Browsing by Author "Burns, John A."
Now showing 1 - 20 of 139
Results Per Page
Sort Options
- An active-constraint logic for nonlinear programmingDas, Alok (Virginia Polytechnic Institute and State University, 1982)The choice of active-constraint logic for screening inequalities has a major impact on the performance of gradient-projection method. It has been found that least-constrained strategies, which keep the number of constraints in the active set as small as possible, are computationally most efficient. However, these strategies are often prone to cycling of constraints between active and inactive status. This occurs mainly due to the violation of some of the constraints, taken as inactive, by the resulting step. This research develops methods for choosing an active set such that constraints in the active set satisfy the Kuhn-Tucker conditions and the resulting step does not violate the linear approximations to any of the constraints satisfied as equalities but considered inactive. Some of the existing active-constraint logics, specifically the dual-violator rule, yield the desired active set when two constraints are satisfied as equalities. However, when three or more constraints are satisfied as equalities, none of the existing logics give the desired active set. A number of general results, which help in the selection of the active set, have been developed in this research. An active-constraint logic has been developed for the case of three constraints. This logic gives the desired active-set. For the general case, when more than three constraints are satisfied as equalities, a separate active-set logic is suggested. This guarantees the nonviolation of the linear approximations to any of the constraints, taken as inactive, by the resulting step. The resulting active-set may not, however, satisfy the Kuhn-Tucker conditions. The efficiency of the proposed logic was tested computationally using quadratic programming problems. Three existing active-set strategies were used for comparision. The proposed logic almost always performed as well or better than the best among the three existing active-set strategies.
- The Algebra of Systems BiologyVeliz-Cuba, Alan A. (Virginia Tech, 2010-07-05)In order to understand biochemical networks we need to know not only how their parts work but also how they interact with each other. The goal of systems biology is to look at biological systems as a whole to understand how interactions of the parts can give rise to complex dynamics. In order to do this efficiently, new techniques have to be developed. This work shows how tools from mathematics are suitable to study problems in systems biology such as modeling, dynamics prediction, reverse engineering and many others. The advantage of using mathematical tools is that there is a large number of theory, algorithms and software available. This work focuses on how algebra can contribute to answer questions arising from systems biology.
- Algorithmic Approaches for Solving the Euclidean Distance Location and Location-Allocation ProblemsAl-Loughani, Intesar Mansour (Virginia Tech, 1997-07-08)This dissertation is concerned with the development of algorithmic approaches for solving the minisum location and location-allocation problems in which the Euclidean metric is used to measure distances. To overcome the nondifferentiability difficulty associated with the Euclidean norm function, specialized solution procedures are developed for both the location and the location-allocation problems. For the multifacility location problem (EMFLP), two equivalent convex differentiable reformulations are proposed. The first of these is formulated directly in the primal space, and relationships between its Karush-Kuhn-Tucker (KKT) conditions and the necessary and sufficient optimality conditions for EMFLP are established in order to explore the use of standard convex differentiable nonlinear programming algorithms that are guaranteed to converge to KKT solutions. The second equivalent differentiable formulation is derived via a Lagrangian dual approach based on the optimum of a linear function over a unit ball (circle). For this dual approach, which recovers Francis and Cabot's (1972) dual problem, we also characterize the recovery of primal location decisions, hence settling an issue that has remained open since 1972. In another approach for solving EMFLP, conjugate or deflected subgradient based algorithms along with suitable line-search strategies are proposed. The subgradient deflection method considered is the Average Direction Strategy (ADS) imbedded within the Variable Target Value Method (VTVM). The generation of two types of subgradients that are employed in conjunction with ADS are investigated. The first type is a simple valid subgradient that assigns zero components corresponding to the nondifferentiable terms in the objective function. The second type expends more effort to derive a low-norm member of the subdifferential in order to enhance the prospect of obtaining a descent direction. Furthermore, a Newton-based line-search is also designed and implemented in order to enhance the convergence behavior of the developed algorithm. Various combinations of the above strategies are composed and evaluated on a set of test problems. Computational results for all the proposed algorithmic approaches are presented, using a set of test problems that include some standard problems from the literature. These results exhibit the relative advantages of employing the new proposed procedures. Finally, we study the capacitated Euclidean distance location-allocation problem. There exists no global optimization algorithm that has been developed and tested for this class of problems, aside from a total enumeration approach. We develop a branch-and-bound algorithm that implicitly/partially enumerates the vertices of the feasible region of the transportation constraints in order to determine a global optimum for this nonconvex problem. For deriving lower bounds on node subproblems, a specialized variant of the Reformulation-Linearization Technique (RLT) is suitably designed which transforms the representation of this nonconvex problem from the original defining space into a higher dimensional space associated with a lower bounding (largely linear) convex program. The maximum of the RLT relaxation based lower bound that is obtained via a deflected subgradient strategy applied to a Lagrangian dual formulation of this problem, and another readily computed lower bound in the projected location space is considered at each node of the branch-and-bound tree for fathoming purposes. In addition, certain cut-set inequalities in the allocation space, and objective function based cuts in the location space are generated to further tighten the lower bounding relaxation. Computational experience is provided on a set of randomly generated test problems to investigate both the RLT-based and the projected location- space lower bounding schemes. The results indicate that the proposed global optimization approach for this class of problem offers a promising viable solution procedure. In fact, for two instances available available in the in the literature, we report significantly improved solutions. The dissertation concludes with recommendations for further research for this challenging class of problems. Data for the collection of test problems is provided in the Appendix to facilitate further testing in this area.
- Algorithms for modeling and simulation of biological systems; applications to gene regulatory networksVera-Licona, Martha Paola (Virginia Tech, 2007-06-06)Systems biology is an emergent field focused on developing a system-level understanding of biological systems. In the last decade advances in genomics, transcriptomics and proteomics have gathered a remarkable amount data enabling the possibility of a system-level analysis to be grounded at a molecular level. The reverse-engineering of biochemical networks from experimental data has become a central focus in systems biology. A variety of methods have been proposed for the study and identification of the system's structure and/or dynamics. The objective of this dissertation is to introduce and propose solutions to some of the challenges inherent in reverse-engineering of biological systems. First, previously developed reverse engineering algorithms are studied and compared using data from a simulated network. This study draws attention to the necessity for a uniform benchmark that enables an ob jective comparison and performance evaluation of reverse engineering methods. Since several reverse-engineering algorithms require discrete data as input (e.g. dynamic Bayesian network methods, Boolean networks), discretization methods are being used for this purpose. Through a comparison of the performance of two network inference algorithms that use discrete data (from several different discretization methods) in this work, it has been shown that data discretization is an important step in applying network inference methods to experimental data. Next, a reverse-engineering algorithm is proposed within the framework of polynomial dynamical systems over finite fields. This algorithm is built for the identification of the underlying network structure and dynamics; it uses as input gene expression data and, when available, a priori knowledge of the system. An evolutionary algorithm is used as the heuristic search method for an exploration of the solution space. Computational algebra tools delimit the search space, enabling also a description of model complexity. The performance and robustness of the algorithm are explored via an artificial network of the segment polarity genes in the D. melanogaster. Once a mathematical model has been built, it can be used to run simulations of the biological system under study. Comparison of simulated dynamics with experimental measurements can help refine the model or provide insight into qualitative properties of the systems dynamical behavior. Within this work, we propose an efficient algorithm to describe the phase space, in particular to compute the number and length of all limit cycles of linear systems over a general finite field. This research has been partially supported by NIH Grant Nr. RO1GM068947-01.
- An alternating direction search algorithm for low dimensional optimization: an application to power flowBurrell, Tinal R. (Virginia Tech, 1993-05-05)Presented in this paper is a scheme for minimizing the cost function of a three-source technique to arrive at an approximation point (I,J) that is within one unit of the true minimum. The Line-Step algorithm is applied to several systems and is also compared to other minimization techniques, including the Equal Incremental Loss Algorithm. Variations are made on the Line-Step Algorithm for faster convergence and also to handle inequality constraints.
- Analysis and Approximation of Viscoelastic and Thermoelastic Joint-Beam SystemsFulton, Brian I. (Virginia Tech, 2006-07-21)Rigidizable/Inflatable space structures have been the focus of renewed interest in recent years due to efficient packaging for transport. In this work, we examine new mathematical systems used to model small-scale joint dynamics for inflatable space truss structures. We investigate the regularity and asymptotic behavior of systems resulting from various damping models, including Kelvin-Voigt, Boltzmann, and thermoelastic damping. Approximation schemes will also be introduced. Finally, we look at optimal control for the Kelvin-Voigt model using a linear feedback regulator.
- An Analysis of Stability Margins for Continuous SystemsAlbanus, Julie C. (Virginia Tech, 1999-05-06)When designing or reviewing control systems, it is important to understand the limitations of the system's design. Many systems today are designed using numerical methods. Although the numerical model may be controllable, stabilizable, or stable, small perturbations of the system parameters can result in the loss of these properties. In this thesis, we investigate these issues for finite element approximations of a thermal convection loop.
- Analysis of time varying load for minimum loss distribution reconfigurationKhan, Asif H. (Virginia Tech, 1992-04-05)A reconfiguration algorithm for electrical distribution system to reduce system losses is presented. The algorithm determines the switching patterns as a function of time. Either seasonal or daily time studies may be performed. Both manual and automatic switches are used to reconfigure the system for seasonal studies, whereas only automatic switches are considered for daily studies. An algorithm for load estimation is developed. The load estimation algorithm provides load information for each time point to be analyzed. The load estimation algorithm can incorporate any or all of the following: spot loads, circuit measurements, and customer time-varying diversified load characteristics. Voltage dependency of loads is considered at the circuit level. It is shown that switching at the system peak can reduce losses but may cause a marginal increase in system peak. Voltage and current constraints are incorporated in the reconfiguration algorithm. Data base tables and data structures used in the algorithm are described. Example problems are provided to illustrate results.
- Analysis, finite element approximation, and computation of optimal and feedback flow control problemsLee, Hyung-Chun (Virginia Tech, 1994-07-05)The analysis, finite element approximation, and numerical simulation of some control problems associated with fluid flows are considered. First, we consider a coupled solid/fluid temperature control problem. This optimization problem is motivated by the desire to remove temperature peaks, i.e., "hot spots", along the bounding surface of containers of fluid flows. The heat equation of the solid container is coupled to the energy equation for the fluid. Control is effected by adjustments to the temperature of the fluid at the inflow boundary. We give a precise statement of the mathematical model, prove the existence and uniqueness of optimal solutions, and derive an optimality system. We study a finite element approximation and provide rigorous error estimates for the error in the approximate solution of the optimality system. We then develop and implement an iterative algorithm to compute the approximate solution. Second, a computational study of the feedback control of the magnitude of the lift in flow around a cylinder is presented. The uncontrolled flow exhibits an unsymmetric Karman vortex street and a periodic lift coefficient. The size of the oscillations in the lift is reduced through an active feedback control system. The control used is the injection and suction of fluid through orifices on the cylinder; the amount of fluid injected or sucked is determined, through a simple feedback law, from pressure measurements at stations along the surface of the cylinder. The results of some computational experiments are given; these indicate that the simple feedback law used is effective in reducing the size of the oscillations in the lift. Finally, some boundary value problems which arise from a feedback control problem are considered. We give a precise statement of the mathematical problems and then prove the existence and uniqueness of solutions to the boundary value problems for the Laplace and Stokes equations by studying the boundary integral equation method.
- Approximation and control of a thermoviscoelastic systemLiu, Zhuangyi (Virginia Polytechnic Institute and State University, 1989)In this paper consider the problem of controlling a thermoviscoelastic system. We present a semigroup setting for this system, and prove the well-posedness by applying a general theorem which is given in this paper. We also study the stability of the system. We give a finite element/averaging scheme to approximate the linear quadratic regulator problem governed by the system. We prove that yields faster convergence. We give a proof of convergence of the simulation problem for singular kernels and of the control problem for L2 kernels. We carry on the numerical computation to investigate the effect of heat transfer on damping and the closed-loop system.
- Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building SystemsHu, Weiwei (Virginia Tech, 2012-05-21)In this thesis we present theoretical and numerical results for a feedback control problem defined by a thermal fluid. The problem is motivated by recent interest in designing and controlling energy efficient building systems. In particular, we show that it is possible to locally exponentially stabilize the nonlinear Boussinesq Equations by applying Neumann/Robin type boundary control on a bounded and connected domain. The feedback controller is obtained by solving a Linear Quadratic Regulator problem for the linearized Boussinesq equations. Applying classical results for semilinear equations where the linear term generates an analytic semigroup, we establish that this Riccati-based optimal boundary feedback control provides a local stabilizing controller for the full nonlinear Boussinesq equations. In addition, we present a finite element Galerkin approximation. Finally, we provide numerical results based on standard Taylor-Hood elements to illustrate the theory.
- Approximation of integro-partial differential equations of hyperbolic typeFabiano, Richard H. (Virginia Polytechnic Institute and State University, 1986)A state space model is developed for a class of integro-partial differential equations of hyperbolic type which arise in viscoelasticity. An approximation scheme is developed based on a spline approximation in the spatial variable and an averaging approximation in the de1ay variable. Techniques from linear semigroup theory are used to discuss the well-posedness of the state space model and the convergence properties of the approximation scheme. We give numerical results for a sample problem to illustrate some properties of the approximation scheme.
- Approximation of the LQR control problem for systems governed by partial functional differential equationsMiller, Robert Edwin (Virginia Polytechnic Institute and State University, 1988)We present an abstract framework for state space formulation and a generalized theorem on well-posedness which can be applied to a class of partial functional differential equations which arise in the modeling of viscoelastic and certain thermo-viscoelastic systems. Examples to which the theory applies include both second- and fourth-order equations with a variety of boundary conditions. The theory presented here allows for singular kernels as well as flexibility in the choice of state space. We discuss an approximation scheme using spline in the spatial variable and an averaging scheme in the delay variable. We compare a uniform mesh to a nonuniform mesh and give numerical results which indicate that the non-uniform mesh, which gives a better approximation of the kernel near the singularity, yields faster convergence. We give a proof of convergence of the simulation problem for singular kernels and of the control problem for bounded kernels. We use techniques of semigroup theory to establish the results on well-posedness and convergence.
- Approximations and Object-Oriented Implementation for a Parabolic Partial Differential EquationCamphouse, Russell C. (Virginia Tech, 1999-01-27)This work is a numerical study of the 2-D heat equation with Dirichlet boundary conditions over a polygonal domain. The motivation for this study is a chemical vapor deposition (CVD) reactor in which a substrate is heated while being exposed to a gas containing precursor molecules. The interaction between the gas and the substrate results in the deposition of a compound thin film on the substrate. Two different numerical approximations are implemented to produce numerical solutions describing the conduction of thermal energy in the reactor. The first method used is a Crank-Nicholson finite difference technique which tranforms the 2-D heat equation into an algebraic system of equations. For the second method, a semi-discrete method is used which transforms the partial differential equation into a system of ordinary differential equations. The goal of this work is to investigate the influence of boundary conditions, domain geometry, and initial condition on thermal conduction throughout the reactor. Once insight is gained with respect to the aforementioned conditions, optimal design and control can be investigated. This work represents a first step in our long term goal of developing optimal design and control of such CVD systems. This work has been funded through Partnerships in Research Excellence and Transition (PRET) grant number F49620-96-1-0329.
- Approximations for Singular Integral EquationsHerdman, Darwin T. (Virginia Tech, 1999-05-12)This work is a numerical study of a class of weakly singular neutral equations. The motivation for this study is an aeroelastic system. Numerical techniques are developed to approximate the singular integral equation component appearing in the complete dynamical model for the elastic motions of a three degree of freedom structure, an airfoil with trailing edge flap, in a two dimensional unsteady flow. The flap can be viewed as an active control surface to dampen vibrations that contribute to flutter. The goal of this work is to provide accurate approximations for weakly singular neutral equations using finite elements as basis functions for the initial function space. Several examples are presented in order to validate the numerical scheme.
- Bifurcation Analysis of a Model of the Frog Egg Cell CycleBorisuk, Mark T. (Virginia Tech, 1997-04-21)Fertilized frog eggs (and cell-free extracts) undergo periodic oscillations in the activity of "M-phase promoting factor" (MPF), the crucial triggering enzyme for mitosis (nuclear division) and cell division. MPF activity is regulated by a complex network of biochemical reactions. Novak and Tyson, and their collaborators, have been studying the qualitative and quantitative properties of a large system of nonlinear ordinary differential equations that describe the molecular details of this system as currently known. Important clues to the behavior of the model are provided by bifurcation theory, especially characterization of the codimension-1 and -2 bifurcation sets of the differential equations. To illustrate this method, I have been studying a system of 9 ordinary differential equations that describe the frog egg cell cycle with some fidelity. I will describe the bifurcation diagram of this system in a parameter space spanned by the rate constants for cyclin synthesis and cycling degradation. My results suggest either that the cell cycle control system should show dynamical behavior considerably more complex than the limit cycles and steady states reported so far, or that the biochemical rate constants of the system are constrained to avoid regions of parameter space where complex bifurcation points unfold.
- Cascade analysis and synthesis of transfer functions of infinite dimensional linear systemsCarpenter, Lon E. (Virginia Tech, 1992-06-13)Problems of cascade connections (synthesis) and decomposition (analysis) are analyzed for two classes of linear systems with infinite dimensional state spaces, namely, 1) admissible systems in the sense of Bart, Gohberg and Kaashoek and 2) regular systems as recently introduced by Weiss. For the class of BGK-admissible systems, it is shown that the product of two admissible systems is again admissible and that a Wiener-Hopf factorization problem can be solved just as in the finite-dimensional case. For the class of regular systems, it is shown that the cascade connection of a rational stable and antistable system has an additive stable-antistable decomposition; this involves giving a distribution interpretation to the solution of a linear Sylvester equation involving unbounded operator coefficients. As an application, some preliminary work is presented toward obtaining a state space solution of the sensitivity minimization problem for a pure delay plant.
- Compensator design for a system of two connected beamsHuang, Wei (Virginia Tech, 1994-08-05)The goal of this paper is to study the LQG problem for a class of infinite dimensional systems. We investigate the convergence of compensator gains for such systems when standard finite element schemes are used to discretize the problem. We are particularly interested in the analysis of the uniformly exponential stability of the corresponding closed - loop systems resulting from the finite dimensional compensators. A specific multiple component flexible structure is used to focus the analysis and to test problem in numerical simulations. An abstract framework for analysis and approximation of the corresponding dynamics system is developed and used to design finite - dimensional compensators. Linear semigroup theory is used to establish that the systems are well posed and to prove the convergence of generic approximation schemes. Approximate solutions of the optimal regulator and optimal observer are constructed via Galerkin - type approximations. Convergence of the scheme is established and numerical results are presented to illustrate the method
- Computational Algorithms for Face Alignment and RecognitionBellino, Kathleen Ann (Virginia Tech, 2002-04-27)Real-time face recognition has recently become available for the government and industry due to developments in face recognition algorithms, human head detection algorithms, and faster/low cost computers. Despite these advances, however, there are still some critical issues that affect the performance of real-time face recognition software. This paper addresses the problem of off-centered and out-of-pose faces in pictures, particularly in regard to the eigenface method for face recognition. We first demonstrate how the representation of faces by the eigenface method, and ultimately the performance of the software depend on the location of the eyes in the pictures. The eigenface method for face recognition is described: specifically, the creation of a face basis using the singular value decomposition, the reduction of dimension, and the unique representation of faces in the basis. Two different approaches for aligning the eyes in images are presented. The first considers the rotation of images using the orthogonal Procrustes Problem. The second approach looks at locating features in images using energy-minimizing active contours. We then conclude with a simple and fast algorithm for locating faces in images. Future research is also discussed.
- Computational Approaches to Improving Room Heating and Cooling for Energy Efficiency in BuildingsMcBee, Brian K. (Virginia Tech, 2011-08-25)With a nation-wide aim toward reducing operational energy costs in buildings, it is important to understand the dynamics of controlled heating, cooling, and air circulation of an individual room, the "One-Room Model Problem." By understanding how one most efficiently regulates a room's climate, one can use this knowledge to help develop overall best-practice power reduction strategies. A key toward effectively analyzing the "One-Room Model Problem" is to understand the capabilities and limitations of existing commercial tools designed for similar problems. In this thesis we develop methodology to link commercial Computational Fluid Dynamics (CFD) software COMSOL with standard computational mathematics software MATLAB, and design controllers that apply inlet airflow and heating or cooling to a room and investigate their effects. First, an appropriate continuum model, the Boussinesq System, is described within the framework of this problem. Next, abstract and weak formulations of the problem are described and tied to a Finite Element Method (FEM) approximation as implemented in the interface between COMSOL and MATLAB. A methodology is developed to design Linear Quadratic Regulator (LQR) controllers and associated functional gains in MATLAB which can be implemented in COMSOL. These "closed-loop" methods are then tested numerically in COMSOL and compared against "open-loop" and average state closed-loop controllers.