex tribological behavior of elastomers. Our simulations reveal that the network transport properties are defined by the network porosity and by the degree of network anisotropy due to mechanical deformations. In particular, the permeability of mechanically deformed networks can be predicted based on the alignment of network filaments that is characterized by a second order orientation tensor. Moreover, our numerical calculations demonstrate that responsive microcapsules can be effectively utilized for steady and pulsatile release of encapsulated solutes. We show that swollen gel capsules allow steady, diffusive release of nanoparticles and polymer chains, whereas gel deswelling causes burst-like discharge of solutes driven by an outward flow of the solvent initially enclosed within a shrinking capsule. We further demonstrate that this hydrodynamic release can be regulated by introducing rigid microscopic rods in the capsule interior. We also probe the effects of velocity, temperature, and normal load on the sliding of elastomers on smooth and corrugated substrates. Our friction simulations predict a bell-shaped curve for the dependence of the friction coefficient on the sliding velocity. Our simulations also illustrate that at low sliding velocities, the friction decreases with an increase in the temperature. Overall, our findings improve the current understanding of the behavior of polymer networks in equilibrium and non-equilibrium conditions, which has important implications for synthesizing new drug delivery agents, designing tissue engineering systems, and developing novel methods for controlling the friction of elastomers.

We present simulation results of a minimal life-like, artificial, molecular aggregate (i.e. protocell) that has been proposed by Steen Rasussen and coworkers and is currently pursued both experimentally and computationally in interdisciplinary international research projects. We develop a space-time continuous physically motivated simulation framework based on the method of dissipative particle dynamics (DPD) which we incrementally extend (most notably by chemical reactions) to cope with the needs of our model. The applicability of the method over the entire length scale of interest is reintroduced, by rejecting a concern that DPD introduces a freezing artifact for any model above the atomistic scale. This is achieved by deriving an alternative scaling procedure for interaction parameters in the model. We perform system-level simulations of the design which attempt to account for theoretical, and experimental knowledge, as well as results from other computational models. This allows us to address key issues of the replicating subsystems container, genome, and metabolism both individually and in mutual coupling. We analyze each step in the life-cycle of the molecular aggregate, and a finnal integrated simulation of the entire life-cycle is prepared. Our simulations confirm most assumptions of the theoretical designs, but also exhibit unanticipated system-level dynamics. These findings are used to revise the original design of the Los Alamos minimal protocell over the course of the analysis. The results support the hypothesis that self-replication and probably other life-like features can be achieved in systems of formerly unanticipated simplicity if these systems exploit physicochemical principles that are immanent to their physical scale.

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