Universiẗat, Diss., 2000--Göttingen.

The system of sparse gridpoints was used to propagate uncertainty forward in time through orbital mechanics simulations. Propagation of initial uncertainty through a nonlinear dynamic model is examined in regards to the uncertainty of orbit estimation. The necessary underlying mechanics of orbital mechanics, probability, and nonlinear estimation theory are reviewed to allow greater understanding of the problem. The sparse grid method itself and its implementation is covered in detail, along with the necessary properties and how to best it to a given problem based on inputs and desired outputs. Three test cases were run in the form of a restricted two-body problem, a perturbed two-body problem, and a three-body problem in which the orbiting body is positioned at a Lagrange point. It is shown that the sparse grid method shows sufficient accuracy for all mean calculations in the given problems and that higher accuracy levels allow for accurate estimation of higher moments such as the covariance.

This thesis addresses the problem of representing few group, homogenised neutron cross sections as a function of state parameters (e.g. burn-up, fuel and moderator temperature, etc.) that describe the conditions in the reactor. The problem is multi-dimensional and the cross section samples, required for building the representation, are the result of expensive transport calculations. At the same time, practical applications require high accuracy. The representation method must therefore be efficient in terms of the number of samples needed for constructing the representation, storage requirements and cross section reconstruction time. Sparse grid methods are proposed for constructing such an efficient representation. Approximation through quasi-regression as well as polynomial interpolation, both based on sparse grids, were investigated. These methods have built-in error estimation capabilities and methods for optimising the representation, and scale well with the number of state parameters. An anisotropic sparse grid integrator based on Clenshaw-Curtis quadrature was implemented, verified and coupled to a pre-existing cross section representation system. Some ways to improve the integrator’s performance were also explored. The sparse grid methods were used to construct cross section representations for various Light Water Reactor fuel assemblies. These reactors have different operating conditions, enrichments and state parameters and therefore pose different challenges to a representation method. Additionally, an example where the cross sections have a different group structure, and were calculated using a different transport code, was used to test the representation method. The built-in error measures were tested on independent, uniformly distributed, quasi-random sample points. In all the cases studied, interpolation proved to be more accurate than approximation for the same number of samples. The primary source of error was found to be the Xenon transient at the beginning of an element’s life (BOL). To address this, the domain was split along the burn-up dimension into “start-up” and “operating” representations. As an alternative, the Xenon concentration was set to its equilibrium value for the whole burn-up range. The representations were also improved by applying anisotropic sampling. It was concluded that interpolation on a sparse grid shows promise as a method for building a cross section representation of sufficient accuracy to be used for practical reactor calculations with a reasonable number of samples. / Thesis (MSc Engineering Sciences (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2013.

This thesis addresses the problem of representing few group, homogenised neutron cross sections as a function of state parameters (e.g. burn-up, fuel and moderator temperature, etc.) that describe the conditions in the reactor. The problem is multi-dimensional and the cross section samples, required for building the representation, are the result of expensive transport calculations. At the same time, practical applications require high accuracy. The representation method must therefore be efficient in terms of the number of samples needed for constructing the representation, storage requirements and cross section reconstruction time. Sparse grid methods are proposed for constructing such an efficient representation. Approximation through quasi-regression as well as polynomial interpolation, both based on sparse grids, were investigated. These methods have built-in error estimation capabilities and methods for optimising the representation, and scale well with the number of state parameters. An anisotropic sparse grid integrator based on Clenshaw-Curtis quadrature was implemented, verified and coupled to a pre-existing cross section representation system. Some ways to improve the integrator’s performance were also explored. The sparse grid methods were used to construct cross section representations for various Light Water Reactor fuel assemblies. These reactors have different operating conditions, enrichments and state parameters and therefore pose different challenges to a representation method. Additionally, an example where the cross sections have a different group structure, and were calculated using a different transport code, was used to test the representation method. The built-in error measures were tested on independent, uniformly distributed, quasi-random sample points. In all the cases studied, interpolation proved to be more accurate than approximation for the same number of samples. The primary source of error was found to be the Xenon transient at the beginning of an element’s life (BOL). To address this, the domain was split along the burn-up dimension into “start-up” and “operating” representations. As an alternative, the Xenon concentration was set to its equilibrium value for the whole burn-up range. The representations were also improved by applying anisotropic sampling. It was concluded that interpolation on a sparse grid shows promise as a method for building a cross section representation of sufficient accuracy to be used for practical reactor calculations with a reasonable number of samples. / Thesis (MSc Engineering Sciences (Nuclear Engineering))--North-West University, Potchefstroom Campus, 2013.

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[pt] Os modelos de simulação de reservatórios estão sujeitos à incerteza presente em uma grande variedade de seus parâmetros de entrada. Esta incerteza é o resultado da heterogeneidade das formações geológicas, erros nas medições dos dados e da modelagem petrofísica, estrutural e do transporte dos fluidos no meio poroso. Uma quantificação precisa da incerteza requer, na maioria dos casos, uma quantidade elevada de simulações, o que é usualmente inviável se considerarmos o tempo consumido para simular modelos de grande escala. Por outro lado, uma avaliação adequada da incerteza aumenta a qualidade e robustez das decisões tomadas para o gerenciamento dos campos de petróleo. Com esta motivação, foi investigado o método das Expansões por Caos Polinomial (PCE, por suas siglas em inglês). PCE é uma técnica de convergência rápida utilizada para analisar como se propaga, na saída de um modelo, a incerteza presente nos parâmetros de entrada. Mediante PCE, pode-se representar a resposta aleatória de um modelo de simulação de reservatórios de petróleo como um polinômio, construído a partir de uma base de funções que dependem da distribuição de probabilidade das variáveis incertas de entrada. Por outro lado, quando a relação entre os parâmetros de entrada e a saída do modelo têm um componente não polinomial, o algoritmo de Programação Genética (PG) pode ser utilizado para representar esta dependência utilizando funções ou operadores mais complexos. PG é um algoritmo de regressão simbólica capaz de encontrar uma expressão aleatória explícita, que aproxime a saída de um modelo de simulação de reservatórios de petróleo, conhecendo-se a priori a distribuição de probabilidade dos parâmetros de entrada. Neste trabalho foram aplicadas as duas técnicas, antes mencionadas, num modelo de simulação de reservatórios baseado no campo PUNQ-S3, considerando até vinte e três parâmetros incertos durante um período de produção de 13 anos. Foi feita uma análise de incerteza, calculando-se a distribuição de probabilidade completa da saída do simulador. Os resultados foram comparados com o método de Monte Carlo, indicando um alto desempenho em termos de custo computacional e acurácia. Ambas as técnicas conseguem níveis de ajuste superiores a 80 porcento com uma quantidade de simulações consideravelmente baixa. / [en] Reservoir simulation models are subject to uncertainty in a wide variety of its inputs. This uncertainty is a result of the heterogeneity of the geological formations, data measurement errors, and petrophysical, structural, and fluid transport in porous media modelling. An accurate uncertainty quantification requires, in most cases, a large number of simulations, which is unviable considering the time it takes to simulate large scale models. On the other hand, a proper uncertainty assessment, increases the robustness of the decision making process for the oil field management. To this end, the method of Polynomial Chaos Expansions (PCE) was studied. PCE is a fast paced convergence technique, used to analyze the uncertainty propagation of the input parameters all the way to the output of the model. Through PCE is possible to represent the response of an oil reservoir simulation model as a polynomial, built from a function basis, that depend on the probability distribution of the uncertain input variables. Furthermore, when the relationship between the input and output parameters of the model has a non-polynomial component, the algorithm of Genetic Programming (GP) can be used to represent this dependency by more elaborate functions or operators. GP is a symbolic regression algorithm, capable of finding an explicit expression that approximates the output of a reservoir simulation model, with prior knowledge of the probability distribution of the input parameters. In this work, the two previously mentioned techniques were applied in a reservoir simulation model, based on the oil field PUNQ-S3, considering up to twenty three uncertain parameters during a simulation period of 13 years. An uncertainty analysis of the output of the simulator was conducted, calculating the entire probability distribution. The results were compared to the Monte Carlo simulation method, presenting a satisfactory performance in terms of accuracy and computational cost. Both techniques show adjustment levels higher than 80 percent, with a considerable small amount simulations.

Compuesta por tres capítulos: El primero es un estudio sobre la implementación the Sparse Grid métodos para es el estudio de modelos económicos con muchas dimensiones. Llevado a cabo mediante aplicaciones noveles del método de Smolyak con el objetivo de favorecer la tratabilidad y obtener resultados preciso. Los resultados muestran mejoras en la eficiencia de la implementación de modelos con múltiples agentes. El segundo capítulo introduce una nueva metodología para la evaluación de políticas económicas, llamada Synthetic Control with Statistical Learning, todo ello aplicado a políticas particulares: a) reducción del número de horas laborales en Portugal en 1996 y b) reducción del coste del despido en España en 2010. La metodología funciona y se erige como alternativa a previos métodos. En términos empíricos se muestra que tras la implementación de la política se produjo una reducción efectiva del desempleo y en el caso de España un incremento del mismo. El tercer capítulo utiliza la metodología utiliza en el segundo capítulo y la aplica para evaluar la implementación del Tercer Programa Europeo para la Seguridad Vial (Third European Road Safety Action Program) entre otras metodologías. Los resultados muestran que la coordinación a nivel europeo de la seguridad vial a supuesto una ayuda complementaria. En el año 2010 se estima una reducción de víctimas mortales de entre 13900 y 19400 personal en toda Europa.