# Browsing by Subject "Applied mathematics"

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Item Open Access A Class of Tetrahedral Finite Elements for Complex Geometry and Nonlinear Mechanics: A Variational Multiscale Approach(2019) abboud, nabilIn this work, a stabilized finite element framework is developed to simulate small and large deformation solid mechanics problems involving complex geometries and complicated constitutive models. In particular, the focus is on solid dynamics problems involving nearly and fully incompressible materials. The work is divided into three main themes, the first is concerned with the development of stabilized finite element algorithms for hyperelastic materials, the second handles the case of viscoelastic materials, and the third focuses on algorithms for J2-plastic materials. For all three cases, problems in the small and large deformation regime are considered, and for the J2-plasticity case, both quasi-static and dynamic problems are examined.

Some of the key features of the algorithms developed in this work is the simplicity of their implementation into an existing finite element code, and their applicability to problems involving complicated geometries. The former is achieved by using a mixed formulation of the solid mechanics equations where the velocity and pressure unknowns are represented by linear shape functions, whereas the latter is realized by using triangular elements which offer numerous advantages compared to quadrilaterals, when meshing complicated geometries. To achieve the stability of the algorithm, a new approach is proposed in which the variational multiscale approach is applied to the mixed form of the solid mechanics equations written down as a first order system, whereby the pressure equation is cast in rate form.

Through a series of numerical simulations, it is shown that the stability properties of the proposed algorithm is invariant to the constitutive model and the time integrator used. By running convergence tests, the algorithm is shown to be second order accurate, in the $L^2$-nrom, for the displacements, velocities, and pressure. Finally, the robustness of the algorithm is showcased by considering realistic test cases involving complicated geometries and very large deformation.

Item Open Access A Geometric Approach to Biomedical Time Series Analysis(2020) Malik, JohnBiomedical time series are non-invasive windows through which we may observe human systems. Although a vast amount of information is hidden in the medical field's growing collection of long-term, high-resolution, and multi-modal biomedical time series, effective algorithms for extracting that information have not yet been developed. We are particularly interested in the physiological dynamics of a human system, namely the changes in state that the system experiences over time (which may be intrinsic or extrinsic in origin). We introduce a mathematical model for a particular class of biomedical time series, called the wave-shape oscillatory model, which quantifies the sense in which dynamics are hidden in those time series. There are two key ideas behind the new model. First, instead of viewing a biomedical time series as a sequence of measurements made at the sampling rate of the signal, we can often view it as a sequence of cycles occurring at irregularly-sampled time points. Second, the "shape" of an individual cycle is assumed to have a one-to-one correspondence with the state of the system being monitored; as such, changes in system state (dynamics) can be inferred by tracking changes in cycle shape. Since physiological dynamics are not random but are well-regulated (except in the most pathological of cases), we can assume that all of the system's states lie on a low-dimensional, abstract Riemannian manifold called the phase manifold. When we model the correspondence between the hidden system states and the observed cycle shapes using a diffeomorphism, we allow the topology of the phase manifold to be recovered by methods belonging to the field of unsupervised manifold learning. In particular, we prove that the physiological dynamics hidden in a time series adhering to the wave-shape oscillatory model can be well-recovered by applying the diffusion maps algorithm to the time series' set of oscillatory cycles. We provide several applications of the wave-shape oscillatory model and the associated algorithm for dynamics recovery, including unsupervised and supervised heartbeat classification, derived respiratory monitoring, intra-operative cardiovascular monitoring, supervised and unsupervised sleep stage classification, and f-wave extraction (a single-channel blind source separation problem).

Item Open Access A New Method for Modeling Free Surface Flows and Fluid-structure Interaction with Ocean Applications(2016) Lee, CurtisThe computational modeling of ocean waves and ocean-faring devices poses numerous challenges. Among these are the need to stably and accurately represent both the fluid-fluid interface between water and air as well as the fluid-structure interfaces arising between solid devices and one or more fluids. As techniques are developed to stably and accurately balance the interactions between fluid and structural solvers at these boundaries, a similarly pressing challenge is the development of algorithms that are massively scalable and capable of performing large-scale three-dimensional simulations on reasonable time scales. This dissertation introduces two separate methods for approaching this problem, with the first focusing on the development of sophisticated fluid-fluid interface representations and the second focusing primarily on scalability and extensibility to higher-order methods.

We begin by introducing the narrow-band gradient-augmented level set method (GALSM) for incompressible multiphase Navier-Stokes flow. This is the first use of the high-order GALSM for a fluid flow application, and its reliability and accuracy in modeling ocean environments is tested extensively. The method demonstrates numerous advantages over the traditional level set method, among these a heightened conservation of fluid volume and the representation of subgrid structures.

Next, we present a finite-volume algorithm for solving the incompressible Euler equations in two and three dimensions in the presence of a flow-driven free surface and a dynamic rigid body. In this development, the chief concerns are efficiency, scalability, and extensibility (to higher-order and truly conservative methods). These priorities informed a number of important choices: The air phase is substituted by a pressure boundary condition in order to greatly reduce the size of the computational domain, a cut-cell finite-volume approach is chosen in order to minimize fluid volume loss and open the door to higher-order methods, and adaptive mesh refinement (AMR) is employed to focus computational effort and make large-scale 3D simulations possible. This algorithm is shown to produce robust and accurate results that are well-suited for the study of ocean waves and the development of wave energy conversion (WEC) devices.

Item Open Access Accelerating the Computation of Density Functional Theory's Correlation Energy under Random Phase Approximations(2019) Thicke, KyleWe propose novel algorithms for the fast computation of density functional theory's exchange-correlation energy in both the particle-hole and particle-particle random phase approximations (phRPA and ppRPA). For phRPA, we propose a new cubic scaling algorithm for the calculation of the RPA correlation energy. Our scheme splits up the dependence between the occupied and virtual orbitals in the density response function by use of Cauchy's integral formula. This introduces an additional integral to be carried out, for which we provide a geometrically convergent quadrature rule. Our scheme also uses the interpolative separable density fitting algorithm to further reduce the computational cost in a way analogous to that of the resolution of identity method.

For ppRPA, we propose an algorithm based on stochastic trace estimation. A contour integral is used to break up the dependence between orbitals. The logarithm is expanded into a polynomial, and a variant of the Hutchinson algorithm is proposed to find the trace of the polynomial. This modification of the Hutchinson algorithm allows us to use the structure of the problem to compute each Hutchinson iteration in only quadratic time. This is a large asymptotic improvement over the previous state-of-the-art quartic-scaling method and over the naive sextic-scaling method.

Item Open Access Adaptive Data Representation and Analysis(2018) Xu, JierenThis dissertation introduces and analyzes algorithms that aim to adaptively handle complex datasets arising in the real-world applications. It contains two major parts. The first part describes an adaptive model of 1-dimensional signals that lies in the field of adaptive time-frequency analysis. It explains a current state-of-the-art work, named the Synchrosqueezed transform, in this field. Then it illustrates two proposed algorithms that use non-parametric regression to reveal the underlying os- cillatory patterns of the targeted 1-dimensional signal, as well as to estimate the instantaneous information, e.g., instantaneous frequency, phase, or amplitude func-

tions, by a statistical pattern driven model.

The second part proposes a population-based imaging technique for human brain

bundle/connectivity recovery. It applies local streamlines as novelly adopted learn- ing/testing features to segment the brain white matter and thus reconstruct the whole brain information. It also develops a module, named as the streamline diffu- sion filtering, to improve the streamline sampling procedure.

Even though these two parts are not related directly, they both rely on an align- ment step to register the latent variables to some coordinate system and thus to facilitate the final inference. Numerical results are shown to validate all the pro- posed algorithms.

Item Open Access Adaptive Sparse Grid Approaches to Polynomial Chaos Expansions for Uncertainty Quantification(2015) Winokur, Justin GregoryPolynomial chaos expansions provide an efficient and robust framework to analyze and quantify uncertainty in computational models. This dissertation explores the use of adaptive sparse grids to reduce the computational cost of determining a polynomial model surrogate while examining and implementing new adaptive techniques.

Determination of chaos coefficients using traditional tensor product quadrature suffers the so-called curse of dimensionality, where the number of model evaluations scales exponentially with dimension. Previous work used a sparse Smolyak quadrature to temper this dimensional scaling, and was applied successfully to an expensive Ocean General Circulation Model, HYCOM during the September 2004 passing of Hurricane Ivan through the Gulf of Mexico. Results from this investigation suggested that adaptivity could yield great gains in efficiency. However, efforts at adaptivity are hampered by quadrature accuracy requirements.

We explore the implementation of a novel adaptive strategy to design sparse ensembles of oceanic simulations suitable for constructing polynomial chaos surrogates. We use a recently developed adaptive pseudo-spectral projection (aPSP) algorithm that is based on a direct application of Smolyak's sparse grid formula, and that allows for the use of arbitrary admissible sparse grids. Such a construction ameliorates the severe restrictions posed by insufficient quadrature accuracy. The adaptive algorithm is tested using an existing simulation database of the HYCOM model during Hurricane Ivan. The {\it a priori} tests demonstrate that sparse and adaptive pseudo-spectral constructions lead to substantial savings over isotropic sparse sampling.

In order to provide a finer degree of resolution control along two distinct subsets of model parameters, we investigate two methods to build polynomial approximations. The two approaches are based with pseudo-spectral projection (PSP) methods on adaptively constructed sparse grids. The control of the error along different subsets of parameters may be needed in the case of a model depending on uncertain parameters and deterministic design variables. We first consider a nested approach where an independent adaptive sparse grid pseudo-spectral projection is performed along the first set of directions only, and at each point a sparse grid is constructed adaptively in the second set of directions. We then consider the application of aPSP in the space of all parameters, and introduce directional refinement criteria to provide a tighter control of the projection error along individual dimensions. Specifically, we use a Sobol decomposition of the projection surpluses to tune the sparse grid adaptation. The behavior and performance of the two approaches are compared for a simple two-dimensional test problem and for a shock-tube ignition model involving 22 uncertain parameters and 3 design parameters. The numerical experiments indicate that whereas both methods provide effective means for tuning the quality of the representation along distinct subsets of parameters, adaptive PSP in the global parameter space generally requires fewer model evaluations than the nested approach to achieve similar projection error.

In order to increase efficiency even further, a subsampling technique is developed to allow for local adaptivity within the aPSP algorithm. The local refinement is achieved by exploiting the hierarchical nature of nested quadrature grids to determine regions of estimated convergence. In order to achieve global representations with local refinement, synthesized model data from a lower order projection is used for the final projection. The final subsampled grid was also tested with two more robust, sparse projection techniques including compressed sensing and hybrid least-angle-regression. These methods are evaluated on two sample test functions and then as an {\it a priori} analysis of the HYCOM simulations and the shock-tube ignition model investigated earlier. Small but non-trivial efficiency gains were found in some cases and in others, a large reduction in model evaluations with only a small loss of model fidelity was realized. Further extensions and capabilities are recommended for future investigations.

Item Open Access Algorithms with Applications to Anthropology(2018) Ravier, Robert JamesIn this dissertation, we investigate several problems in shape analysis. We start by discussing the shape matching problem. Given that homeomorphisms of shapes are computed in practice by interpolating sparse correspondence, we give an algorithm to refine pairwise mappings in a collection by employing a simple metric condition to obtain partial correspondences of points chosen in a manner that outlines the shapes of interest in a relatively small number of points. We then use this mapping algorithm in two separate applications. First, we investigate the extent to which classical assumptions and methods in statistical shape analysis hold for near continuous discretizations of surfaces spanning different species and genus groups. We find that these methods yield biologically meaningful information, and that resulting operations with these correspondences, including averaging and linear interpolation, yield biologically meaningful surfaces even for distinct geometries. As a second application, we discuss the problem of dictionary learning on shapes in an effort to find sparse decompositions of shapes in a collection. To this end, we define a construction of wavelet-like and ridgelet-like objects that are easily computable at the level of the discretization, both of which have natural interpretation in the smooth case. We then use these in tandem with feature points to create a sparse dictionary, and show that standard sparsification practices still retain biological information.

Item Open Access An Investigation into the Multiscale Nature of Turbulence and its Effect on Particle Transport(2022) Tom, JosinWe study the effect of the multiscale properties of turbulence on particle transport, specifically looking at the physical mechanisms by which different turbulent flow scales impact the settling speeds of particles in turbulent flows. The average settling speed of small heavy particles in turbulent flows is important for many environmental problems such as water droplets in clouds and atmospheric aerosols. The traditional explanation for enhanced particle settling speeds in turbulence for a one-way coupled (1WC) system is the preferential sweeping mechanism proposed by Maxey (1987, J. Fluid Mech.), which depends on the preferential sampling of the fluid velocity gradient field by the inertial particles. However, Maxey's analysis does not shed light on role of different turbulent flow scales contributing to the enhanced settling, partly since the theoretical analysis was restricted to particles with weak inertia.

In the first part of the work, we develop a new theoretical result, valid for particles of arbitrary inertia, that reveals the multiscale nature of the preferential sweeping mechanism. In particular, the analysis shows how the range of scales at which the preferential sweeping mechanism operates depends on particle inertia. This analysis is complemented by results from Direct Numerical Simulations (DNS) where we examine the role of different flow scales on the particle settling speeds by coarse-graining (filtering) the underlying flow. The results explain the dependence of the particle settling speeds on Reynolds number and show how the saturation of this dependence at sufficiently large Reynolds number depends upon particle inertia. We also explore how particles preferentially sample the fluid velocity gradients at various scales and show that while rapidly settling particles do not preferentially sample the fluid velocity gradients, they do preferentially sample the fluid velocity gradients coarse-grained at scales outside of the dissipation range.

Inspired by our finding that the effectiveness of the preferential sweeping mechanism depends on how particles interact with the strain and vorticity fields at different scales, we next shed light on the multiscale dynamics of turbulence by exploring the properties of the turbulent velocity gradients at different scales. We do this by analyzing the evolution equations for the filtered velocity gradient tensor (FVGT) in the strain-rate eigenframe. However, the pressure Hessian and viscous stress are unclosed in this frame of reference, requiring in-depth modelling. Using data from DNS of the forced Navier-Stokes equation, we consider the relative importance of local and non-local terms in the FVGT eigenframe equations across the scales using statistical analysis. We show that the anisotropic pressure Hessian (which is one of the unclosed terms) exhibits highly non-linear behavior at low values of normalized local gradients, with important modeling implications. We derive a generalization of the classical Lumley triangle that allows us to show that the pressure Hessian has a preference for two-component axisymmetric configurations at small scales, with a transition to a more isotropic state at larger scales. We also show that the current models fail to capture a number of subtle features observed in our results and provide useful guidelines for improving Lagrangian models of the FVGT.

In the final part of the work, we look at how two-way coupling (2WC) modifies the multiscale preferential sweeping mechanism. We comment on the the applicability of the theoretical analysis developed in the first part of the work for 2WC flows. Monchaux & Dejoan (2017, Phys. Rev. Fluids) showed using DNS that while for low particle loading the effect of 2WC on the global flow statistics is weak, 2WC enables the particles to drag the fluid in their vicinity down with them, significantly enhancing their settling, and they argued that two-way coupling suppresses the preferential sweeping mechanism. We explore this further by considering the impact of 2WC on the contribution made by eddies of different sizes on the particle settling. In agreement with Monchaux & Dejoan, we show that even for low loading, 2WC strongly enhances particle settling. However, contrary to their study, we show that preferential sweeping remains important in 2WC flows. In particular, for both 1WC and 2WC flows, the settling enhancement due to turbulence is dominated by contributions from particles in straining regions of the flow, but for the 2WC case, the particles also drag the fluid down with them, leading to an enhancement of their settling compared to the 1WC case. Overall, the novel results presented here not only augments the current understanding of the different physical mechanisms in producing enhanced settling speeds from a fundamental physics perspective, but can also be used to improve predictive capabilities in large-scale atmospheric modeling.

Item Unknown Analytical and Numerical Study of Lindblad Equations(2020) Cao, YuLindblad equations, since introduced in 1976 by Lindblad, and by Gorini, Kossakowski, and Sudarshan, have received much attention in many areas of scientific research. Around the past fifty years, many properties and structures of Lindblad equations have been discovered and identified. In this dissertation, we study Lindblad equations from three aspects: (I) physical perspective; (II) numerical perspective; and (III) information theory perspective.

In Chp. 2, we study Lindblad equations from the physical perspective. More specifically, we derive a Lindblad equation for a simplified Anderson-Holstein model arising from quantum chemistry. Though we consider the classical approach (i.e., the weak coupling limit), we provide more explicit scaling for parameters when the approximations are made. Moreover, we derive a classical master equation based on the Lindbladian formalism.

In Chp. 3, we consider numerical aspects of Lindblad equations. Motivated by the dynamical low-rank approximation method for matrix ODEs and stochastic unraveling for Lindblad equations, we are curious about the relation between the action of dynamical low-rank approximation and the action of stochastic unraveling. To address this, we propose a stochastic dynamical low-rank approximation method. In the context of Lindblad equations, we illustrate a commuting relation between the dynamical low-rank approximation and the stochastic unraveling.

In Chp. 4, we investigate Lindblad equations from the information theory perspective. We consider a particular family of Lindblad equations: primitive Lindblad equations with GNS-detailed balance. We identify Riemannian manifolds in which these Lindblad equations are gradient flow dynamics of sandwiched Rényi divergences. The necessary condition for such a geometric structure is also studied. Moreover, we study the exponential convergence behavior of these Lindblad equations to their equilibria, quantified by the whole family of sandwiched Rényi divergences.

Item Unknown Applications of Topological Data Analysis and Sliding Window Embeddings for Learning on Novel Features of Time-Varying Dynamical Systems(2017) Ghadyali, Hamza MustafaThis work introduces geometric and topological data analysis (TDA) tools that can be used in conjunction with sliding window transformations, also known as delay-embeddings, for discovering structure in time series and dynamical systems in an unsupervised or supervised learning framework. For signals of unknown period, we introduce an intuitive topological method to discover the period, and we demonstrate its use in synthetic examples and real temperature data. Alternatively, for almost-periodic signals of known period, we introduce a metric called Geometric Complexity of an Almost Periodic signal (GCAP), based on a topological construction, which allows us to continuously measure the evolving variation of its periods. We apply this method to temperature data collected from over 200 weather stations in the United States and describe the novel patterns that we observe. Next, we show how geometric and TDA tools can be used in a supervised learning framework. Seizure-detection using electroencephalogram (EEG) data is formulated as a binary classification problem. We define new collections of geometric and topological features of multi-channel data, which utilizes temporal and spatial context of EEG, and show how it results in better overall performance of seizure detection than using the usual time-domain and frequency domain features. Finally, we introduce a novel method to sonify persistence diagrams, and more generally any planar point cloud, using a modified version of the harmonic table. This auditory display can be useful for finding patterns that visual analysis alone may miss.

Item Open Access Compressive Sensing in Transmission Electron Microscopy(2018) Stevens, AndrewElectron microscopy is one of the most powerful tools available in observational science. Magnifications of 10,000,000x have been achieved with picometer precision. At this high level of magnification, individual atoms are visible. This is possible because the wavelength of electrons is much smaller than visible light, which also means that the highly focused electron beams used to perform imaging contain significantly more energy than visible light. The beam energy is high enough that it can cause radiation damage to metal specimens. Reducing radiation dose while maintaining image quality has been a central research topic in electron microscopy for several decades. Without the ability to reduce the dose, most organic and biological specimens cannot be imaged at atomic resolution. Fundamental processes in materials science and biology arise at the atomic level, thus understanding these processes can only occur if the observational tools can capture information with atomic resolution.

The primary objective of this research is to develop new techniques for low dose and high resolution imaging in (scanning) transmission electron microscopy (S/TEM). This is achieved through the development of new machine learning based compressive sensing algorithms and microscope hardware for acquiring a subset of the pixels in an image. Compressive sensing allows recovery of a signal from significantly fewer measurements than total signal size (under certain conditions). The research objective is attained by demonstrating application of compressive sensing to S/TEM in several simulations and real microscope experiments. The data types considered are images, videos, multispectral images, tomograms, and 4-dimensional ptychographic data. In the simulations, image quality and error metrics are defined to verify that reducing dose is possible with a small impact on image quality. In the microscope experiments, images are acquired with and without compressive sensing so that a qualitative verification can be performed.

Compressive sensing is shown to be an effective approach to reduce dose in S/TEM without sacrificing image quality. Moreover, it offers increased acquisition speed and reduced data size. Research leading to this dissertation has been published in 25 articles or conference papers and 5 patent applications have been submitted. The published papers include contributions to machine learning, physics, chemistry, and materials science. The newly developed pixel sampling hardware is being productized so that other microscopists can use compressive sensing in their experiments. In the future, scientific imaging devices (e.g., scanning transmission x-ray microscopy (STXM) and secondary-ion mass spectrometry (SIMS)) could also benefit from the techniques presented in this dissertation.

Item Open Access Data Transfer between Meshes for Large Deformation Frictional Contact Problems(2013) Kindo, Temesgen MarkosIn the finite element simulation of problems with contact there arises

the need to change the mesh and continue the simulation on a new mesh.

This is encountered when the mesh has to be changed because the original mesh experiences severe distortion or the mesh is adapted to minimize errors in the solution. In such instances a crucial component is the transfer of data from the old mesh to the new one.

This work proposes a strategy by which such remeshing can be accomplished in the presence of mortar-discretized contact,

focusing in particular on the remapping of contact variables which must occur to make the method robust and efficient.

By splitting the contact stress into normal and tangential components and transferring the normal component as a scalar and the tangential component by parallel transporting on the contact surface an accurate and consistent transfer scheme is obtained. Penalty and augmented Lagrangian formulations are considered. The approach is demonstrated by a number of two and three dimensional numerical examples.

Item Open Access Designing Quantum Channels Induced by Diagonal Gates(2023) Hu, JingzhenThe challenge of quantum computing is to combine error resilience with universal computation. Diagonal gates such as the transversal T gate play an important role in implementing a universal set of quantum operations. We introduce a framework that describes the process of preparing a code state, applying a diagonal physical gate, measuring a code syndrome, and applying a Pauli correction that may depend on the measured syndrome (the average logical channel induced by an arbitrary diagonal gate). The framework describes the interaction of code states and physical gates in terms of generator coefficients determined by the induced logical operator. The interaction of code states and diagonal gates depends on the signs of Z-stabilizers in the CSS code, and the proposed generator coefficient framework explicitly includes this degree of freedom. We derive necessary and sufficient conditions for an arbitrary diagonal gate to preserve the code space of a stabilizer code, and provide an explicit expression of the induced logical operator. When the diagonal gate is a quadratic form diagonal gate, the conditions can be expressed in terms of divisibility of weights in the two classical codes that determine the CSS code. These codes find applications in magic state distillation and elsewhere. When all the signs are positive, we characterize all possible CSS codes, invariant under transversal Z-rotation through π/2^l, that are constructed from classical Reed-Muller codes by deriving the necessary and sufficient constraints on the level l. According to the divisibility conditions, we construct new families of CSS codes using cosets of the first order Reed-Muller code defined by quadratic forms. The generator coefficient framework extends to arbitrary stabilizer codes but the more general class of non-degenerate stabilizer codes does not bring advantages when designing the code parameters.

Relying on the generator coefficient framework, we introduce a method of synthesizing CSS codes that realizes a target logical diagonal gate at some level l in the Clifford hierarchy. The method combines three basic operations: concatenation, removal of Z-stabilizers, and addition of X-stabilizers. It explicitly tracks the logical gate induced by a diagonal physical gate that preserves a CSS code. The first step is concatenation, where the input is a CSS code and a physical diagonal gate at level l inducing a logical diagonal gate at the same level. The output is a new code for which a physical diagonal gate at level l+1 induces the original logical gate. The next step is judicious removal of Z-stabilizers to increase the level of the induced logical operator. We identify three ways of climbing the logical Clifford hierarchy from level l to level l+1, each built on a recursive relation on the Pauli coefficients of the induced logical operators. Removal of Z-stabilizers may reduce distance, and the purpose of the third basic operation, addition of X-stabilizers, is to compensate for such losses. Our approach to logical gate synthesis is demonstrated by two proofs of concept: the [[2^(l+1) − 2, 2, 2]] triorthogonal code family, and the [[2^m, (m choose r) , 2^(min{r, m-r})]] quantum Reed-Muller code family.

Item Open Access Dynamics and Steady-states of Thin Film Droplets on Homogeneous and Heterogeneous Substrates(2019) Liu, WeifanIn this dissertation, we study the dynamics and steady-states of thin liquid films on solid substrates using lubrication equations. Steady-states and bifurcation of thin films on chemically patterned substrates have been previously studied for thin films on infinite domains with periodic boundary conditions. Inspired by previous work, we study the steady-state thin film on a chemically heterogeneous 1-D domain of finite length, subject to no-flux boundary conditions. Based on the structure of the bifurcation diagram, we classify the 1-D steady-state solutions that could exist on such substrates into six different branches and develop asymptotic approximation of steady-states on each branch. We show that using perturbation expansions, the leading order solutions provide a good prediction of steady-state thin film on a stepwise-patterned substrate. We also show that all of the analysis in 1-D can be easily extended to axisymmetric solutions in 2-D, which leads to qualitatively the same results.

Subject to long-wave instability, thin films break up and form droplets. In presence of small fluxes, these droplets move and exchange mass. In 2002, Glasner and Witelski proposed a simplified model that predicts the pressure and position evolution of droplets in 1-D on homogeneous substrates when fluxes are small. While the model is capable of giving accurate prediction of the dynamics of droplets in presence of small fluxes, the model becomes less accurate as fluxes increase. We present a refined model that computes the pressure and position of a single droplet on a finite domain. Through numerical simulations, we show that the refined model captures single-droplet dynamics with higher accuracy than the previous model.

Item Open Access Efficient Algorithms for High-dimensional Eigenvalue Problems(2020) Wang, ZheThe eigenvalue problem is a traditional mathematical problem and has a wide applications. Although there are many algorithms and theories, it is still challenging to solve the leading eigenvalue problem of extreme high dimension. Full configuration interaction (FCI) problem in quantum chemistry is such a problem. This thesis tries to understand some existing algorithms of FCI problem and propose new efficient algorithms for the high-dimensional eigenvalue problem. In more details, we first establish a general framework of inexact power iteration and establish the convergence theorem of full configuration interaction quantum Monte Carlo (FCIQMC) and fast randomized iteration (FRI). Second, we reformulate the leading eigenvalue problem as an optimization problem, then compare the show the convergence of several coordinate descent methods (CDM) to solve the leading eigenvalue problem. Third, we propose a new efficient algorithm named Coordinate Descent Full Configuration Interaction (CDFCI) based on coordinate descent methods to solve the FCI problem, which produces some state-of-the-art results. Finally, we conduct various numerical experiments to fully test the algorithms.

Item Open Access EM Scattering from Perforated Films: Transmission and Resonance(2012) Jackson, Aaron DavidWe calculate electromagnetic transmission through periodic gratings using a mode-matching method for solving Maxwell's equations. We record the derivation of the equations involved for several variations of the problem, including one- and two- dimensionally periodic films, one-sided films, films with complicated periodicity, and a simpler formula for the case of a single contributing waveguide mode. We demonstrate the effects of the Rayleigh anomaly, which causes energy transmission to be very low compared to nearby frequencies, and the associated transmission maxima which may be as high as 100% for certain energy frequencies. Finally we present further variations of the model to account for the effects of conductivity, finite hole arrays, and collimation. We find that assuming the film is perfectly conducting with infinite periodicity does not change the transmission sufficiently to explain the difference between experimental and theoretical results. However, removing the assumption that the incident radiation is in the form of a plane wave brings the transmission much more in agreement with experimental results.

Item Open Access From Spectral Theorem to Spectral Statistics of Large Random Matrices with Spatio-Temporal Dependencies(2023) Naeem, Muhammad AbdullahHigh dimensional random dynamical systems are ubiquitous, including-but not limited to- cyber-physical systems, daily return on different stocks of S\&P 1500 and velocity profile of interacting particle systems around McKeanVlasov limit. Mathematically speaking, observed time series data can be captured via a stable $n-$ dimensional linear transformation `$A$' and additive randomness. System identification aims at extracting useful information about underlying dynamical system, given a length $N$ trajectory from it (corresponds to an $n \times N$ dimensional data matrix). We use spectral theorem for non-Hermitian operators to show that spatio-temperal correlations are dictated by the \emph{discrepancy between algebraic andgeometric multiplicity of distinct eigenvalues} corresponding to state transition matrix. Small discrepancies imply that original trajectory essentially comprises of multiple \emph{lower dimensional random dynamical systems living on $A$ invariant subspaces and are statistically independent of each other}. In the process, we provide first quantitative handle on decay rate of finite powers of state transition matrix $\|A^{k}\|$ . It is shown that when a stable dynamical system has only one distinct eigenvalue and discrepancy of $n-1$: $\|A\|$ has a dependence on $n$, resulting dynamics are \emph{spatially inseparable} and consequently there exist at least one row with covariates of typical size $\Theta\big(\sqrt{N-n+1}$ $e^{n}\big)$ i.e., even under stability assumption, covariates can \emph{suffer from curse of dimensionality }.

In the light of these findings we set the stage for non-asymptotic error analysis in estimation of state transition matrix $A$ via least squares regression on observed trajectory by showing that element-wise error is essentially a variant of well-know Littlewood-Offord problem and(can be extremely sensitive to dimension of the state space and number of iterations). We also show that largest singular value of the data matrix can be cursed by dimensionality even when state-transition matrix is stable. Overarching theme of this thesis is new theoretical results on spectral theorem for non-Hermitian operators, non-asymptotic behavior of high dimensional dynamical systems , which we incorporate with the work of Talagrand on concentration of measure phenomenon to better understand behavior of the structured random matrices(data matrix) and subsequently the performance of different learning algorithms with dependent data. Besides, we also show that there exists stable linear Gaussians with process level Talagrands' inequality linear in dimension of the state space(previously an open problem), along with deterioration of mixing times with increase in discrepancy between algebraic and geometric multiplicity of $A$.

Item Open Access General and Efficient Bayesian Computation through Hamiltonian Monte Carlo Extensions(2017) Nishimura, AkihikoHamiltonian Monte Carlo (HMC) is a state-of-the-art sampling algorithm for Bayesian computation. Popular probabilistic programming languages Stan and PyMC rely on HMC’s generality and efficiency to provide automatic Bayesian inference platforms for practitioners. Despite its wide-spread use and numerous success stories, HMC has several well known pitfalls. This thesis presents extensions of HMC that overcome its two most prominent weaknesses: inability to handle discrete parameters and slow mixing on multi-modal target distributions.

Discontinuous HMC (DHMC) presented in Chapter 2 extends HMC to discontinuous target distributions – and hence to discrete parameter distributions through embedding them into continuous spaces — using an idea of event-driven Monte Carlo from the computational physics literature. DHMC is guaranteed to outperform a Metropolis-within-Gibbs algorithm since, as it turns out, the two algorithms coincide under a specific (and sub-optimal) implementation of DHMC. The theoretical justification of DHMC extends an existing theory of non-smooth Hamiltonian mechanics and of measure-valued differential inclusions.

Geometrically tempered HMC (GTHMC) presented in Chapter 3 improves HMC’s performance on multi-modal target distributions. The efficiency improvement is achieved through differential geometric techniques, relating a target distribution to

another distribution with less severe multi-modality. We establish a geometric theory behind Riemannian manifold HMC to motivate our geometric tempering methods. We then develop an explicit variable stepsize reversible integrator for simulating

Hamiltonian dynamics to overcome a stability issue of the usual Stormer-Verlet integrator. The integrator is of independent interest, being the first of its kind designed specifically for HMC variants.

In addition to the two extensions described above, Chapter 4 describes a variable trajectory length algorithm that generalizes the acceptance and rejection procedure of HMC — and in fact of any reversible dynamics based samplers — to allow for more flexible choices of trajectory lengths. The algorithm in particular enables an effective application of a variable stepsize integrator to HMC extensions, including GTHMC. The algorithm is widely applicable and provides a recipe for constructing valid dynamics based samplers beyond the known HMC variants. Chapter 5 concludes the thesis with a simple and practical algorithm to improve computational efficiencies of HMC and related algorithms over their traditional implementations.

Item Open Access Geometric Multimedia Time Series(2017) Tralie, Christopher JohnThis thesis provides a new take on problems in multimedia times series analysis by using a shape-based perspective to quantify patterns in time, which is complementary to more traditional analysis-based time series techniques. Inspired by the dynamical systems community, we turn time series into shapes via sliding window embeddings, which we refer to as ``time-ordered point clouds'' (TOPCs). This framework has traditionally been used on a single 1D observation function for deterministic systems, but we generalize the sliding window technique so that it not only applies to multivariate data (e.g. videos), but that it also applies to data which is not stationary (e.g. music).

The geometry of our time-ordered point clouds can be quite informative. For periodic signals, the point clouds fill out topological loops, which, depending on harmonic content, reside on various high dimensional tori. For quasiperiodic signals, the point clouds are dense on a torus. We use modern tools from topological data analysis (TDA) to quantify degrees of periodicity and quasiperiodicity by looking at these shapes, and we show that this can be used to detect anomalies in videos of vibrating vocal folds. In the case of videos, this has the advantage of substantially reducing the amount of preprocessing, as no motion tracking is needed, and the technique operates on raw pixels. This is also one of the first known uses of persistent H2 in a high dimensional setting.

Periodic processes represent only a sliver of possible dynamics, and we also show that sequences of arbitrary normalized sliding window point clouds are approximately isometric between ``cover songs,'' or different versions of the same song, possibly with radically different spectral content. Surprisingly, in this application, an incredibly simple geometric descriptor based on self-similarity matrices performs the best, and it also enables us to use MFCC features for this task, which was previously thought not to be possible due to significant timbral differences that can exist between versions. When combined with traditional pitch-based features using similarity metric fusion, we obtain state of the art results on automatic cover song identification.

In addition to being used as a geometric descriptor, self-similarity matrices provide a unifying description of phenomena in time-ordered point clouds throughout our work, and we use them to illustrate properties such as recurrence, mirror symmetry in time, and harmonics in periodic processes. They also provide the base representation for designing isometry blind time warping algorithms, which we use to synchronize time-ordered point clouds that are shifted versions of each other in space without ever having to do a spatial alignment. In particular, we devise an algorithm that lower bounds the 1-stress between two time-ordered point clouds, which is related to the Gromov-Hausdorff distance.

Overall, we show a proof-of-concept and promise of the nascent field of geometric signal processing, which is worthy of further study in applications of music structure, multimodal data analysis, and video analysis.

Item Open Access Homeostasis-Bifurcation Singularities and Identifiability of Feedforward Networks(2020) Duncan, WilliamThis dissertation addresses two aspects of dynamical systems arising from biological networks: homeostasis-bifurcation and identifiability.

Homeostasis occurs when a biological quantity does not change very much as a parameter is varied over a wide interval. Local bifurcation occurs when the multiplicity or stability of equilibria changes at a point. Both phenomena can occur simultaneously and as the result of a single mechanism. We show that this is the case in the feedback inhibition network motif. In addition we prove that longer feedback inhibition networks are less stable. Towards understanding interactions between homeostasis and bifurcations, we define a new type of singularity, the homeostasis-bifurcation point. Using singularity theory, the behavior of dynamical systems with homeostasis-bifurcation points is characterized. In particular, we show that multiple homeostatic plateaus separated by hysteretic switches and homeostatic limit cycle periods and amplitudes are common when these singularities occur.

Identifiability asks whether it is possible to infer model parameters from measurements. We characterize the structural identifiability properties for feedforward networks with linear reaction rate kinetics. Interestingly, the set of reaction rates corresponding to the edges of the graph are identifiable, but the assignment of rates to edges is not; Permutations of the reaction rates leads to the same measurements. We show how the identifiability results for linear kinetics can be extended to Michaelis-Menten kinetics using asymptotics.

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