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Virginia Tech
1.
D'Augustine, Anthony Frank.
MATLODE: A MATLAB ODE Solver and Sensitivity Analysis Toolbox.
Degree: MS, Computer Science and Applications, 2018, Virginia Tech
URL: http://hdl.handle.net/10919/83081 
► Sensitivity analysis quantifies the effect that of perturbations of the model inputs have on the model's outputs. Some of the key insights gained using sensitivity…
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▼ Sensitivity analysis quantifies the effect that of perturbations of the model inputs have on the model's outputs. Some of the key insights gained using sensitivity analysis are to understand the robustness of the model with respect to perturbations, and to select the most important parameters for the model. MATLODE is a tool for sensitivity analysis of models described by ordinary differential equations (ODEs). MATLODE implements two distinct approaches for sensitivity analysis: direct (via the tangent linear model) and adjoint. Within each approach, four families of numerical methods are implemented, namely explicit Runge-Kutta, implicit Runge-Kutta, Rosenbrock, and single diagonally implicit Runge-Kutta. Each approach and family has its own strengths and weaknesses when applied to real world problems. MATLODE has a multitude of options that allows users to find the best approach for a wide range of initial value problems. In spite of the great importance of sensitivity analysis for models governed by differential equations, until this work there was no MATLAB ordinary differential equation sensitivity analysis toolbox publicly available. The two most popular sensitivity analysis packages, CVODES [8] and FATODE [10], are geared toward the high performance modeling space; however, no native MATLAB toolbox was available. MATLODE fills this need and offers sensitivity analysis capabilities in MATLAB, one of the most popular programming languages within scientific communities such as chemistry, biology, ecology, and oceanogra- phy. We expect that MATLODE will prove to be a useful tool for these communities to help facilitate their research and fill the gap between theory and practice.
Advisors/Committee Members: Sandu, Adrian (committeechair), Zietsman, Lizette (committee member), Cao, Yang (committee member).
Subjects/Keywords: ODE Solver; Tangent Linear Model; Adjoint Model; Sensitivity Analysis; Software
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APA (6th Edition):
D'Augustine, A. F. (2018). MATLODE: A MATLAB ODE Solver and Sensitivity Analysis Toolbox. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/83081
Chicago Manual of Style (16th Edition):
D'Augustine, Anthony Frank. “MATLODE: A MATLAB ODE Solver and Sensitivity Analysis Toolbox.” 2018. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/83081.
MLA Handbook (7th Edition):
D'Augustine, Anthony Frank. “MATLODE: A MATLAB ODE Solver and Sensitivity Analysis Toolbox.” 2018. Web. 15 Apr 2021.
Vancouver:
D'Augustine AF. MATLODE: A MATLAB ODE Solver and Sensitivity Analysis Toolbox. [Internet] [Masters thesis]. Virginia Tech; 2018. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/83081.
Council of Science Editors:
D'Augustine AF. MATLODE: A MATLAB ODE Solver and Sensitivity Analysis Toolbox. [Masters Thesis]. Virginia Tech; 2018. Available from: http://hdl.handle.net/10919/83081
Virginia Tech
2.
Munster, Drayton William.
Sensitivity Enhanced Model Reduction.
Degree: MS, Mathematics, 2013, Virginia Tech
URL: http://hdl.handle.net/10919/23169
► In this study, we numerically explore methods of coupling sensitivity analysis to the reduced model in order to increase the accuracy of a proper orthogonal…
(more)
▼ In this study, we numerically explore methods of coupling sensitivity analysis to the reduced model in order to increase the accuracy of a proper orthogonal decomposition (POD) basis across a wider range of parameters. Various techniques based on polynomial interpolation and basis alteration are compared. These techniques are performed on a 1-dimensional reaction-diffusion equation and 2-dimensional incompressible Navier-Stokes equations solved using the finite element method (FEM) as the full scale model. The expanded model formed by expanding the POD basis with the orthonormalized basis sensitivity vectors achieves the best mixture of accuracy and computational efficiency among the methods compared.
Advisors/Committee Members: Zietsman, Lizette (committeechair), Borggaard, Jeffrey T. (committee member), Gugercin, Serkan (committee member).
Subjects/Keywords: Model Reduction; Sensitivity Analysis
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APA (6th Edition):
Munster, D. W. (2013). Sensitivity Enhanced Model Reduction. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/23169
Chicago Manual of Style (16th Edition):
Munster, Drayton William. “Sensitivity Enhanced Model Reduction.” 2013. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/23169.
MLA Handbook (7th Edition):
Munster, Drayton William. “Sensitivity Enhanced Model Reduction.” 2013. Web. 15 Apr 2021.
Vancouver:
Munster DW. Sensitivity Enhanced Model Reduction. [Internet] [Masters thesis]. Virginia Tech; 2013. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/23169.
Council of Science Editors:
Munster DW. Sensitivity Enhanced Model Reduction. [Masters Thesis]. Virginia Tech; 2013. Available from: http://hdl.handle.net/10919/23169
Virginia Tech
3.
Oremland, Matthew Scott.
Optimization and Optimal Control of Agent-Based Models.
Degree: MS, Mathematics, 2011, Virginia Tech
URL: http://hdl.handle.net/10919/78119
► Agent-based models are computer models made up of agents that can exist in a finite number of states. The state of the system at any…
(more)
▼ Agent-based models are computer models made up of agents that can exist in a finite number of states. The state of the system at any given time is determined by rules governing agents' interaction. The rules may be deterministic or stochastic. Optimization is the process of finding a solution that optimizes some value that is determined by simulating the model. Optimal control of an agent-based model is the process of determining a sequence of control inputs to the model that steer the system to a desired state in the most efficient way. In large and complex models, the number of possible control inputs is too large to be enumerated by computers; hence methods must be developed for use with these models in order to find solutions without searching the entire solution space. Heuristic algorithms have been applied to such models with some success. Such algorithms are discussed; case studies of examples from biology are presented. The lack of a standard format for agent-based models is a major issue facing the study of agent-based models; presentation as polynomial dynamical systems is presented as a viable option. Algorithms are adapted and presented for use in this framework.
Advisors/Committee Members: Laubenbacher, Reinhard C. (committeechair), Loehr, Nicholas A. (committee member), Zietsman, Lizette (committee member).
Subjects/Keywords: optimization; optimal control; individual-based model; polynomial dynamical system; agent-based model; bioinformatics; heuristic algorithm; discrete model; systems biology
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APA (6th Edition):
Oremland, M. S. (2011). Optimization and Optimal Control of Agent-Based Models. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/78119
Chicago Manual of Style (16th Edition):
Oremland, Matthew Scott. “Optimization and Optimal Control of Agent-Based Models.” 2011. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/78119.
MLA Handbook (7th Edition):
Oremland, Matthew Scott. “Optimization and Optimal Control of Agent-Based Models.” 2011. Web. 15 Apr 2021.
Vancouver:
Oremland MS. Optimization and Optimal Control of Agent-Based Models. [Internet] [Masters thesis]. Virginia Tech; 2011. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/78119.
Council of Science Editors:
Oremland MS. Optimization and Optimal Control of Agent-Based Models. [Masters Thesis]. Virginia Tech; 2011. Available from: http://hdl.handle.net/10919/78119
Virginia Tech
4.
KusterJr, George Emil.
H-Infinity Norm Calculation via a State Space Formulation.
Degree: MS, Mathematics, 2013, Virginia Tech
URL: http://hdl.handle.net/10919/49544
► There is much interest in the design of feedback controllers for linear systems that minimize the H-infty norm of a specific closed-loop transfer function. The…
(more)
▼ There is much interest in the design of feedback controllers for linear systems that minimize the H-infty norm of a specific closed-loop transfer function. The H-infty optimization problem initiated by Zames (1981), \, has received a lot of interest since its formulation. In H-infty control theory one uses the H-infty norm of a stable transfer function as a performance measure. One typically uses approaches in either the frequency domain or a state space formulation to tackle this problem. Frequency domain approaches use operator theory, J-spectral factorization or polynomial methods while in the state space approach one uses ideas similar to LQ theory and differential games. One of the key computational issues in the design of H-infty optimal controllers is the determination of the optimal H-infty norm. That is, determining the infimum of r for which the H-infty norm of the associated transfer function matrix is less than r. Doyle et al (1989), presented a state space characterization for the sub-optimal H-infty control problem. This characterization requires that the unique stabilizing solutions to two Algebraic Riccati Equations are positive semi definite as well as satisfying a spectral radius coupling condition. In this work, we describe an algorithm by Lin et al(1999), used to calculate the H-infty norm for the state feedback and output feedback control problems. This algorithm only relies on standard assumptions and divides the problem into three sub-problems. The first two sub-problems rely on algorithms for the state feedback problem formulated in the frequency domain as well as a characterization of the optimal value in terms of the singularity of the upper-half of a matrix created by the stacked basis vectors of the invariant sub-space of the associated Hamiltonian matrix. This characterization is verified through a bisection or secant method. The third sub-problem relies on the geometric nature of the spectral radius of the product of the two solutions to the Algebraic Riccati Equations associated with the first two sub-problems. Doyle makes an intuitive argument that the spectral radius condition will fail before the conditions involving the Algebraic Riccati Equations fail. We present numerical results where we demonstrate that the Algebraic Riccati Equation conditions fail before the spectral radius condition fails.
Advisors/Committee Members: Zietsman, Lizette (committeechair), Borggaard, Jeffrey T. (committee member), Wawro, Megan (committee member).
Subjects/Keywords: H-infinity control
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APA (6th Edition):
KusterJr, G. E. (2013). H-Infinity Norm Calculation via a State Space Formulation. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/49544
Chicago Manual of Style (16th Edition):
KusterJr, George Emil. “H-Infinity Norm Calculation via a State Space Formulation.” 2013. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/49544.
MLA Handbook (7th Edition):
KusterJr, George Emil. “H-Infinity Norm Calculation via a State Space Formulation.” 2013. Web. 15 Apr 2021.
Vancouver:
KusterJr GE. H-Infinity Norm Calculation via a State Space Formulation. [Internet] [Masters thesis]. Virginia Tech; 2013. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/49544.
Council of Science Editors:
KusterJr GE. H-Infinity Norm Calculation via a State Space Formulation. [Masters Thesis]. Virginia Tech; 2013. Available from: http://hdl.handle.net/10919/49544
Virginia Tech
5.
Lattimer, Alan Martin.
Model Reduction of Nonlinear Fire Dynamics Models.
Degree: PhD, Mathematics, 2016, Virginia Tech
URL: http://hdl.handle.net/10919/70870
► Due to the complexity, multi-scale, and multi-physics nature of the mathematical models for fires, current numerical models require too much computational effort to be useful…
(more)
▼ Due to the complexity, multi-scale, and multi-physics nature of the mathematical models for fires, current numerical models require too much computational effort to be useful in design and real-time decision making, especially when dealing with fires over large domains. To reduce the computational time while retaining the complexity of the domain and physics, our research has focused on several reduced-order modeling techniques. Our contributions are improving wildland fire reduced-order models (ROMs), creating new ROM techniques for nonlinear systems, and preserving optimality when discretizing a continuous-time ROM. Currently, proper orthogonal decomposition (POD) is being used to reduce wildland fire-spread models with limited success. We use a technique known as the discrete empirical interpolation method (DEIM) to address the slowness due to the nonlinearity. We create new methods to reduce nonlinear models, such as the Burgers' equation, that perform better than POD over a wider range of input conditions. Further, these ROMs can often be constructed without needing to capture full-order solutions a priori. This significantly reduces the off-line costs associated with creating the ROM. Finally, we investigate methods of time-discretization that preserve the optimality conditions in a certain norm associated with the input to output mapping of a dynamical system. In particular, we are able to show that the Crank-Nicholson method preserves the optimality conditions, but other single-step methods do not. We further clarify the need for these discrete-time ROMs to match at infinity in order to ensure local optimality.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Gugercin, Serkan (committeechair), Burns, John A. (committee member), Zietsman, Lizette (committee member).
Subjects/Keywords: Model Reduction; Fire Models; IRKA; POD; Discrete-Time Systems
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APA (6th Edition):
Lattimer, A. M. (2016). Model Reduction of Nonlinear Fire Dynamics Models. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/70870
Chicago Manual of Style (16th Edition):
Lattimer, Alan Martin. “Model Reduction of Nonlinear Fire Dynamics Models.” 2016. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/70870.
MLA Handbook (7th Edition):
Lattimer, Alan Martin. “Model Reduction of Nonlinear Fire Dynamics Models.” 2016. Web. 15 Apr 2021.
Vancouver:
Lattimer AM. Model Reduction of Nonlinear Fire Dynamics Models. [Internet] [Doctoral dissertation]. Virginia Tech; 2016. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/70870.
Council of Science Editors:
Lattimer AM. Model Reduction of Nonlinear Fire Dynamics Models. [Doctoral Dissertation]. Virginia Tech; 2016. Available from: http://hdl.handle.net/10919/70870
Virginia Tech
6.
Erwin, Samantha H.
Modeling of Passive Chilled Beams for use in Efficient Control of Indoor-Air Environments.
Degree: MS, Mathematics, 2013, Virginia Tech
URL: http://hdl.handle.net/10919/23301
► This work is done as a small facet of a much larger study on efficient control of indoor air environments. Halton passive chilled beams are…
(more)
▼ This work is done as a small facet of a much larger study on efficient control of indoor air environments. Halton passive chilled beams are used to cool rooms and the focus of this work is to model the beams. This work also reviews the mesh making process in Gmsh. ANSYS Fluent was used throughout the entire research and this thesis describes the software and a careful description of the case study.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Herdman, Terry L. (committee member), Zietsman, Lizette (committee member).
Subjects/Keywords: Computational Fluid Dynamics; Chilled Beams; Fluent
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APA (6th Edition):
Erwin, S. H. (2013). Modeling of Passive Chilled Beams for use in Efficient Control of Indoor-Air Environments. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/23301
Chicago Manual of Style (16th Edition):
Erwin, Samantha H. “Modeling of Passive Chilled Beams for use in Efficient Control of Indoor-Air Environments.” 2013. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/23301.
MLA Handbook (7th Edition):
Erwin, Samantha H. “Modeling of Passive Chilled Beams for use in Efficient Control of Indoor-Air Environments.” 2013. Web. 15 Apr 2021.
Vancouver:
Erwin SH. Modeling of Passive Chilled Beams for use in Efficient Control of Indoor-Air Environments. [Internet] [Masters thesis]. Virginia Tech; 2013. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/23301.
Council of Science Editors:
Erwin SH. Modeling of Passive Chilled Beams for use in Efficient Control of Indoor-Air Environments. [Masters Thesis]. Virginia Tech; 2013. Available from: http://hdl.handle.net/10919/23301
Virginia Tech
7.
Glaws, Andrew Taylor.
Finite Element Simulations of Two Dimensional Peridynamic Models.
Degree: MS, Mathematics, 2014, Virginia Tech
URL: http://hdl.handle.net/10919/48121
► This thesis explores the science of solid mechanics via the theory of peridynamics. Peridynamics has several key advantages over the classical theory of elasticity. The…
(more)
▼ This thesis explores the science of solid mechanics via the theory of peridynamics. Peridynamics has several key advantages over the classical theory of elasticity. The most notable of which is the ease with which fractures in the the material are handled. The goal here is to study the two theories and how they relate for problems in which the classical method is known to work well. While it is known that state-based peridynamic models agree with classical elasticity as the horizon radius vanishes, similar results for bond-based models have yet to be developed. In this study, we use numerical simulations to investigate the behavior of bond-based peridynamic models under this limit for a number of cases where analytic solutions of the classical elasticity problem are known. To carry out this study, the integral-based peridynamic model is solved using the finite element method in two dimensions and compared against solutions using the classical approach.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Zietsman, Lizette (committee member), Lin, Tao (committee member).
Subjects/Keywords: Peridynamics; Elasticity; Solid Mechanics
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APA (6th Edition):
Glaws, A. T. (2014). Finite Element Simulations of Two Dimensional Peridynamic Models. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/48121
Chicago Manual of Style (16th Edition):
Glaws, Andrew Taylor. “Finite Element Simulations of Two Dimensional Peridynamic Models.” 2014. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/48121.
MLA Handbook (7th Edition):
Glaws, Andrew Taylor. “Finite Element Simulations of Two Dimensional Peridynamic Models.” 2014. Web. 15 Apr 2021.
Vancouver:
Glaws AT. Finite Element Simulations of Two Dimensional Peridynamic Models. [Internet] [Masters thesis]. Virginia Tech; 2014. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/48121.
Council of Science Editors:
Glaws AT. Finite Element Simulations of Two Dimensional Peridynamic Models. [Masters Thesis]. Virginia Tech; 2014. Available from: http://hdl.handle.net/10919/48121
Virginia Tech
8.
Grimm, Alexander Rudolf.
Taming of Complex Dynamical Systems.
Degree: MS, Mathematics, 2013, Virginia Tech
URL: http://hdl.handle.net/10919/24775
► The problem of establishing local existence and uniqueness of solutions to systems of differential equations is well understood and has a long history. However, the…
(more)
▼ The problem of establishing local existence and uniqueness of solutions to systems of differential equations is well understood and has a long history. However, the problem of proving global existence and uniqueness is more difficult and fails even for some very simple ordinary differential equations. It is still not known if the 3D Navier-Stokes equation have global unique solutions and this open problem is one of the Millennium Prize Problems. However, many of these mathematical models are extremely useful in the understanding of complex physical systems. For years people have considered methods for modifying these equations in order to obtain models that still capture the observed fundamental physics, but for which one can rigorously establish global results. In this thesis we focus on a taming method to achieve this goal and apply taming to modeling and numerical problems. The method is also applied to a class of nonlinear differential equations with conservative nonlinearities and to Burgers’ Equation with Neumann boundary conditions. Numerical results are presented to illustrate the ideas.
Advisors/Committee Members: Burns, John A. (committeechair), Zietsman, Lizette (committee member), Gugercin, Serkan (committee member).
Subjects/Keywords: Taming; Dynamical Systems; Navier Stokes; Burgers Equation; Finite Elements
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APA (6th Edition):
Grimm, A. R. (2013). Taming of Complex Dynamical Systems. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/24775
Chicago Manual of Style (16th Edition):
Grimm, Alexander Rudolf. “Taming of Complex Dynamical Systems.” 2013. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/24775.
MLA Handbook (7th Edition):
Grimm, Alexander Rudolf. “Taming of Complex Dynamical Systems.” 2013. Web. 15 Apr 2021.
Vancouver:
Grimm AR. Taming of Complex Dynamical Systems. [Internet] [Masters thesis]. Virginia Tech; 2013. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/24775.
Council of Science Editors:
Grimm AR. Taming of Complex Dynamical Systems. [Masters Thesis]. Virginia Tech; 2013. Available from: http://hdl.handle.net/10919/24775
9.
May, Thomas Joseph.
Minimally Corrective, Approximately Recovering Priors to Correct Expert Judgement in Bayesian Parameter Estimation.
Degree: MS, Mathematics, 2015, Virginia Tech
URL: http://hdl.handle.net/10919/54593
► Bayesian parameter estimation is a popular method to address inverse problems. However, since prior distributions are chosen based on expert judgement, the method can inherently…
(more)
▼ Bayesian parameter estimation is a popular method to address inverse problems. However, since prior distributions are chosen based on expert judgement, the method can inherently introduce bias into the understanding of the parameters. This can be especially relevant in the case of distributed parameters where it is difficult to check for error. To minimize this bias, we develop the idea of a minimally corrective, approximately recovering prior (MCAR prior) that generates a guide for the prior and corrects the expert supplied prior according to that guide. We demonstrate this approach for the 1D elliptic equation or the elliptic partial differential equation and observe how this method works in cases with significant and without any expert bias. In the case of significant expert bias, the method substantially reduces the bias and, in the case with no expert bias, the method only introduces minor errors. The cost of introducing these small errors for good judgement is worth the benefit of correcting major errors in bad judgement. This is particularly true when the prior is only determined using a heuristic or an assumed distribution.
Advisors/Committee Members: Zietsman, Lizette (committeechair), Borggaard, Jeffrey T. (committee member), Rossi, John F. (committee member).
Subjects/Keywords: Bayesian Parameter Estimation; Minimally Corrective Priors; Distributed Parameters; Elliptic Equation; Karhunen-Loeve Theorem.
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APA (6th Edition):
May, T. J. (2015). Minimally Corrective, Approximately Recovering Priors to Correct Expert Judgement in Bayesian Parameter Estimation. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/54593
Chicago Manual of Style (16th Edition):
May, Thomas Joseph. “Minimally Corrective, Approximately Recovering Priors to Correct Expert Judgement in Bayesian Parameter Estimation.” 2015. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/54593.
MLA Handbook (7th Edition):
May, Thomas Joseph. “Minimally Corrective, Approximately Recovering Priors to Correct Expert Judgement in Bayesian Parameter Estimation.” 2015. Web. 15 Apr 2021.
Vancouver:
May TJ. Minimally Corrective, Approximately Recovering Priors to Correct Expert Judgement in Bayesian Parameter Estimation. [Internet] [Masters thesis]. Virginia Tech; 2015. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/54593.
Council of Science Editors:
May TJ. Minimally Corrective, Approximately Recovering Priors to Correct Expert Judgement in Bayesian Parameter Estimation. [Masters Thesis]. Virginia Tech; 2015. Available from: http://hdl.handle.net/10919/54593
10.
Saul, April Leigh.
The Role of e-Antigen in Hepatitis B Infection.
Degree: MS, Mathematics, 2015, Virginia Tech
URL: http://hdl.handle.net/10919/53957
► Mathematical modeling of biological systems has improved the knowledge of scientists for many years. In virology, particularly in the study of hepatitis B virus, mathematical…
(more)
▼ Mathematical modeling of biological systems has improved the knowledge of scientists for many years. In virology, particularly in the study of hepatitis B virus, mathematical models were used to explain interactions between hepatitis B virus and the human host in the absence and presence of interventions such as drug therapy and vaccines. This thesis seeks to explain the role of e-Antigen, a particle produced by hepatitis B virus, in the pathogenesis of hepatitis B infection. To accomplish this goal, I will provide biological background as well as previous modeling work on the role of e-Antigen in hepatitis B virus infection, before finally developing a new model adapted specifically for connecting hepatitis B progression with e-Antigen and drug therapy. I will analyze the model both analytically and numerically, fit it to virus data from humans chronically infected with hepatitis B that undergo drug therapy, and draw conclusions about the relation between drugs, immune activation, and loss of e-Antigen.
Advisors/Committee Members: Ciupe, Mihaela Stanca (committeechair), Zietsman, Lizette (committee member), Chung, Matthias (committee member).
Subjects/Keywords: Mathematical modeling; Hepatitis B Virus; e-Antigen
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APA (6th Edition):
Saul, A. L. (2015). The Role of e-Antigen in Hepatitis B Infection. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/53957
Chicago Manual of Style (16th Edition):
Saul, April Leigh. “The Role of e-Antigen in Hepatitis B Infection.” 2015. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/53957.
MLA Handbook (7th Edition):
Saul, April Leigh. “The Role of e-Antigen in Hepatitis B Infection.” 2015. Web. 15 Apr 2021.
Vancouver:
Saul AL. The Role of e-Antigen in Hepatitis B Infection. [Internet] [Masters thesis]. Virginia Tech; 2015. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/53957.
Council of Science Editors:
Saul AL. The Role of e-Antigen in Hepatitis B Infection. [Masters Thesis]. Virginia Tech; 2015. Available from: http://hdl.handle.net/10919/53957
11.
Hasanyan, Jalil Davresh.
Modeling and Analysis of a Moving Conductive String in a Magnetic Field.
Degree: MS, Mathematics, 2019, Virginia Tech
URL: http://hdl.handle.net/10919/87530
► A wide range of physical systems are modeled as axially moving strings; such examples are belts, tapes, wires and fibers with applied electromagnetic fields. In…
(more)
▼ A wide range of physical systems are modeled as axially moving strings; such examples are belts, tapes, wires and fibers with applied electromagnetic fields. In this study, we propose a model that describes the motion of a current-carrying conductive string in a lateral magnetic field, while it is being pulled axially. This model is a generalization of past studies that have neglected one or more properties featured in our system. It is assumed that the string is moving with a constant velocity between two rings that are a finite distance apart. Directions of the magnetic field and the motion of the string coincide. The problem is first considered in a static setting. Stability critical values of the magnetic field, pulling speed, and current are shown to exist when the uniform motion (along a string line) of the string buckles into spiral forms. In the dynamic setting, conditions for stability of certain solutions are presented and discussed. It is shown that there is a divergence between the critical values in the linear dynamic and static cases. Furthermore, traveling wave solutions are examined for certain cases of our general system. We develop an approximate solution for a nonlinear moving string when a periodic nonstationary current flows through the string. Domains of
parameters are defined when the string falls into a pre-chaotic state, i.e., the frequency of vibrations is doubled.
Advisors/Committee Members: Zietsman, Lizette (committeechair), Burns, John A. (committee member), Embree, Mark Partick (committee member).
Subjects/Keywords: Current-carrying string; stability; modeling; magnetic field; resonance vibrations
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APA (6th Edition):
Hasanyan, J. D. (2019). Modeling and Analysis of a Moving Conductive String in a Magnetic Field. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/87530
Chicago Manual of Style (16th Edition):
Hasanyan, Jalil Davresh. “Modeling and Analysis of a Moving Conductive String in a Magnetic Field.” 2019. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/87530.
MLA Handbook (7th Edition):
Hasanyan, Jalil Davresh. “Modeling and Analysis of a Moving Conductive String in a Magnetic Field.” 2019. Web. 15 Apr 2021.
Vancouver:
Hasanyan JD. Modeling and Analysis of a Moving Conductive String in a Magnetic Field. [Internet] [Masters thesis]. Virginia Tech; 2019. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/87530.
Council of Science Editors:
Hasanyan JD. Modeling and Analysis of a Moving Conductive String in a Magnetic Field. [Masters Thesis]. Virginia Tech; 2019. Available from: http://hdl.handle.net/10919/87530
12.
Kurdila, Hannah Robertshaw.
Gappy POD and Temporal Correspondence for Lizard Motion Estimation.
Degree: MS, Mathematics, 2018, Virginia Tech
URL: http://hdl.handle.net/10919/83603
► With the maturity of conventional industrial robots, there has been increasing interest in designing robots that emulate realistic animal motions. This discipline requires careful and…
(more)
▼ With the maturity of conventional industrial robots, there has been increasing interest in designing robots that emulate realistic animal motions. This discipline requires careful and systematic investigation of a wide range of animal motions from biped, to quadruped, and even to serpentine motion of centipedes, millipedes, and snakes. Collecting optical motion capture data of such complex animal motions can be complicated for several reasons. Often there is the need to use many high-quality cameras for detailed subject tracking, and self-occlusion, loss of focus, and contrast variations challenge any imaging experiment. The problem of self-occlusion is especially pronounced for animals. In this thesis, we walk through the process of collecting motion capture data of a running lizard. In our collected raw video footage, it is difficult to make temporal correspondences using interpolation methods because of prolonged blurriness, occlusion, or the limited field of vision of our cameras. To work around this, we first make a model data set by making our best guess of the points' locations through these corruptions. Then, we randomly eclipse the data, use Gappy POD to repair the data and then see how closely it resembles the initial set, culminating in a test case where we simulate the actual corruptions we see in the raw video footage.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Gugercin, Serkan (committee member), Zietsman, Lizette (committee member).
Subjects/Keywords: Gappy proper orthogonal decomposition; lizard locomotion; motion capture; occlusion; pose estimation; temporal correspondence; tracking
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APA (6th Edition):
Kurdila, H. R. (2018). Gappy POD and Temporal Correspondence for Lizard Motion Estimation. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/83603
Chicago Manual of Style (16th Edition):
Kurdila, Hannah Robertshaw. “Gappy POD and Temporal Correspondence for Lizard Motion Estimation.” 2018. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/83603.
MLA Handbook (7th Edition):
Kurdila, Hannah Robertshaw. “Gappy POD and Temporal Correspondence for Lizard Motion Estimation.” 2018. Web. 15 Apr 2021.
Vancouver:
Kurdila HR. Gappy POD and Temporal Correspondence for Lizard Motion Estimation. [Internet] [Masters thesis]. Virginia Tech; 2018. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/83603.
Council of Science Editors:
Kurdila HR. Gappy POD and Temporal Correspondence for Lizard Motion Estimation. [Masters Thesis]. Virginia Tech; 2018. Available from: http://hdl.handle.net/10919/83603
13.
Letona Bolivar, Cristina Felicitas.
On a Class of Parametrized Domain Optimization Problems with Mixed Boundary Condition Types.
Degree: PhD, Mathematics, 2016, Virginia Tech
URL: http://hdl.handle.net/10919/73308
► The methods for solving domain optimization problems depends on the case of study. There are methods that have been developed for the discretized problem, but…
(more)
▼ The methods for solving domain optimization problems depends on the case of study. There are methods that have been developed for the discretized problem, but not much is done in the infinite dimensional case. We analyze the theoretical aspects of the infinite dimensional case for a particular domain optimization problem where a portion of the boundary is parametrized, these results involve the existence of the solution to our problem and the calculation of the derivative of the shape functional.
Shape optimization problems have a long history of mathematical study and a wide range of applications. In recent decades there has been an interest in solving these problems with partial differential equation (PDE) constraints. We consider a special class of PDE-constrained shape optimization problems where different boundary condition types (Dirichlet and Neumann) are imposed on the same boundary segment. We also consider the case where the interface between these different boundary condition types may also be parameter dependent. This study also includes special cases where the shape of the region where the PDE is imposed does not change, but the domain of the partial differential operator is parameter dependent, due to the change in boundary condition type. Our treatment centers on the infinite dimensional formulation of the optimization problem. We consider existence of solutions as well as the calculation of derivatives of the associated shape functionals via adjoint solutions. These derivative formulations serve as a starting point for practical numerical approximations.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Zietsman, Lizette (committee member), Iliescu, Traian (committee member), Lin, Tao (committee member).
Subjects/Keywords: Domain Optimization; Shape Derivatives; PDE Constraints; Mixed Boundary Conditions.
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APA (6th Edition):
Letona Bolivar, C. F. (2016). On a Class of Parametrized Domain Optimization Problems with Mixed Boundary Condition Types. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/73308
Chicago Manual of Style (16th Edition):
Letona Bolivar, Cristina Felicitas. “On a Class of Parametrized Domain Optimization Problems with Mixed Boundary Condition Types.” 2016. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/73308.
MLA Handbook (7th Edition):
Letona Bolivar, Cristina Felicitas. “On a Class of Parametrized Domain Optimization Problems with Mixed Boundary Condition Types.” 2016. Web. 15 Apr 2021.
Vancouver:
Letona Bolivar CF. On a Class of Parametrized Domain Optimization Problems with Mixed Boundary Condition Types. [Internet] [Doctoral dissertation]. Virginia Tech; 2016. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/73308.
Council of Science Editors:
Letona Bolivar CF. On a Class of Parametrized Domain Optimization Problems with Mixed Boundary Condition Types. [Doctoral Dissertation]. Virginia Tech; 2016. Available from: http://hdl.handle.net/10919/73308
14.
Erwin, Samantha H.
Mathematical Models of Immune Responses to Infectious Diseases.
Degree: PhD, Mathematics, 2017, Virginia Tech
URL: http://hdl.handle.net/10919/77026
► In this dissertation, we investigate the mechanisms behind diseases and the immune responses required for successful disease resolution in three projects: i) A study of…
(more)
▼ In this dissertation, we investigate the mechanisms behind diseases and the immune responses required for successful disease resolution in three projects: i) A study of HIV and HPV co-infection, ii) A germinal center dynamics model, iii) A study of monoclonal antibody therapy. We predict that the condition leading to HPV persistence during HIV/HPV co-infection is the permissive immune environment created by HIV, rather than the direct HIV/HPV interaction. In the second project, we develop a germinal center model to understand the mechanisms that lead to the formation of potent long-lived plasma. We predict that the T follicular helper cells are a limiting resource and present possible mechanisms that can revert this limitation in the presence of non-mutating and mutating antigen. Finally, we develop a pharmacokinetic model of 3BNC117 antibody dynamics and HIV viral dynamics following antibody therapy. We fit the models to clinical trial data and conclude that antibody binding is delayed and that the combined effects of initial CD4 T cell count, initial HIV levels, and virus production are strong indicators of a good response to antibody immunotherapy.
Advisors/Committee Members: Ciupe, Mihaela Stanca (committeechair), Chung, Matthias (committee member), Zietsman, Lizette (committee member), Childs, Lauren M. (committee member).
Subjects/Keywords: Mathematical biology; theoretical immunology; ordinary differential equations
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APA (6th Edition):
Erwin, S. H. (2017). Mathematical Models of Immune Responses to Infectious Diseases. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/77026
Chicago Manual of Style (16th Edition):
Erwin, Samantha H. “Mathematical Models of Immune Responses to Infectious Diseases.” 2017. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/77026.
MLA Handbook (7th Edition):
Erwin, Samantha H. “Mathematical Models of Immune Responses to Infectious Diseases.” 2017. Web. 15 Apr 2021.
Vancouver:
Erwin SH. Mathematical Models of Immune Responses to Infectious Diseases. [Internet] [Doctoral dissertation]. Virginia Tech; 2017. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/77026.
Council of Science Editors:
Erwin SH. Mathematical Models of Immune Responses to Infectious Diseases. [Doctoral Dissertation]. Virginia Tech; 2017. Available from: http://hdl.handle.net/10919/77026
15.
Kramer, Boris Martin Josef.
Model and Data Reduction for Control, Identification and Compressed Sensing.
Degree: PhD, Mathematics, 2015, Virginia Tech
URL: http://hdl.handle.net/10919/75179
► This dissertation focuses on problems in design, optimization and control of complex, large-scale dynamical systems from different viewpoints. The goal is to develop new algorithms…
(more)
▼ This dissertation focuses on problems in design, optimization and control of complex, large-scale dynamical systems from different viewpoints. The goal is to develop new algorithms and methods, that solve real problems more efficiently, together with providing mathematical insight into the success of those methods. There are three main contributions in this dissertation.
In Chapter 3, we provide a new method to solve large-scale algebraic Riccati equations, which arise in optimal control, filtering and model reduction. We present a projection based algorithm utilizing proper orthogonal decomposition, which is demonstrated to produce highly accurate solutions at low rank. The method is parallelizable, easy to implement for practitioners, and is a first step towards a matrix free approach to solve AREs. Numerical examples for n >= 100,000 unknowns are presented.
In Chapter 4, we develop a system identification method which is motivated by tangential interpolation. This addresses the challenge of fitting linear time invariant systems to input-output responses of complex dynamics, where the number of inputs and outputs is relatively large. The method reduces the computational burden imposed by a full singular value decomposition, by carefully choosing directions on which to project the impulse response prior to assembly of the Hankel matrix. The identification and model reduction step follows from the eigensystem realization algorithm. We present three numerical examples, a mass spring damper system, a heat transfer problem, and a fluid dynamics system. We obtain error bounds and stability results for this method.
Chapter 5 deals with control and observation design for parameter dependent dynamical systems. We address this by using local parametric reduced order models, which can be used online. Data available from simulations of the system at various configurations (parameters, boundary conditions) is used to extract a sparse basis to represent the dynamics (via dynamic mode decomposition). Subsequently, a new compressed sensing based classification algorithm is developed which incorporates the extracted dynamic information into the sensing basis. We show that this augmented classification basis makes the method more robust to noise, and results in superior identification of the correct parameter. Numerical examples consist of a Navier-Stokes, as well as a Boussinesq flow application.
Advisors/Committee Members: Burns, John A. (committeechair), Gugercin, Serkan (committee member), Cliff, Eugene M. (committee member), Zietsman, Lizette (committee member), Borggaard, Jeffrey T. (committee member).
Subjects/Keywords: Model and Data Reduction; Dynamical Systems; System Identification; Optimal Control; Compressed Sensing
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APA (6th Edition):
Kramer, B. M. J. (2015). Model and Data Reduction for Control, Identification and Compressed Sensing. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/75179
Chicago Manual of Style (16th Edition):
Kramer, Boris Martin Josef. “Model and Data Reduction for Control, Identification and Compressed Sensing.” 2015. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/75179.
MLA Handbook (7th Edition):
Kramer, Boris Martin Josef. “Model and Data Reduction for Control, Identification and Compressed Sensing.” 2015. Web. 15 Apr 2021.
Vancouver:
Kramer BMJ. Model and Data Reduction for Control, Identification and Compressed Sensing. [Internet] [Doctoral dissertation]. Virginia Tech; 2015. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/75179.
Council of Science Editors:
Kramer BMJ. Model and Data Reduction for Control, Identification and Compressed Sensing. [Doctoral Dissertation]. Virginia Tech; 2015. Available from: http://hdl.handle.net/10919/75179
Virginia Tech
16.
McBee, Brian K.
Computational Approaches to Improving Room Heating and Cooling for Energy Efficiency in Buildings.
Degree: PhD, Mathematics, 2011, Virginia Tech
URL: http://hdl.handle.net/10919/28911
► 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…
(more)
▼ 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.
Advisors/Committee Members: Burns, John A. (committeechair), Zietsman, Lizette (committee member), Cliff, Eugene M. (committee member), Borggaard, Jeffrey T. (committee member).
Subjects/Keywords: COMSOL; Finite Elements; Building Energy Efficiency; Boundary Control; Boussinesq
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APA (6th Edition):
McBee, B. K. (2011). Computational Approaches to Improving Room Heating and Cooling for Energy Efficiency in Buildings. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/28911
Chicago Manual of Style (16th Edition):
McBee, Brian K. “Computational Approaches to Improving Room Heating and Cooling for Energy Efficiency in Buildings.” 2011. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/28911.
MLA Handbook (7th Edition):
McBee, Brian K. “Computational Approaches to Improving Room Heating and Cooling for Energy Efficiency in Buildings.” 2011. Web. 15 Apr 2021.
Vancouver:
McBee BK. Computational Approaches to Improving Room Heating and Cooling for Energy Efficiency in Buildings. [Internet] [Doctoral dissertation]. Virginia Tech; 2011. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/28911.
Council of Science Editors:
McBee BK. Computational Approaches to Improving Room Heating and Cooling for Energy Efficiency in Buildings. [Doctoral Dissertation]. Virginia Tech; 2011. Available from: http://hdl.handle.net/10919/28911
Virginia Tech
17.
Hu, Weiwei.
Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems.
Degree: PhD, Mathematics, 2012, Virginia Tech
URL: http://hdl.handle.net/10919/38664
► 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…
(more)
▼ 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.
Advisors/Committee Members: Burns, John A. (committeechair), Cliff, Eugene M. (committee member), Zietsman, Lizette (committee member), Ball, Joseph A. (committee member), Borggaard, Jeffrey T. (committee member).
Subjects/Keywords: Taylor-Hood Elements; Analytic Semigroup; Algebraic Riccati Equation; LQR Control; Boundary Feedback Control; Boussinesq Equations
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APA (6th Edition):
Hu, W. (2012). Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/38664
Chicago Manual of Style (16th Edition):
Hu, Weiwei. “Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems.” 2012. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/38664.
MLA Handbook (7th Edition):
Hu, Weiwei. “Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems.” 2012. Web. 15 Apr 2021.
Vancouver:
Hu W. Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems. [Internet] [Doctoral dissertation]. Virginia Tech; 2012. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/38664.
Council of Science Editors:
Hu W. Approximation and Control of the Boussinesq Equations with Application to Control of Energy Efficient Building Systems. [Doctoral Dissertation]. Virginia Tech; 2012. Available from: http://hdl.handle.net/10919/38664
Virginia Tech
18.
Leite Dos Santos Nunes, Vitor Manuel.
Fréchet Sensitivity Analysis and Parameter Estimation in Groundwater Flow Models.
Degree: PhD, Mathematics, 2013, Virginia Tech
URL: http://hdl.handle.net/10919/50653
► In this work we develop and analyze algorithms motivated by the parameter estimation problem corresponding to a multilayer aquifer/interbed groundwater flow model. The parameter estimation…
(more)
▼ In this work we develop and analyze algorithms motivated by the parameter estimation problem corresponding to a multilayer aquifer/interbed groundwater flow model. The parameter estimation problem is formulated as an optimization problem, then addressed with algorithms based on adjoint equations, quasi-Newton schemes, and multilevel optimization. In addition to the parameter estimation problem, we consider properties of the parameter to solution map. This includes invertibility (known as identifiability) and differentiability properties of the map. For differentiability, we expand existing results on Fréchet sensitivity analysis to convection diffusion equations and groundwater flow equations. This is achieved by proving that the Fréchet derivative of the solution operator is Hilbert-Schmidt, under smoothness assumptions for the parameter space. In addition, we approximate this operator by time dependent matrices, where their singular values and singular vectors converge to their infinite dimension peers. This decomposition proves to be very useful as it provides vital information as to which perturbations in the distributed parameters lead to the most significant changes in the solutions, as well as applications to uncertainty quantification. Numerical results complement our theoretical findings.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Zietsman, Lizette (committee member), Burns, John A. (committee member), Adjerid, Slimane (committee member).
Subjects/Keywords: Fréchet derivative operators; groundwater flow models; parameter estimation; parameter zonation; sensitivity analysis
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APA (6th Edition):
Leite Dos Santos Nunes, V. M. (2013). Fréchet Sensitivity Analysis and Parameter Estimation in Groundwater Flow Models. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/50653
Chicago Manual of Style (16th Edition):
Leite Dos Santos Nunes, Vitor Manuel. “Fréchet Sensitivity Analysis and Parameter Estimation in Groundwater Flow Models.” 2013. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/50653.
MLA Handbook (7th Edition):
Leite Dos Santos Nunes, Vitor Manuel. “Fréchet Sensitivity Analysis and Parameter Estimation in Groundwater Flow Models.” 2013. Web. 15 Apr 2021.
Vancouver:
Leite Dos Santos Nunes VM. Fréchet Sensitivity Analysis and Parameter Estimation in Groundwater Flow Models. [Internet] [Doctoral dissertation]. Virginia Tech; 2013. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/50653.
Council of Science Editors:
Leite Dos Santos Nunes VM. Fréchet Sensitivity Analysis and Parameter Estimation in Groundwater Flow Models. [Doctoral Dissertation]. Virginia Tech; 2013. Available from: http://hdl.handle.net/10919/50653
Virginia Tech
19.
Wang, Zhu.
Reduced-Order Modeling of Complex Engineering and Geophysical Flows: Analysis and Computations.
Degree: PhD, Mathematics, 2012, Virginia Tech
URL: http://hdl.handle.net/10919/27504
► Reduced-order models are frequently used in the simulation of complex flows to overcome the high computational cost of direct numerical simulations, especially for three-dimensional nonlinear…
(more)
▼ Reduced-order models are frequently used in the simulation of complex flows to overcome the high computational cost of direct numerical simulations, especially for three-dimensional nonlinear problems.
Proper orthogonal decomposition, as one of the most commonly used tools to generate reduced-order models, has been utilized in many engineering and scientific applications.
Its original promise of computationally efficient, yet accurate approximation of coherent structures in high Reynolds number turbulent flows, however, still remains to be fulfilled. To balance the low computational cost required by reduced-order modeling and the complexity of the targeted flows, appropriate closure modeling strategies need to be employed.
In this dissertation, we put forth two new closure models for the proper orthogonal decomposition reduced-order modeling of structurally dominated turbulent flows: the dynamic subgrid-scale model and the variational multiscale model.
These models, which are considered state-of-the-art in large eddy simulation, are carefully derived and numerically investigated.
Since modern closure models for turbulent flows generally have non-polynomial nonlinearities, their efficient numerical discretization within a proper orthogonal decomposition framework is challenging. This dissertation proposes a two-level method for an efficient and accurate numerical discretization of general nonlinear proper orthogonal decomposition closure models. This method computes the nonlinear terms of the reduced-order model on a coarse mesh. Compared with a brute force computational approach in which the nonlinear terms are evaluated on the fine mesh at each time step, the two-level method attains the same level of accuracy while dramatically reducing the computational cost. We numerically illustrate these improvements in the two-level method by using it in three settings: the one-dimensional Burgers equation with a small diffusion parameter, a two-dimensional flow past a cylinder at Reynolds number Re = 200, and a three-dimensional flow past a cylinder at Reynolds number Re = 1000.
With the help of the two-level algorithm, the new nonlinear proper orthogonal decomposition closure models (i.e., the dynamic subgrid-scale model and the variational multiscale model), together with the mixing length and the Smagorinsky closure models, are tested in the numerical simulation of a three-dimensional turbulent flow past a cylinder at Re = 1000. Five criteria are used to judge the performance of the proper orthogonal decomposition reduced-order models: the kinetic energy spectrum, the mean velocity, the Reynolds stresses, the root mean square values of the velocity fluctuations, and the time evolution of the proper orthogonal decomposition basis coefficients. All the numerical results are benchmarked against a direct numerical simulation. Based on these numerical results, we conclude that the dynamic subgrid-scale and the variational multiscale models are the most accurate.
We present a rigorous numerical analysis for the…
Advisors/Committee Members: Iliescu, Traian (committeechair), Burns, John A. (committee member), Borggaard, Jeffrey T. (committee member), Lin, Tao (committee member), Zietsman, Lizette (committee member).
Subjects/Keywords: variational multiscale; dynamic subgrid-scale model; two-level algorithm; approximate deconvolution; finite elements; numerical analysis; Proper orthogonal decomposition; reduced-order modeling; large eddy simulation; eddy viscosity; turbulence
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APA ·
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Export
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APA (6th Edition):
Wang, Z. (2012). Reduced-Order Modeling of Complex Engineering and Geophysical Flows: Analysis and Computations. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/27504
Chicago Manual of Style (16th Edition):
Wang, Zhu. “Reduced-Order Modeling of Complex Engineering and Geophysical Flows: Analysis and Computations.” 2012. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/27504.
MLA Handbook (7th Edition):
Wang, Zhu. “Reduced-Order Modeling of Complex Engineering and Geophysical Flows: Analysis and Computations.” 2012. Web. 15 Apr 2021.
Vancouver:
Wang Z. Reduced-Order Modeling of Complex Engineering and Geophysical Flows: Analysis and Computations. [Internet] [Doctoral dissertation]. Virginia Tech; 2012. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/27504.
Council of Science Editors:
Wang Z. Reduced-Order Modeling of Complex Engineering and Geophysical Flows: Analysis and Computations. [Doctoral Dissertation]. Virginia Tech; 2012. Available from: http://hdl.handle.net/10919/27504
Virginia Tech
20.
Jarvis, Christopher Hunter.
Parameter Dependent Model Reduction for Complex Fluid Flows.
Degree: PhD, Mathematics, 2014, Virginia Tech
URL: http://hdl.handle.net/10919/47357
► When applying optimization techniques to complex physical systems, using very large numerical models for the solution of a system of parameter dependent partial differential equations…
(more)
▼ When applying optimization techniques to complex physical systems, using very large numerical models for the solution of a system of parameter dependent partial differential equations (PDEs) is usually intractable. Surrogate models are used to provide an approximation to the high fidelity models while being computationally cheaper to evaluate. Typically, for time dependent nonlinear problems a reduced order model is built using a basis obtained through proper orthogonal decomposition (POD) and Galerkin projection of the system dynamics. In this thesis we present theoretical and numerical results for parameter dependent model reduction techniques. The methods are motivated by the need for surrogate models specifically designed for nonlinear parameter dependent systems. We focus on methods in which the projection basis also depends on the parameter through extrapolation and interpolation. Numerical examples involving 1D Burgers' equation, 2D Navier-Stokes equations and 2D Boussinesq equations are presented. For each model problem comparison to traditional POD reduced order models will also be presented.
Advisors/Committee Members: Burns, John A. (committeechair), Zietsman, Lizette (committee member), Borggaard, Jeffrey T. (committee member), Cliff, Eugene M. (committee member).
Subjects/Keywords: Model Reduction; Nonlinear Systems; Fluid Flows
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APA (6th Edition):
Jarvis, C. H. (2014). Parameter Dependent Model Reduction for Complex Fluid Flows. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/47357
Chicago Manual of Style (16th Edition):
Jarvis, Christopher Hunter. “Parameter Dependent Model Reduction for Complex Fluid Flows.” 2014. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/47357.
MLA Handbook (7th Edition):
Jarvis, Christopher Hunter. “Parameter Dependent Model Reduction for Complex Fluid Flows.” 2014. Web. 15 Apr 2021.
Vancouver:
Jarvis CH. Parameter Dependent Model Reduction for Complex Fluid Flows. [Internet] [Doctoral dissertation]. Virginia Tech; 2014. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/47357.
Council of Science Editors:
Jarvis CH. Parameter Dependent Model Reduction for Complex Fluid Flows. [Doctoral Dissertation]. Virginia Tech; 2014. Available from: http://hdl.handle.net/10919/47357
Virginia Tech
21.
van Wyk, Hans-Werner.
A Variational Approach to Estimating Uncertain Parameters in Elliptic Systems.
Degree: PhD, Mathematics, 2012, Virginia Tech
URL: http://hdl.handle.net/10919/27635
► As simulation plays an increasingly central role in modern science and engineering research, by supplementing experiments, aiding in the prototyping of engineering systems or informing…
(more)
▼ As simulation plays an increasingly central role in modern science and engineering research, by supplementing experiments, aiding in the prototyping of engineering systems or informing decisions on safety and reliability, the need to quantify uncertainty in model outputs due to uncertainties in the model parameters becomes critical. However, the statistical characterization of the model parameters is rarely known. In this thesis, we propose a variational approach to solve the stochastic inverse problem of obtaining a statistical description of the diffusion coefficient in an elliptic partial differential equation, based noisy measurements of the model output. We formulate the parameter identification problem as an infinite dimensional constrained optimization problem for which we establish existence of minimizers as well as first order necessary conditions. A spectral approximation of the uncertain observations (via a truncated Karhunen-Loeve expansion) allows us to estimate the infinite dimensional problem by a smooth, albeit high dimensional, deterministic optimization problem, the so-called 'finite noise' problem, in the space of functions with bounded mixed derivatives. We prove convergence of 'finite noise' minimizers to the appropriate infinite dimensional ones, and devise a gradient based, as well as a sampling based strategy for locating these numerically. Lastly, we illustrate our methods by means of numerical examples.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Herdman, Terry L. (committee member), Zietsman, Lizette (committee member), Day, Martin V. (committee member).
Subjects/Keywords: uncertainty quantification; parameter identification; elliptic systems; stochastic collocation methods
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APA ·
Chicago ·
MLA ·
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to Zotero / EndNote / Reference
Manager
APA (6th Edition):
van Wyk, H. (2012). A Variational Approach to Estimating Uncertain Parameters in Elliptic Systems. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/27635
Chicago Manual of Style (16th Edition):
van Wyk, Hans-Werner. “A Variational Approach to Estimating Uncertain Parameters in Elliptic Systems.” 2012. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/27635.
MLA Handbook (7th Edition):
van Wyk, Hans-Werner. “A Variational Approach to Estimating Uncertain Parameters in Elliptic Systems.” 2012. Web. 15 Apr 2021.
Vancouver:
van Wyk H. A Variational Approach to Estimating Uncertain Parameters in Elliptic Systems. [Internet] [Doctoral dissertation]. Virginia Tech; 2012. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/27635.
Council of Science Editors:
van Wyk H. A Variational Approach to Estimating Uncertain Parameters in Elliptic Systems. [Doctoral Dissertation]. Virginia Tech; 2012. Available from: http://hdl.handle.net/10919/27635
Virginia Tech
22.
Kuster Jr, George Emil.
On the role of student understanding of function and rate of change in learning differential equations.
Degree: PhD, Mathematics, 2016, Virginia Tech
URL: http://hdl.handle.net/10919/71827
► In this research, I utilize the theoretical perspective Knowledge In Pieces to identify the knowledge resources students utilize while in the process of completing various…
(more)
▼ In this research, I utilize the theoretical perspective Knowledge In Pieces to identify the knowledge resources students utilize while in the process of completing various differential equations tasks. In addition I explore how this utilization changes over the course of a semester, and how resources related to the concepts of function and rate of change supported the students in completing the tasks. I do so using data collected from a series of task-based individual interviews with two students enrolled in separate differential equations courses. The results provide a fine-grained description of the knowledge students consider to be productive with regard to completing various differential equations tasks. Further the analysis resulted in the identification of five ways students interpret differential equations tasks and how these interpretations are related to the knowledge resources students utilize while completing the various tasks. Lastly, this research makes a contribution to mathematics education by illuminating the knowledge concerning function and rate of change students utilize and how this knowledge comes together to support students in drawing connections between differential equations and their solutions, structuring those solutions, and reasoning with relationships present in the differential equations.
Advisors/Committee Members: Wawro, Megan (committeechair), Wagner, Joseph F. (committee member), Zietsman, Lizette (committee member), Johnson, Estrella (committee member), Ulrich, Catherine Louise (committee member).
Subjects/Keywords: Differential equations; Knowledge in Pieces; Undergraduate Mathematics Education
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APA (6th Edition):
Kuster Jr, G. E. (2016). On the role of student understanding of function and rate of change in learning differential equations. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/71827
Chicago Manual of Style (16th Edition):
Kuster Jr, George Emil. “On the role of student understanding of function and rate of change in learning differential equations.” 2016. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/71827.
MLA Handbook (7th Edition):
Kuster Jr, George Emil. “On the role of student understanding of function and rate of change in learning differential equations.” 2016. Web. 15 Apr 2021.
Vancouver:
Kuster Jr GE. On the role of student understanding of function and rate of change in learning differential equations. [Internet] [Doctoral dissertation]. Virginia Tech; 2016. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/71827.
Council of Science Editors:
Kuster Jr GE. On the role of student understanding of function and rate of change in learning differential equations. [Doctoral Dissertation]. Virginia Tech; 2016. Available from: http://hdl.handle.net/10919/71827
23.
Krometis, Justin.
A Bayesian Approach to Estimating Background Flows from a Passive Scalar.
Degree: PhD, Mathematics, 2018, Virginia Tech
URL: http://hdl.handle.net/10919/83783
► We consider the statistical inverse problem of estimating a background flow field (e.g., of air or water) from the partial and noisy observation of a…
(more)
▼ We consider the statistical inverse problem of estimating a background flow field (e.g., of air or water) from the partial and noisy observation of a passive scalar (e.g., the concentration of a pollutant). Here the unknown is a vector field that is specified by large or infinite number of degrees of freedom. We show that the inverse problem is ill-posed, i.e., there may be many or no background flows that match a given set of observations. We therefore adopt a Bayesian approach, incorporating prior knowledge of background flows and models of the observation error to develop probabilistic estimates of the fluid flow. In doing so, we leverage frameworks developed in recent years for infinite-dimensional Bayesian inference. We provide conditions under which the inference is consistent, i.e., the posterior measure converges to a Dirac measure on the true background flow as the number of observations of the solute concentration grows large. We also define several computationally-efficient algorithms adapted to the problem. One is an adjoint method for computation of the gradient of the log likelihood, a key ingredient in many numerical methods. A second is a particle method that allows direct computation of point observations of the solute concentration, leveraging the structure of the inverse problem to avoid approximation of the full infinite-dimensional scalar field. Finally, we identify two interesting example problems with very different posterior structures, which we use to conduct a large-scale benchmark of the convergence of several Markov Chain Monte Carlo methods that have been developed in recent years for infinite-dimensional settings.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Chung, Matthias (committee member), Zietsman, Lizette (committee member), Glatt-Holtz, Nathan (committee member).
Subjects/Keywords: Bayesian Statistical Inversion; Bayesian Consistency; Markov Chain Monte Carlo (MCMC); Passive Scalars; Fluid Turbulence
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Manager
APA (6th Edition):
Krometis, J. (2018). A Bayesian Approach to Estimating Background Flows from a Passive Scalar. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/83783
Chicago Manual of Style (16th Edition):
Krometis, Justin. “A Bayesian Approach to Estimating Background Flows from a Passive Scalar.” 2018. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/83783.
MLA Handbook (7th Edition):
Krometis, Justin. “A Bayesian Approach to Estimating Background Flows from a Passive Scalar.” 2018. Web. 15 Apr 2021.
Vancouver:
Krometis J. A Bayesian Approach to Estimating Background Flows from a Passive Scalar. [Internet] [Doctoral dissertation]. Virginia Tech; 2018. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/83783.
Council of Science Editors:
Krometis J. A Bayesian Approach to Estimating Background Flows from a Passive Scalar. [Doctoral Dissertation]. Virginia Tech; 2018. Available from: http://hdl.handle.net/10919/83783
24.
Swirydowicz, Katarzyna.
Strategies For Recycling Krylov Subspace Methods and Bilinear Form Estimation.
Degree: PhD, Mathematics, 2017, Virginia Tech
URL: http://hdl.handle.net/10919/78695
► The main theme of this work is effectiveness and efficiency of Krylov subspace methods and Krylov subspace recycling. While solving long, slowly changing sequences of…
(more)
▼ The main theme of this work is effectiveness and efficiency of Krylov subspace methods and Krylov subspace recycling. While solving long, slowly changing sequences of large linear systems, such as the ones that arise in engineering, there are many issues we need to consider if we want to make the process reliable (converging to a correct solution) and as fast as possible. This thesis is built on three main components. At first, we target bilinear and quadratic form estimation. Bilinear form c
TA-1b is often associated with long sequences of linear systems, especially in optimization problems. Thus, we devise algorithms that adapt cheap bilinear and quadratic form estimates for Krylov subspace recycling. In the second part, we develop a hybrid recycling method that is inspired by a complex CFD application. We aim to make the method robust and cheap at the same time. In the third part of the thesis, we optimize the implementation of Krylov subspace methods on Graphic Processing Units (GPUs). Since preconditioners based on incomplete matrix factorization (ILU, Cholesky) are very slow on the GPUs, we develop a preconditioner that is effective but well suited for GPU implementation.
Advisors/Committee Members: De Sturler, Eric (committeechair), Roy, Christopher John (committee member), Zietsman, Lizette (committee member), Embree, Mark Partick (committee member).
Subjects/Keywords: Bilinear form estimation; quadratic form estimation; sparse approximate inverse preconditioning; high performance computing; Krylov subspace recycling; diffuse optical tomography; topology optimization; computational fluid dynamics
…developed in the
Department of Mechanical Engineering at Virginia Tech [91]. The code is…
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APA ·
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MLA ·
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CSE |
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to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Swirydowicz, K. (2017). Strategies For Recycling Krylov Subspace Methods and Bilinear Form Estimation. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/78695
Chicago Manual of Style (16th Edition):
Swirydowicz, Katarzyna. “Strategies For Recycling Krylov Subspace Methods and Bilinear Form Estimation.” 2017. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/78695.
MLA Handbook (7th Edition):
Swirydowicz, Katarzyna. “Strategies For Recycling Krylov Subspace Methods and Bilinear Form Estimation.” 2017. Web. 15 Apr 2021.
Vancouver:
Swirydowicz K. Strategies For Recycling Krylov Subspace Methods and Bilinear Form Estimation. [Internet] [Doctoral dissertation]. Virginia Tech; 2017. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/78695.
Council of Science Editors:
Swirydowicz K. Strategies For Recycling Krylov Subspace Methods and Bilinear Form Estimation. [Doctoral Dissertation]. Virginia Tech; 2017. Available from: http://hdl.handle.net/10919/78695
Virginia Tech
25.
Boyce, Steven James.
Modeling Students' Units Coordinating Activity.
Degree: PhD, Mathematics, 2014, Virginia Tech
URL: http://hdl.handle.net/10919/50430
► Primarily via constructivist teaching experiment methodology, units coordination (Steffe, 1992) has emerged as a useful construct for modeling students' psychological constructions pertaining to several mathematical…
(more)
▼ Primarily via constructivist teaching experiment methodology, units coordination (Steffe, 1992) has emerged as a useful construct for modeling students' psychological constructions pertaining to several mathematical domains, including counting sequences, whole number multiplicative conceptions, and fractions schemes. I describe how consideration of units coordination as a Piagetian (1970b) structure is useful for modeling units coordination across contexts. In this study, I extend teaching experiment methodology (Steffe and Thompson, 2000) to model the dynamics of students' units coordinating activity across contexts within a teaching experiment, using the construct of propensity to coordinate units. Two video-recorded teaching experiments involving pairs of sixth-grade students were analyzed to form a model of the dynamics of students' units coordinating activity. The modeling involved separation of transcriptions into chunks that were coded dichotomously for the units coordinating activity of a single student in each dyad. The two teaching experiments were used to form 5 conjectures about the output of the model that were then tested with a third teaching experiment. The results suggest that modeling units coordination activity via the construct of propensity to coordinate units was useful for describing patterns in the students' perturbations during the teaching sessions. The model was moderately useful for identifying sequences of
interactions that support growth in units coordination. Extensions, modifications, and implications of the modeling approach are discussed.
Advisors/Committee Members: Norton, Anderson H. III (committeechair), Wilkins, Jesse L. M. (committee member), Wawro, Megan (committee member), Zietsman, Lizette (committee member), Ulrich, Catherine Louise (committee member).
Subjects/Keywords: Units coordination; Teaching experiment methodology; Fractions; Multiplicative reasoning
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APA ·
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MLA ·
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Export
to Zotero / EndNote / Reference
Manager
APA (6th Edition):
Boyce, S. J. (2014). Modeling Students' Units Coordinating Activity. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/50430
Chicago Manual of Style (16th Edition):
Boyce, Steven James. “Modeling Students' Units Coordinating Activity.” 2014. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/50430.
MLA Handbook (7th Edition):
Boyce, Steven James. “Modeling Students' Units Coordinating Activity.” 2014. Web. 15 Apr 2021.
Vancouver:
Boyce SJ. Modeling Students' Units Coordinating Activity. [Internet] [Doctoral dissertation]. Virginia Tech; 2014. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/50430.
Council of Science Editors:
Boyce SJ. Modeling Students' Units Coordinating Activity. [Doctoral Dissertation]. Virginia Tech; 2014. Available from: http://hdl.handle.net/10919/50430
Virginia Tech
26.
Newbury, Golnar.
A Numerical Study of a Delay Differential Equation Model for Breast Cancer.
Degree: MS, Mathematics, 2007, Virginia Tech
URL: http://hdl.handle.net/10919/34420
► In this thesis we construct a new model of the immune response to the growth of breast cancer cells and investigate the impact of certain…
(more)
▼ In this thesis we construct a new model of the immune response to the growth of breast cancer cells and investigate the impact of certain drug therapies on the cancer. We use delay differential equations to model the interaction of breast cancer cells with the immune system. The new model is constructed by combining two previous models. The first model accounts for different cell cycles and includes terms to evaluate drug treatments, but ignores quiescent tumor cells. The second model includes quiescent cells, but ignores the immune response and drug treatments. The new model is obtained by combining and modifying these two models to account for quiescent cells, immune cells and includes drug intervention terms. This new model is used to evaluate the effects of pulsed applications of the drug Paclitaxel for models with and without quiescent cells. We use sensitivity equation methods to analyze the sensitivity of the model with respect to the initial number of immune cytotoxic T-cells. Numerical experiments are conducted to compare the model predictions to observed behavior.
Advisors/Committee Members: Burns, John A. (committeechair), Zietsman, Lizette (committee member), Herdman, Terry L. (committee member).
Subjects/Keywords: delay differential equations; cancer model; parameter sensitivity; ordinary differential equations; Paclitaxel; cell cycle; cycle specific chemotherapy; proliferating cells; quiescent cells
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Manager
APA (6th Edition):
Newbury, G. (2007). A Numerical Study of a Delay Differential Equation Model for Breast Cancer. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/34420
Chicago Manual of Style (16th Edition):
Newbury, Golnar. “A Numerical Study of a Delay Differential Equation Model for Breast Cancer.” 2007. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/34420.
MLA Handbook (7th Edition):
Newbury, Golnar. “A Numerical Study of a Delay Differential Equation Model for Breast Cancer.” 2007. Web. 15 Apr 2021.
Vancouver:
Newbury G. A Numerical Study of a Delay Differential Equation Model for Breast Cancer. [Internet] [Masters thesis]. Virginia Tech; 2007. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/34420.
Council of Science Editors:
Newbury G. A Numerical Study of a Delay Differential Equation Model for Breast Cancer. [Masters Thesis]. Virginia Tech; 2007. Available from: http://hdl.handle.net/10919/34420
Virginia Tech
27.
Stoyanov, Miroslav Karolinov.
Optimal Linear Feedback Control for Incompressible Fluid Flow.
Degree: MS, Mathematics, 2006, Virginia Tech
URL: http://hdl.handle.net/10919/33454
► Linear feedback control is considered for large systems of differential algebraic equations arising from discretization of saddle point problems. Necessary conditions are derived by applying…
(more)
▼ Linear feedback control is considered for large systems of differential algebraic equations
arising from discretization of saddle point problems. Necessary conditions are derived by
applying the Maximum Principle and have the form of constrained Riccati equations. We
consider two approaches for solving the feedback control problem as well as practical numerical
methods. Numerical studies using examples derived from a constrained heat equation
and Stokes equation confirms the effectiveness of the approaches we consider.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Zietsman, Lizette (committee member), Burns, John A. (committee member).
Subjects/Keywords: control; feedback; incompressible; fluid; DAE
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APA (6th Edition):
Stoyanov, M. K. (2006). Optimal Linear Feedback Control for Incompressible Fluid Flow. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/33454
Chicago Manual of Style (16th Edition):
Stoyanov, Miroslav Karolinov. “Optimal Linear Feedback Control for Incompressible Fluid Flow.” 2006. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/33454.
MLA Handbook (7th Edition):
Stoyanov, Miroslav Karolinov. “Optimal Linear Feedback Control for Incompressible Fluid Flow.” 2006. Web. 15 Apr 2021.
Vancouver:
Stoyanov MK. Optimal Linear Feedback Control for Incompressible Fluid Flow. [Internet] [Masters thesis]. Virginia Tech; 2006. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/33454.
Council of Science Editors:
Stoyanov MK. Optimal Linear Feedback Control for Incompressible Fluid Flow. [Masters Thesis]. Virginia Tech; 2006. Available from: http://hdl.handle.net/10919/33454
Virginia Tech
28.
Boyce, Steven James.
The Distance to Uncontrollability via Linear Matrix Inequalities.
Degree: MS, Mathematics, 2010, Virginia Tech
URL: http://hdl.handle.net/10919/36138
► The distance to uncontrollability of a controllable linear system is a measure of the degree of perturbation a system can undergo and remain controllable. The…
(more)
▼ The distance to uncontrollability of a controllable linear system is a measure of the degree of
perturbation a system can undergo and remain controllable. The deï¬ nition of the distance
to uncontrollability leads to a non-convex optimization problem in two variables. In 2000
Gu proposed the ï¬ rst polynomial time algorithm to compute this distance. This algorithm
relies heavily on efficient eigenvalue solvers.
In this work we examine two alternative algorithms that result in linear matrix inequalities.
For the ï¬ rst algorithm, proposed by Ebihara et. al., a semideï¬ nite programming problem
is derived via the Kalman-Yakubovich-Popov (KYP) lemma. The dual formulation is also
considered and leads to rank conditions for exactness veriï¬ cation of the approximation.
For the second algorithm, by Dumitrescu, Å icleru and Å tefan, a semideï¬ nite programming
problem is derived using a sum-of-squares relaxation of an associated matrix-polynomial and
the associated Gram matrix parameterization. In both cases the optimization problems are
solved using primal-dual-interior point methods that retain positive semideï¬ niteness at each
iteration.
Numerical results are presented to compare the three algorithms for a number of bench-
mark examples. In addition, we also consider a system that results from a ï¬ nite element
discretization of the one-dimensional advection-diffusion equation. Here our objective is to
test these algorithms for larger problems that originate in PDE-control.
Advisors/Committee Members: Zietsman, Lizette (committeechair), Borggaard, Jeffrey T. (committee member), Norton, Anderson H. III (committee member), Day, Martin V. (committee member).
Subjects/Keywords: sensor location; LaGrange multipliers; SDP; numerical; unobservability
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Manager
APA (6th Edition):
Boyce, S. J. (2010). The Distance to Uncontrollability via Linear Matrix Inequalities. (Masters Thesis). Virginia Tech. Retrieved from http://hdl.handle.net/10919/36138
Chicago Manual of Style (16th Edition):
Boyce, Steven James. “The Distance to Uncontrollability via Linear Matrix Inequalities.” 2010. Masters Thesis, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/36138.
MLA Handbook (7th Edition):
Boyce, Steven James. “The Distance to Uncontrollability via Linear Matrix Inequalities.” 2010. Web. 15 Apr 2021.
Vancouver:
Boyce SJ. The Distance to Uncontrollability via Linear Matrix Inequalities. [Internet] [Masters thesis]. Virginia Tech; 2010. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/36138.
Council of Science Editors:
Boyce SJ. The Distance to Uncontrollability via Linear Matrix Inequalities. [Masters Thesis]. Virginia Tech; 2010. Available from: http://hdl.handle.net/10919/36138
Virginia Tech
29.
Stoyanov, Miroslav.
Reduced Order Methods for Large Scale Riccati Equations.
Degree: PhD, Mathematics, 2009, Virginia Tech
URL: http://hdl.handle.net/10919/27832
► Solving the linear quadratic regulator (LQR) problem for partial differential equa- tions (PDEs) leads to many computational challenges. The primary challenge comes from the fact…
(more)
▼ Solving the linear quadratic regulator (LQR) problem for partial differential equa-
tions (PDEs) leads to many computational challenges. The primary challenge comes
from the fact that discretization methods for PDEs typically lead to very large sys-
tems of differential or differential algebraic equations. These systems are used to form
algebraic Riccati equations involving high rank matrices. Although we restrict our
attention to control problems with small numbers of control inputs, we allow for po-
tentially high order control outputs. Problems with this structure appear in a number
of practical applications yet no suitable algorithm exists. We propose and analyze so-
lution strategies based on applying model order reduction methods to Chandrasekhar
equations, Lyapunov/Sylvester equations, or combinations of these equations. Our nu-
merical examples illustrate improvements in computational time up to several orders of
magnitude over standard tools (when these tools can be used). We also present exam-
ples that cannot be solved using existing methods. These cases are motivated by flow
control problems that are solved by computing feedback controllers for the linearized
system.
Advisors/Committee Members: Borggaard, Jeffrey T. (committeechair), Gugercin, Serkan (committee member), Burns, John A. (committee member), Zietsman, Lizette (committee member).
Subjects/Keywords: Large Scale; High Rank; Navier-Stokes; Riccati Equations
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MLA ·
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Manager
APA (6th Edition):
Stoyanov, M. (2009). Reduced Order Methods for Large Scale Riccati Equations. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/27832
Chicago Manual of Style (16th Edition):
Stoyanov, Miroslav. “Reduced Order Methods for Large Scale Riccati Equations.” 2009. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/27832.
MLA Handbook (7th Edition):
Stoyanov, Miroslav. “Reduced Order Methods for Large Scale Riccati Equations.” 2009. Web. 15 Apr 2021.
Vancouver:
Stoyanov M. Reduced Order Methods for Large Scale Riccati Equations. [Internet] [Doctoral dissertation]. Virginia Tech; 2009. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/27832.
Council of Science Editors:
Stoyanov M. Reduced Order Methods for Large Scale Riccati Equations. [Doctoral Dissertation]. Virginia Tech; 2009. Available from: http://hdl.handle.net/10919/27832
Virginia Tech
30.
Krueger, Denise A.
Stabilized Finite Element Methods for Feedback Control of Convection Diffusion Equations.
Degree: PhD, Mathematics, 2004, Virginia Tech
URL: http://hdl.handle.net/10919/11214
► We study the behavior of numerical stabilization schemes in the context of linear quadratic regulator (LQR) control problems for convection diffusion equations. The motivation for…
(more)
▼ We study the behavior of numerical stabilization schemes in the context of linear quadratic regulator (LQR) control problems for convection diffusion equations. The motivation for this effort comes from the observation that when linearization is applied to fluid flow control problems the resulting equations have the form of a convection diffusion equation. This effort is focused on the specific problem of computing the feedback functional gains that are the kernels of the feedback operators defined by solutions of operator Riccati equations. We develop a stabilization scheme based on the Galerkin Least Squares (GLS) method and compare this scheme to the standard Galerkin finite element method. We use cubic B-splines in order to keep the higher order terms that occur in GLS formulation. We conduct a careful numerical investigation into the convergence and accuracy of the functional gains computed using stabilization. We also conduct numerical studies of the role that the stabilization parameter plays in this convergence. Overall, we discovered that stabilization produces much better approximations to the functional gains on coarse meshes than the unstabilized method and that adjustments in the stabilization parameter greatly effects the accuracy and convergence rates. We discovered that the optimal stabilization parameter for simulation and steady state analysis is not necessarily optimal for solving the Riccati equation that defines the functional gains. Finally, we suggest that the stabilized GLS method might provide good initial values for iterative schemes on coarse meshes.
Advisors/Committee Members: King, Belinda B. (committeechair), Burns, John A. (committee member), Borggaard, Jeffrey T. (committee member), Iliescu, Traian (committee member), Zietsman, Lizette (committee member).
Subjects/Keywords: Stabilized Finite Elements; Convection-Diffusion Equation; Linear Quadratic Regulator Problems; Non-normal
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APA (6th Edition):
Krueger, D. A. (2004). Stabilized Finite Element Methods for Feedback Control of Convection Diffusion Equations. (Doctoral Dissertation). Virginia Tech. Retrieved from http://hdl.handle.net/10919/11214
Chicago Manual of Style (16th Edition):
Krueger, Denise A. “Stabilized Finite Element Methods for Feedback Control of Convection Diffusion Equations.” 2004. Doctoral Dissertation, Virginia Tech. Accessed April 15, 2021.
http://hdl.handle.net/10919/11214.
MLA Handbook (7th Edition):
Krueger, Denise A. “Stabilized Finite Element Methods for Feedback Control of Convection Diffusion Equations.” 2004. Web. 15 Apr 2021.
Vancouver:
Krueger DA. Stabilized Finite Element Methods for Feedback Control of Convection Diffusion Equations. [Internet] [Doctoral dissertation]. Virginia Tech; 2004. [cited 2021 Apr 15].
Available from: http://hdl.handle.net/10919/11214.
Council of Science Editors:
Krueger DA. Stabilized Finite Element Methods for Feedback Control of Convection Diffusion Equations. [Doctoral Dissertation]. Virginia Tech; 2004. Available from: http://hdl.handle.net/10919/11214
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