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Numerical Analysis and Computation Commons™
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Articles 1 - 13 of 13
Full-Text Articles in Numerical Analysis and Computation
Investigating An Energy-Preserving Spectral Finite Difference Numerical Method For A Wave Equation On The Metric Graph, Tue Minh Tran
Investigating An Energy-Preserving Spectral Finite Difference Numerical Method For A Wave Equation On The Metric Graph, Tue Minh Tran
University Honors Theses
To investigate the accuracy and long-term energy conservation of a spectral finite difference numerical method for a wave equation on metric graphs. In conservative systems, numerical methods should preserve total energy. However, explicit finite difference methods require impractically small space steps and exhibit energy drift at end points. To address these limitations, a spectral finite difference method is implemented using a Fourier transformation. This semi-spectral method improves stability at endpoints while maintaining second-order accuracy, achieving an overall error of O(∆t2). We implement the semi-spectral method on the IEEE14 metric graph and provide visuals showing the initial condition …
Pt-Symmetry And Eigenmodes, Tamara Gratcheva
Pt-Symmetry And Eigenmodes, Tamara Gratcheva
University Honors Theses
Spectra of systems with balanced gain and loss, described by Hamiltonians with parity and time-reversal (PT) symmetry is a rich area of research. This work studies by means of numerical techniques, how eigenvalues and eigenfunctions of a Schrodinger operator change as a gain-loss parameter changes. Two cases on a disk with zero boundary conditions are considered. In the first case, within the enclosing disk, we place a parity (P) symmetric configuration of three smaller disks containing gain and loss media, which does not have PT-symmetry. In the second case, we study a PT-symmetric configuration …
Exploring Food Deserts And Environmental Impacts On Health In Chicago And Oregon, Sivasomasundari Arunarasu, Paulina Grzybowicz
Exploring Food Deserts And Environmental Impacts On Health In Chicago And Oregon, Sivasomasundari Arunarasu, Paulina Grzybowicz
altREU Projects
Food deserts are defined as, “an impoverished area where residents lack access to healthy foods”. This lack of access can be due to a combination of socioeconomic, geographic, and food-related variables, and has been proven to impact the health of residents in the area. In this project, several statistical and machine learning techniques are used to model the impact of food desserts and various other factors on health outcomes, including diabetes and obesity rates, in both the different neighborhoods in the City of Chicago and the various counties in the state of Oregon. The models are then used to determine …
Combating Covid On College Campuses: The Impact Of Structural Changes On Viral Transmissions, Jared Knofczynski, Aria Killebrew Bruehl, Ben Warner, Ryne Shelton
Combating Covid On College Campuses: The Impact Of Structural Changes On Viral Transmissions, Jared Knofczynski, Aria Killebrew Bruehl, Ben Warner, Ryne Shelton
altREU Projects
One of the most significant issues in the COVID-19 pandemic is the reopening of schools while minimizing the transmission of coronavirus. Opportunities for evaluating the effectiveness of policies that might be utilized at such institutions are limited, as the necessary empirical data has not been gathered yet. Agent-based modeling, where various entities within an environment are simulated as agents, offers an opportunity to examine the effectiveness of various policies in a way that drastically minimizes the health and economic risks involved. Agent-based modeling is common within biology, ecology and other fields; and has seen some use within the coronavirus literature. …
Analyzing Network Topology For Ddos Mitigation Using The Abelian Sandpile Model, Bhavana Panchumarthi, Monroe Ame Stephenson
Analyzing Network Topology For Ddos Mitigation Using The Abelian Sandpile Model, Bhavana Panchumarthi, Monroe Ame Stephenson
altREU Projects
A Distributed Denial of Service (DDoS) is a cyber attack, which is capable of triggering a cascading failure in the victim network. While DDoS attacks come in different forms, their general goal is to make a network's service unavailable to its users. A common, but risky, countermeasure is to blackhole or null route the source, or the attacked destination. When a server becomes a blackhole, or referred to as the sink in the paper, the data that is assigned to it "disappears" or gets deleted. Our research shows how mathematical modeling can propose an alternative blackholing strategy that could improve …
Trefftz Finite Elements On Curvilinear Polygons, Akash Anand, Jeffrey S. Ovall, Samuel E. Reynolds, Steffen Weisser
Trefftz Finite Elements On Curvilinear Polygons, Akash Anand, Jeffrey S. Ovall, Samuel E. Reynolds, Steffen Weisser
Mathematics and Statistics Faculty Publications and Presentations
We present a Trefftz-type finite element method on meshes consisting of curvilinear polygons. Local basis functions are computed using integral equation techniques that allow for the efficient and accurate evaluation of quantities needed in the formation of local stiffness matrices. To define our local finite element spaces in the presence of curved edges, we must also properly define what it means for a function defined on a curved edge to be "polynomial" of a given degree on that edge. We consider two natural choices, before settling on the one that yields the inclusion of complete polynomial spaces in our local …
Spectral Discretization Errors In Filtered Subspace Iteration, Jay Gopalakrishnan, Luka Grubišić, Jeffrey S. Ovall
Spectral Discretization Errors In Filtered Subspace Iteration, Jay Gopalakrishnan, Luka Grubišić, Jeffrey S. Ovall
Mathematics and Statistics Faculty Publications and Presentations
We consider filtered subspace iteration for approximating a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. The algorithm is motivated by a quadrature approximation of an operator-valued contour integral of the resolvent. Resolvents on infinite dimensional spaces are discretized in computable finite-dimensional spaces before the algorithm is applied. This study focuses on how such discretizations result in errors in the eigenspace approximations computed by the algorithm. The computed eigenspace is then used to obtain approximations of the eigenvalue cluster. Bounds for the Hausdorff distance between the computed and exact eigenvalue clusters …
The Auxiliary Space Preconditioner For The De Rham Complex, Jay Gopalakrishnan, Martin Neumüller, Panayot S. Vassilevski
The Auxiliary Space Preconditioner For The De Rham Complex, Jay Gopalakrishnan, Martin Neumüller, Panayot S. Vassilevski
Portland Institute for Computational Science Publications
We generalize the construction and analysis of auxiliary space preconditioners to the n-dimensional finite element subcomplex of the de Rham complex. These preconditioners are based on a generalization of a decomposition of Sobolev space functions into a regular part and a potential. A discrete version is easily established using the tools of finite element exterior calculus. We then discuss the four-dimensional de Rham complex in detail. By identifying forms in four dimensions (4D) with simple proxies, form operations are written out in terms of familiar algebraic operations on matrices, vectors, and scalars. This provides the basis for our implementation of …
Filtered Subspace Iteration For Selfadjoint Operators, Jay Gopalakrishnan, Luka Grubišić, Jeffrey S. Ovall
Filtered Subspace Iteration For Selfadjoint Operators, Jay Gopalakrishnan, Luka Grubišić, Jeffrey S. Ovall
Portland Institute for Computational Science Publications
We consider the problem of computing a cluster of eigenvalues (and its associated eigenspace) of a (possibly unbounded) selfadjoint operator in a Hilbert space. A rational function of the operator is constructed such that the eigenspace of interest is its dominant eigenspace, and a subspace iteration procedure is used to approximate this eigenspace. The computed space is then used to obtain approximations of the eigenvalues of interest. An eigenvalue and eigenspace convergence analysis that considers both iteration error and dis- cretization error is provided. A realization of the proposed approach for a model second-order elliptic operator is based on a …
Computational Algorithms For Improved Representation Of The Model Error Covariance In Weak-Constraint 4d-Var, Jeremy A. Shaw
Computational Algorithms For Improved Representation Of The Model Error Covariance In Weak-Constraint 4d-Var, Jeremy A. Shaw
Dissertations and Theses
Four-dimensional variational data assimilation (4D-Var) provides an estimate to the state of a dynamical system through the minimization of a cost functional that measures the distance to a prior state (background) estimate and observations over a time window. The analysis fit to each information input component is determined by the specification of the error covariance matrices in the data assimilation system (DAS). Weak-constraint 4D-Var (w4D-Var) provides a theoretical framework to account for modeling errors in the analysis scheme. In addition to the specification of the background error covariance matrix, the w4D-Var formulation requires information on the model error statistics and …
Discrete Stability Of Dpg Methods, Ammar Harb
Discrete Stability Of Dpg Methods, Ammar Harb
Dissertations and Theses
This dissertation presents a duality theorem of the Aubin-Nitsche type for discontinuous Petrov Galerkin (DPG) methods. This explains the numerically observed higher convergence rates in weaker norms. Considering the specific example of the mild-weak (or primal) DPG method for the Laplace equation, two further results are obtained. First, for triangular meshes, the DPG method continues to be solvable even when the test space degree is reduced, provided it is odd. Second, a non-conforming method of analysis is developed to explain the numerically observed convergence rates for a test space of reduced degree. Finally, for rectangular meshes, the test space is …
A Tent Pitching Scheme Motivated By Friedrichs Theory, Jay Gopalakrishnan, Peter Monk, Paulina Sepulveda
A Tent Pitching Scheme Motivated By Friedrichs Theory, Jay Gopalakrishnan, Peter Monk, Paulina Sepulveda
Mathematics and Statistics Faculty Publications and Presentations
Certain Friedrichs systems can be posed on Hilbert spaces normed with a graph norm. Functions in such spaces arising from advective problems are found to have traces with a weak continuity property at points where the inflow and outflow boundaries meet. Motivated by this continuity property, an explicit space-time finite element scheme of the tent pitching type, with spaces that conform to the continuity property, is designed. Numerical results for a model one-dimensional wave propagation problem are presented.
Polynomial Extension Operators. Part Ii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl
Polynomial Extension Operators. Part Ii, Leszek Demkowicz, Jay Gopalakrishnan, Joachim Schöberl
Mathematics and Statistics Faculty Publications and Presentations
Consider the tangential trace of a vector polynomial on the surface of a tetrahedron. We construct an extension operator that extends such a trace function into a polynomial on the tetrahedron. This operator can be continuously extended to the trace space of H(curl ). Furthermore, it satisfies a commutativity property with an extension operator we constructed in Part I of this series. Such extensions are a fundamental ingredient of high order finite element analysis.