Physical Sciences and Mathematics Commons^™

A C0 Finite Element Method For The Biharmonic Problem In A Polygonal Domain, Charuka Dilhara Wickramasinghe

Wayne State University Dissertations

This dissertation studies the biharmonic equation with Dirichlet boundary conditions in a polygonal domain. The biharmonic problem appears in various real-world applications, for example in plate problems, human face recognition, radar imaging, and hydrodynamics problems. There are three classical approaches to discretizing the biharmonic equation in the literature: conforming finite element methods, nonconforming finite element methods, and mixed finite element methods. We propose a mixed finite element method that effectively decouples the fourth-order problem into a system of one steady-state Stokes equation and one Poisson equation. As a generalization to the above-decoupled formulation, we propose another decoupled formulation using a …

Sequences Of Random Matrices Modulated By A Discrete-Time Markov Chain, Huy Nguyen

Wayne State University Dissertations

In this dissertation, we consider a number of matrix-valued random sequences that are modulated by a discrete-time Markov chain having a finite space.Assuming that the state space of the Markov chain is large, our main effort in this work is devoted to reducing the complexity. To achieve this goal, our formulation uses time-scale separation of the Markov chain. The state-space of the Markov chain is split into subspaces. Next, the states of the Markov chain in each subspace are aggregated into a ``super'' state. Then we normalize the matrix-valued sequences that are modulated by the two-time-scale Markov chain. Under simple …

Solving And Applications Of Multi-Facility Location Problems, Anuj Bajaj

Wayne State University Dissertations

This thesis is devoted towards the study and solving of a new class of multi-facility location problems. This class is of a great theoretical interest both in variational analysis and optimization while being of high importance to a variety of practical applications. Optimization problems of this type cannot be reduced to convex programming like, the much more investigated facility location problems with only one center. In contrast, such classes of multi-facility location problems can be described by using DC (difference of convex) programming, which are significantly more involved from both theoretical and numerical viewpoints.In this thesis, we present a new …

Variational Analysis Of Composite Optimization, Ashkan Mohammadi

Wayne State University Dissertations

The dissertation is devoted to the study of the first- and second-order variational analysis of the composite functions with applications to composite optimization. By considering a fairly general composite optimization problem, our analysis covers numerous classes of optimization problems such as constrained optimization; in particular, nonlinear programming, second-order cone programming and semidefinite programming(SDP). Beside constrained optimization problems our framework covers many important composite optimization problems such as the extended nonlinear programming and eigenvalue optimization problem. In first-order analysis we develop the exact first-order calculus via both subderivative and subdifferential. For the second-order part we develop calculus rules via second-order subderivative …

Second-Order Generalized Differentiation Of Piecewise Linear-Quadratic Functions And Its Applications, Hong Do

Wayne State University Dissertations

The area of second-order variational analysis has been rapidly developing during the recent years with many important applications in optimization. This dissertation is devoted to the study and applications of the second-order generalized differentiation of a remarkable

class of convex extended-real-valued functions that is highly important in many aspects of nonlinear and variational analysis, specifically those related to optimization and stability.

The first goal of this dissertation is to compute the second-order subdifferential of the functions described above, which will be applied in the study of the stability of composite optimization problems associated with piecewise linear-quadratic functions, known as extended …

Switching Diffusions: Applications To Ecological Models, And Numerical Methods For Games In Insurance, Trang Thi-Huyen Bui

Wayne State University Dissertations

Recently, a class of dynamic systems called ``hybrid systems" containing both continuous dynamics and discrete events has been adapted to treat a wide variety of situations arising in many real-world situations. Motivated by such development, this dissertation is devoted to the study of dynamical systems involving a Markov chain as the randomly switching process. The systems studied include hybrid competitive Lotka-Volterra ecosystems and non-zero-sum stochastic differential games between two insurance companies with regime-switching.

The first part is concerned with competitive Lotka-Volterra model with Markov switching. A novelty of the contribution is that the Markov chain has a countable state space. …

Switching Diffusion Systems With Past-Dependent Switching Having A Countable State Space, Hai Dang Nguyen

Wayne State University Dissertations

Emerging and existing applications in wireless communications, queueing networks, biological models, financial engineering, and social networks demand the

mathematical modeling and analysis of hybrid models in which continuous dynamics and discrete events coexist.

Assuming that the systems are in continuous times,

stemming from stochastic-differential-equation-based models and random discrete events,

switching diffusions come into being. In such systems, continuous states and discrete events

(discrete states)

coexist and interact.

A switching diffusion is a two-component process $(X(t),\alpha(t))$, a continuous component and a discrete component taking values in a discrete set (a set consisting of isolated points).

When the discrete component takes a …

Monotonicity Of Set-Valued Mappings And Full Stability Of General Parametrical Variational Systems, Dat Pham

Wayne State University Dissertations

The dissertation introduces and studies the notions of Lipschitzian and Holderian full stability of solutions to three-parametric variational systems described in the generalized equation formalism involving nonsmooth base mappings and partial subgradients of prox-regular functions acting in Hilbert spaces. Employing advanced tools and techniques of second-order variational analysis allows us to establish complete characterizations of, as well as directly variable sufficient conditions for, such full stability properties under mild assumptions. Furthermore, we derive exact formulas and effective quantitative estimates for the corresponding moduli.

Product Development Resilience Through Set-Based Design, Stephen H. Rapp

Wayne State University Dissertations

Often during a system Product Development program external factors or requirements change, forcing system design change. This uncertainty adversely affects program outcome, adding to development time and cost, production cost, and compromise to system performance. We present a development approach that minimizes the impacts, by considering the possibility of changes in the external factors and the implications of mid-course design changes. The approach considers the set of alternative designs and the burdens of a mid-course change from one design to another in determining the relative value of a specific design. The approach considers and plans parallel development of alternative designs …

Stochastic Approximation Algorithms With Applications To Particle Swarm Optimization, Adaptive Optimization, And Consensus, Quan Yuan

Wayne State University Dissertations

In this dissertation, we present three problems arising in recent applications of stochastic approximation methods. In Chapter 2, we use stochastic approximation to analyze Particle Swarm Optimization (PSO) algorithm. We introduce four coefficients and rewrite the PSO procedure as a stochastic approximation type iterative algorithm. Then we analyze its convergence using weak convergence method. It is proved that a suitably scaled sequence of swarms converge to the solution of an ordinary differential equation. We also establish certain stability results. Moreover, convergence rates are ascertained by using weak convergence method. A centered and scaled sequence of the estimation errors is shown …

Consensus-Type Stochastic Approximation Algorithms, Yu Sun

Wayne State University Dissertations

This work is concerned with asymptotic properties of consensus-type algorithms for networked systems whose topologies switch randomly. The regime-switching process is modeled as a discrete-time Markov chain with a nite state space. The consensus control is achieved by designing stochastic approximation algorithms. In the setup, the regime-switching process (the Markov chain) contains a rate parameter

"Ε> 0 in the transition probability matrix that characterizes how frequently the topology switches. On the other hand, the consensus control algorithm uses a step-size Μ that denes how fast the network states are updated. Depending on their relative values, three distinct scenarios emerge. Under …

Analytical And Experimental Investigations Of Ships Impact Interaction With One-Sided Barrier, Ihab M. Grace

Wayne State University Dissertations

This study deals with impact interaction of ships with one-sided ice barrier during roll dynamics. An analytical model of ship roll motion interacting with ice is developed based on Zhuravlev and Ivanov non-smooth coordinate transformations. These transformations have the advantage of converting the vibro-impact oscillator into an oscillator without barriers such that the corresponding equation of motion does not contain any impact term. Such approaches, however, account for the energy loss at impact times in different ways. The present work, in particular, brings to the attention the fact that the impact dynamics may have qualitatively different response characteristics to different …

Asymptotic Expansions And Stability Of Hybrid Systems With Two-Time Scales, Dung Tien Nguyen

Wayne State University Dissertations

In this dissertation, we consider solutions of hybrid systems in which both continuous dynamics and discrete events coexists. One

of the main ingredients of our models is the two-time-scale formulation. Under broad conditions, asymptotic expansions are developed for the solutions of the systems of backward equations for switching diffusion in two classes of models, namely, fast switching systems and fast diffusion systems. To prove the validity of the asymptotic expansions, uniform error bounds are obtained.

In the second part of the dissertation, a singular linear system is considered. Again a two-time-scale formulation is used. Under suitable conditions, the system has …

Methods Of Variational Analysis In Pessimistic Bilevel Programming, Samarathunga M. Dassanayaka

Wayne State University Dissertations

Bilevel programming problems are of growing interest both from theoretical and practical points of view. These models are used in various applications, such as economic planning, network design, and so on. The purpose of this dissertation is to study the pessimistic (or strong) version of bilevel programming problems in finite-dimensional spaces. Problems of this type are intrinsically nonsmooth (even for smooth initial data) and can be treated by using appropriate tools of modern variational analysis and generalized differentiation developed by B. Mordukhovich.

This dissertation begins with analyzing pessimistic bilevel programs, formulation of the problems, literature review, practical application, existence of …

Spectral Methods For The Hamiltonian Systems, Nairat Kanyamee

Wayne State University Dissertations

We conduct a systematic comparison of spectral methods with some

symplectic methods in solving Hamiltonian dynamical systems. Our

main emphasis is on the non-linear problems. Numerical evidence has

demonstrated that the proposed spectral collocation method preserves

both energy and symplectic structure up to the machine error in each

time (large) step, and therefore has a better long time behavior.