In this paper, we present an approach to discretizing multivariate continuous data while learning the structure of a graphical model. We derive the joint scoring function from the principle of predictive accuracy, which inherently ensures the optimal trade-off between goodness of fit and model complexity (including the number of discretization levels). Using the so-called finest grid implied by the data, our scoring function depends only on the number of data points in the various discretization levels. Not only can it be computed efficiently, but it is also independent of the metric used in the continuous space. Our experiments with gene expression data show that discretization plays a crucial role regarding the resulting network structure.

Better knowledge in the areas of aerodynamics and optimization has allowed designers to develop efficient wingtip structures in recent years. However, the requirements faced by wingtip devices can be considerably different amongst an aircraft’s flight regimes. Traditional static wingtip devices are then a compromise between conflicting requirements, resulting in less than optimal performance within each regime. Alternatively, a morphing wingtip can reconfigure leading to improved performance over a range of dissimilar flight conditions. Developed within this thesis, is a modular morphing wingtip concept that centers on the use of variable geometry truss mechanisms to permit morphing. A conceptual design framework is established to aid in the development of the concept. The framework uses a metaheuristic optimization procedure to determine optimal continuous wingtip configurations. The configurations are then discretized for the modular concept. The functionality of the framework is demonstrated through a design study on a hypothetical wing/winglet within the thesis.

Better knowledge in the areas of aerodynamics and optimization has allowed designers to develop efficient wingtip structures in recent years. However, the requirements faced by wingtip devices can be considerably different amongst an aircraft’s flight regimes. Traditional static wingtip devices are then a compromise between conflicting requirements, resulting in less than optimal performance within each regime. Alternatively, a morphing wingtip can reconfigure leading to improved performance over a range of dissimilar flight conditions. Developed within this thesis, is a modular morphing wingtip concept that centers on the use of variable geometry truss mechanisms to permit morphing. A conceptual design framework is established to aid in the development of the concept. The framework uses a metaheuristic optimization procedure to determine optimal continuous wingtip configurations. The configurations are then discretized for the modular concept. The functionality of the framework is demonstrated through a design study on a hypothetical wing/winglet within the thesis.

Made available in DSpace on 2014-06-11T19:22:18Z (GMT). No. of bitstreams: 0 Previous issue date: 2012-02-17Bitstream added on 2014-06-13T20:28:18Z : No. of bitstreams: 1 porto_d_me_sjrp.pdf: 388104 bytes, checksum: ef6b41d53662a13bad28fcc9baed7ad9 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / O objetivo deste trabalho é o estudo do sistema de controle impulsivo de [Wolenski e Zabi´c 2007] para o caso em que o sistema é dado por uma igualdade e modificado pela adição de dois controles abstratos. Tal estudo foi feito utilizando duas abordagens. Na primeira, reparametrizamos o sistema inicial a partir da função distribuição relacionada à medida atômica e, através da discretização de Euler do sistema reparametrizado, obtemos uma sequência de soluções que converge no gráfico para a solução do sistema original, sob algumas hipóteses. Na segunda abordagem, definimos um novo sistema associado a uma sequência de medidas absolutamente contínuas que converge no gráfico para a medida atômica. A partir desse novo sistema, obtemos uma sequência de soluções com a propriedade de convergência no gráfico da solução do sistema original / The aim of this work is to study the impulsive control system of [ Wolenski e Zabic 2007] to the case where system is given by an equality and modified by addition of two abstract controls. The study was done using two approaches. At first, we've reparameterized the initial system from distribution function related to atomic measure and, through Euler's discretization of reparameterized system, we've obtained a sequence of solutions which graph converge to the solution of original system, under some hypothesis. In the second approach, we've defined a new system associated with a sequence of absolutely continuous measures which graph converge to atomic measure. From this new system, we've obtained a sequence of solutions with the graph convergence prop erty of the solution of the original system

Dissertation (Ph.D.)--Auburn University, 2006. / Abstract. Vita. Includes bibliographic references.

The thesis is concerned with a detailed examination of the most familiar short-rate models.Furthermore,it contains some author's own derivations of formulas for prices of interest rate derivatives and some relationships between certain discretizations of these short-rate models. These formulas are then used for calibration of ceratain chosen models to the actual market data.All the calculations are performed in R using author's own functions,which are along with the other more involved derivations placed in the appendix.

No description available.

Page IV was not included in the thesis, and thus not included in the page count. / <p> Discretization is an important tool to transfer optimization problems that include differentiations and integrals into standard optimization problems with a finite number of variables and a finite number of constraints. Recently, Betts and Campbell proposed a heat-transfer optimization problem that includes the heat partial differential equation as one of its constraints, and the objective function includes integrals of the temperature function squared. Using discretization methods, this problem can be converted to a convex quadratic optimization problem, which can be solved by standard interior point method solvers in polynomial time.</p> <p> The discretized model of the one dimensional problem is further analyzed, and some of its variants are studied. Extensive numerical testing is performed to demonstrate the power of the "discretize then optimize". Then the heat transfer optimization problem is generalized to two dimensions, and the discretized model and computational comparisons for this variant are included.</p> <p> Flexibility of discretization methods allow us to apply the same "diseretize then optimize" methodology to solve optimization problems that include differential and integral functions as constraints or objectives.</p> / Thesis / Master of Science (MSc)

Computational fluid dynamics (CFD) is a tool that is widely used in most industries today. It is important to have rigorous techniques to estimate the error produced when using CFD. This thesis develops techniques to estimate discretization error for unsteady flows using the unsteady error transport equation (ETE) as well as defect correction. A framework to obtain exact truncation error and estimated truncation error is also presented. The technique and results for the steady-state cases are given and the algorithm used for the steady case is extended for the unsteady case. Numerical results are presented for the steady viscous Burgers' equation, unsteady viscous Burgers' equation, steady quasi-1D Euler equations, and unsteady 1D Euler equations when applied to a shock tube. Cases using either defect correction or ETE are shown to give higher orders of accuracy for the corrected discretization error estimates when compared to the discretization error of the primal solution. / Master of Science / Computational fluid dynamics (CFD) is a tool that is widely used in most industries today. It is used to understand complex flows that are difficult to replicate using experimental techniques or by theoretical methods. It is important to have rigorous techniques to estimate the error produced when using CFD even when the exact solution is not available for comparison. This paper develops techniques to estimate discretization error for unsteady flows. Discretization error has one of the largest error magnitudes in CFD solutions. The exact physics dictates the use of continuous equations but to apply CFD techniques, the continuous equations have to be converted to discrete equations. Truncation error is, the error obtained when converting the continuous equations to discrete equations. This truncation error is in turn, the local source term for discretization error. To reduce the discretization error in the discrete equations, the exact or estimated truncation error is either added as a source term to the discrete equations or is used along with the error transport equation to get a better estimate of the solutions. A framework to obtain exact truncation error and estimated truncation error is also presented. The framework is first applied to the steady equations and is verified with results from previous studies and is then extended to the unsteady flows.

In this thesis, we study the contractivity preserving, high order, time discretization methods for solving non-stiff ordinary differential equations. We construct a class of one-step, explicit, contractivity preserving, multi-stage, multi-derivative, Hermite-Birkhoff-Taylor methods of order p=5,6, ..., 15, that we denote by CPHBT, with nonnegative coefficients by casting s-stage Runge-Kutta methods of order 4 and 5 with Taylor methods of order p-3 and p-4, respectively. The constructed CPHBT methods are implemented using an efficient variable step algorithm and are compared to other well-known methods on a variety of initial value problems. The results show that CPHBT methods have larger regions of absolute stability, require less function evaluations and hence they require less CPU time to achieve the same accuracy requirements as other methods in the literature. Also, we show that the contractivity preserving property of CPHBT is very efficient in suppressing the effect of the propagation of discretization errors when a long-term integration of a standard N-body problem is considered. The formulae of 49 CPHBT methods of various orders are provided in Butcher form.