Applied Mathematics Commons^™

Modeling And Numerical Analysis Of The Cholesteric Landau-De Gennes Model, Andrew L. Hicks

LSU Doctoral Dissertations

This thesis gives an analysis of modeling and numerical issues in the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs) with cholesteric effects. We derive various time-step restrictions for a (weighted) $L^2$ gradient flow scheme to be energy decreasing. Furthermore, we prove a mesh size restriction, for finite element discretizations, that is critical to avoid spurious numerical artifacts in discrete minimizers that is not well-known in the LC literature, particularly when simulating cholesteric LCs that exhibit ``twist''. Furthermore, we perform a computational exploration of the model and present several numerical simulations in 3-D, on both slab geometries and spherical …

Finite Element Methods For Elliptic Optimal Control Problems With General Tracking, Seonghee Jeong

LSU Doctoral Dissertations

This dissertation concerns a linear-quadratic elliptic distributed optimal control problem with pointwise state constraints in two spatial dimensions, where the cost function tracks the state at points, curves and regions of a domain.

First we explore the elliptic optimal control problem subject to pointwise control constraints. This problem is reduced into a problem that only involves the control. The solution of the reduced problem is characterized by a variational inequality. Then we introduce the elliptic optimal control problem with general tracking and pointwise state constraints. Here we reformulate the optimal control problem into a problem that only involves the state, …

First-Order Algorithms For Nonlinear Structured Optimization, Miao Zhang

LSU Doctoral Dissertations

Nonlinear optimization is a critical branch in applied mathematics and has attracted wide attention due to its popularity in practical applications. In this work, we present two methods which use first-order information to solve two typical classes of nonlinear structured optimization problems.

For a class of unconstrained nonconvex composite optimization problems where the objective is the sum of a smooth but possibly nonconvex function and a convex but possibly nonsmooth function, we propose a unified proximal gradient method with extrapolation, which provides unified treatment to convex and nonconvex problems. The method achieves the best-known convergence rate for first-order methods when …

An Investigation Of The Effects Of Variable Magnetic Field Gradients On Soot And Co Emissions From Non-Premixed Hydrocarbon Flames, Edison Ekperechukwu Chukwuemeka

LSU Doctoral Dissertations

The interaction of the paramagnetic species in a combustion process with the mag- netic field placed in the vicinity of non-premixed flames affects the characteristics of the non-premixed flames - flame height and flame lift-off height. However, the effect of this magnetic interaction on the pollutants generated by the flame is unknown.

In general, pollutant formation is promoted in most combustion systems due to in- complete combustion of the hydrocarbon due to improper mixing. Since paramagnetic combustion species such as O2, O, OH, etc interacts with magnetic fields and possess a preferential motion direction, imposing magnetic field on non-premixed flames …

A Phase-Field Approach To Diffusion-Driven Fracture, Friedrich Wilhelm Alexander Dunkel

LSU Doctoral Dissertations

In recent years applied mathematicians have used modern analysis to develop variational phase-field models of fracture based on Griffith's theory. These variational phase-field models of fracture have gained popularity due to their ability to predict the crack path and handle crack nucleation and branching.

In this work, we are interested in coupled problems where a diffusion process drives the crack propagation. We extend the variational phase-field model of fracture to account for diffusion-driving fracture and study the convergence of minimizers using gamma-convergence. We will introduce Newton's method for the constrained optimization problem and present an algorithm to solve the diffusion-driven …

Multigrid Methods For Elliptic Optimal Control Problems, Sijing Liu

LSU Doctoral Dissertations

In this dissertation we study multigrid methods for linear-quadratic elliptic distributed optimal control problems.

For optimal control problems constrained by general second order elliptic partial differential equations, we design and analyze a $P_1$ finite element method based on a saddle point formulation. We construct a $W$-cycle algorithm for the discrete problem and show that it is uniformly convergent in the energy norm for convex domains. Moreover, the contraction number decays at the optimal rate of $m^{-1}$, where $m$ is the number of smoothing steps. We also prove that the convergence is robust with respect to a regularization parameter. The robust …

General Stochastic Integral And Itô Formula With Application To Stochastic Differential Equations And Mathematical Finance, Jiayu Zhai

LSU Doctoral Dissertations

A general stochastic integration theory for adapted and instantly independent stochastic processes arises when we consider anticipative stochastic differential equations. In Part I of this thesis, we conduct a deeper research on the general stochastic integral introduced by W. Ayed and H.-H. Kuo in 2008. We provide a rigorous mathematical framework for the integral in Chapter 2, and prove that the integral is well-defined. Then a general Itô formula is given. In Chapter 3, we present an intrinsic property, near-martingale property, of the general stochastic integral, and Doob-Meyer's decomposition for near-submartigales. We apply the new stochastic integration theory to several …

Information Theoretic Study Of Gaussian Graphical Models And Their Applications, Ali Moharrer

LSU Doctoral Dissertations

In many problems we are dealing with characterizing a behavior of a complex stochastic system or its response to a set of particular inputs. Such problems span over several topics such as machine learning, complex networks, e.g., social or communication networks; biology, etc. Probabilistic graphical models (PGMs) are powerful tools that offer a compact modeling of complex systems. They are designed to capture the random behavior, i.e., the joint distribution of the system to the best possible accuracy. Our goal is to study certain algebraic and topological properties of a special class of graphical models, known as Gaussian graphs. First, …