This work is aimed at development of systematic approaches to the design of multiscale systems. Specifically four problems are addressed: environmental impact assessment (EIA) of new and retrofitted industrial processes, integration of process effluents with the macroscopic environmental systems, eco-industrial parks (EIP), and advanced life support (ALS) systems for planetary habitation. While design metrics and specific natures of each problem poses different challenges, there are common themes in the devised solution strategies: a. An integrated approach provides insights unseen by addressing the individual components of the system and, therefore, better understanding and superior results. b. Instead of dealing with multiple scales simultaneously, the design problem is addressed through interconnected stages without infringing upon the optimization degrees of freedom in each stage. This is possible through the concept of targeting. c. Mathematical programming techniques can be used effectively to systematize the integration concepts, the target identification, and the design of multi-scale systems. The dissertation also introduces the following specific contributions: i. For EIA, a new procedure is developed to overcome the limitations of conventional approaches. The introduced procedure is based on three concepts: process synthesis for systematic generation of alternatives and targeting for benchmarking environmental impact ahead of detailed design, integration of alternative with rest of the process, and reverse problem formulation for targeting. ii. For integrating process effluents with macroscopic environmental systems, focus is given to the impact of wastewater discharges on macroscopic watersheds and drainage systems. A reverse problem formulation is introduced to determine maximum allowable process discharges that will meet overall environmental requirements of the watershed. iii. For EIPs, a new design procedure is developed to allow multiple processes to share a common environmental infrastructure, exchange materials, and jointly utilize interception systems that treat waste materials and byproducts. A source-interception-sink representation is developed and modeled through an optimization formulation. Optimal interactions among the various processes and shared infrastructure to be installed are identified. iv. A computational metric is introduced to compare various alternatives in ALS and planetary habitation systems. A selection criterion identifies the alternative which contributes to the maximum reduction of the total ESM of the system.

In this thesis, a framework is laid for holistic uncertainty management for simulation-based design of multiscale systems. The work is founded on uncertainty management for microstructure mediated design (MMD) of material and product, which is a representative example of a system over multiple length and time scales, i.e., a multiscale system. The characteristics and challenges for uncertainty management for multiscale systems are introduced context of integrated material and product design. This integrated approach results in different kinds of uncertainty, i.e., natural uncertainty (NU), model parameter uncertainty (MPU), model structure uncertainty (MSU) and propagated uncertainty (PU). We use the Inductive Design Exploration Method to reach feasible sets of robust solutions against MPU, NU and PU. MMD of material and product is performed for the product autonomous underwater vehicle (AUV) employing the material in-situ metal matrix composites using IDEM to identify robust ranged solution sets. The multiscale system results in decision nodes for MSU consideration at hierarchical levels, termed as multilevel design. The effectiveness of using game theory to model strategic interaction between the different levels to facilitate decision making for mitigating MSU in multilevel design is illustrated using the compromise decision support problem (cDSP) technique. Information economics is identified as a research gap to address holistic uncertainty management in simulation-based multiscale systems, i.e., to address the reduction or mitigation of uncertainty considering the current design decision and scope for further simulation model refinement in order to reach better robust solutions. It necessitates development of an improvement potential (IP) metric based on value of information which suggests the scope of improvement in a designer's decision making ability against modeled uncertainty (MPU) in simulation models in multilevel design problem. To address the research gap, the integration of robust design (using IDEM), information economics (using IP) and game theoretic constructs (using cDSP) is proposed. Metamodeling techniques and expected value of information are critically reviewed to facilitate efficient integration. Robust design using IDEM and cDSP are integrated to improve MMD of material and product and address all four types of uncertainty simultaneously. Further, IDEM, cDSP and IP are integrated to assist system level designers in allocating resources for simulation model refinement in order to satisfy performance and robust process requirements. The approach for managing MPU, MSU, NU and PU while mitigating MPU is presented using the MMD of material and product. The approach presented in this article can be utilized by system level designers for managing all four types of uncertainty and reducing model parameter uncertainty in any multiscale system.

In a first part, we are interested in the behavior of a system of Stochastic PDEs with two time-scales- more precisely, we focus on the approximation of the slow component thanks to an efficient numerical scheme. We first prove an averaging principle, which states that the slow component converges to the solution of the so-called averaged equation. We then show that a numerical scheme of Euler type provides a good approximation of an unknown coefficient appearing in the averaged equation. Finally, we build and we analyze a discretization scheme based on the previous results, according to the HMM methodology (Heterogeneous Multiscale Method). We precise the orders of convergence with respect to the time-scale parameter and to the parameters of the numerical discretization- we study the convergence in a strong sense - approximation of the trajectories - and in a weak sense - approximation of the laws. In a second part, we study a method for approximating solutions of parabolic PDEs, which combines a semi-lagrangian approach and a Monte-Carlo discretization. We first show in a simplified situation that the variance depends on the discretization steps. We then provide numerical simulations of solutions, in order to show some possible applications of such a method.

Dans une première partie, on s'intéresse à un système d'EDP stochastiques variant selon deux échelles de temps, et plus particulièrement à l'approximation de la composante lente à l'aide d'un schéma numérique efficace. On commence par montrer un principe de moyennisation, à savoir la convergence de la composante lente du système vers la solution d'une équation dite moyennée. Ensuite on prouve qu'un schéma numérique de type Euler fournit une bonne approximation d'un coefficient inconnu apparaissant dans cette équation moyennée. Finalement, on construit et on analyse un schéma de discrétisation du système à partir des résultats précédents, selon la méthodologie dite HMM (Heterogeneous Multiscale Method). On met en évidence l'ordre de convergence par rapport au paramètre d'échelle temporelle et aux différents paramètres du schéma numérique- on étudie les convergences au sens fort (approximation des trajectoires) et au sens faible (approximation des lois). Dans une seconde partie, on étudie une méthode d'approximation de solutions d'EDP paraboliques, en combinant une approche semi-lagrangienne et une discrétisation de type Monte-Carlo. On montre d'abord dans un cas simplifié que la variance dépend des pas de discrétisation- enfin on fournit des simulations numériques de solutions, afin de mettre en avant les applications possibles d'une telle méthode. / In a first part, we are interested in the behavior of a system of Stochastic PDEs with two time-scales- more precisely, we focus on the approximation of the slow component thanks to an efficient numerical scheme. We first prove an averaging principle, which states that the slow component converges to the solution of the so-called averaged equation. We then show that a numerical scheme of Euler type provides a good approximation of an unknown coefficient appearing in the averaged equation. Finally, we build and we analyze a discretization scheme based on the previous results, according to the HMM methodology (Heterogeneous Multiscale Method). We precise the orders of convergence with respect to the time-scale parameter and to the parameters of the numerical discretization- we study the convergence in a strong sense - approximation of the trajectories - and in a weak sense - approximation of the laws. In a second part, we study a method for approximating solutions of parabolic PDEs, which combines a semi-lagrangian approach and a Monte-Carlo discretization. We first show in a simplified situation that the variance depends on the discretization steps. We then provide numerical simulations of solutions, in order to show some possible applications of such a method.