Analysis Commons^™

Data-Assimilation-Enabled Fast Convergence Of The Uzawa Scheme For Navier-Stokes Equations With Large Reynolds Numbers, Jessica C. Franklin

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This work studies efficient techniques utilized to solve the Navier-Stokes equations that model incompressible Newtonian flow in the setting where partial solution data is available. First, we analyze the Picard iteration and Picard with continuous data assimilation applied and test convergence. Then, we analyze the Arrow-Hurwicz (AH) and Uzawa iterative procedures and test convergence. Finally, we apply continuous data assimilation to Uzawa, and analytical and numerical results show that CDA improves Uzawa convergence for the NSE, similar to CDA-Picard but it is much more efficient.

The Spectral Asymptotics Of Toeplitz Operators On Hilbert Spaces Of Analytic Functions, Trevor Camper

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Many physical systems, whether they are ocean waves or particles moving through space, can be described using the mathematical language of “partial differ- ential equations.” In many circumstances, it is useful to study how these equations amplify an input to the equation, in which case the amplification factor is called an “eigenvalue.” The usefulness of these amplification factors is that they can be used to describe properties of the physical system. In this dissertation, I have studied this amplification factor for a related set of equations called “Toeplitz operators.” In par- ticular, I have studied eigenvalues using statistical techniques. The …

Computational Representation, Analysis And Verification Of Requirements In Engineering Design And Systems Engineering, Chandan Kumar Sahu

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Systems are developed to satisfy a set of requirements derived from stakeholders’ needs, defining the problem space for which the system is created as a feasible solution. The system design process begins with eliciting these requirements and concludes with validating whether the created system meets them. Requirements engineering (RE) encompasses elicitation, representation, analysis, documentation, verification, and validation. However, challenges in RE, such as imprecision in natural language (NL), proprietary restrictions, and a lack of standardized quality metrics, hinder the creation of well-formed and comprehensive requirements. These challenges complicate formalization and analysis of requirements.

This dissertation addresses these challenges by proposing …

Probabilistic Frames And Concepts From Optimal Transport, Dongwei Chen

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As the generalization of frames in the Euclidean space $\mathbb{R}^n$, a probabilistic frame is a probability measure on $\mathbb{R}^n$ that has a finite second moment and whose support spans $\mathbb{R}^n$. The p-Wasserstein distance with $p \geq 1$ from optimal transport is often used to compare probabilistic frames. It is particularly useful to compare frames of various cardinalities in the context of probabilistic frames. We show that the 2-Wasserstein distance appears naturally in the fundamental objects of frame theory and draws consequences leading to a geometric viewpoint of probabilistic frames.

We convert the classic lower bound estimates of 2-Wasserstein distance \cite{Gelbrich90, …

Aspects Of Stochastic Geometric Mechanics In Molecular Biophysics, David Frost

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In confocal single-molecule FRET experiments, the joint distribution of FRET efficiency and donor lifetime distribution can reveal underlying molecular conformational dynamics via deviation from their theoretical Forster relationship. This shift is referred to as a dynamic shift. In this study, we investigate the influence of the free energy landscape in protein conformational dynamics on the dynamic shift by simulation of the associated continuum reaction coordinate Langevin dynamics, yielding a deeper understanding of the dynamic and structural information in the joint FRET efficiency and donor lifetime distribution. We develop novel Langevin models for the dye linker dynamics, including rotational dynamics, based …

Concentration Theorems For Orthonormal Sequences In A Reproducing Kernel Hilbert Space, Travis Alvarez

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Let H be a reproducing kernel Hilbert space with reproducing kernel elements {Kx} indexed by a measure space {X,mu}. If H can be embedded in L2(X,mu), then H can be viewed as a framed Hilbert space. We study concentration of orthonormal sequences in such reproducing kernel Hilbert spaces.

Defining different versions of concentration, we find quantitative upper bounds on the number of orthonormal functions that can be classified by such concentrations. Examples are shown to prove sharpness of the bounds. In the cases that we can add "concentrated" orthonormal vectors indefinitely, the growth rate of doing so is shown.

Asymptotic Cones Of Quadratically Defined Sets And Their Applications To Qcqps, Alexander Joyce

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Quadratically constrained quadratic programs (QCQPs) are a set of optimization problems defined by a quadratic objective function and quadratic constraints. QCQPs cover a diverse set of problems, but the nonconvexity and unboundedness of quadratic constraints lead to difficulties in globally solving a QCQP. This thesis covers properties of unbounded quadratic constraints via a description of the asymptotic cone of a set defined by a single quadratic constraint. A description of the asymptotic cone is provided, including properties such as retractiveness and horizon directions.

Using the characterization of the asymptotic cone, we generalize existing results for bounded quadratically defined regions with …

Recovering Coefficients Of Second-Order Hyperbolic And Plate Equations Via Finite Measurements On The Boundary, Scott Randall Scruggs

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Abstract In this dissertation, we consider the inverse problem for a second-order hyperbolic equation of recovering n + 3 unknown coefficients defined on an open bounded domain with a smooth enough boundary. We also consider the inverse problem of recovering an unknown coefficient on the Euler- Bernoulli plate equation on a lower-order term again defined on an open bounded domain with a smooth enough boundary. For the second-order hyperbolic equation, we show that we can uniquely and (Lipschitz) stably recover all these coefficients from only using half of the corresponding boundary measurements of their solutions, and for the plate equation, …