Thesis (Ph. D.)--University of Washington, 2000. / Vita. Includes bibliographical references (p. 188-196).

The inversion method for generating non-uniform random variates has some advantages compared to other generation methods, since it monotonically transforms uniform random numbers into non-uniform random variates. Hence it is the method of choice in the simulation literature. However, except for some simple cases where the inverse of the cumulative distribution function is a simple function we need numerical methods. Often inversion by ``brute force" is used, applying either very slow iterative methods or linear interpolation of the CDF and huge tables. But then the user has to accept unnecessarily large errors or excessive memory requirements, that slow down the algorithm. In this paper we demonstrate that with Hermite interpolation of the inverse CDF we can obtain very small error bounds close to machine precision. Using our adaptive interval splitting method this accuracy is reached with moderately sized tables that allow for a fast and simple generation procedure. The algorithms described in this paper have been implemented in ANSI C in a library called UNURAN which is available via anonymous ftp. (author's abstract) / Series: Preprint Series / Department of Applied Statistics and Data Processing

Obtaining the distribution of the profit and loss (PL) of a portfolio is a key problem in market risk measurement. However, existing methods, such as those based on the Normal distribution, and historical simulation methods, which use empirical distribution of risk factors, face difficulties in dealing with at least one of the following three problems: describing the distributional properties of risk factors appropriately (description problem); deriving distributions of risk factors with time horizon longer than one day (time aggregation problem); and deriving the distribution of the PL given the distributional properties of the risk factors (risk aggregation problem). Here, we show that expansion methods can provide reasonable solutions to all three problems. Expansion methods approximate a probability density function by a sum of orthogonal polynomials multiplied by an associated weight function. One of the most important advantages of expansion methods is that they only require moments of the target distribution up to some order to obtain an approximation. Therefore they have the potential to be applied in a wide range of situations, including in attempts to solve the three problems listed above. On the other hand, it is also known that expansions lack robustness: they often exhibit unignorable negative density and their approximation quality can be extremely poor. This limits applications of expansion methods in existing studies. In this thesis, we firstly develop techniques to provide robustness, with which expansion methods result in a practical approximation quality in a wider range of examples than investigated to date. Specifically, we investigate three techniques: standardisation, use of Laguerre expansion and optimisation. Standardisation applies expansion methods to a variable which is transformed so that its first and second moments are the same as those of the weight function. Use of Laguerre expansions applies those expansions to a risk factor so that heavy tails can be captured better. Optimisation considers expansions with coefficients of polynomials optimised so that the difference between the approximation and the target distribution is minimised with respect to mean integrated squared error. We show, by numerical examples using data sets of stock index returns and log differences of implied volatility, and GARCH models, that expansions with our techniques are more robust than conventional expansion methods. As such, marginal distributions of risk factors can be approximated by expansion methods. This solves a part of the description problem: the information on the marginal distributions of risk factors can be summarised by their moments. Then we show that the dependence structure among risk factors can be summarised in terms of their cross-moments. This solves the other part of the description problem. We also use the fact that moments of risk factors can be aggregated using their moments and cross-moments, to show that expansion methods can be applied to both the time and risk aggregation problems. Furthermore, we introduce expansion methods for multivariate distributions, which can also be used to approximate conditional expectations and copula densities by rational functions.

Thesis (Ph.D.)--University of Wollongong, 2004. / Typescript. Includes bibliographical references: leaf 230-241.

Techn. Universiẗat, Diss., 2000--Darmstadt.

A distribuição de massa molar (DMM) e seus parâmetros são de fundamental importância na caracterização dos polímeros. Por este motivo, o desenvolvimento de técnicas que permitam a determinação da DMM de forma mais rápida e a menor custo é de grande importância prática. Os principais objetivos deste trabalho foram a implementação de alguns dos modelos baseados da teoria da reptação dupla propostos na literatura para descrever o mecanismo de relaxação das cadeias poliméricas, a avaliação dessas implementações e a análise de dois passos fundamentais na obtenção da DMM a partir de dados reológicos que são a metodologia de cálculo do espectro de relaxação baseado no modelo de Maxwell e a estratégia para a avaliação numérica das integrais que aparecem nos modelos de relaxação. Foi resolvido o problema denominado problema inverso, ou seja, a determinação da DMM a partir de dados reológicos usando um modelo de relaxação especificado e uma função de distribuição imposta. Foi usada a função Exponencial Generalizada (GEX) para representar a probabilidade de distribuição, sendo consideradas duas abordagens: i) cálculo explícito do espectro de relaxação e ii) aproximações paramétricas de Schwarzl, que evitam a necessidade do cálculo explícito do espectro de relaxação. A metodologia de determinação da DMM foi aplicada para amostras de polietileno e foram estimadas distribuições com boa representação dos dados experimentais do GPC, ao considerarem-se amostras com polidispersões inferiores a 10. Com relação a metodologia de cálculo do espectro de relaxação, foi realizado um estudo comparativo da aplicação de espectros de relaxação discreto e contínuo, com o objetivo de estabelecer critérios para especificação do número ótimo de modos de Maxwell a serem considerados. Ao efetuar-se a comparação entre as técnicas, verificou-se o espectro discreto apresenta como um sistema melhor condicionado, permitindo assim obter maior confiabilidade dos parâmetros estimados. Também é proposta uma modificação da metodologia de determinação da DMM, em que é aplicada a quadratura de Gauss-Hermite para a resolução numérica da integral dos modelos de relaxação. / The molecular weight distribution (MWD) and its parameters are of the fundamental importance in the characterization of polymers. Therefore, the development of techniques for faster and less time consuming determination of the MWD is of great practical relevance. The goals of this work were the implementation of some of the relaxation models from double reptation theory proposed in the literature, the evaluation of these implementations and the analysis of two key points in the recovery of the MWD from rheological data which are the methodology for calculation of the relaxation spectrum based on the Maxwell model and the numeric strategy for the evaluation of the integrals appearing in the relaxation models. The inverse problem, i.e., the determination of the MWD from rheological data using a specified relaxation model and an imposed distribution function, was solved. In the analysis of the inverse problem, the Generalized Exponential (GEX) was used as distribution function and two approaches were considered: i) explicit calculation of the relaxation spectrum and ii) use of the parametric method proposed by Schwarzl to avoid the explicit calculation of the relaxation spectrum. In the test of commercial samples of polyethylene with polidispersity less than 10, the application of this methodology led to MWD curves which provided good fit of the experimental SEC data. Regarding the methodology for calculation of the relaxation spectrum, a comparison between the performance of discrete and continuous relaxation spectrum was performed and some possible a criteria to determine the appropriate number of relaxation modes of Maxwell to be used were evaluated. It was found that the technique of discrete spectrum leads to better conditioned systems and, consequently, greater reliability of the estimated parameters. With relation to the numeric strategy for the evaluation of the integrals appearing in the relaxation models, the use of Gauss-Hermite quadrature using a new change of variables was proposed.

A distribuição de massa molar (DMM) e seus parâmetros são de fundamental importância na caracterização dos polímeros. Por este motivo, o desenvolvimento de técnicas que permitam a determinação da DMM de forma mais rápida e a menor custo é de grande importância prática. Os principais objetivos deste trabalho foram a implementação de alguns dos modelos baseados da teoria da reptação dupla propostos na literatura para descrever o mecanismo de relaxação das cadeias poliméricas, a avaliação dessas implementações e a análise de dois passos fundamentais na obtenção da DMM a partir de dados reológicos que são a metodologia de cálculo do espectro de relaxação baseado no modelo de Maxwell e a estratégia para a avaliação numérica das integrais que aparecem nos modelos de relaxação. Foi resolvido o problema denominado problema inverso, ou seja, a determinação da DMM a partir de dados reológicos usando um modelo de relaxação especificado e uma função de distribuição imposta. Foi usada a função Exponencial Generalizada (GEX) para representar a probabilidade de distribuição, sendo consideradas duas abordagens: i) cálculo explícito do espectro de relaxação e ii) aproximações paramétricas de Schwarzl, que evitam a necessidade do cálculo explícito do espectro de relaxação. A metodologia de determinação da DMM foi aplicada para amostras de polietileno e foram estimadas distribuições com boa representação dos dados experimentais do GPC, ao considerarem-se amostras com polidispersões inferiores a 10. Com relação a metodologia de cálculo do espectro de relaxação, foi realizado um estudo comparativo da aplicação de espectros de relaxação discreto e contínuo, com o objetivo de estabelecer critérios para especificação do número ótimo de modos de Maxwell a serem considerados. Ao efetuar-se a comparação entre as técnicas, verificou-se o espectro discreto apresenta como um sistema melhor condicionado, permitindo assim obter maior confiabilidade dos parâmetros estimados. Também é proposta uma modificação da metodologia de determinação da DMM, em que é aplicada a quadratura de Gauss-Hermite para a resolução numérica da integral dos modelos de relaxação. / The molecular weight distribution (MWD) and its parameters are of the fundamental importance in the characterization of polymers. Therefore, the development of techniques for faster and less time consuming determination of the MWD is of great practical relevance. The goals of this work were the implementation of some of the relaxation models from double reptation theory proposed in the literature, the evaluation of these implementations and the analysis of two key points in the recovery of the MWD from rheological data which are the methodology for calculation of the relaxation spectrum based on the Maxwell model and the numeric strategy for the evaluation of the integrals appearing in the relaxation models. The inverse problem, i.e., the determination of the MWD from rheological data using a specified relaxation model and an imposed distribution function, was solved. In the analysis of the inverse problem, the Generalized Exponential (GEX) was used as distribution function and two approaches were considered: i) explicit calculation of the relaxation spectrum and ii) use of the parametric method proposed by Schwarzl to avoid the explicit calculation of the relaxation spectrum. In the test of commercial samples of polyethylene with polidispersity less than 10, the application of this methodology led to MWD curves which provided good fit of the experimental SEC data. Regarding the methodology for calculation of the relaxation spectrum, a comparison between the performance of discrete and continuous relaxation spectrum was performed and some possible a criteria to determine the appropriate number of relaxation modes of Maxwell to be used were evaluated. It was found that the technique of discrete spectrum leads to better conditioned systems and, consequently, greater reliability of the estimated parameters. With relation to the numeric strategy for the evaluation of the integrals appearing in the relaxation models, the use of Gauss-Hermite quadrature using a new change of variables was proposed.

In this thesis we deal with two problems in harmonic analysis. In the first problem we discuss weighted norm inequalities for Weyl multipliers satisfying Mauceri’s condition. As an application, we prove certain multiplier theorems on the Heisenberg group and also show in the context of a theorem of Weis on operator valued Fourier multipliers that the R-boundedness of the derivative of the multiplier is not necessary for the boundedness of the multiplier transform. In the second problem we deal with a variation of a theorem of Mauceri concerning the Lp bound-edness of operators Mwhich are known to be bounded on L2 .We obtain sufficient conditions on the kernel of the operaor Mso that it satisfies weighted Lp estimates. As an application we prove Lp boundedness of Hermite pseudo-multipliers.

A distribuição de massa molar (DMM) e seus parâmetros são de fundamental importância na caracterização dos polímeros. Por este motivo, o desenvolvimento de técnicas que permitam a determinação da DMM de forma mais rápida e a menor custo é de grande importância prática. Os principais objetivos deste trabalho foram a implementação de alguns dos modelos baseados da teoria da reptação dupla propostos na literatura para descrever o mecanismo de relaxação das cadeias poliméricas, a avaliação dessas implementações e a análise de dois passos fundamentais na obtenção da DMM a partir de dados reológicos que são a metodologia de cálculo do espectro de relaxação baseado no modelo de Maxwell e a estratégia para a avaliação numérica das integrais que aparecem nos modelos de relaxação. Foi resolvido o problema denominado problema inverso, ou seja, a determinação da DMM a partir de dados reológicos usando um modelo de relaxação especificado e uma função de distribuição imposta. Foi usada a função Exponencial Generalizada (GEX) para representar a probabilidade de distribuição, sendo consideradas duas abordagens: i) cálculo explícito do espectro de relaxação e ii) aproximações paramétricas de Schwarzl, que evitam a necessidade do cálculo explícito do espectro de relaxação. A metodologia de determinação da DMM foi aplicada para amostras de polietileno e foram estimadas distribuições com boa representação dos dados experimentais do GPC, ao considerarem-se amostras com polidispersões inferiores a 10. Com relação a metodologia de cálculo do espectro de relaxação, foi realizado um estudo comparativo da aplicação de espectros de relaxação discreto e contínuo, com o objetivo de estabelecer critérios para especificação do número ótimo de modos de Maxwell a serem considerados. Ao efetuar-se a comparação entre as técnicas, verificou-se o espectro discreto apresenta como um sistema melhor condicionado, permitindo assim obter maior confiabilidade dos parâmetros estimados. Também é proposta uma modificação da metodologia de determinação da DMM, em que é aplicada a quadratura de Gauss-Hermite para a resolução numérica da integral dos modelos de relaxação. / The molecular weight distribution (MWD) and its parameters are of the fundamental importance in the characterization of polymers. Therefore, the development of techniques for faster and less time consuming determination of the MWD is of great practical relevance. The goals of this work were the implementation of some of the relaxation models from double reptation theory proposed in the literature, the evaluation of these implementations and the analysis of two key points in the recovery of the MWD from rheological data which are the methodology for calculation of the relaxation spectrum based on the Maxwell model and the numeric strategy for the evaluation of the integrals appearing in the relaxation models. The inverse problem, i.e., the determination of the MWD from rheological data using a specified relaxation model and an imposed distribution function, was solved. In the analysis of the inverse problem, the Generalized Exponential (GEX) was used as distribution function and two approaches were considered: i) explicit calculation of the relaxation spectrum and ii) use of the parametric method proposed by Schwarzl to avoid the explicit calculation of the relaxation spectrum. In the test of commercial samples of polyethylene with polidispersity less than 10, the application of this methodology led to MWD curves which provided good fit of the experimental SEC data. Regarding the methodology for calculation of the relaxation spectrum, a comparison between the performance of discrete and continuous relaxation spectrum was performed and some possible a criteria to determine the appropriate number of relaxation modes of Maxwell to be used were evaluated. It was found that the technique of discrete spectrum leads to better conditioned systems and, consequently, greater reliability of the estimated parameters. With relation to the numeric strategy for the evaluation of the integrals appearing in the relaxation models, the use of Gauss-Hermite quadrature using a new change of variables was proposed.

The main goal of the thesis is to construct explicit, s-stage, strong-stability-preserving (SSP) Hermite–Birkhoff (HB) time discretization methods of order p with nonnegative coefficients for the integration of hyperbolic conservation laws. The Shu–Osher form and the canonical Shu–Osher form by means of the vector formulation for SSP Runge–Kutta (RK) methods are extended to SSP HB methods. The SSP coefficients of k-step, s-stage methods of order p, HB(k,s,p), as combinations of k-step methods of order (p − 3) with s-stage explicit RK methods of order 4, and k-step methods of order (p-4) with s-stage explicit RK methods of order 5, respectively, for s = 4, 5,..., 10 and p = 4, 5,..., 12, are constructed and compared with other methods. The good efficiency gains of the new, optimal, SSP HB methods over other SSP methods, such as Huang’s hybrid methods and RK methods, are numerically shown by means of their effective SSP coefficients and largest effective CFL numbers. The formulae of these new, optimal methods are presented in their Shu–Osher form.