Chip temperature is increasing with continued technology scaling due to increased power density and decreased device feature sizes. Since temperature has significant impact on performance and reliability, accurate thermal and circuit analysis are of great importance. Due to the shrinking device feature size, effects occurring at the nanometre scale, such as ballistic transport of energy carriers and electron tunneling, have become increasingly important and must be considered. However, many existing thermal and circuit analysis methods are not able to consider these effects efficiently, if at all. This thesis presents methods for accurate and efficient multi-scale thermal and circuit analysis. For circuit analysis, the simulation of single-electron device circuits is specifically studied. To target thermal analysis, in this work, ThermalScope, a multi-scale thermal analysis method for nanometre-scale IC design is developed. It unifies microscopic and macroscopic thermal physics modeling methods, i.e., the Boltzmann transport and Fourier modeling methods. Moreover, it supports adaptive multi-resolution modeling. Together, these ideas enable efficient and accurate characterization of nanometre-scale heat transport as well as chip-package level heat flow. ThermalScope is designed for full chip thermal analysis of billion-transistor nanometre-scale IC designs, with accuracy at the scale of individual devices. ThermalScope has been implemented in software and used for full chip thermal analysis and temperature-dependent leakage analysis of an IC design with more than 150 million transistors. To target circuit analysis, in this work, SEMSIM, a multi-scale single-electron device simulator is developed with an adaptive simulation technique based on the Monte Carlo method. This technique significantly improves the time efficiency while maintaining accuracy for single-electron device and circuit simulation. It is shown that it is possible to reduce simulation time up to nearly 40 times and maintain an average propagation delay error of under 5% compared to a non-adaptive Monte Carlo method. This simulator has been used to handle large circuit benchmarks with more than 6000 junctions, showing efficiency comparable to SPICE, with much better accuracy. In addition, the simulator can characterize important secondary effects including cotunneling and Cooper pair tunneling, which are critical for device research. / Thesis (Master, Electrical & Computer Engineering) -- Queen's University, 2008-09-26 13:33:12.389

Cette thèse traite de la solution numérique de deux types de problèmes stochastiques. Premièrement, nous nous intéressons aux EDS fortement oscillantes, c'est-à-dire, les systèmes composés de variables ergodiques évoluant rapidement par rapport aux autres. Nous proposons un algorithme basé sur des résultats d'homogénéisation. Il est défini par un schéma d'Euler appliqué aux variables lentes couplé avec un estimateur à pas décroissant pour approcher la limite ergodique des variables rapides. Nous prouvons la convergence forte de l'algorithme et montrons que son erreur normalisée satisfait un résultat du type théorème limite centrale généralisé. Nous proposons également une version extrapolée de l'algorithme ayant une meilleure complexité asymptotique en satisfaisant les mêmes propriétés que la version originale. Ensuite, nous étudions la solution des EDS de type McKean-Vlasov (EDSPR-MKV) associées à la solution de certains problèmes de contrôle sous un environnement formé d'un grand nombre de particules ayant des interactions du type champ-moyen. D'abord, nous présentons un nouvel algorithme, basé sur la méthode de cubature sur l'espace de Wiener, pour approcher faiblement la solution d'une EDS du type McKean-Vlasov. Il est déterministe et peut être paramétré pour atteindre tout ordre de convergence souhaité. Puis, en utilisant ce nouvel algorithme, nous construisons deux schémas pour résoudre les EDSPR-MKV découplées et nous montrons que ces schémas ont des convergences d'ordres un et deux. Enfin, nous considérons le problème de réduction de la complexité de la méthode présentée tout en respectant la vitesse de convergence énoncée. / This Ph.D. thesis deals with the numerical solution of two types of stochastic problems. First, we investigate the numeric solution to strongly oscillating SDEs, i.e. systems in which some ergodic state variables evolve quickly with respect to the remaining ones. We propose an algorithm that uses homogenization results and consists of an Euler scheme for the slow scale variables coupled with a decreasing step estimator for the ergodic averages of the fast variables. We prove the strong convergence of the algorithm as well as a generalized central limit theorem result for the normalized error distribution. In addition, we propose an extrapolated version applicable under stronger regularity assumptions and which satisfies the same properties of the original algorithm with lower asymptotic complexity. Then, we treat the problem of solving decoupled Forward Backward Stochastic Differential equations of McKean-Vlasov type (MKV-FBSDE) which appear in some stochastic control problems in an environment of a large number of particles with mean field interactions. As a first step, we propose a new algorithm, based on the cubature method on Wiener spaces, to weakly approach the solution of a McKean-Vlasov SDE. It is deterministic and can be parametrized to obtain any given order of convergence. Using this first forward approximation algorithm, we construct two procedures to solve the decoupled MKV-FBSDE and show that they converge with orders one and two under appropriate regularity conditions. Finally, we consider the problem of reducing the complexity of the presented method while preserving the presented convergence rates.

As an important endogenous source of reactive oxygen species (ROS), NADPH oxidase 1 (Nox1) has received tremendous attention in the past few decades. It has been identified to play a key role as the initial "kindle," whose activation is crucial for amplifying ROS production through several propagation mechanisms in the vascular system. As a consequence, Nox1 has been implicated in the initiation and genesis of many cardiovascular diseases and has therefore been the subject of detailed investigations. The literature on experimental studies of the Nox1 system is extensive. Numerous investigations have identified essential features of the Nox1 system in vasculature and characterized key components, possible regulatory signals and/or signaling pathways, potential activation mechanisms, a variety of Nox1 stimuli, and its potential physiological and pathophysiological functions. While these experimental studies have greatly enhanced our understanding of the Nox1 system, many open questions remain regarding the overall functionality and dynamic behavior of Nox1 in response to specific stimuli. Such questions include the following. What are the main regulatory and/or activation mechanisms of Nox1 systems in different types of vascular cells? Once Nox1 is activated, how does the system return to its original, unstimulated state, and how will its subunits be recycled? What are the potential disassembly pathways of Nox1? Are these pathways equally important for effectively reutilizing Nox1 subunits? How does Nox1 activity change in response to dynamic signals? Are there generic features or principles within the Nox1 system that permit optimal performance? These types of questions have not been answered by experiments, and they are indeed quite difficult to address with experiments. I demonstrate in this dissertation that one can pose such questions and at least partially answer them with mathematical and computational methods. Two specific cell types, namely endothelial cells (ECs) and vascular smooth muscle cells (VSMCs), are used as "templates" to investigate distinct modes of regulation of Nox1 in different vascular cells. By using a diverse array of modeling methods and computer simulations, this research identifies different types of regulation and their distinct roles in the activation process of Nox1. In the first study, I analyze ECs stimulated by mechanical stimuli, namely shear stresses of different types. The second study uses different analytical and simulation methods to reveal generic features of alternative disassembly mechanisms of Nox1 in VSMCs. This study leads to predictions of the overall dynamic behavior of the Nox1 system in VSMCs as it responds to extracellular stimuli, such as the hormone angiotensin II. The studies and investigations presented here improve our current understanding of the Nox1 system in the vascular system and might help us to develop potential strategies for manipulation and controlling Nox1 activity, which in turn will benefit future experimental and clinical studies.