Thesis (M.S.)--Brigham Young University. Dept of Mathematics, 2006. / Includes bibliographical references (p. 30).

The method of characteristics (MOC) is a deterministic transport method that has been applied to large-scale problems including those in reactor physics and radiative transfer. Long characteristics, (LC) methods, have been used extensively to discretize and solve transport problems in the spatial domain. There is a need for an equally adequate time-dependent discretization for these transport problems. The new contributions from this research include the development of a space-time long characteristic (STLC) method with various source approximations including several that employ a piece-wise linear (PWL) approximation spatially. In the prism-PWL (PPWL) method the coefficient of each PWL spatial function is linear in time in each space-time cell. Along with STLC, a PWL-LC method is developed for steady-state problems in (x, y) and (x, y, z). The methods developed in this work use least-squares projections to determine the coefficients of their source approximations. This work presents a detailed asymptotic analysis of the PWL-LC and STLC methods in the thick diffusion limit, which is of special interest in radiative transfer problems. This is the first such analysis reported for LC methods and these new methods are the first that are expected to perform well in this limit. Results from test problems executed with a modified version of the Parallel Deterministic Transport code, PDT, show the PWL-LC and STLC methods are more accurate than current methods for streaming problems. From asymptotic analysis and test problems, it is found that the steady-state PWL-LC method is accurate in the thick diffusion limit with solutions similar to those of analogous discontinuous finite element method, DFEM, solutions. Similarly, the PPWL-STLC method is found to be accurate in time-dependent thick diffusive problems. STLC is also a promising method for massively parallel applications because it permits the use of track-based sweeping, which appears to have significant advantages over cell-based sweeping. This is a key topic recommended for further research.

We consider a portfolio selection problem in the presence of transaction costs. Transaction costs on each asset are assumed to be a convex function of the amount sold or bought. This function can be nondifferentiable in a finite number of points. The objective function of this problem is a sum of a convex twice differentiable function and a separable convex nondifferentiable function. We first consider the problem in the presence of linear constraints and later generalize the results to the case when the constraints are given by the convex piece-wise linear functions. <br /><br /> Due to the special structure, this problem can be replaced by an equivalent differentiable problem in a higher dimension. It's main drawback is efficiency since the higher dimensional problem is computationally expensive to solve. <br /><br /> We propose several alternative ways to solve this problem which do not require introducing new variables or constraints. We derive the optimality conditions for this problem using subdifferentials. First, we generalize an active set method to this class of problems. We solve the problem by considering a sequence of equality constrained subproblems, each subproblem having a twice differentiable objective function. Information gathered at each step is used to construct the subproblem for the next step. We also show how the nonsmoothness can be handled efficiently by using spline approximations. The problem is then solved using a primal-dual interior-point method. <br /><br /> If a higher accuracy is needed, we do a crossover to an active set method. Our numerical tests show that we can solve large scale problems efficiently and accurately.

We consider a portfolio selection problem in the presence of transaction costs. Transaction costs on each asset are assumed to be a convex function of the amount sold or bought. This function can be nondifferentiable in a finite number of points. The objective function of this problem is a sum of a convex twice differentiable function and a separable convex nondifferentiable function. We first consider the problem in the presence of linear constraints and later generalize the results to the case when the constraints are given by the convex piece-wise linear functions. <br /><br /> Due to the special structure, this problem can be replaced by an equivalent differentiable problem in a higher dimension. It's main drawback is efficiency since the higher dimensional problem is computationally expensive to solve. <br /><br /> We propose several alternative ways to solve this problem which do not require introducing new variables or constraints. We derive the optimality conditions for this problem using subdifferentials. First, we generalize an active set method to this class of problems. We solve the problem by considering a sequence of equality constrained subproblems, each subproblem having a twice differentiable objective function. Information gathered at each step is used to construct the subproblem for the next step. We also show how the nonsmoothness can be handled efficiently by using spline approximations. The problem is then solved using a primal-dual interior-point method. <br /><br /> If a higher accuracy is needed, we do a crossover to an active set method. Our numerical tests show that we can solve large scale problems efficiently and accurately.

In many digital signal processing applications, we often need some special function units that can compute complicated arithmetic functions such as reciprocal, logarithm, power of 2, trigonometric functions, etc. The most popular designs are based on look-up tables with polynomial approximation. However, the table size will increase significantly in accordance with precision. In this thesis, we propose a method called remapping to reduce the table size by using non-uniform segmentation. When we obtain the coefficients for all segments, we do not store them in order. By sorting the coefficients in the ROM ,we design a efficient hardware mapping. The method can reduce the ROM size with lower extra cost spent in address mapping for non-uniform segmentation.

In today¡¦s modern society, our latest up-to-date technology contains various types of multimedia applications. These applications don¡¦t necessarily have to be executed with the most precise accuracy. In short, they are fault tolerant. As a consequence, this thesis proposes a multi-precision iterative floating-point special function unit, which can be executed under different modes to meet the error requirements of each specific application, and also achieve power reduction during the process. In order to minimize the area of our design, we have developed two iterative architectures to implement the multi-precision floating point special function unit. The first proposed architecture can perform three kinds of operations, which include a reciprocal operation, a reciprocal square root operation, and last but not least, a logarithm operation. After deciding which function is to be performed, the user can choose four precision modes to execute the special function unit. According to each mode from lowest precision to highest, we name them the first mode, the second mode, the third mode, and the fourth mode. During implementation, a C model has also been designed to evaluate the maximum error of each mode by making comparisons with the most accurate software result, which is the 23 bit result. When the reciprocal function is chosen, and the user defines that application to be performed in full precision, the multi-precision special function operator needs to be executed twice, and it has the error rate of approximately 0.0001%. When less precision is required, we can choose from two intermediate modes, one offers 15 bit accuracy, and the other can guarantee a 12 bit precision. The former precision mode also required the hardware to be executed twice, but the latter only once. The 15 bit accuracy mode has an error rate around 0.01¢H, and the 12 bit mode has the error rate roughly around 0.05¢H. In addition, when visual effects or even audio effects are not our greatest concern, we provide a least accurate mode for the users to pick to execute the special function operator. This mode can maintain 8 bit accuracy, and has the error rate of approximately 0.8%. Other operations including the reciprocal square root, and the logarithm also have four precision modes to choose from. The reciprcocal square root operation can guarantee the same accuracy in each mode as the reciprocal operation, and their error rates are 0.004%, 0.01%, 0.06%, and 0.5% from the highest precision mode to the lowest one. The precisions the logarithm operation can guarantee from highest accuracy to the lowest one are 23, 16, 12, and 8 bits, respectively, and have error rates including 0.00003%, 0.002%, 0.06%, and 0.3%. These different precision choices are built in the proposed structure mainly to reduce the power consumption. The main concept is to pick a low precision mode in order shut down some components in our design. In addition to switching modes, we¡¦ve also added tri-state buffers in certain components as another means to decrease power. Through experimental results we¡¦ve discovered that the proposed architecture¡¦s affect on power reduction was not as we¡¦ve expected. Due to the integration of the Newton Raphson Method and the Piecewise Polynomial Approximation Method, our architecture¡¦s delay and area have largely increased, and causing a bad influence on saving power. As a consequence, we¡¥ve developed a second architecture to meet our demands. This architecture is mainly based on the Piecewise Polynomial Approximation Method. From this method, we¡¦ve implemented an iterative design which also supports three kinds of operations, the same as the first architecture. It also provides three precision modes for the user to choose. The lowest precision mode provides 8 bit accuracy. The second mode provides 14 bit accuracy, and the third mode, which is the most precise mode, can provide 22 bit accuracy. According to our C model, we can specify our maximum error rate in each function while executing under different modes. When the reciprocal function is executed, the largest error rate in from the lowest mode to the highest mode is 0.19%, 0.00006% and 0.000015% , and the error rate for reciprocal square root from lowest precision mode to the highest is 0.09%, 0.000022% and 0.000014%, and the error rate for the logarithm function is 0.33%, 0.000043% and 0.000015%, from the lowest to the highest. From experimental results we can discover that the newly proposed architecture is better in comparison with the traditional Piecewise Polynomial Approximation architecture. The proposed architecture has a smaller area, and a faster delay, and most important of all, it reduces power and energy affectively.

In this work, we discuss the accuracy of measurements for electromagnetic. The two kinds of antenna we use are Dipole antenna and Broadband antenna. In general, if the antenna measurements we recorded at different frequencies do not exceed the ideal value $pm 4$dB, we would regard this site as a normalized site, otherwise it is not a normalized site(just a measurement exceeds the range). Traditionally, all we use is Dipole antenna, but due to difficulty of operation and inaccuracy of Dipole antenna, we investigate by statistical methods if we may use the Broadband antenna to replace the traditional Dipole antenna to measure. First of all, we introduce the data and procedure in the experiments, and fit a statistical regression model to predict the measurements at different frequencies in different test setups. Then, according to the data we collected, use the change point models to modify the statistical models. Our goal is to find a suitable statistical model for the measurements. Finally, we compare the measurements of Broadband antenna with Dipole antenna in the other experimental conditions keep the same.

In this thesis, an efficiency method for reducing the rotation ROM size in table-based architecture is proposed. The original rotation can be divided into two stages, coarse stage and fine stage. Our approach modifies the previous two-stage rotation method and proposes a multi-stage architecture and discuses three-stage phase calculation. The effect of table reduction is more apparently for higher accuracy requirement in the three-stage architecture. The total area of the previous two-stage architecture is larger than the proposed table-reduced three-stage architecture because the table size takes a significant ratio of the total area especially when the required bit accuracy is large. In the proposed three-stage design, there are two different types of architectures, depending on the rotation angles in the first and second rotation stages. Comparison of different types of architecture with the previous method shows that our designs indeed reduce the table size and the total area significantly.

We studied invariant measures and invariant densities for dynamical systems with random switching (switching systems, in short). These switching systems can be described by a two-component Markov process whose first component is a stochastic process on a finite-dimensional smooth manifold and whose second component is a stochastic process on a finite collection of smooth vector fields that are defined on the manifold. We identified sufficient conditions for uniqueness and absolute continuity of the invariant measure associated to this Markov process. These conditions consist of a Hoermander-type hypoellipticity condition and a recurrence condition. In the case where the manifold is the real line or a subset of the real line, we studied regularity properties of the invariant densities of absolutely continuous invariant measures. We showed that invariant densities are smooth away from critical points of the vector fields. Assuming in addition that the vector fields are analytic, we derived the asymptotically dominant term for invariant densities at critical points.

Made available in DSpace on 2014-06-11T19:27:08Z (GMT). No. of bitstreams: 0 Previous issue date: 2012Bitstream added on 2014-06-13T18:30:56Z : No. of bitstreams: 1 scarabello_mc_me_prud.pdf: 7894728 bytes, checksum: 7780bf65e553d805c887201bc480e587 (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Em um artigo publicado em maio de 2008 na revista Nature [17], um grupo de pesquisadores da Hewllet-Packard Company (HP) anunciou a fabricação de um componente eletrônico chamado memristor, uma contração para “memory resistor”. A existência teórica dos memristores havia sido prevista em 1971, pelo Engenheiro da Universidade da Califórnia em Berkeley, Leon Chua, com base em propriedades de simetria de certos circuitos elétricos, porém até 2008 sua existência física não havia sido comprovada. Tal componente é considerado o quarto componente eletrônico fundamental, ao lado do resistor, do capacitor e do indutor, pois possui propriedades que não podem ser duplicadas por nenhuma combinação desses três outros componentes. A construção física do memristor atraiu grande interesse no mundo todo, devido ao grande potencial de aplicações deste componente. No presente trabalho fazemos um estudo das bifurcações que ocorrem em um sistema de equações diferenciais ordinárias, que serve como modelo matemático de um circuito elétrico formado pelos quatro elementos fundamentais: um memristor, um capacitor, um indutor e um resistor. O circuito estudado foi proposto por Itoh e Chua em [9]... / In the present work we make a bifurcation analysis of a system of ordinary differential equations, which serves as a mathematical model of an electric circuit formed by the four fundamental elements: one memristor, one capacitor, one inductor and one resistor. The studied circuit was proposed by Itoh and Chua in [9] and was constructed based on the well-known Chua's oscillators. The studied model is given by a discontinuous piecewise-linear system, defined on three zones in R 3, determined by the following inequalities: |z|<1 (called central zone) and |z|>1 (called external zones). The z-axis is composed by equilibrium points of the system. The local normal stability of these equilibira in each zone is analyzed. We show that, due to the existence of this line of equilibria, the phase space R 3 is foliated by invariant planes transversal to the z-axis and parallel to each other, in each zone. The solutions of the system are contained in a combination of three of these invariant planes: one of them in the central zone and the other two in the external zones. We also show that the system may present nonlinear oscillations due to the existence of periodic orbits passing through two of the three zones or passing by three zones. The analysis developed here has analytical and numerical parts. The analytical part was developed based on the study of planar piecewise-linear systems with three zones presented by Freire et al. in [5]... (Complete abstract click electronic access below)