Browsing by Author "Embree, Mark P."
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- Approximation of Parametric Dynamical SystemsCarracedo Rodriguez, Andrea (Virginia Tech, 2020-09-02)Dynamical systems are widely used to model physical phenomena and, in many cases, these physical phenomena are parameter dependent. In this thesis we investigate three prominent problems related to the simulation of parametric dynamical systems and develop the analysis and computational framework to solve each of them. In many cases we have access to data resulting from simulations of a parametric dynamical system for which an explicit description may not be available. We introduce the parametric AAA (p-AAA) algorithm that builds a rational approximation of the underlying parametric dynamical system from its input/output measurements, in the form of transfer function evaluations. Our algorithm generalizes the AAA algorithm, a popular method for the rational approximation of nonparametric systems, to the parametric case. We develop p-AAA for both scalar and matrix-valued data and study the impact of parameter scaling. Even though we present p-AAA with parametric dynamical systems in mind, the ideas can be applied to parametric stationary problems as well, and we include such examples. The solution of a dynamical system can often be expressed in terms of an eigenvalue problem (EVP). In many cases, the resulting EVP is nonlinear and depends on a parameter. A common approach to solving (nonparametric) nonlinear EVPs is to approximate them with a rational EVP and then to linearize this approximation. An existing algorithm can then be applied to find the eigenvalues of this linearization. The AAA algorithm has been successfully applied to this scheme for the nonparametric case. We generalize this approach by using our p-AAA algorithm to find a rational approximation of parametric nonlinear EVPs. We define a corresponding linearization that fits the format of the compact rational Krylov (CORK) algorithm for the approximation of eigenvalues. The simulation of dynamical systems may be costly, since the need for accuracy may yield a system of very large dimension. This cost is magnified in the case of parametric dynamical systems, since one may be interested in simulations for many parameter values. Interpolatory model order reduction (MOR) tackles this problem by creating a surrogate model that interpolates the original, is of much smaller dimension, and captures the dynamics of the quantities of interest well. We generalize interpolatory projection MOR methods from parametric linear to parametric bilinear systems. We provide necessary subspace conditions to guarantee interpolation of the subsystems and their first and second derivatives, including the parameter gradients and Hessians. Throughout the dissertation, the analysis is illustrated via various benchmark numerical examples.
- Continual Traveling waves in Finite Structures: Theory, Simulations, and ExperimentsMalladi, Vijaya Venkata Narasimha Sriram (Virginia Tech, 2016-07-06)A mechanical wave is generated as a result of an external force interacting with the well-defined medium and it propagates through that medium transferring energy from one location to another. The ability to generate and control the motion of the mechanical waves through the finite medium opens up the opportunities for creating novel actuation mechanisms not possible before. However, any impedance to the path of these waves, especially in the form of finite boundaries, disperses this energy in the form of reflections. Therefore, it is impractical to achieve steady state traveling waves in finite structures without any reflections. In-spite of all these conditions, is it possible to generate waveforms that travel despite reflections at the boundaries? The work presented in this thesis develops a framework to answer this question by leveraging the dynamics of the finite structures without any active control. Therefore, this work investigates how mechanical waves are developed in finite structures and identifies the factors that influence steady state wave characteristics. Theoretical and experimental analysis is conducted on 1D and 2D structures to realize different type of traveling waves. Owing to the robust characteristics of the piezo-ceramics (PZTs) in vibrational studies, we developed piezo-coupled structures to develop traveling waves through experiments.The results from this study provided the fundamental physics behind the generation of mechanical waves and their propagation through finite mediums. This research will consolidate the outcomes and develop a structural framework that will aid with the design of adaptable structural systems built for the purpose. The present work aims to generate and harness structural traveling waves for various applications.
- Deep Time: Deep Learning Extensions to Time Series Factor Analysis with Applications to Uncertainty Quantification in Economic and Financial ModelingMiller, Dawson Jon (Virginia Tech, 2022-09-12)This thesis establishes methods to quantify and explain uncertainty through high-order moments in time series data, along with first principal-based improvements on the standard autoencoder and variational autoencoder. While the first-principal improvements on the standard variational autoencoder provide additional means of explainability, we ultimately look to non-variational methods for quantifying uncertainty under the autoencoder framework. We utilize Shannon's differential entropy to accomplish the task of uncertainty quantification in a general nonlinear and non-Gaussian setting. Together with previously established connections between autoencoders and principal component analysis, we motivate the focus on differential entropy as a proper abstraction of principal component analysis to this more general framework, where nonlinear and non-Gaussian characteristics in the data are permitted. Furthermore, we are able to establish explicit connections between high-order moments in the data to those in the latent space, which induce a natural latent space decomposition, and by extension, an explanation of the estimated uncertainty. The proposed methods are intended to be utilized in economic and financial factor models in state space form, building on recent developments in the application of neural networks to factor models with applications to financial and economic time series analysis. Finally, we demonstrate the efficacy of the proposed methods on high frequency hourly foreign exchange rates, macroeconomic signals, and synthetically generated autoregressive data sets.
- Efficient Algorithms for Data Analytics in Geophysical ImagingKump, Joseph Lee (Virginia Tech, 2021-06-14)Modern sensing systems such as distributed acoustic sensing (DAS) can produce massive quantities of geophysical data, often in remote locations. This presents significant challenges with regards to data storage and performing efficient analysis. To address this, we have designed and implemented efficient algorithms for two commonly utilized techniques in geophysical imaging: cross-correlations, and multichannel analysis of surface waves (MASW). Our cross-correlation algorithms operate directly in the wavelet domain on compressed data without requiring a reconstruction of the original signal, reducing memory costs and improving scalabiliy. Meanwhile, our MASW implementations make use of MPI parallelism and GPUs, and present a novel problem for the GPU.
- Efficient Fock Space Configuration Interaction Approaches For Large Strongly Correlated SystemsHouck, Shannon Elizabeth (Virginia Tech, 2021-07-07)Over the past few decades, single-molecule magnets (SMMs) have been an area of significant interest due to their plethora of potential uses, including possible applications to quantum computing and compact data storage devices. Although theoretical chemistry calculations could aid our understanding of the magnetic couplings present in these types of systems, they are often multiconfigurational in nature, making them difficult to model with tradi- tional single-reference approaches. Methods to handle these types of strongly correlated systems have been developed but often have significant drawbacks, and so these molecules remain difficult to model computationally. In this work, we discuss the application of Fock-space CI approaches to large transition metal complexes. First, we introduce a novel formalism which combines the spin-flip (SF), ioniza- tion potential (IP), and electron affinity (EA) approaches. This redox spin-flip approach, the restricted active space spin-flip and ionization potential/electron affinity (RAS-SF-IP/EA) method, is applied to several molecules exhibiting double exchange behavior. Model Hamil- tonian parameters are extracted from energy gaps and found to be in qualitative agreement with experiment. Having shown the efficacy of this approach, we move on to optimization, using a diagrammatic approach to derive equations for several RAS-1SF-IP/EA schemes. These equations allow direct construction of the most expensive intermediates in the David- son algorithm and should provide significant speedup, allowing application of Fock-space CI approaches to larger systems than ever before. The derived equations are implemented in the LibRASSF package in Q-Chem, as well as in an open-source PyFockCI code, avail- able on GitHub. A Bloch effective Hamiltonian formalism is also utilized to extract model Hamiltonian parameters from RAS-1SF calculations, allowing more nuanced studies of the Heisenberg J couplings present in many molecules with magnetically coupled sites. Over- all, our work with Fock-space CI provides a way to study magnetic couplings in very large strongly correlated systems at relatively low computational cost. This work was supported by a grant from the U.S. Department of Energy: DE-SC0018326.
- Efficient 𝐻₂-Based Parametric Model Reduction via Greedy SearchCooper, Jon Carl (Virginia Tech, 2021-01-19)Dynamical systems are mathematical models of physical phenomena widely used throughout the world today. When a dynamical system is too large to effectively use, we turn to model reduction to obtain a smaller dynamical system that preserves the behavior of the original. In many cases these models depend on one or more parameters other than time, which leads to the field of parametric model reduction. Constructing a parametric reduced-order model (ROM) is not an easy task, and for very large parametric systems it can be difficult to know how well a ROM models the original system, since this usually involves many computations with the full-order system, which is precisely what we want to avoid. Building off of efficient 𝐻-infinity approximations, we develop a greedy algorithm for efficiently modeling large-scale parametric dynamical systems in an 𝐻₂-sense. We demonstrate the effectiveness of this greedy search on a fluid problem, a mechanics problem, and a thermal problem. We also investigate Bayesian optimization for solving the optimization subproblem, and end with extending this algorithm to work with MIMO systems.
- Energy-based Footstep Localization using Floor Vibration Measurements from AccelerometersAlajlouni, Sa'ed Ahmad (Virginia Tech, 2017-11-30)This work addresses the problem of localizing an impact in a dispersive medium (waveguide) using a network of vibration sensors (accelerometers), distributed at various locations in the waveguide, measuring (and detecting the arrival of) the impact-generated seismic wave. In particular, the last part of this document focuses on the problem of localizing footsteps using underfloor accelerometers. The author believes the outcomes of this work pave the way for realizing real-time indoor occupant tracking using underfloor accelerometers; a system that is tamper-proof and non-intrusive compared to occupant tracking systems that rely on video image processing. A dispersive waveguide (e.g., a floor) causes the impact-generated wave to distort with the traveled distance and renders conventional time of flight localization methods inaccurate. Therefore, this work focuses on laying the foundation of a new alternative approach to impact localization in dispersive waveguides. In this document, localization algorithms, including wave-signal detection and signal processing, are developed utilizing the fact that the generated wave's energy is attenuated with the traveled distance. The proposed localization algorithms were evaluated using simulations and experiments of hammer impacts, in addition to occupant tracking experiments. The experiments were carried out on an instrumented floor section inside a smart building. As will be explained in this document, energy-based localization will turn out to be computationally cheap and more accurate than conventional time of flight techniques.
- Finite Horizon Optimality and Operator Splitting in Model Reduction of Large-Scale Dynamical SystemSinani, Klajdi (Virginia Tech, 2020-07-15)Simulation, design, and control of dynamical systems play an important role in numerous scientific and industrial tasks. The need for detailed models leads to large-scale dynamical systems, posing tremendous computational difficulties when employed in numerical simulations. In order to overcome these challenges, we perform model reduction, replacing the large-scale dynamics with high-fidelity reduced representations. There exist a plethora of methods for reduced order modeling of linear systems, including the Iterative Rational Krylov Algorithm (IRKA), Balanced Truncation (BT), and Hankel Norm Approximation. However, these methods generally target stable systems and the approximation is performed over an infinite time horizon. If we are interested in a finite horizon reduced model, we utilize techniques such as Time-limited Balanced Truncation (TLBT) and Proper Orthogonal Decomposition (POD). In this dissertation we establish interpolation-based optimality conditions over a finite horizon and develop an algorithm, Finite Horizon IRKA (FHIRKA), that produces a locally optimal reduced model on a specified time-interval. Nonetheless, the quantities being interpolated and the interpolant are not the same as in the infinite horizon case. Numerical experiments comparing FHIRKA to other algorithms further support our theoretical results. Next, we discuss model reduction for nonlinear dynamical systems. For models with unstructured nonlinearities, POD is the method of choice. However, POD is input dependent and not optimal with respect to the output. Thus, we use operator splitting to integrate the best features of system theoretic approaches with trajectory based methods such as POD in order to mitigate the effect of the control inputs for the approximation of nonlinear dynamical systems. We reduce the linear terms with system theoretic methods and the nonlinear terms terms via POD. Evolving the linear and nonlinear terms separately yields the reduced operator splitting solution. We present an error analysis for this method, as well as numerical results that illustrate the effectiveness of our approach. While in this dissertation we only pursue the splitting of linear and nonlinear terms, this approach can be implemented with Quadratic Bilinear IRKA or Balanced Truncation for Quadratic Bilinear systems to further diminish the input dependence of the reduced order modeling.
- Frequency-Domain Learning of Dynamical Systems From Time-Domain DataAckermann, Michael Stephen (Virginia Tech, 2022-06-21)Dynamical systems are useful tools for modeling many complex physical phenomena. In many situations, we do not have access to the governing equations to create these models. Instead, we have access to data in the form of input-output measurements. Data-driven approaches use these measurements to construct reduced order models (ROMs), a small scale model that well approximates the true system, directly from input/output data. Frequency domain data-driven methods, which require access to values (and in some cases to derivatives) of the transfer function, have been very successful in constructing high-fidelity ROMs from data. However, at times this frequency domain data can be difficult to obtain or one might have only access to time-domain data. Recently, Burohman et al. [2020] introduced a framework to approximate transfer function values using only time-domain data. We first discuss improvements to this method to allow a more efficient and more robust numerical implementation. Then, we develop an algorithm that performs optimal-H2 approximation using purely time-domain data; thus significantly extending the applicability of H2-optimal approximation without a need for frequency domain sampling. We also investigate how well other established frequency-based ROM techniques (such as the Loewner Framework, Adaptive Anderson-Antoulas Algorithm, and Vector Fitting) perform on this identified data, and compare them to the optimal-H2 model.
- Gaussian Processes for Power System Monitoring, Optimization, and PlanningJalali, Mana (Virginia Tech, 2022-07-26)The proliferation of renewables, electric vehicles, and power electronic devices calls for innovative approaches to learn, optimize, and plan the power system. The uncertain and volatile nature of the integrated components necessitates using swift and probabilistic solutions. Gaussian process regression is a machine learning paradigm that provides closed-form predictions with quantified uncertainties. The key property of Gaussian processes is the natural ability to integrate the sensitivity of the labels with respect to features, yielding improved accuracy. This dissertation tailors Gaussian process regression for three applications in power systems. First, a physics-informed approach is introduced to infer the grid dynamics using synchrophasor data with minimal network information. The suggested method is useful for a wide range of applications, including prediction, extrapolation, and anomaly detection. Further, the proposed framework accommodates heterogeneous noisy measurements with missing entries. Second, a learn-to-optimize scheme is presented using Gaussian process regression that predicts the optimal power flow minimizers given grid conditions. The main contribution is leveraging sensitivities to expedite learning and achieve data efficiency without compromising computational efficiency. Third, Bayesian optimization is applied to solve a bi-level minimization used for strategic investment in electricity markets. This method relies on modeling the cost of the outer problem as a Gaussian process and is applicable to non-convex and hard-to-evaluate objective functions. The designed algorithm shows significant improvement in speed while attaining a lower cost than existing methods.
- An Implementation-Based Exploration of HAPOD: Hierarchical Approximate Proper Orthogonal DecompositionBeach, Benjamin Josiah (Virginia Tech, 2018-01-25)Proper Orthogonal Decomposition (POD), combined with the Method of Snapshots and Galerkin projection, is a popular method for the model order reduction of nonlinear PDEs. The POD requires the left singular vectors from the singular value decomposition (SVD) of an n-by-m "snapshot matrix" S, each column of which represents the computed state of the system at a given time. However, the direct computation of this decomposition can be computationally expensive, particularly for snapshot matrices that are too large to fit in memory. Hierarchical Approximate POD (HAPOD) (Himpe 2016) is a recent method for the approximate truncated SVD that requires only a single pass over S, is easily parallelizable, and can be computationally cheaper than direct SVD, all while guaranteeing the requested accuracy for the resulting basis. This method processes the columns of S in blocks based on a predefined rooted tree of processors, concatenating the outputs from each stage to form the inputs for the next. However, depending on the selected parameter values and the properties of S, the performance of HAPOD may be no better than that of direct SVD. In this work, we numerically explore the parameter values and snapshot matrix properties for which HAPOD is computationally advantageous over the full SVD and compare its performance to that of a parallelized incremental SVD method (Brand 2002, Brand 2003, and Arrighi2015). In particular, in addition to the two major processor tree structures detailed in the initial publication of HAPOD (Himpe2016), we explore the viability of a new structure designed with an MPI implementation in mind.
- The Importance of Data in RF Machine LearningClark IV, William Henry (Virginia Tech, 2022-11-17)While the toolset known as Machine Learning (ML) is not new, several of the tools available within the toolset have seen revitalization with improved hardware, and have been applied across several domains in the last two decades. Deep Neural Network (DNN) applications have contributed to significant research within Radio Frequency (RF) problems over the last decade, spurred by results in image and audio processing. Machine Learning (ML), and Deep Learning (DL) specifically, are driven by access to relevant data during the training phase of the application due to the learned feature sets that are derived from vast amounts of similar data. Despite this critical reliance on data, the literature provides insufficient answers on how to quantify the data training needs of an application in order to achieve a desired performance. This dissertation first aims to create a practical definition that bounds the problem space of Radio Frequency Machine Learning (RFML), which we take to mean the application of Machine Learning (ML) as close to the sampled baseband signal directly after digitization as is possible, while allowing for preprocessing when reasonably defined and justified. After constraining the problem to the Radio Frequency Machine Learning (RFML) domain space, an understanding of what kinds of Machine Learning (ML) have been applied as well as the techniques that have shown benefits will be reviewed from the literature. With the problem space defined and the trends in the literature examined, the next goal aims at providing a better understanding for the concept of data quality through quantification. This quantification helps explain how the quality of data: affects Machine Learning (ML) systems with regard to final performance, drives required data observation quantity within that space, and impacts can be generalized and contrasted. With the understanding of how data quality and quantity can affect the performance of a system in the Radio Frequency Machine Learning (RFML) space, an examination of the data generation techniques and realizations from conceptual through real-time hardware implementations are discussed. Consequently, the results of this dissertation provide a foundation for estimating the investment required to realize a performance goal within a Deep Learning (DL) framework as well as a rough order of magnitude for common goals within the Radio Frequency Machine Learning (RFML) problem space.
- In Pursuit of Local Correlation for Reduced-Scaling Electronic Structure Methods in Molecules and Periodic SolidsClement, Marjory Carolena (Virginia Tech, 2021-08-05)Over the course of the last century, electronic structure theory (or, alternatively, computational quantum chemistry) has grown from being a fledgling field to being a "full partner with experiment" [Goddard Science 1985, 227 (4689), 917--923]. Numerous instances of theory matching experiment to very high accuracy abound, with one excellent example being the high-accuracy ab initio thermochemical data laid out in the 2004 work of Tajti and co-workers [Tajti et al. J. Chem. Phys. 2004, 121, 11599] and another being the heats of formation and molecular structures computed by Feller and co-workers in 2008 [Feller et al. J. Chem. Phys. 2008, 129, 204105]. But as the authors of both studies point out, this very high accuracy comes at a very high cost. In fact, at this point in time, electronic structure theory does not suffer from an accuracy problem (as it did in its early days) but a cost problem; or, perhaps more precisely, it suffers from an accuracy-to-cost ratio problem. We can compute electronic energies to nearly any precision we like, as long as we are willing to pay the associated cost. And just what are these high computational costs? For the purposes of this work, we are primarily concerned with the way in which the computational cost of a given method scales with the system size; for notational purposes, we will often introduce a parameter, N, that is proportional to the system size. In the case of Hartree-Fock, a one-body wavefunction-based method, the scaling is formally N⁴, and post-Hartree-Fock methods fare even worse. The coupled cluster singles, doubles, and perturbative triples method [CCSD(T)], which is frequently referred to as the "gold standard" of quantum chemistry, has an N⁷ scaling, making it inapplicable to many systems of real-world import. If highly accurate correlated wavefunction methods are to be applied to larger systems of interest, it is crucial that we reduce their computational scaling. One very successful means of doing this relies on the fact that electron correlation is fundamentally a local phenomenon, and the recognition of this fact has led to the development of numerous local implementations of conventional many-body methods. One such method, the DLPNO-CCSD(T) method, was successfully used to calculate the energy of the protein crambin [Riplinger, et al. J. Chem. Phys 2013, 139, 134101]. In the following work, we discuss how the local nature of electron correlation can be exploited, both in terms of the occupied orbitals and the unoccupied (or virtual) orbitals. In the case of the former, we highlight some of the historical developments in orbital localization before applying orbital localization robustly to infinite periodic crystalline systems [Clement, et al. 2021, Submitted to J. Chem. Theory Comput.]. In the case of the latter, we discuss a number of different ways in which the virtual space can be compressed before presenting our pioneering work in the area of iteratively-optimized pair natural orbitals ("iPNOs") [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589]. Concerning the iPNOs, we were able to recover significant accuracy with respect to traditional PNOs (which are unchanged throughout the course of a correlated calculation) at a comparable truncation level, indicating that our improved PNOs are, in fact, an improved representation of the coupled cluster doubles amplitudes. For example, when studying the percent errors in the absolute correlation energies of a representative sample of weakly bound dimers chosen from the S66 test suite [Řezác, et al. J. Chem. Theory Comput. 2011, 7 (8), 2427--2438], we found that our iPNO-CCSD scheme outperformed the standard PNO-CCSD scheme at every truncation threshold (τPNO) studied. Both PNO-based methods were compared to the canonical CCSD method, with the iPNO-CCSD method being, on average, 1.9 times better than the PNO-CCSD method at τPNO = 10⁻⁷ and more than an order of magnitude better for τPNO < 10⁻¹⁰ [Clement, et al. J. Chem. Theory Comput 2018, 14 (9), 4581--4589]. When our improved PNOs are combined with the PNO-incompleteness correction proposed by Neese and co-workers [Neese, et al. J. Chem. Phys. 2009, 130, 114108; Neese, et al. J. Chem. Phys. 2009, 131, 064103], the results are truly astounding. For a truncation threshold of τPNO = 10⁻⁶, the mean average absolute error in binding energy for all 66 dimers from the S66 test set was 3 times smaller when the incompleteness-corrected iPNO-CCSD method was used relative to the incompleteness-corrected PNO-CCSD method [Clement, et al. J. Chem. Theory Comput. 2018, 14 (9), 4581--4589]. In the latter half of this work, we present our implementation of a limited-memory Broyden-Fletcher-Goldfarb-Shanno (BFGS) based Pipek-Mezey Wannier function (PMWF) solver [Clement, et al. 2021 }, Submitted to J. Chem. Theory Comput.]. Although orbital localization in the context of the linear combination of atomic orbitals (LCAO) representation of periodic crystalline solids is not new [Marzari, et al. Rev. Mod. Phys. 2012, 84 (4), 1419--1475; Jònsson, et al. J. Chem. Theory Comput. 2017, 13} (2), 460--474], to our knowledge, this is the first implementation to be based on a BFGS solver. In addition, we are pleased to report that our novel BFGS-based solver is extremely robust in terms of the initial guess and the size of the history employed, with the final results and the time to solution, as measured in number of iterations required, being essentially independent of these initial choices. Furthermore, our BFGS-based solver converges much more quickly and consistently than either a steepest ascent (SA) or a non-linear conjugate gradient (CG) based solver, with this fact demonstrated for a number of 1-, 2-, and 3-dimensional systems. Armed with our real, localized Wannier functions, we are now in a position to pursue the application of local implementations of correlated many-body methods to the arena of periodic crystalline solids; a first step toward this goal will, most likely, be the study of PNOs, both conventional and iteratively-optimized, in this context.
- Iterative Rational Krylov Algorithm for Unstable Dynamical Systems and Genaralized Coprime FactorizationsSinani, Klajdi (Virginia Tech, 2016-01-08)Generally, large-scale dynamical systems pose tremendous computational difficulties when applied in numerical simulations. In order to overcome these challenges we use several model reduction techniques. For stable linear models these techniques work very well and provide good approximations for the full model. However, large-scale unstable systems arise in many applications. Many of the known model reduction methods are not very robust, or in some cases, may not even work if we are dealing with unstable systems. When approximating an unstable system by a reduced order model, accuracy is not the only concern. We also need to consider the structure of the reduced order model. Often, it is important that the number of unstable poles in the reduced system is the same as the number of unstable poles in the original system. The Iterative Rational Krylov Algorithm (IRKA) is a robust model reduction technique which is used to locally reduce stable linear dynamical systems optimally in the ℋ₂-norm. While we cannot guarantee that IRKA reduces an unstable model optimally, there are no numerical obstacles to the reduction of an unstable model via IRKA. In this thesis, we investigate IRKA's behavior when it is used to reduce unstable models. We also consider systems for which we cannot obtain a first order realization of the transfer function. We can use Realization-independent IRKA to obtain a reduced order model which does not preserve the structure of the original model. In this paper, we implement a structure preserving algorithm for systems with nonlinear frequency dependency.
- Mathematical Modeling and Dynamic Recovery of Power SystemsGarcia Hilares, Nilton Alan (Virginia Tech, 2023-05-19)Power networks are sophisticated dynamical systems whose stable operation is essential to modern society. We study the swing equation for networks and its linearization (LSEN) as a tool for modeling power systems. Nowadays, phasor measurement units (PMUs) are used across power networks to measure the magnitude and phase angle of electric signals. Given the abundant data that PMUs can produce, we study applications of the dynamic mode decomposition (DMD) and Loewner framework to power systems. The matrices that define the LSEN model have a particular structure that is not recovered by DMD. We thus propose a novel variant of DMD, called structure-preserving DMD (SPDMD), that imposes the LSEN structure upon the recovered system. Since the solution of the LSEN can potentially exhibit interesting transient dynamics, we study the transient growth for the exponential matrix related to the LSEN. We follow Godunov's approach to get upper bounds for the transient growth and also analyze the relationship of such bounds with classical bounds based on the spectrum, numerical range, and pseudospectra. We show how Godunov's bounds can be optimized to bound the solution operator at a given time. The Loewner framework provides a tool for identifying a dynamical system from tangential measurements. The singular values of Loewner matrices guide the discovery of the true order of the underlying system. However, these singular values can exhibit rapid decay when the interpolation points are far from the poles of the system. We establish a range of bounds for this decay of singular values and apply this analysis to power systems.
- Modeling and Analysis of a Moving Conductive String in a Magnetic FieldHasanyan, Jalil Davresh (Virginia Tech, 2019-02-07)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.
- Modeling spider webs as multilinked structures using a Chebyshev pseudospectral collocation methodGreen, Jennifer Neal (Virginia Tech, 2018-06-19)Spiders weave intricate webs for catching prey, providing shelter and setting mating rituals; arguably the most notable of these creations is the orb web. In this thesis we model the essential vibrations of orb webs by first considering a spider web as a multilinked structure of elastic strings. We then solve the associated eigenvalue problem using a Chebyshev pseudospectral collocation method to discretize the system. This thesis first examines the vibrations of webs with uniform material properties throughout, then investigates the effects of using biologically realistic material parameters for silks within a single web. Understanding how spiders detect and react to the vibrations produced by their webs is of interest for both biological and engineering applications.
- Numerical Methods for Separable Nonlinear Inverse Problems with Constraint and Low RankCho, Taewon (Virginia Tech, 2017-11-20)In this age, there are many applications of inverse problems to lots of areas ranging from astronomy, geoscience and so on. For example, image reconstruction and deblurring require the use of methods to solve inverse problems. Since the problems are subject to many factors and noise, we can't simply apply general inversion methods. Furthermore in the problems of interest, the number of unknown variables is huge, and some may depend nonlinearly on the data, such that we must solve nonlinear problems. It is quite different and significantly more challenging to solve nonlinear problems than linear inverse problems, and we need to use more sophisticated methods to solve these kinds of problems.
- On the Tightness of the Balanced Truncation Error Bound with an Application to Arrowhead SystemsReiter, Sean Joseph (Virginia Tech, 2022-01-28)Balanced truncation model reduction for linear systems yields reduced-order models that satisfy a well-known error bound in terms of a system's Hankel singular values. This bound is known to hold with equality under certain conditions, such as when the full-order system is state-space symmetric. In this work, we derive more general conditions in which the balanced truncation error bound holds with equality. We show that this holds for single-input, single-output systems that exhibit a generalized type of state-space symmetry based on the sign parameters corresponding to a system's Hankel singular values. We prove an additional result that shows how to determine this state-space symmetry from the arrowhead realization of a system, if available. In particular, we provide a formula for the sign parameters of an arrowhead system in terms of the off-diagonal entries of its arrowhead realization. We then illustrate these results with an example of an arrowhead system arising naturally in power systems modeling that motivated our study.
- A Parallel Aggregation Algorithm for Inter-Grid Transfer Operators in Algebraic MultigridGarcia Hilares, Nilton Alan (Virginia Tech, 2019-09-13)As finite element discretizations ever grow in size to address real-world problems, there is an increasing need for fast algorithms. Nowadays there are many GPU/CPU parallel approaches to solve such problems. Multigrid methods can be used to solve large-scale problems, or even better they can be used to precondition the conjugate gradient method, yielding better results in general. Capabilities of multigrid algorithms rely on the effectiveness of the inter-grid transfer operators. In this thesis we focus on the aggregation approach, discussing how different aggregation strategies affect the convergence rate. Based on these discussions, we propose an alternative parallel aggregation algorithm to improve convergence. We also provide numerous experimental results that compare different aggregation approaches, multigrid methods, and conjugate gradient iteration counts, showing that our proposed algorithm performs better in serial and parallel.