Boundary Approximation Method (BAM), or the Collocation Trefftz Method called in the literature, is the most efficient method to solve elliptic boundary value problems with singularities. There are several versions of BAM in practical computation, including the Numerical BAM, Symbolic BAM and their variants. It is known that the Symbolic BAM is much slower than Numerical counterpart. In this thesis, we improve the Symbolic BAM to become the fastest method among all versions of BAM. We prove several important lemmas to reduce the computing time, and a recursive procedure is found to expedite the evaluation of major integrals. Another drawback of the Symbolic BAM is its large condition number. We find a good and easy preconditioner to significantly reduce the condition number. The numerical experiments and comparison are also provided for the Motz problem, a prototype of Laplace boundary value problem with singularity, and the Schiff's Model, a prototype of biharmonic boundary value problem with singularity.

The position-dependent effective mass (PDEM) problem is of enormous importance to the realization of the extended Wentzel-Kramers-Brillouin (WKB) approximation in bound state calculations for semiconductor heterostructures. By studying some model problems, we show that the extended WKB method provides good approximations for the bound states with the high eigenenergies. In addition, the effect of the smoothness of the effective mass distribution functions and potential barrier in the PDEM problems is discussed in our work. We found the precision can be affected by the effective mass and potential barrier in the PDEM.

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.

none

none

In the ¡Krst chapter, separation of variables is used to derive the explicit particular solutions for a class of singularly perturbed di¤erential equations with constant coe¢ cients on a rectangular domain. Although only the Dirichlet boundary condition is taken into account; it can be similarly extended to other boundary conditions. Based on these results, the behavior of the solutions and their derivatives can be easily illustrated. Moreover, we have proposed a model with exact solution, which can be used to explore the behavior of layer and to test numerical methods. Hence, these analytic solutions are very important to the study in this ¡Keld. In the second chapter, we study the model of Shi¤ et al. [20]. It is a biharmonic equation on the rectangular domain [¡ a; a]£ [0; b] with clamped boundary condition. We compute its most accurate numerical solution by boundary approximation method (BAM), which is a special version of spectral method or collocation method. Its convergence unfortunately is not as good as the usual spectral method with exponential decay rate. We discover that the slowdown is due to the very mild singularity at two corners not considered by BAM. We further simplify the basis functions and their partial derivatives. Using these functions we can construct several models useful for testing numerical methods. We also explore how the stress intensity factor depends on the sizes of domain a and b, and the load ¸ by reducing the original problem with three parameters lambda, a, b to that with only one parameter t.

In the present work, we study the propositional proof complexity. First, we prove an exponential lower bound on the complexity of resolution applying directly Razborov's approximation method, which was previously used only for bounds on the size of monotone circuits. Next, we use the approximation method for a new proof of an exponential lower bound on the complexity of random resolution refutations. That should have further applications in separating some theories in bounded arithmetic. In both cases, we use a problem from the graph theory called Broken Mosquito Screens. Finally, we state a conjecture that the approximation method is applicable even for stronger propositional proof systems, for example Cutting Plane Proofs. Powered by TCPDF (www.tcpdf.org)

Simultaneous Perturbation Stochastic Approximation method has attracted considerable application in many different areas such as statistical parameter estimation, feedback control, simulation-based optimization, signal & image processing, and experimental design. In this paper, its performance as a viable optimization tool is demonstrated by applying it first to a simple wing geometry design problem for which the objective function is described by an empirical formula from aircraft design practice and then it is used in a transonic fan blade design problem in which the objective function is not represented by any explicit function but is estimated at each design iteration by a computational fluid dynamics algorithm for solving the Navier-Stokes equations / Singapore-MIT Alliance (SMA)

A spectral element algorithm for solution of the incompressible Navier-Stokes and heat transfer equations is developed, with an emphasis on extending the classical conforming Galerkin formulations to nonconforming spectral elements. The new algorithm employs both the Constrained Approximation Method (CAM), and the Mortar Element Method (MEM) for p-and h-type nonconforming elements. Detailed descriptions, and formulation steps for both methods, as well as the performance comparisons between CAM and MEM, are presented. This study fills an important gap in the literature by providing a detailed explanation for treatment of p-and h-type nonconforming interfaces. A comparative eigenvalue spectrum analysis of diffusion and convection operators is provided for CAM and MEM. Effects of consistency errors due to the nonconforming formulations on the convergence of steady and time dependent problems are studied in detail. Incompressible flow solvers that can utilize these nonconforming formulations on both p- and h-type nonconforming grids are developed and validated. Engineering use of the developed solvers are demonstrated by detailed parametric analyses of oscillatory flow forced convection heat transfer in two-dimensional channels.

A spectral element algorithm for solution of the incompressible Navier-Stokes and heat transfer equations is developed, with an emphasis on extending the classical conforming Galerkin formulations to nonconforming spectral elements. The new algorithm employs both the Constrained Approximation Method (CAM), and the Mortar Element Method (MEM) for p-and h-type nonconforming elements. Detailed descriptions, and formulation steps for both methods, as well as the performance comparisons between CAM and MEM, are presented. This study fills an important gap in the literature by providing a detailed explanation for treatment of p-and h-type nonconforming interfaces. A comparative eigenvalue spectrum analysis of diffusion and convection operators is provided for CAM and MEM. Effects of consistency errors due to the nonconforming formulations on the convergence of steady and time dependent problems are studied in detail. Incompressible flow solvers that can utilize these nonconforming formulations on both p- and h-type nonconforming grids are developed and validated. Engineering use of the developed solvers are demonstrated by detailed parametric analyses of oscillatory flow forced convection heat transfer in two-dimensional channels.