Browsing by Author "Embree, Mark"
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Item A DEIM Induced CUR Factorization(SIAM, 2016) Sorensen, D.C.; Embree, MarkWe derive a CUR approximate matrix factorization based on the discrete empirical interpolation method (DEIM). For a given matrix ${\bf A}$, such a factorization provides a low-rank approximate decomposition of the form ${\bf A} \approx \bf C \bf U \bf R$, where ${\bf C}$ and ${\bf R}$ are subsets of the columns and rows of ${\bf A}$, and ${\bf U}$ is constructed to make $\bf C\bf U \bf R $ a good approximation. Given a low-rank singular value decomposition ${\bf A} \approx \bf V \bf S \bf W^T$, the DEIM procedure uses ${\bf V}$ and ${\bf W}$ to select the columns and rows of ${\bf A}$ that form ${\bf C}$ and ${\bf R}$. Through an error analysis applicable to a general class of CUR factorizations, we show that the accuracy tracks the optimal approximation error within a factor that depends on the conditioning of submatrices of ${\bf V}$ and ${\bf W}$. For very large problems, ${\bf V}$ and ${\bf W}$ can be approximated well using an incremental QR algorithm that makes only one pass through ${\bf A}$. Numerical examples illustrate the favorable performance of the DEIM-CUR method compared to CUR approximations based on leverage scores.Item Adaptive Reduction of Large Spiking Neurons(2013-11-21) Du, Bosen; Sorensen, Danny C.; Cox, Steven J.; Embree, Mark; Antoulas, Athanasios C.This thesis develops adaptive reduction approaches for various models of large spiking neurons. Most neurons are like dendritic trees with many branches, and they communicate by nonlinear spiking behaviors. However, with the exception of Kellems' Strong-Weak model, most existing reduction approaches compromise the active ionic mechanisms that cause the spiking dynamics. The Strong-Weak model can predict the spikes caused by suprathreshold input traveling from the dendritic branches to the spike initiation zone (SIZ), but it is not able to reproduce the back propagation from SIZ to the dendritic branches after spikes. This thesis develops a new model called QAact, the mechanisms to incorporate QAact into the hybrid model to capture the back propagation behavior, and different model reduction techniques for each part of the new hybrid model where they are most advantageous. Computational tests of QAact and the new hybrid model as well as corresponding model reduction techniques on FitzHugh-Nagumo system, active nonuniform cable, and branched cell LGMD, demonstrate a significant reduction of dimension, computational complexity and running time.Item Aggregate Economic Implications of New Technologies in Energy Industry(2013-09-16) Zhang, Xinya; Hartley, Peter R.; Medlock, Kenneth B., III; Loch-Temzelides, Ted; Embree, MarkThis thesis studies technological progress in the energy sector and the transition path from fossil fuels to renewable energy, with a particular emphasis on the conse- quences to the whole economy. Currently, there is an active discussion regarding sub- sidizing renewable energy sources, which are often portrayed as the sole future source of energy and the driver of signi cant employment and economic growth. However, innovation in the fossil fuel sector and its continuing development can also be a game changer and should not be ignored. In the rst chapter, we use a dynamic general equilibrium model with endogenous technological progress in energy production to study the optimal transition from fossil fuels to renewable energy in a neoclassical growth economy. We emphasize the importance of modeling technology innovation in the fossil fuel sector, as well as in the renewable energy industry. Advancements in the development of shale oil and gas increase the supply of fossil fuel. This implies that the \parity cost target" for renewables is a moving one. We believe that this important observation is often neglected in policy discussions. Our quantitative analysis nds that these advancements allow fossil fuels to remain competitive for a longer period of time. While technological breakthroughs in the fossil fuel sector have postponed the full transition to renewable energy, they have also created many jobs and stimulated local economies. In the third chapter, we use an econometric analysis to compare job creation in the shale gas and oil sectors with that in the wind power sector in Texas. The results show that shale development and well drilling activities have brought strong employment and wage growth to Texas, while the impact of wind industry development on employment and wages statewide has been either not statistically signi cant or quite small. The rst and third chapters question the current enthusiasm in policy circles for only focusing on alternative energy. Chapter 2 provides some theoretical support for subsidizing renewable energy development. Here we develop a decentralized ver- sion of the model in Chapter 1 and allow for technological externalities. We analyze the e ciency of the competitive equilibrium solution and discuss in particular dif- ferent scenarios whereby externalities can result in an ine cient outcome. We show that the decentralized economy with externalities leads to under-investment in R&D, lower investment and consumption, and delayed transition to the renewable economy. This may provide an opportunity for government action to improve private sector outcomes.Item Applying the Short-Time Direct Directed Transfer Function to Human Electrocorticographic Recordings from a Language Task(2013-06-28) Whaley, Meagan; Cox, Steven J.; Embree, Mark; Tandon, Nitin; Dabaghian, YuriThis thesis applied the short-time direct directed transfer function (SdDTF) to time series data recordings from intracranial electrodes that measure the brain's electrical activity to determine the causal influences that occurred between brain regions during a speech production task. The combination of high temporal and spatial resolution of the electrocorticography (ECoG) recordings directly from the cortex render these measurements of brain activity desirable, particularly when analyzing the fine cognitive dynamics involved in word generation. This research applied a new method to characterize the SdDTF results by compressing across time and high gamma frequencies, generating adjacency matrices, and graphing them to visualize the influences between anatomical regions over the duration of the entire task. This consolidated SdDTF analysis technique allowed for data from a total of seven patients to be combined, generating results which were consistent with current speech production models. The results from this thesis contribute to the expansion of language research by identifying areas relevant to word generation, providing information that will help surgeons avoid irreparable damage to crucial cortex during brain surgery.Item Convergence of Polynomial Restart Krylov Methods for Eigenvalue Computation(2003-08) Beattie, Christopher A.; Embree, Mark; Sorensen, D.C.The convergence of Krylov subspace eigenvalue algorithms can be robustly measured by the angle the approximating Krylov space makes with a desired invariant subspace. This paper describes a new bound on this angle that handles the complexities introduced by non-Hermitian matrices, yet has a simpler derivation than similar previous bounds. The new bound reveals that ill-conditioning of the desired eigenvalues has little impact on convergence, while instability of unwanted eigenvalues plays an essential role. Practical computations usually require the approximating Krylov space to be restarted for efficiency, whereby the starting vector that generates the subspace is improved via a polynomial filter. Such filters dynamically steer a low-dimensional Krylov space toward a desired invariant subspace. We address the design of these filters, and illustrate with examples the subtleties involved in restarting non-Hermitian iterations.Item Energy bounds on point-wise damped wave operators(2006) Hardesty, Sean S.; Embree, MarkOne plays a harmonic by gently touching a finger to a vibrating string so as to divide it into two segments whose lengths form the ratio of small integers. The only prominent frequencies in the sound thereby produced correspond to those modes of the undamped string that have nodes at the contact point; the others are damped by the action of the finger. Bamberger, Rauch, and Taylor [1] modeled the phenomenon with point-wise damping and suggested that the "correct touch" (force applied by the finger) is that which causes the damped modes to decay most rapidly. Cox and Henrot [3] investigated the spectral properties of the associated operator, and identified the correct touch as that which minimizes its spectral abscissa. We give bounds on the total energy associated with the damped modes, and assess their utility in helping us understand the correct touch.Item Explicit Discontinuous Galerkin Methods for Linear Hyperbolic Problems(2013-11-14) Atcheson, Thomas; Warburton, Timothy; Symes, William W.; Embree, MarkDiscontinuous Galerkin methods have many features which make them a natural candidate for the solution of hyperbolic problems. One feature is flexibility with the order of approximation; a user with knowledge of the solution's regularity can increase the spatial order of approximation by increasing the polynomial order of the discontinuous Galerkin method. A marked increase in time-stepping difficulty, known as stiffness, often accompanies this increase in spatial order however. This thesis analyzes two techniques for reducing the impact of this stiffness on total time of simulation. The first, operator modification, directly modifies the high order method in a way that retains the same formal order of accuracy, but reduces the stiffness. The second, optimal Runge-Kutta methods, adds additional stages to Runge-Kutta methods and modifies them to customize their stability region to the problem. Three operator modification methods are analyzed analytically and numerically, the mapping technique of Kosloff/Tal-Ezer the covolume filtering technique of Warburton/Hagstrom , and the flux filtering technique of Chalmers, et al. . The covolume filtering and flux filtering techniques outperform mapping in that they negligibly impact accuracy but yield a reasonable improvement in efficiency. For optimal Runge-Kutta methods this thesis considers five top performing methods from the literature on hyperbolic problems and applies them to an unmodified method, a flux filtered method, and a covolume filtered method. Gains of up to 80\% are seen for covolume filtered solutions applied with optimal Runge-Kutta methods, showing the potential for efficient high order solutions of unsteady systems.Item Functional inference of conductances in the LGMD neuron(2013-08-27) Ackermann, Etienne; Cox, Steven J.; Embree, Mark; Sorensen, Danny C.; Dabaghian, YuriThis thesis develops an approach to determine spatially-varying ionic channel conductances throughout the dendrites of the LGMD neuron from distal transmembrane potential recordings in response to distributed subthreshold current injections. In particular this approach is demonstrated on a straight cable approximation to the LGMD neuron with leak and hyperpolarization-activated h-currents. Knowledge of the underlying channel conductances can help neuroscientists to characterize, better understand, and predict neuronal behavior---and topographic integration in the LGMD neuron in particular---but it is extremely difficult to measure these conductances directly. As a consequence, these conductances are commonly estimated by searching for several parameters that lead to simulated responses that are consistent with recorded behavior. In contrast, the approach presented here uses the method of moments to directly recover the underlying conductances, eliminating the need to simulate responses, making this approach both faster and more robust than typical optimization approaches since the solution cannot get trapped in local minima.Item gNek: A GPU Accelerated Incompressible Navier Stokes Solver(2013-09-16) Stilwell, Nichole; Warburton, Timothy; Riviere, Beatrice M.; Embree, MarkThis thesis presents a GPU accelerated implementation of a high order splitting scheme with a spectral element discretization for the incompressible Navier Stokes (INS) equations. While others have implemented this scheme on clusters of processors using the Nek5000 code, to my knowledge this thesis is the first to explore its performance on the GPU. This work implements several of the Nek5000 algorithms using OpenCL kernels that efficiently utilize the GPU memory architecture, and achieve massively parallel on chip computations. These rapid computations have the potential to significantly enhance computational fluid dynamics (CFD) simulations that arise in areas such as weather modeling or aircraft design procedures. I present convergence results for several test cases including channel, shear, Kovasznay, and lid-driven cavity flow problems, which achieve the proven convergence results.Item Improved Spectral Calculations for Discrete Schroedinger Operators(2013-09-16) Puelz, Charles; Embree, Mark; Sorensen, Danny C.; Damanik, DavidThis work details an O(n^2) algorithm for computing the spectra of discrete Schroedinger operators with periodic potentials. Spectra of these objects enhance our understanding of fundamental aperiodic physical systems and contain rich theoretical structure of interest to the mathematical community. Previous work on the Harper model led to an O(n^2) algorithm relying on properties not satisfied by other aperiodic operators. Physicists working with the Fibonacci Hamiltonian, a popular quasicrystal model, have instead used a problematic dynamical map approach or a sluggish O(n^3) procedure for their calculations. The algorithm presented in this work, a blend of well-established eigenvalue/vector algorithms, provides researchers with a more robust computational tool of general utility. Application to the Fibonacci Hamiltonian in the sparsely studied intermediate coupling regime reveals structure in canonical coverings of the spectrum that will prove useful in motivating conjectures regarding band combinatorics and fractal dimensions.Item Lagrange rational interpolation and its applications to approximation of large-scale dynamical systems(2013-11-06) Ionita, Antonio; Antoulas, Athanasios C.; Zhong, Lin; Embree, MarkWe present several new, efficient algorithms that extract low complexity models from frequency response measurements of large-scale dynamical systems. Our work is motivated by the fact that, in many applications, analytical models of a dynamical system are seldom available. Instead, we may only have access to its frequency response measurements. For example, for a system with multiple inputs and outputs, we may only have access to data sets of S-parameters. In this setting, our new approach extracts models that interpolate the given measurements. The extracted models have low complexity (or reduced order) and, thus, lead to short simulation times and low data storage requirements. The main tool used by our approach is Lagrange rational interpolation -- a generalization of the classic result of Lagrange polynomial interpolation. We present an in-depth look at Lagrange rational interpolation and provide several new insights and simplified proofs. This analysis leads to new algorithms that rely on the singular value decomposition (SVD) of the Loewner matrix pencil formed directly from the measurements. We show several new results on rational interpolation for measurements of linear, bi-linear and quadratic-linear systems. Furthermore, we generalize these results to parametrized measurements, that is, we show how to interpolate frequency response measurements that depend on parameters. We showcase this new approach through a series of relevant numerical examples such as n-port systems and parametrized partial differential equations.Item Magnetic Control in Crystal Growth from a Melt(2012-09-05) Huang, Yue; Houchens, Brent C.; Akin, John Edward.; Embree, MarkControl of bulk melt crystal growth techniques is desirable for producing semiconductors with the highest purity and ternary alloys with tunable electrical properties. Because these molten materials are electrically conducting, external magnetic fields are often employed to regulate the flow in the melt. However, complicated by the coupled flow, thermal, electromagnetic and chemical physics, such magnetic control is typically empirical or even an educated guess. Two magnetic flow control mechanisms: flow damping by steady magnetic fields, and flow stirring by alternating magnetic fields, are investigated numerically. Magnetic damping during optically-heated float-zone crystal growth is modeled using a spectral collocation method. The Marangoni convection at the free melt-gas interface is suppressed by applying a steady magnetic field, measured by the Hartmann number Ha. Using normal mode linear stability analyses, suppression of detrimental flow instabilities is quantitatively determined in a range applicable to experiments (up to Ha = 300 for Pr = 0.02, and up to Ha = 500 for Pr = 0.001). The hydrodynamic flow instability for small Prandtl number P r float-zone is confirmed by energy analyses. Rotating magnetic field stirring during confined crystal growth in an ampoule is also modeled. Decoupled from the flow field at small magnetic Reynolds number, the electromagnetic field is solved in a finite element solver. At low AC frequencies, the force is only in the azimuthal direction but penetrates deep into the melt. In contrast, the magnetic shielding effect is observed at high alternating current (AC) frequencies, where the external magnetic field penetrates only by a skin depth into the electrically conducting media within the short AC cycle. As a result, the electromagnetic body force is primarily confined to the ampoule surface. At these high AC frequencies the magnetic flux lines are drastically distorted within the melt. The body force is fully three-dimensional and is much stronger than at low AC frequencies, but is confined to near the ampoule surface due to the magnetic shielding effect. These models promote fundamental understanding of flow dynamics regulated by electromagnetic body forces. They provide quantitative guidance for crystal growth to minimize trial and error experimentation that is slow and expensive.Item Magnetic damping of an elastic conductor(2009) Hokanson, Jeffrey M.; Embree, Mark; Cox, Steven J.Many applications call for a design that maximizes the rate of energy decay. Typical problems of this class include one dimensional damped wave operators, where energy dissipation is caused by a damping operator acting on the velocity. Two damping operators are well understood: a multiplication operator (known as viscous damping) and a scaled Laplacian (known as Kelvin---Voigt damping). Paralleling the analysis of viscous damping, this thesis investigates energy decay for a novel third operator known as magnetic damping, where the damping is expressed via a rank-one self-adjoint operator, dependent on a function a. This operator describes a conductive monochord embedded in a spatially varying magnetic field perpendicular to the monochord and proportional to a. Through an analysis of the spectrum, this thesis suggests that unless a has a singularity at one boundary for any finite time, there exist initial conditions that give arbitrarily small energy decay at any time.Item Modeling competition in natural gas markets(2013-09-16) Cigerli, Burcu; Hartley, Peter R.; Medlock, Kenneth B., III; Embree, MarkThis dissertation consists of three chapters; each models competition in natural gas markets. These models provide insight into interactions between changes in market conditions/policies and market players’ strategic behavior. In all three chapters, we apply our models to a natural gas trade network formed by using BP’s Statistical Review of World Energy 2010 major trade flows. In the first chapter, we develop a model for the world natural gas market where buyers and sellers are connected by a trading network. Each natural gas producer is a Cournot player with a fixed supply capacity. Each of them is also connected to a unique set of importing markets. We show that this constrained noncooperative Cournot game is a potential game and its potential function has a unique maximizer. In the scenario analysis, we find that any exogenous change affecting Europe also has an effect in the Asia Pacific. The reason is that two big producers, Russia and the Middle East, are connected to both markets. We also find that a collusive agreement between Russia and the Middle East leads them to specialize in supply to markets based on their marginal costs of exporting natural gas. The second chapter is devoted to analyzing the impacts of North American shale gas on the world natural gas market. To better represent the North American natural gas market, this chapter also allows for perfect competition in that market. We find that North America exports natural gas when its supply curve is highly elastic and hence the domestic price impact of its exports is very small. Even so, the price impacts on the importing markets are substantial. We also find that shale gas development in North America decreases dominant producers’ market power elsewhere in the world and hence decreases the incentive of any parties to form a natural gas cartel. In the third chapter, we relax the assumption of fixed supply capacities and allow for natural gas producers to invest in their supply capacities. We assume a two period model with no uncertainty and show that there is a unique Cournot-Nash equilibrium and the open-loop Cournot-Nash equilibrium and closed-loop Cournot-Nash equilibrium investments coincide.Item Moment Matching and Modal Truncation for Linear Systems(2013-07-24) Hergenroeder, AJ; Embree, Mark; Sorensen, Danny C.; Warburton, TimWhile moment matching can effectively reduce the dimension of a linear, time-invariant system, it can simultaneously fail to improve the stable time-step for the forward Euler scheme. In the context of a semi-discrete heat equation with spatially smooth forcing, the high frequency modes are virtually insignificant. Eliminating such modes dramatically improves the stable time-step without sacrificing output accuracy. This is accomplished by modal filtration, whose computational cost is relatively palatable when applied following an initial reduction stage by moment matching. A bound on the norm of the difference between the transfer functions of the moment-matched system and its modally-filtered counterpart yields an intelligent choice for the mode of truncation. The dual-stage algorithm disappoints in the context of highly nonnormal semi-discrete convection-diffusion equations. There, moment matching can be ineffective in dimension reduction, precluding a cost-effective modal filtering step.Item Nonnormality in Lyapunov Equations(2016-04-22) Baker, Jonathan; Sorensen, Danny; Embree, MarkThe singular values of the solution to a Lyapunov equation determine the potential accuracy of the low-rank approximations constructed by iterative methods. Low- rank solutions are more accurate if most of the singular values are small, so a priori bounds that describe coefficient matrix properties that correspond to rapid singular value decay are valuable. Previous bounds take similar forms, all of which weaken (quadratically) as the coefficient matrix departs from normality. Such bounds suggest that the more nonnormal the coefficient matrix becomes, the slower the singular values of the solution will decay. However, simple examples typically exhibit an eventual acceleration of decay if the coefficient becomes very nonnormal. We will show that this principle is universal: decay always improves as departure from normality increases beyond a given threshold, specifically as the numerical range of the coefficient matrix extends farther into the right half-plane. We also give examples showing that similar behavior can occur for general Sylvester equations, though the right-hand side plays a more important role.Item Numerically Stable and Statistically Efficient Algorithms for Large Scale Exponential Fitting(2013-12-06) Hokanson, Jeffrey; Embree, Mark; Cox, Steven J.; Antoulas, Athanasios C.; Heinkenschloss, MatthiasThe exponential fitting problem appears in diverse applications such as magnetic resonance spectroscopy, mechanical resonance, chemical reactions, system identification, and radioactive decay. In each application, the exponential fitting problem decomposes measurements into a sum of exponentials with complex coefficients plus noise. Although exponential fitting algorithms have existed since the invention of Prony's Method in 1795, the modern challenge is to build algorithms that stably recover statistically optimal estimates of these complex coefficients while using millions of measurements in the presence of noise. Existing variants of Prony's Method prove either too expensive, most scaling cubically in the number of measurements, or too unstable. Nonlinear least squares methods scale linearly in the number of measurements, but require well-chosen initial estimates lest these methods converge slowly or find a spurious local minimum. We provide an analysis connecting the many variants of Prony's Method that have been developed in different fields over the past 200 years. This provides a unified framework that extends our understanding of the numerical and statistical properties of these algorithms. We also provide two new algorithms for exponential fitting that overcome several practical obstacles. The first algorithm is a modification of Prony's Method that can recover a few exponential coefficients from measurements containing thousands of exponentials, scaling linearly in the number of measurements. The second algorithm compresses measurements onto a subspace that minimizes the covariance of the resulting estimates and then recovers the exponential coefficients using an existing nonlinear least squares algorithm restricted to this subspace. Numerical experiments suggest that small compression spaces can be effective; typically we need fewer than 20 compressed measurements per exponential to recover the parameters with 90% efficiency. We demonstrate the efficacy of this approach by applying these algorithms to examples from magnetic resonance spectroscopy and mechanical vibration. Finally, we use these new algorithms to help answer outstanding questions about damping in mechanical systems. We place a steel string inside vacuum chamber and record the free response at multiple pressures. Analyzing these measurements with our new algorithms, we recover eigenvalue estimates as a function of pressure that illuminate the mechanism behind damping.Item One Can Hear the Composition of a String: Experiments with an Inverse Eigenvalue Problem(2008-07) Cox, Steven J.; Embree, Mark; Hokanson, Jeffrey M.To what extent do the vibrations of a mechanical system reveal its composition? Despite innumerable applications and mathematical elegance, this question often slips through those cracks that separate courses in mechanics, differential equations, and linear algebra. We address this omission by detailing a classical nite dimensional example: the use of frequencies of vibration to recover positions and masses of beads vibrating on a string. First we derive the equations of motion, then compare the eigenvalues of the resulting linearized model against vibration data measured from our laboratory's monochord. More challenging is the recovery of masses and positions of the beads from spectral data, a problem elegantly solved, through application of continued fractions, by Mark Krein. After presenting Krein's algorithm in a manner suitable for advanced undergraduates, we confirm its effcacy through physical experiment. We encourage readers to conduct their own explorations using data sets we provide on the web.Item Preconditioned iterative methods for inhomogeneous acoustic scattering applications(2010) Sifuentes, Josef; Embree, MarkThis thesis develops and analyzes efficient iterative methods for solving discretizations of the Lippmann--Schwinger integral equation for inhomogeneous acoustic scattering. Analysis and numerical illustrations of the spectral properties of the scattering problem demonstrate that a significant portion of the spectrum is approximated well on coarse grids. To exploit this, I develop a novel restarted GMRES method with adaptive deflation preconditioning based on spectral approximations on multiple grids. Much of the literature in this field is based on exact deflation, which is not feasible for most practical computations. This thesis provides an analytical framework for general approximate deflation methods and suggests a way to rigorously study a host of inexactly-applied preconditioners. Approximate deflation algorithms are implemented for scattering through thin inhomogeneities in photonic band gap problems. I also develop a short term recurrence for solving the one dimensional version of the problem that exploits the observation that the integral operator is a low rank perturbation of a self-adjoint operator. This method is based on strategies for solving Schur complement problems, and provides an alternative to a recent short term recurrence algorithm for matrices with such structure that we show to be numerically unstable for this application. The restarted GMRES method with adaptive deflation preconditioning over multiple grids, as well as the short term recurrence method for operators with low rank skew-adjoint parts, are very effective for reducing both the computational time and computer memory required to solve acoustic scattering problems. Furthermore, the methods are sufficiently general to be applicable to a wide class of problems.Item Preconditioning the integral formulation of the Helmholtz equation via deflation(2006) Sifuentes, Josef; Embree, MarkIn this thesis we propose methods for preconditioning Krylov subspace methods for solving the integral equation formulation of the Helmholtz partial differential equation for modeling scattered waves. An advantage of using an integral formulation is that only the scattering obstacle is discretized and the outgoing boundary conditions are automatically satisfied. Furthermore, convergence is dictated by the wave number kappa with only a mild dependence on the discretization. However such methods are increasingly computationally expensive for increasing values of kappa. This cost is due to GMRES iteration counts that increase like O(kappa2), for a linear system that is dense with dimension N = O(kappa 4). GMRES is slow due to a small subset of the spectrum that is well separated, a part of which approaches the origin as kappa increases. The troublesome subset corresponds to low frequency eigenfunctions which can be approximated using coarse meshes. We propose a preconditioner based on deflating this subset of the spectrum which we evaluate by interpolating coarse mesh approximations of the spectrum. We show that for discretizations of less than one node per wavelength, we can effectively precondition the full problem over a sufficiently resolved mesh.