L'objectif de la thèse est l'étude mathématique et l'analyse numérique du problème non linéaire de Sobolev. Un premier chapitre est consacré à l'analyse a priori pour le problème de Sobolev où on utilise des méthodes de semi-discrétisation explicite en temps. Des estimations d'erreurs ont été obtenues assurant que les schémas numériques utilisés convergent lorsque le pas de discrétisation en temps et le pas de discrétisation en espace tendent vers zéro. Dans le second chapitre, on s'intéresse au problème de Sobolev singulièrement perturbé. En vue de la stabilité des schémas numériques, on utilise dans cette partie des méthodes numériques implicites (la méthode d'Euler et la méthode de Crank- Nicolson) pour discrétiser le problème par rapport au temps. Dans le troisième chapitre, on présente des applications et des illustrations où on utilise le logiciel "FreeFem++". Dans le dernier chapitre, on considère une équation de type Sobolev et on s'intéresse à la dérivation d'estimations d'erreur a posteriori pour la discrétisation de cette équation par la méthode des éléments finis conforme en espace et un schéma d'Euler implicite en temps. La borne supérieure est globale en espace et en temps et permet le contrôle effectif de l'erreur globale. A la fin du chapitre, on propose un algorithme adaptatif qui permet d'atteindre une précision relative fixée par l'utilisateur en raffinant les maillages adaptativement et en équilibrant les contributions en espace et en temps de l'erreur. On présente également des essais numériques. / The purpose of this work is the mathematical study and the numerical analysis of the nonlinear Sobolev problem. A first chapter is devoted to the a priori analysis for the Sobolev problem, where we use an explicit semidiscretization in time. A priori error estimates were obtained ensuring that the used numerical schemes converge when the time step discretization and the spatial step discretization tend to zero. In a second chapter, we are interested in the singularly perturbed Sobolev problem. For the stability of numerical schemes, we used in this part implicit semidiscretizations in time (the Euler method and the Crank-Nicolson method). Our estimates of Chapters 1 and 2 are confirmed in the third chapter by some numerical experiments. In the last chapter, we consider a Sobolev equation and we derive a posteriori error estimates for the discretization of this equation by a conforming finite element method in space and an implicit Euler scheme in time. The upper bound is global in space and time and allows effective control of the global error. At the end of the chapter, we propose an adaptive algorithm which ensures the control of the total error with respect to a user-defined relative precision by refining the meshes adaptively, equilibrating the time and space contributions of the error. We also present numerical experiments.

The diploma thesis describes the bootstrap method and its applications in the estimate accuracy statement, in the confidence intervals generation and in the testing of statistical hypotheses. Further the method of the discrete probability estimation of the categorical quantity is presented, making use the gradient of the quasi-norm hereof distribution. On concrete examples the bootstrap method is applied in the confidence intervals forming of the categorical quantity probability function. The diploma thesis was supported by the project of MŠMT of the Czech Republic no. 1M06047 "Centre for Quality and Reliability of Production", by the grant of Grant Agency of the Czech Republic (Czech Science Foundation) reg. no. 103/08/1658 "Advanced optimum design of composed concrete structures" and by the research plan of MŠMT of the Czech Republic no. MSM0021630519 "Progressive reliable and durable structures".

The diploma thesis describes the bootstrap method and its applications in the confidence intervals generation, in the testing of statistical hypotheses and in the regression analysis. We present the confidence interval for individual value. Further the method of discrete probability estimation of the categorical quantity is presented, making use the gradient and the line estimate.

Nos anos noventa, com o desenvolvimento da informática, as variadas alternativas de comunicação mediadas pela Internet (rede internacional de computadores) surgem como um importante instrumento para pesquisas em populações. Essas possibilidades estão associadas fundamentalmente com a rapidez de troca de informações e com o grande número de usuários que podem acessar a esse meio. O objetivo do presente estudo foi a elaboração e a validade relativa de um questionário semiquantitativo de freqüência alimentar on-line (QSFA) para avaliar ingestão de Ca e Fe. O site (sítio) de acesso à pesquisa e as interações foram realizadas por meio do jornal eletrônico, Folha Online através do qual foram selecionados 30 Internautas (usuários da rede), de ambos os sexos, residentes no Estado de São Paulo, com idades de 21 a 45 anos. Para a validação do QSFA on-line, foram comparadas as informações obtidas do registro de quatro dias (R4) aplicados em dois meses seguidos, com aquelas obtidas do QSFA on-line aplicado no terceiro mês subseqüente. Foram realizados dois testes de validade relativa, correlação e teste do sinal. Os dois testes demonstraram que o QSFA on-line pode ser utilizado para avaliar média de ingestão de Ca da mesma forma que o R4. Em relação ao Fe, o QSFA on-line deve ser reavaliado em uma amostra maior. O coeficiente de correlação de Pearson, depois do ajuste e da correção pela variância foi de 0,57 e -0,13 para Ca e Fe, respectivamente. Os resultados indicam que o QSFA on-line apresentou validade relativa para avaliar média de ingestão de Ca. / The development of computers in the 90\'s, brought different alternatives of communication which are now available through the Internet and have become an important tool for possible researches in population. These possibilities are essentially associated with the speed that information can be exchanged and with the number of users that can access this means.The objective of this study was to develop and validate a semi-quantitative food-frequency questionnaire on-line (QSFA) to evaluate the intake of Calcium and Iron. The site of access to the research and the interaction were obtained through a newspaper entitled Folha Online. Thirty internet users were selected, of both sexes, living in the state of São Paulo, ages 21 to 45. To validate the QSFA on-line, the information obtained from a 4-day period (R4) during two consecutive months, was compared to the information obtained from the QSFA on-line applied during the third month thereafter. In order to check the adequate intake of calcium and iron, two relative validity tests were used: correlations and sign test and both tests showed that the QSFA on-line can be used to evaluate the average intake of Ca, as well as the R4. With regards to the Iron nutrients, the QSFA on-line should be reevaluated on a larger sampling. The Pearson\'s correlation coefficients, after adjustment and correction for variance was 0,57 and -0,13 for calcium and iron respectively. The results indicated that the QSFA on-line, presented relative validity to evaluate the average calcium intake.

Many physical problems lead to boundary value problems for partial differential equations, which can be solved with the finite element method. In order to construct adaptive solution algorithms or to measure the error one aims at reliable a posteriori error estimators. Many such estimators are known, as well as their theoretical foundation. Some boundary value problems yield so-called anisotropic solutions (e.g. with boundary layers). Then anisotropic finite element meshes can be advantageous. However, the common error estimators for isotropic meshes fail when applied to anisotropic meshes, or they were not investigated yet. For rectangular or cuboidal anisotropic meshes a modified error estimator had already been derived. In this paper error estimators for anisotropic tetrahedral or triangular meshes are considered. Such meshes offer a greater geometrical flexibility. For the Poisson equation we introduce a residual error estimator, an estimator based on a local problem, several Zienkiewicz-Zhu estimators, and an L_2 error estimator, respectively. A corresponding mathematical theory is given.For a singularly perturbed reaction-diffusion equation a residual error estimator is derived as well. The numerical examples demonstrate that reliable and efficient error estimation is possible on anisotropic meshes. The analysis basically relies on two important tools, namely anisotropic interpolation error estimates and the so-called bubble functions. Moreover, the correspondence of an anisotropic mesh with an anisotropic solution plays a vital role. AMS(MOS): 65N30, 65N15, 35B25

Arnold and Strauss (1991) 探討2x2列聯表中的3個方格 (cell) 有相同機率θ的問題，他們比較了參數θ的最大概似估計值與最大擬概似估計值，發現參數θ的最大概似估計值與最大擬概似估計值是不相同的。在本論文中，我們將2x2列聯表中的3個方格的參數值 (機率值)，從限制為相同θ，放寬為成某種比例，並證明了在一般情況下參數θ的最大概似估計值與最大擬概似估計值也不相同。我們也提出一些使參數θ的最大概似估計值及最大擬概似估計值相同的特殊條件，諸如三個方格內的觀察值跟機率值成比例或格子內的觀察值有某些特定值。本論文也透過電腦模擬的結果，發現最大概似估計式較最大擬概似估計式來得精確，而且當參數θ在參數空間之中點附近時，最大概似估計值與最大擬概似估計值的差異為最大。 / Arnold and Strauss (1991) study the cases that three of the four cells in the 2x2 contingency table have the same cell probability θ. In particular, Arnold and Strauss (1991) compare the maximum likelihood estimate (MLE) and maximum pseudolikelihood estimate (MPLE) of the parameter θ. They find that MLE and MPLE of the parameter are not the same. In this thesis, we relax the assumptions so that those three cell probabilities may not be the same and each is proportional to a parameter θ. We find that, in general, MLE’s of θ are still not the same as MPLE’s of θ. Some special cases that make MLE the same as MPLE are also given. We also find, through computer simulations, that MLE’s are accurate than MPLE’s and that the difference between MLE and MPLE is getting larger when the parameter θ is closer to the midpoint of its space.

Conselho Nacional de Desenvolvimento Científico e Tecnológico / This dissertation presents a methodology of capacitance estimation and power consumption in CMOS circuits combinational constituted basically of complex logic gates at logic level. The main objective in the development of this method is to provide a fast estimate of the power consumption of circuits at the logical design gates. Of this form, the considered method allows to the application of techniques to the reduction of power consumption or the alteration of the design before being prototyped. The consumed dynamic power in complex logic gates depends on the following factors: switching activity of each circuit node, voltage of supplies, parasite capacitance and clock frequency. With the exception of the parasite capacitance, all other parameters are easily determined. The analysis proposed in this dissertation, treats estimative of the dynamic power consumption of complex logic gates, through the estimate of the parasite capacitance CMOS devices. The model considered here concentrates all internal capacitances on the external gate nodes depending on the combinations of the input signals. The resulting capacitance in an only external node of an input of the gate is resulted of the transitions of inputs of the too much nodes on the node that if wants to determine. The results obtained in this work, regarding the estimate of power consumption of the complex logic gates, had been considered satisfactory, once they had presented a maximum error of 10% when compared with to the electric simulation result preformed with ELDO. Moreover, the method supplies significant reduction in the simulation time of the circuits, being able esteem the power consumption of a circuit up to 200 times faster than gotten to the simulated electric level with ELDO tool. / Este trabalho apresenta uma metodologia de estimativa de capacitâncias e de consumo de potência de circuitos CMOS constituídos basicamente por portas lógicas complexas no nível lógico. O principal objetivo no desenvolvimento deste método é fazer uma rápida previsão do consumo de potência de circuitos ainda na fase de projeto lógico composto de portas complexas. Desta forma, o método proposto permite a aplicação de técnicas de redução de consumo de potência ou a alteração de todo o projeto antes de ser prototipado. A potência dinâmica consumida em portas lógicas complexas depende dos seguintes fatores: atividade de comutação de cada nó do circuito, tensão de alimentação, freqüência de operação e da capacitância parasita. Com a exceção da capacitância parasita, todos os demais parâmetros são facilmente determinados. A análise proposta nesta dissertação, trata da aproximação (cálculo aproximado) do consumo de potência dinâmica de portas lógicas complexas, através da estimativa da capacitância parasita dos dispositivos CMOS. O modelo aqui proposto concentra as capacitâncias nos nós externos das portas, que variam em função das combinações dos sinais de entrada. A capacitância resultante, representada em um único nó externo da entrada da porta analisada, é resultado das transições dos sinais das demais entradas que agem sobre o nó que se quer determinar. Os resultados obtidos neste trabalho a respeito da estimativa de consumo potência das portas lógicas complexas foram considerados satisfatórios, pois apresentaram um erro máximo de 10% quando comparados às simulações elétricas pelo uso da ferramenta ELDO. Além disso, o método fornece significante redução no tempo de simulação dos circuitos, podendo estimar o consumo de potência de um circuito até 200 vezes mais rápido que obtido ao nível elétrico simulado com a ferramenta ELDO.

Nos anos noventa, com o desenvolvimento da informática, as variadas alternativas de comunicação mediadas pela Internet (rede internacional de computadores) surgem como um importante instrumento para pesquisas em populações. Essas possibilidades estão associadas fundamentalmente com a rapidez de troca de informações e com o grande número de usuários que podem acessar a esse meio. O objetivo do presente estudo foi a elaboração e a validade relativa de um questionário semiquantitativo de freqüência alimentar on-line (QSFA) para avaliar ingestão de Ca e Fe. O site (sítio) de acesso à pesquisa e as interações foram realizadas por meio do jornal eletrônico, Folha Online através do qual foram selecionados 30 Internautas (usuários da rede), de ambos os sexos, residentes no Estado de São Paulo, com idades de 21 a 45 anos. Para a validação do QSFA on-line, foram comparadas as informações obtidas do registro de quatro dias (R4) aplicados em dois meses seguidos, com aquelas obtidas do QSFA on-line aplicado no terceiro mês subseqüente. Foram realizados dois testes de validade relativa, correlação e teste do sinal. Os dois testes demonstraram que o QSFA on-line pode ser utilizado para avaliar média de ingestão de Ca da mesma forma que o R4. Em relação ao Fe, o QSFA on-line deve ser reavaliado em uma amostra maior. O coeficiente de correlação de Pearson, depois do ajuste e da correção pela variância foi de 0,57 e -0,13 para Ca e Fe, respectivamente. Os resultados indicam que o QSFA on-line apresentou validade relativa para avaliar média de ingestão de Ca. / The development of computers in the 90\'s, brought different alternatives of communication which are now available through the Internet and have become an important tool for possible researches in population. These possibilities are essentially associated with the speed that information can be exchanged and with the number of users that can access this means.The objective of this study was to develop and validate a semi-quantitative food-frequency questionnaire on-line (QSFA) to evaluate the intake of Calcium and Iron. The site of access to the research and the interaction were obtained through a newspaper entitled Folha Online. Thirty internet users were selected, of both sexes, living in the state of São Paulo, ages 21 to 45. To validate the QSFA on-line, the information obtained from a 4-day period (R4) during two consecutive months, was compared to the information obtained from the QSFA on-line applied during the third month thereafter. In order to check the adequate intake of calcium and iron, two relative validity tests were used: correlations and sign test and both tests showed that the QSFA on-line can be used to evaluate the average intake of Ca, as well as the R4. With regards to the Iron nutrients, the QSFA on-line should be reevaluated on a larger sampling. The Pearson\'s correlation coefficients, after adjustment and correction for variance was 0,57 and -0,13 for calcium and iron respectively. The results indicated that the QSFA on-line, presented relative validity to evaluate the average calcium intake.

Durant les dernières décennies, l’évolution de la puissance de calcul a conduit au développement de codes de simulation en physique des réacteurs de plus en plus prédictifs pour la modélisation du comportement d’un réacteur nucléaire en situation de fonctionnement normal et accidentel. Un cadre d’analyse d’incertitudes cohérent avec l’utilisation de modélisations Best Estimate (BE) a été développé. On parle d’approche Best Estimate Plus Uncertain-ties (BEPU) et cette approche donne lieu `a de nombreux travaux de R&D à l’international en simulation numérique. Dans cette thèse, on étudie la quantification d’incertitudes multi-physiques dans le cas d’un transitoire d’ éjection de Grappe de contrôle (REA- Rod Ejection Accident) dans un Réacteur à Eau Pressurisée (REP). La modélisation BE actuellement disponible au CEA est réalisée en couplant les codes APOLLO3 R (netronique) et FLICA4 (thermohydraulique-thermique du combustible) dans l’environnement SALOME/CORPUS. Dans la première partie de la thèse, on examine différents outils statistiques disponibles dans la littérature scientifique dont la réduction de dimension, l’analyse de sensibilité globale, des modèles de substitution et la construction de plans d’expérience. On utilise ces outils pour développer une méthodologie de quantification d’incertitudes. Dans la deuxième partie de la thèse, on améliore la modélisation du comportement du combustible. Un couplage Best Effort pour la simulation d’un transitoire REA est disponible au CEA. Il comprend le code ALCYONE V1.4 qui permet une modélisation fine du comportement thermomécanique du combustible. Cependant, l’utilisation d’une telle modélisation conduit à une augmentation significative du temps de calcul ce qui rend actuellement difficile la réalisation d’une analyse d’incertitudes. Pour cela, une méthodologie de calibrage d’un modèle analytique simplifié pour le transfert de chaleur pastille-gaine basé sur des calculs ALCYONE V1.4 découplés a été développée. Le modèle calibré est finalement intégré dans la modélisation BE pour améliorer sa prédictivité. Ces deux méthodologies sont maquettées initialement sur un cœur de petite échelle représentatif d’un REP puis appliquées sur un cœur REP à l’échelle 1 dans le cadre d’une analyse multi-physique d’un transitoire REA. / The computational advancements of the last decades lead to the development of numerical codes for simulating the reactor physics with increa-sing predictivity allowing the modeling of the beha-vior of a nuclear reactor under both normal and acci-dental conditions. An uncertainty analysis framework consistent with Best Estimate (BE) codes was develo-ped in order to take into account the different sources of uncertainties. This framework is called Best Esti-mate Plus Uncertainties (BEPU) and is currently a field of increasing research internationally. In this the-sis we study the multi-physics uncertainty quantifi-cation for Rod Ejection Accident (REA) in Pressuri-zed Water Reactors (PWR). The BE modeling avai-lable in CEA is used with a coupling of APOLLO3 (neutronics) and FLICA4 (thermal-hydraulics and fuel-thermal) in the framework of SALOME/CORPUS tool. In the first part of the thesis, we explore different statistical tools available in the scientific literature including: dimension reduction, global sensitivity analy-sis, surrogate modeling and design of experiments. We then use them in order to develop an uncer-tainty quantification methodology. In the second part of the thesis, we improve the BE modeling in terms of its uncertainty representation. A Best Effort coupling scheme for REA analysis is available at CEA. This in-cludes ALCYONE V1.4 code for a detailed modeling of fuel-thermomechanics behavior. However, the use of such modeling increases significantly the compu-tational cost for a REA transient rendering the uncer-tainty analysis prohibited. To this purpose, we deve-lop a methodology for calibrating a simplified analytic gap heat transfer model using decoupled ALCYONE V1.4 REA calculations. The calibrated model is finally used to improve the previous BE modeling. Both de-veloped methodologies are tested initially on a small scale core representative of a PWR and then applied on a large scale PWR core.

Mit deterministischen Sicherheitsanalysen werden Auslegungsstörfälle bei Kernkraftwerken anhand von Rechenmodellen am Computer berechnet, um damit die Funktionalität der installierten Sicherheitssysteme eines jeden Kraftwerkes zu überprüfen. Allerdings sind bei solchen Störfalluntersuchungen stets Unsicherheiten vorhanden, die den zu ermittelnden Störfallablauf stark beeinflussen können. Beispielsweise können aufgrund technisch bedingter Fertigungstoleranzen schwankende Geometrie- und Materialdaten entstehen, die bei der Modellierung des Rechenmodells zu Unsicherheiten führen. Weitere Unsicherheiten können auf die physikalischen Modelle eines Störfall-analyseprogrammes zurückgeführt werden. Insbesondere haben die empirischen Beziehungen Unsicherheiten, da diese aus experimentellen Daten ermittelt wurden. In der vorliegenden Arbeit wurden daher die empirischen Beziehungen des Programmes TRACE analysiert und dessen Unsicherheiten quantifiziert. Mit der entwickelten „Dynamic Best-Estimate Safety Analysis“-Methode (DYBESA-Methode) lässt sich diese programmspezifische Unsicherheit bei der Störfalluntersuchung berücksichtigen. Es wurden 13 verschiedene „Correlation, Identification and Ranking Table“ (CIRT) erstellt, die die relevanten Unsicherheiten bei den unterschiedlichen Auslegungsstörfällen für Druckwasserreaktoren kategorisieren. Damit können Unsicherheitsanalysen basierend auf dem statistischen Verfahren nach S. S. Wilks durchgeführt werden. Schlussendlich werden die sicherheitsrelevanten Rechenergebnisse realistisch und vor allem mit einer hohen Zuverlässigkeit, im Vergleich zu einer herkömmlichen konservativen Berechnungsmethode, ermittelt.:1 Einleitung 2 Sicherheitsanforderungen an Kernkraftwerke 2.1 Einschluss radioaktiver Stoffe und Abschirmung ionisierender Strahlung 2.2 Störfallkategorien 2.3 Thermohydraulische Nachweiskriterien 3 Analysetechniken für die Durchführung von Störfalluntersuchungen 3.1 Das Störfallanalyseprogramm TRACE 3.2 Identifikation von Unsicherheiten bei Störfalluntersuchungen 3.3 Konservative Methode und Best-Estimate-Methode mit Unsicherheitsanalyse 3.4 Stand von Wissenschaft und Technik der wesentlichen Best-Estimate-Methoden 3.4.1 CSAU-Methode 3.4.2 UMAE-Methode 3.4.3 CIAU-Methode 3.4.4 GRS-Methode 3.4.5 ASTRU-Methode 4 Entwicklung der DYBESA-Methode für DWR-Störfalluntersuchungen 4.1 Thermohydraulische Phänomene beim Störfallablauf 4.2 Regressionsverfahren für die experimentelle Datenanalyse 4.2.1 Vertrauens- und Vorhersagebereich 4.2.2 Statistische Toleranzgrenzen 4.3 Identifikation von empirischen Beziehungen und deren Bewertung 4.4 Erzeugen und Kombinieren von geeigneten Stichproben 4.5 Programm zur Vorbereitung und Auswertung von Unsicherheitsanalysen 4.6 Modifikation des Störfallanalyseprogramms TRACE für die Berücksichtigung der programmspezifischen Unsicherheit 4.7 Verifikation der DYBESA-Methode 5 Ergebnisse 5.1 FEBA 5.1.1 Verifikation Rechenmodell 5.1.2 Verifikation CIRT 5.2 Marviken-Test-Station 5.2.1 Verifikation Rechenmodell 5.2.2 Verifikation CIRT 5.3 Druckwasserreaktor 5.3.1 Mittlerer Bruch einer Hauptkühlmittelleitung 5.3.2 Notstromfall 6 Diskussion und Ausblick 7 Zusammenfassung 8 Quellenverzeichnis 9 Anlagenverzeichnis