The aim of this thesis is to estimate the volatility function of continuoustime stochastic models. The estimation of the volatility of the following wellknown international stock market indexes is presented as an application: Dow Jones Industrial Average, Standard and Poor’s 500, NIKKEI 225, CAC 40, DAX 30, FTSE 100 and IBEX 35. This estimation is studied from two different perspectives: a) assuming that the volatility of the stock market indexes displays shortrange dependence (SRD), and b) extending the previous model for processes with longrange dependence (LRD), intermediaterange dependence (IRD) or SRD. Under the efficient market hypothesis (EMH), the compatibility of the Vasicek, the CIR, the Anh and Gao, and the CKLS models with the stock market indexes is being tested. Nonparametric techniques are presented to test the affinity of these parametric volatility functions with the volatility observed from the data. Under the assumption of possible statistical patterns in the volatility process, a new estimation procedure based on the Whittle estimation is proposed. This procedure is theoretically and empirically proven. In addition, its application to the stock market indexes provides interesting results.

Thesis (Ph.D.)--University of Adelaide, Dept. of Electrical and Electronic Engineering, 2003. / "January 2003" Includes bibliographical references (leaves 120-129).

Dans cette Thèse, la conception robuste avec un modèle incertain d'un système électromécanique avec vibro-impact est fait. Le système électromécanique est constitué d'un chariot, dont le mouvement est excité par un moteur à courant continu et un marteau embarqué dans ce chariot. Le marteau est relié au chariot par un ressort non linéaire et par un amortisseur linéaire, de façon qu'un mouvement relatif existe entre eux. Une barrière flexible linéaire, placé à l'extérieur du chariot limite les mouvements de marteau. En raison du mouvement relatif entre le marteau et la barrière, impacts peuvent se produire entre ces deux éléments. Le modèle du système développé prend en compte l'influence du courant continu moteur dans le comportement dynamique du système. Certains paramètres du système sont incertains, tels comme les coefficients de rigidité et d'amortissement de la barrière flexible. L'objectif de la Thèse est de réaliser une optimisation de ce système électromécanique par rapport aux paramètres de conception afin de maximiser l'impact puissance sous la contrainte que la puissance électrique consommée par le moteur à courant continu est inférieure à une valeur maximale. Pour choisir les paramètres de conception dans le problème d'optimisation, une analyse de sensibilité a été réalisée afin de définir les paramètres du système les plus sensibles. L'optimisation est formulée dans le cadre de la conception robuste en raison de la présence d'incertitudes dans le modèle. Les lois de probabilités liées aux variables aléatoires du problème sont construites en utilisant le Principe du Maximum l'Entropie et les statistiques de la réponse stochastique du système sont calculées en utilisant la méthode de Monte Carlo. L'ensemble d'équations non linéaires sont présentés, et un solveur temporel adapté est développé. Le problème d'optimisation non linéaire stochastique est résolu pour différents niveaux d'incertitudes, et aussi pour le cas déterministe. Les résultats sont différents, ce qui montre l'importance de la modélisation stochastique / In this Thesis, the robust design with an uncertain model of a vibro-impact electromechanical system is done. The electromechanical system is composed of a cart, whose motion is excited by a DC motor (motor with continuous current), and an embarked hammer into this cart. The hammer is connected to the cart by a nonlinear spring component and by a linear damper, so that a relative motion exists between them. A linear flexible barrier, placed outside of the cart, constrains the hammer movements. Due to the relative movement between the hammer and the barrier, impacts can occur between these two elements. The developed model of the system takes into account the influence of the DC motor in the dynamic behavior of the system. Some system parameters are uncertain, such as the stiffness and the damping coefficients of the flexible barrier. The objective of the Thesis is to perform an optimization of this electromechanical system with respect to design parameters in order to maximize the impact power under the constraint that the electric power consumed by the DC motor is lower than a maximum value. To chose the design parameters in the optimization problem, an sensitivity analysis was performed in order to define the most sensitive system parameters. The optimization is formulated in the framework of robust design due to the presence of uncertainties in the model. The probability distributions of random variables are constructed using the Maximum Entropy Principle and statistics of the stochastic response of the system are computed using the Monte Carlo method. The set of nonlinear equations are presented, and an adapted time domain solver is developed. The stochastic nonlinear constrained design optimization problem is solved for different levels of uncertainties, and also for the deterministic case. The results are different and this show the importance of the stochastic modeling

A produção de frango de corte tem impressionado pelo dinamismo e pela competência conquistada nas últimas décadas, destacando-se o Brasil como o segundo maior produtor dessa proteína animal. O ganho de produtividade, associado à coordenação da cadeia avícola, colocou o País como um dos mais eficientes produtores. Entretanto, a significativa especialização da atividade tende a excluir do processo produtivo os pequenos avicultores e os produtores menos eficientes. Assim, o estudo buscou medir a eficiência econômica dos produtores de frango de corte das regiões Sul e Centro-Oeste do Brasil, identificando quais fatores influenciam essa medida de desempenho. Para isso, usou-se uma função fronteira de lucro estocástica em um estágio (modelo 2) em que os coeficientes da fronteira e os efeitos da ineficiência são obtidos simultaneamente, assumindo que os termos de erro não são identicamente distribuídos. Concluiu-se que para a região Sul, o preço da mão-de-obra contratada interfere significativamente na lucratividade das unidades produtivas, o que seria um dos fatores associados ao maior uso do trabalho familiar no desenvolvimento das atividades. Além disso, os resultados sugerem a presença de uma melhor utilização das áreas ocupadas com a produção avícola. Os efeitos da ineficiência são sentidos principalmente no baixo nível de educação dos que tomam as decisões e nos índices elevados de conversão alimentar. Já no Centro-Oeste, os coeficientes indicaram que o maior uso de mão-de-obra familiar poderia elevar a lucratividade dos produtores. Pelo fato de ser uma região relativamente nova e possuir condições favoráveis ao investimento em capital e tecnologia, estaria indicando um maior lucro na atividade. Não se identificou ganho de escala por meio do modelo estocástico. Embora não se observa uma tendência contínua associada à escala de eficiência, no Centro-Oeste parece existirem ganhos de eficiência relevante nos estratos de médio e alto escala de produção para os padrões regionais. / The dynamism and ability acquired through the last decade by the broiler production is very impressive. The productivity rate gain associated to the good management of poultry chain in Brazil led the country to be the second biggest producer of this animal’s protein. However, the significant specialization of this activity tend to exclude the smaller producers and those who are lesser efficient in the productive process. In view of that, this study intends to precise the economic efficiency of the broiler producers in the Southerner and Center-Southerner regions of Brazil. For that, the main factors that influence the economic efficiency were identified. To reach those results, it was considered the stochastic profit function in a stage (model 2) where the coefficients of the frontier and the inefficiency effects are obtained simultaneously, since the terms of error are not identical distributing. It was possible to conclude that in the Southerner region of Brazil, the contracted labor force influence significantly the profitability of the productive farms. That’s one of the factors that explain the great use of the familiar labor force in that region. Moreover, the results pointed a better utilization of the broiler producing areas in Brazil. The effects of the inefficiency are present mainly in the producers and players with low educational level, and also in the high food conversion ratio. In the Center-Westerner of Brazil, the coefficients pointed that the better use of the familiar manpower could raise the probability of the producers. For been a new region that present positive conditions for the investments in capital and technology, it also obtained a larger profit with the activity. It was not identified gains by the stochastic model. None successively trend associated to the efficiency scale were pointed in the Center-Westerner of Brazil. Even so, the region shows efficiency gains in the middle and high scales of production in regional terms.

Orientador: Pedro Jose Catuogno / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-09T00:50:27Z (GMT). No. of bitstreams: 1 Stelmastchuk_SimaoNicolau_D.pdf: 537546 bytes, checksum: f06c81c8cd3b758c84d267af8373abdd (MD5) Previous issue date: 2007 / Resumo: Neste trabalho estudamos os martingales no fibrado de bases e suas relações com os martingales no fibrado tangente. Caracterizamos as aplicações harmônicas a valores no fibrado de bases e as relacionamos com as aplicações harmônicas a valores no fibrado tangente. Numa segunda parte estudamos a harmonicidade das seções de um fibrado via geometria estocástica. Seja P(M;G) um fibrado principal e E(M;N; G; P) um fibrado associado a P(M;G). Entre outros resultados obtemos que: uma seção s : M - E é harmônica se, e somente se, o seu levantamento eqüivariante Fs : P - N é horizontalmente harmônico; e se a ação à esquerda de G × N em N não fixa pontos então não existe seção s : M - E harmônica ou toda seção harmônica é nula / Abstract: Neste trabalho estudamos os martingales no fibrado de bases e suas relações com os martingales no fibrado tangente. Caracterizamos as aplicações harmônicas a valores no fibrado de bases e as relacionamos com as aplicações harmônicas a valores no fibrado tangente. Numa segunda parte estudamos a harmonicidade das seções de um fibrado via geometria estocástica. Seja P(M;G) um fibrado principal e E(M;N; G; P) um fibrado associado a P(M;G). Entre outros resultados obtemos que: uma seção s : M - E é harmônica se, e somente se, o seu levantamento eqüivariante Fs : P - N é horizontalmente harmônico; e se a ação à esquerda de G × N em N não fixa pontos então não existe seção s : M - E harmônica ou toda seção harmônica é nula / Doutorado / Geometria Estocastica / Doutor em Matemática

Dans cette Thèse, la conception robuste avec un modèle incertain d'un système électromécanique avec vibro-impact est fait. Le système électromécanique est constitué d'un chariot, dont le mouvement est excité par un moteur à courant continu et un marteau embarqué dans ce chariot. Le marteau est relié au chariot par un ressort non linéaire et par un amortisseur linéaire, de façon qu'un mouvement relatif existe entre eux. Une barrière flexible linéaire, placé à l'extérieur du chariot limite les mouvements de marteau. En raison du mouvement relatif entre le marteau et la barrière, impacts peuvent se produire entre ces deux éléments. Le modèle du système développé prend en compte l'influence du courant continu moteur dans le comportement dynamique du système. Certains paramètres du système sont incertains, tels comme les coefficients de rigidité et d'amortissement de la barrière flexible. L'objectif de la Thèse est de réaliser une optimisation de ce système électromécanique par rapport aux paramètres de conception afin de maximiser l'impact puissance sous la contrainte que la puissance électrique consommée par le moteur à courant continu est inférieure à une valeur maximale. Pour choisir les paramètres de conception dans le problème d'optimisation, une analyse de sensibilité a été réalisée afin de définir les paramètres du système les plus sensibles. L'optimisation est formulée dans le cadre de la conception robuste en raison de la présence d'incertitudes dans le modèle. Les lois de probabilités liées aux variables aléatoires du problème sont construites en utilisant le Principe du Maximum l'Entropie et les statistiques de la réponse stochastique du système sont calculées en utilisant la méthode de Monte Carlo. L'ensemble d'équations non linéaires sont présentés, et un solveur temporel adapté est développé. Le problème d'optimisation non linéaire stochastique est résolu pour différents niveaux d'incertitudes, et aussi pour le cas déterministe. Les résultats sont différents, ce qui montre l'importance de la modélisation stochastique / In this Thesis, the robust design with an uncertain model of a vibro-impact electromechanical system is done. The electromechanical system is composed of a cart, whose motion is excited by a DC motor (motor with continuous current), and an embarked hammer into this cart. The hammer is connected to the cart by a nonlinear spring component and by a linear damper, so that a relative motion exists between them. A linear flexible barrier, placed outside of the cart, constrains the hammer movements. Due to the relative movement between the hammer and the barrier, impacts can occur between these two elements. The developed model of the system takes into account the influence of the DC motor in the dynamic behavior of the system. Some system parameters are uncertain, such as the stiffness and the damping coefficients of the flexible barrier. The objective of the Thesis is to perform an optimization of this electromechanical system with respect to design parameters in order to maximize the impact power under the constraint that the electric power consumed by the DC motor is lower than a maximum value. To chose the design parameters in the optimization problem, an sensitivity analysis was performed in order to define the most sensitive system parameters. The optimization is formulated in the framework of robust design due to the presence of uncertainties in the model. The probability distributions of random variables are constructed using the Maximum Entropy Principle and statistics of the stochastic response of the system are computed using the Monte Carlo method. The set of nonlinear equations are presented, and an adapted time domain solver is developed. The stochastic nonlinear constrained design optimization problem is solved for different levels of uncertainties, and also for the deterministic case. The results are different and this show the importance of the stochastic modeling

No description available.

<div>In the past two decades, the pharmaceutical industry has been engaged in modernization of its drug development and manufacturing strategies, spurred onward by changing market pressures, regulatory encouragement, and technological advancement. Concomitant with these changes has been a shift toward new modalities of manufacturing in support of patient-centric medicine and on-demand production. To achieve these objectives requires manufacturing platforms which are both flexible and scalable, hence the interest in development of small-scale, continuous processes for synthesis, purification and drug product production. Traditionally, the downstream steps begin with a crystalline drug powder – the effluent of the final purification steps – and convert this to tablets or capsules through a series of batch unit operations reliant on powder processing. As an alternative, additive manufacturing technologies provide the means to circumvent difficulties associated with dry powder rheology, while being inherently capable of flexible production.</div><div>Through the combination of physical knowledge, experimental work, and data-driven methods, a framework was developed for ink formulation and process operation in drop-on-demand manufacturing with non-Brownian suspensions. Motivated by the challenges at hand, application of novel computational image analysis techniques yielded insight into the effects of non-Brownian particles and fluid properties on rheology. Furthermore, the extraction of modal and statistical information provided insight into the stochastic events which appear to play a notable role in drop formation from such suspensions. These computer vision algorithms can readily be applied by other researchers interested in the physics of drop coalescence and breakup in order to further modeling efforts.</div><div>Returning to the realm of process development to deal with challenges of monitoring and quality control initiated by suspension-based manufacturing, these machine vision algorithms were combined with Bayesian modeling to enact a probabilistic control strategy at the level of each dosage unit by utilizing the real-time image data acquired by an online process image sensor. Drawing upon a large historical database which spanned a wide range of conditions, a hierarchical modeling approach was used to incorporate the various sources of uncertainty inherent to the manufacturing process and monitoring technology, therefore providing more reliable predictions for future data at in-sample and out-of-sample conditions.</div><div>This thesis thus contributes advances in three closely linked areas: additive manufacturing of solid oral drug products, computer vision methods for event recognition in drop formation, and Bayesian hierarchical modeling to predict the probability that each dosage unit produced is within specifications.</div><div><br></div>

[ES] El tema de la investigación se centra en técnicas avanzadas para manejar problemas de aguas subterráneas y superficiales relacionados con métodos inversos y cambio climático. Los filtros de Kalman, con especial atención en Ensemble Smoother with Multiple Data Assimilation (ES-MDA), se analizan y mejoran para la solución de diferentes tipos de problemas inversos. En particular, la principal novedad es la aplicación de estos métodos para la identificación de series temporales. La primera parte de la tesis, luego de la descripción del método, presenta el desarrollo de un software escrito en Python para la aplicación de la metodología propuesta. El software cuenta con un flujo de trabajo flexible que puede adaptarse fácilmente para implementar diferentes variantes del filtro de Kalman y ser aplicado para la solución de varios tipos de problemas. Un paquete de herramientas proporciona varias funcionalidades que permiten de configurar el algoritmo de acuerdo con el problema específico analizado. La primera aplicación se refiere a la solución del problema inverso de flujo en ríos. Este es un procedimiento inverso destinado a estimar el flujo de entrada a un sistema hidráulico en función de información recopilada abajo. El procedimiento se prueba mediante dos ejemplos sintéticos y un estudio de caso real; se investiga el impacto de los tamaños de los conjuntos y la aplicación de técnicas de localización e inflación de covarianzas. Los resultados muestran la capacidad del método propuesto de resolver este tipo de problemas; el rendimiento de ES-MDA mejora, especialmente para tamaños de conjuntos pequeños, cuando se aplican técnicas de inflación y localización de covarianza. La segunda aplicación en el campo de las aguas superficiales se refiere a la calibración de un modelo hidrológico-hidráulico que simula los mecanismos de formación de eventos de inundación. ES-MDA se acopla al modelo numérico de forma paralela para la estimación de los coeficientes de rugosidad e infiltración en base al conocimiento de un hidrograma de flujo en una sección del dominio. Los resultados de dos casos sintéticos y un estudio de caso real demuestran la capacidad del método propuesto para calibrar el modelo hidrológico-hidráulico con un tiempo computacional razonable. En el campo de aguas subterráneas, ES-MDA se aplica por primera vez para identificar simultáneamente la ubicación de la fuente y el historial de liberación de un contaminante en un acuífero a partir de datos de concentración detectados en diferentes puntos del dominio. Se realizaron numerosas pruebas para evaluar la influencia de la distribución espacial y temporal de los datos de concentración, el número del conjunto y el uso de técnicas de localización e inflación; además, se presenta un nuevo procedimiento para realizar una localización iterativa espacio-temporal. La metodología se valida mediante un ejemplo analítico y un estudio de caso que utiliza datos obtenidos en el laboratorio mediante una caja de arena. ES-MDA conduce a una buena estimación de los parámetros investigados; una red de monitoreo bien diseñada y la aplicación de correcciones de covarianza mejoran el rendimiento del método y ayudan a mitigar el posible problema de no unicidad de la solución. Otro propósito de la tesis es investigar el efecto del cambio climático en las aguas subterráneas. Se presenta un modelo simplificado que describe la respuesta de los niveles de agua subterránea a las variables meteorológicas hasta 2100. Es un enfoque estadístico sencillo basado en las correlaciones entre los niveles de agua subterránea y dos índices de sequía que dependen de los datos de precipitación y temperatura. El método se utiliza para evaluar el impacto del cambio climático en los recursos de agua subterránea en un área de estudio ubicada en el norte de Italia utilizando datos históricos y de modelos climáticos regionales. Los resultados m / [CA] El tema de la investigació se centra en tècniques avançades per a manejar problemes d'aigües subterrànies i superficials relacionats amb mètodes inversos i canvi climàtic. Els filtres de Kalman, amb especial atenció en Ensemble Smoother with Multiple Data Assimilation (ES-MDA), s'analitzen i milloren per a la solució de diferents tipus de problemes inversos. En particular, la principal novetat és l'aplicació d'aquests mètodes per a la identificació de sèries temporals. La primera part de la tesi presenta el desenvolupament d'un programari escrit en Python per l'aplicació de la metodologia presentada. El programari compta amb un flux de treball flexible que pot adaptar-se fàcilment per a implementar diferents variants del filtre de Kalman i ser aplicat per a la solució de diversos tipus de problemes. Un paquet complementar d'eines proporciona diverses funcionalitats que permeten de configurar l'algorisme d'acord amb el problema específic analitzat. La primera aplicació es un nou enfocament per la solució del problema invers de flux en rius. Aquest és un procediment invers destinat a estimar el flux d'entrada a un sistema hidràulic en funció d'informació recopilada aigües avall. El procediment es prova mitjançant dos exemples sintètics i un estudi de cas real; s'investiga l'impacte de les grandàries dels conjunts i l'aplicació de tècniques de localització i inflació de covariàncies. Els resultats mostren la capacitat del mètode proposat de resoldre aquest tipus de problemes; el rendiment de ES-MDA millora, especialment per a grandàries de conjunts xicotets, quan s'apliquen tècniques d'inflació i localització de covariància. La segona aplicació en el camp de les aigües superficials es refereix al calibratge d'un model hidrològic-hidràulic que simula els mecanismes de formació d'esdeveniments d'inundació a partir de sollicitació hidrometeorológicas i la seua posterior propagació. ES-MDA s'acobla al model numèric de manera paral·lela per l'estimació dels coeficients de rugositat i infiltració sobre la base del coneixement d'un hidrograma de flux en una secció del domini. Els resultats de dos casos sintètics i un estudi de cas real demostren la capacitat del mètode proposat per calibrar el model hidrològic-hidràulic amb un temps computacional raonable. En el camp d'aigües subterrànies, ES-MDA s'aplica per primera vegada per identificar simultàniament la ubicació de la font i l'historial d'alliberament d'un contaminant en un aqüífer a partir d'un conjunt de dades de concentració detectats en diferents punts del domini. Es van realitzar nombroses proves per avaluar la influència de la distribució espacial i temporal de les dades de concentració, el número del conjunt i l'ús de tècniques de localització i inflació; a més, es presenta un nou procediment per realitzar una localització iterativa espaciotemporal. La metodologia es valguda mitjançant un exemple analític i un estudi de cas per al qual s'utilitzen dades obtingudes en el laboratori mitjançant una caixa d'arena. ES-MDA condueix a una bona reconstrucció dels paràmetres investigats; una xarxa de monitoratge ben dissenyada i l'aplicació de correccions de covariància milloren el rendiment del mètode i ajuden a mitigar el possible problema de no unicitat de la solució. Un altre propòsit de la tesi és investigar l'efecte del canvi climàtic en les aigües subterrànies. Es presenta un model simplificat que descriu la resposta dels nivells d'aigua subterrània a les variables meteorològiques fins a 2100. És un enfocament estadístic senzill basat en les correlacions entre els nivells d'aigua subterrània i dos índexs de sequera que depenen de les dades de precipitació i temperatura. El mètode s'utilitza per a avaluar l'impacte del canvi climàtic en els recursos d'aigua subterrània en una àrea d'estudi situada en el nord d'Itàlia utilitzant dades històriques i de models climàtics regionals. / [EN] This work focuses on the investigation of advanced techniques to handle groundwater and surface water problems in the framework of inverse methods and climate change. The Ensemble Kalman filter methods, with particular attention to the Ensemble Smoother with Multiple Data Assimilation (ES-MDA), are extensively analyzed and improved for the solution of different types of inverse problems. In particular, the main novelty is the application of these methods for the identification of time series function. In the first part of the thesis, after the description of the ES-MDA method, the development of a Python software package for the application of the proposed methodology is presented. It is designed with a flexible workflow that can be easily adapted to implement different variants of the Ensemble Kalman filter and to be applied for the solution of various types of inverse problems. A complemented tool package provides several functionalities that allow to setup the algorithm configuration suiting the specific analyzed problem. The first novelty application of the ES-MDA method aimed at solving the reverse flow routing problem. The objective of the inverse procedure is the estimation of an unknown inflow hydrograph to a hydraulic system on the basis of information collected downstream and a given forward routing model that relates inflow hydrograph and downstream observations. The procedure is tested by means of two synthetic examples and a real case study; the impact of ensemble sizes and the application of covariance localization and inflation techniques are also investigated. The tests show the capability of the proposed method to solve this type of problem; the performance of ES-MDA improves, especially for small ensemble sizes, when covariance localization and inflation techniques are applied. The second application, in the context of surface water, concerns the calibration of a hydrological-hydraulic model that simulates rainfall-runoff processes. The ES-MDA is coupled with the numerical model by parallel way for the estimation of roughness and infiltration coefficients based on the knowledge of a discharge hydrograph at the basin outlet. The results of two synthetic tests and a real case study demonstrate the capability of the proposed method to calibrate the hydrological-hydraulic model with a reasonable computational time. In the groundwater field, ES-MDA is applied for the first time to simultaneously identify the source location and the release history of a contaminant spill in an aquifer from a sparse set of concentration data collected in few points of the aquifer. The impacts of the concentration sampling scheme, the ensemble size and the use of covariance localization and covariance inflation techniques are tested; furthermore, a new procedure to perform a spatiotemporal iterative localization is presented. The methodology is tested by means of an analytical example and a study case that uses real data collected in a laboratory sandbox. ES-MDA leads to a good estimation of the investigated parameters; a well-designed monitoring network and the use of covariance corrections improve the performance of the method and help to minimize ill-posedness and equifinality. A part of the thesis investigates the impact of climate change on the groundwater availability. A surrogate model that describes the response of groundwater levels to meteorological variables up to 2100 is presented. It is a simple statistical approach based on the correlations between groundwater levels and two drought indices that depend on precipitation and temperature data. The presented method is used to evaluate the impact of climate change on groundwater resources in a study area located in Northern Italy using historical and regional climate model data. The results denote a progressive increase of groundwater droughts in the investigated area. / Todaro, V. (2021). Advanced techniques for solving groundwater and surface water problems in the context of inverse methods and climate change [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/166439 / TESIS

<div>Uncertainty quantification techniques based on the spectral approach have been studied extensively in the literature to characterize and quantify, at low computational cost, the impact that uncertainties may have on large-scale engineering problems. One such technique is the <i>generalized polynomial chaos</i> (gPC) which utilizes a time-independent orthogonal basis to expand a stochastic process in the space of random functions. The method uses a specific Askey-chaos system that is concordant with the measure defined in the probability space in order to ensure exponential convergence to the solution. For nearly two decades, this technique has been used widely by several researchers in the area of uncertainty quantification to solve stochastic problems using the spectral approach. However, a major drawback of the gPC method is that it cannot be used in the resolution of problems that feature strong nonlinear dependencies over the probability space as time progresses. Such downside arises due to the time-independent nature of the random basis, which has the undesirable property to lose unavoidably its optimality as soon as the probability distribution of the system's state starts to evolve dynamically in time.</div><div><br></div><div>Another technique is the <i>time-dependent generalized polynomial chaos</i> (TD-gPC) which utilizes a time-dependent orthogonal basis to better represent the stochastic part of the solution space (aka random function space or RFS) in time. The development of this technique was motivated by the fact that the probability distribution of the solution changes with time, which in turn requires that the random basis is frequently updated during the simulation to ensure that the mean-square error is kept orthogonal to the discretized RFS. Though this technique works well for problems that feature strong nonlinear dependencies over the probability space, the TD-gPC method possesses a serious issue: it suffers from the curse of dimensionality at the RFS level. This is because in all gPC-based methods the RFS is constructed using a tensor product of vector spaces with each of these representing a single RFS over one of the dimensions of the probability space. As a result, the higher the dimensionality of the probability space, the more vector spaces needed in the construction of a suitable RFS. To reduce the dimensionality of the RFS (and thus, its associated computational cost), gPC-based methods require the use of versatile sparse tensor products within their numerical schemes to alleviate to some extent the curse of dimensionality at the RFS level. Therefore, this curse of dimensionality in the TD-gPC method alludes to the need of developing a more compelling spectral method that can quantify uncertainties in long-time response of dynamical systems at much lower computational cost.</div><div><br></div><div>In this work, a novel numerical method based on the spectral approach is proposed to resolve the curse-of-dimensionality issue mentioned above. The method has been called the <i>flow-driven spectral chaos</i> (FSC) because it uses a novel concept called <i>enriched stochastic flow maps</i> to track the evolution of a finite-dimensional RFS efficiently in time. The enriched stochastic flow map does not only push the system's state forward in time (as would a traditional stochastic flow map) but also its first few time derivatives. The push is performed this way to allow the random basis to be constructed using the system's enriched state as a germ during the simulation and so as to guarantee exponential convergence to the solution. It is worth noting that this exponential convergence is achieved in the FSC method by using only a few number of random basis vectors, even when the dimensionality of the probability space is considerably high. This is for two reasons: (1) the cardinality of the random basis does not depend upon the dimensionality of the probability space, and (2) the cardinality is bounded from above by <i>M+n+1</i>, where <i>M</i> is the order of the stochastic flow map and <i>n</i> is the order of the governing stochastic ODE. The boundedness of the random basis from above is what makes the FSC method be curse-of-dimensionality free at the RFS level. For instance, for a dynamical system that is governed by a second-order stochastic ODE (<i>n=2</i>) and driven by a stochastic flow map of fourth-order (<i>M=4</i>), the maximum number of random basis vectors to consider within the FSC scheme is just 7, independent whether the dimensionality of the probability space is as low as 1 or as high as 10,000.</div><div><br></div><div>With the aim of reducing the complexity of the presentation, this dissertation includes three levels of abstraction for the FSC method, namely: a <i>specialized version</i> of the FSC method for dealing with structural dynamical systems subjected to uncertainties (Chapter 2), a <i>generalized version</i> of the FSC method for dealing with dynamical systems governed by (nonlinear) stochastic ODEs of arbitrary order (Chapter 3), and a <i>multi-element version</i> of the FSC method for dealing with dynamical systems that exhibit discontinuities over the probability space (Chapter 4). This dissertation also includes an implementation of the FSC method to address the dynamics of large-scale stochastic structural systems more effectively (Chapter 5). The implementation is done via a modal decomposition of the spatial function space as a means to reduce the number of degrees of freedom in the system substantially, and thus, save computational runtime.</div>