Theses and Dissertations - Department of Mathematics
Permanent URI for this collection
Browse
Browsing Theses and Dissertations - Department of Mathematics by Author "Brooks, Robert Edwin"
Now showing 1 - 2 of 2
Results Per Page
Sort Options
Item Efficient algorithms for solving three dimensional parabolic interface problem with variable coefficients(University of Alabama Libraries, 2018) Wei, Zhihan; Zhao, Shan; University of Alabama TuscaloosaThe dissertation consists of two parts, in the first part, a new matched alternating direction implicit (ADI) method is proposed for solving three-dimensional (3D) parabolic interface problems with discontinuous jumps, piecewise constant diffusion coefficients and complex interfaces. This scheme inherits the merits of its ancestor of two-dimensional problems, while possesses several novel features, such as a non-orthogonal local coordinate system for decoupling the jump conditions, two-side estimation of tangential derivatives at an interface point, and a new Douglas-Rachford ADI formulation that minimizes the number of perturbation terms, to attack more challenging 3D problems. In time discretization, this new ADI method is found to be first order and stable in numerical experiments. In space discretization, the matched ADI method achieves a second order of accuracy based on simple Cartesian grids for various irregularly-shaped surfaces and spatial-temporal dependent jumps. Computationally, the matched ADI method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3D parabolic equations, in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom $N$, i.e., $O(N)$. Furthermore, unlike iterative methods, the ADI method is an exact or non-iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed $N$. Therefore, the proposed matched ADI method provides an efficient tool for solving 3D parabolic interface problems. In the second part, instead of constant diffusion coefficients, improved schemes for variable diffusion coefficient are also performed in the work. A comparison of proposed ADI method with different other time splitting methods, including locally one-dimensional implicit Euler(LOD-IE), locally one-dimensional Crank-Nicolson(LOD-CN) and Trapezoidal Splitting(TS) method will be implemented, coupled with different variation of matched interface and boundary (MIB) method in spatial discretization. These large scale computational studies facilitate the further development of matched ADI algorithms for 3D parabolic interface problems.Item Regression models with a universal penalized function and applications in economics and finance(University of Alabama Libraries, 2018) Sun, Mingwei; Wang, Pu; University of Alabama TuscaloosaVariable selection is an important topic in linear regression analysis and attracts considerable research in this era of big data. It is fundamental to high-dimensional statistical modeling, including nonparametric regression. Some classic techniques include stepwise deletion and subset selection. These procedures, however, ignore stochastic errors inherited in the stages of variable selections, and the resulting subset suffers from lack of stability and low prediction accuracy. Penalized least squares provide new approaches to the variable selection problems with high-dimensional data. The least absolute shrinkage and selection operator (LASSO), which imposes an L$_1$-penalty on the regression coefficients, and the Elastic Net which combines an L$_1$ and an L$_2$ penalties are popular members of the penalized regressions. In this dissertation, we develop penalized linear regression with a universal penalty function, which includes the widely used ridge and Lasso penalty functions as special cases. A Monte Carlo simulation approach is developed to illustrate that the Elastic Net is also a special case of our model. The structure and properties of the universal penalty are studied, and the corresponding algorithm to solve the regression coefficients is developed. Furthermore, we apply our model to a real U.S. economic and financial data example. Simulation studies and real-data support the advantageous performance of the proposed method.