Na presente dissertação fazemos um resumo das idéias fundamentais do método de Quantização Estocástica de Parisi e Wu, com aplicações a teorias de campo Escalares, de Gauge e Fermiônicas. Em particular, nós utilizamos o esquema de Regularização Analítica Estocástica no cálculo do tensor de polarização para a Eletrodinâmica Quântica com Bósons ou Fêrmions de Dirac. A influência da regularização na invariância de Gauge e estudada para ambas as teorias, e é sugerida uma extensão do método para alguns modelos supersimétricos. / In the present dissertation, we made a survey of the fundamental ideas about Parisi-Wu\'s Stochastic Quantization Method, with applications to Scalar, Gauge and Fermionic theories. In particular, we use the Analytic Stochastic Regularization Scheme to calculate the polarization tensor for Quantum Electrodynamics with bosons or Dirac Fermions. The regularization influence is studied for both theories and an extension of this method for some supersymmetrical models is suggested.

Na presente dissertação fazemos um resumo das idéias fundamentais do método de Quantização Estocástica de Parisi e Wu, com aplicações a teorias de campo Escalares, de Gauge e Fermiônicas. Em particular, nós utilizamos o esquema de Regularização Analítica Estocástica no cálculo do tensor de polarização para a Eletrodinâmica Quântica com Bósons ou Fêrmions de Dirac. A influência da regularização na invariância de Gauge e estudada para ambas as teorias, e é sugerida uma extensão do método para alguns modelos supersimétricos. / In the present dissertation, we made a survey of the fundamental ideas about Parisi-Wu\'s Stochastic Quantization Method, with applications to Scalar, Gauge and Fermionic theories. In particular, we use the Analytic Stochastic Regularization Scheme to calculate the polarization tensor for Quantum Electrodynamics with bosons or Dirac Fermions. The regularization influence is studied for both theories and an extension of this method for some supersymmetrical models is suggested.

With the current demand for mixed-signal SoCs, an increasing number of designers are looking for ADC architectures that can be easily implemented over digital substrates. Since ADC performance is strongly dependent upon physical and electrical features, it gets more difficult for them to benefit from more recent technologies, where these features are more variable. This way, analog signal acquisition is not allowed to follow an evolutionary trend compatible with Moore’s Law. In fact, such trend shall get worst, since newer technologies are expected to have more variable characteristics. Also, for a matter of economy of scale, many times a mixed-signal SoC presents a good amount of idle processing power. In such systems it is advantageous to employ more costly digital signal processing provided that it allows a reduction in the analog area demanded or the use of less expensive analog blocks, able to cope with process variations and uncertainty. Besides the technological concerns, other factors that impact the cost of the design also advise to transfer problems from the analog to the digital domain whenever possible: design automation and self-test requirements, for instance. Recent surveys indicate that the total cost in designer hours for the analog blocks of a mixed-signal system can be up to three times the cost of the digital ones. This manuscript explores the concept of bottom-up analog acquisition design, using statistical sampling as a way to reduce the analog area demanded in the design of ADCs within mixed-signal systems. More particularly, it investigates the possibility of using digital modeling and digital compensation of non-idealities to ease the design of ADCs. The work is developed around three axes: the definition of target applications, the development of digital compensation algorithms and the exploration of architectural possibilities. New methods and architectures are defined and validated. The main notions behind the proposal are analyzed and it is shown that the approach is feasible, opening new paths of future research. Keywords:

With the current demand for mixed-signal SoCs, an increasing number of designers are looking for ADC architectures that can be easily implemented over digital substrates. Since ADC performance is strongly dependent upon physical and electrical features, it gets more difficult for them to benefit from more recent technologies, where these features are more variable. This way, analog signal acquisition is not allowed to follow an evolutionary trend compatible with Moore’s Law. In fact, such trend shall get worst, since newer technologies are expected to have more variable characteristics. Also, for a matter of economy of scale, many times a mixed-signal SoC presents a good amount of idle processing power. In such systems it is advantageous to employ more costly digital signal processing provided that it allows a reduction in the analog area demanded or the use of less expensive analog blocks, able to cope with process variations and uncertainty. Besides the technological concerns, other factors that impact the cost of the design also advise to transfer problems from the analog to the digital domain whenever possible: design automation and self-test requirements, for instance. Recent surveys indicate that the total cost in designer hours for the analog blocks of a mixed-signal system can be up to three times the cost of the digital ones. This manuscript explores the concept of bottom-up analog acquisition design, using statistical sampling as a way to reduce the analog area demanded in the design of ADCs within mixed-signal systems. More particularly, it investigates the possibility of using digital modeling and digital compensation of non-idealities to ease the design of ADCs. The work is developed around three axes: the definition of target applications, the development of digital compensation algorithms and the exploration of architectural possibilities. New methods and architectures are defined and validated. The main notions behind the proposal are analyzed and it is shown that the approach is feasible, opening new paths of future research. Keywords:

With the current demand for mixed-signal SoCs, an increasing number of designers are looking for ADC architectures that can be easily implemented over digital substrates. Since ADC performance is strongly dependent upon physical and electrical features, it gets more difficult for them to benefit from more recent technologies, where these features are more variable. This way, analog signal acquisition is not allowed to follow an evolutionary trend compatible with Moore’s Law. In fact, such trend shall get worst, since newer technologies are expected to have more variable characteristics. Also, for a matter of economy of scale, many times a mixed-signal SoC presents a good amount of idle processing power. In such systems it is advantageous to employ more costly digital signal processing provided that it allows a reduction in the analog area demanded or the use of less expensive analog blocks, able to cope with process variations and uncertainty. Besides the technological concerns, other factors that impact the cost of the design also advise to transfer problems from the analog to the digital domain whenever possible: design automation and self-test requirements, for instance. Recent surveys indicate that the total cost in designer hours for the analog blocks of a mixed-signal system can be up to three times the cost of the digital ones. This manuscript explores the concept of bottom-up analog acquisition design, using statistical sampling as a way to reduce the analog area demanded in the design of ADCs within mixed-signal systems. More particularly, it investigates the possibility of using digital modeling and digital compensation of non-idealities to ease the design of ADCs. The work is developed around three axes: the definition of target applications, the development of digital compensation algorithms and the exploration of architectural possibilities. New methods and architectures are defined and validated. The main notions behind the proposal are analyzed and it is shown that the approach is feasible, opening new paths of future research. Keywords:

Le projet de thèse comporte différentes directions possibles: a) Améliorer la compréhension des relations entre la théorie des structures de régularité développée par M. Hairer et la méthode des Distributions Paracontrolées développée par Gubinelli, Imkeller et Perkowski, et éventuellement fournir une synthèse des deux. C'est très spéculatif et, pour le moment, il n'y a pas de chemin clair vers cet objectif à long terme. b) Utiliser la théorie des Distributions Paracontrolées pour étudier différents types d'équations aux dérivés partiels: équations de transport et équations générales d'évolution hyperbolique, équations dispersives, systèmes de lois de conservation. Ces EDP ne sont pas dans le domaine des méthodes actuelles qui ont été développées principalement pour gérer les équations d'évolution semi-linéaire parabolique. c) Une fois qu'une théorie pour l'équation de transport perturbée par un signal irregulier a été établie, il sera possible de se dédier à l'étude des phénomènes de régularisation par le bruit qui, pour le moment, n'ont étés étudiés que dans le contexte des équations de transport perturbées par le mouvement brownien, en utilisant des outils standard d'analyse stochastique. d) Les techniques du Groupe de Renormalisation (GR) et les développements multi-échelles ont déjà été utilisés à la fois pour aborder les EDP et pour définir des champs quantiques euclidiens. La théorie des Distributions Paracontrolées peut être comprise comme une sorte d'analyse multi-échelle des fonctionnels non linéaires et il serait intéressant d'explorer l'interaction des techniques paradifférentielles avec des techniques plus standard, comme les "cluster expansions" et les méthodes liées au GR. / The thesis project has various possible directions: a) Improve the understanding of the relations between the theory of Regularity Structures developed by M.Hairer and the method of Paracontrolled Distributions developed by Gubinelli, Imkeller and Perkowski, and eventually to provide a synthesis. This is highly speculative and at the moment there are no clear path towards this long term goal. b) Use the theory of Paracontrolled Distributions to study different types of PDEs: transport equations and general hyperbolic evolution equation, dispersive equations, systems of conservation laws. These PDEs are not in the domain of the current methods which were developed mainly to handle parabolic semilinear evolution equations. c) Once a theory of transport equation driven by rough signals have been established it will become possible to tackle the phenomena of regularization by transport noise which for the moment has been studied only in the context of transport equations driven by Brownian motion, using standard tools of stochastic analysis. d) Renormalization group (RG) techniques and multi-scale expansions have already been used both to tackle PDE problems and to define Euclidean Quantum Field Theories. Paracontrolled Distributions theory can be understood as a kind of mul- tiscale analysis of non-linear functionals and it would be interesting to explore the interplay of paradifferential techniques with more standard techniques like cluster expansions and RG methods.