Numerical Analysis and Computation Commons^™

Solution Of Preconditioned Nonsymmetric Saddle Point Systems Through Modified Conjugate Gradient Iteration, Samson Ayo

Dissertations

In this dissertation, we present an iterative method (Preconditioned Nonsymmetric Saddle Point Conjugate Gradient) for simultaneously solving forward ($A{\bf x}={\bf b}$) and adjoint ($A^T{\bf y}={\bf g}$) linear systems. Our approach involves constructing an augmented nonsymmetric saddle point matrix that has a real positive spectrum and developing a conjugate gradient-like iteration for this matrix. We investigate the use of Schur Complement preconditioners with block-diagonal factorization computed by an incomplete QR factorization of $A$ to speed up the convergence of our method and compare the results to the preconditioned generalized least squares residual (GLSQR) and quasi-minimal residual (QMR) methods. We develop quadrature …

The Numerical Method For Finding All The Zeros Of A Function F (X) On An Interval [A,B], Franissa Simon

Honors Theses

In this study, we develop a simple mathematical method for finding all the roots of a function on a specified interval. Existing classical numerical methods, including the Bisection Method, Secant Method, and Newton Method, cannot find all the zeros of a function f (x) on an interval [a,b]. Popular numerical root solvers like Matlab’s ‘fzero’, Maple’s ‘fsolve’, and SageMath’s ‘find_root’ typically yield only a single zero. This thesis explores a proposed interval computation bisection method, a systematic approach based on the traditional Bisection Method and interval computation. Unlike traditional bisection, which relies on the intermediate value theorem, this approach uses …

Sliding Window Method For Simulating Action Potentials In Axons, Hayden Reed

Honors Theses

ABSTRACT Hodgkin and Huxley’s nonlinear partial differential equations model the excitation and propagation of action potentials in neurons, and there have been numerous attempts at finding the best numerical solution method. This thesis proposes a novel approach to solving these equations: the Sliding Window method, in which a fixed sub-interval is found through capturing the signal’s head and tail. The system is then solved on the sub-interval instead of the entire interval. Using the Sliding Window technique also involves implementing the backward and forward Euler methods and the finite difference method. It will be demonstrated that, in utilizing the Sliding …

A Spectroscopic Survey Of Atmospheres Of Super-Earths, Luke Brust

Honors Theses

Recent research has made it possible to use spectroscopy to analyze the composition of the atmospheres of exoplanets, planets that orbit other stars. Most projects thus far have focused on the atmospheres of gas giants, as it is less challenging to observe them with available equipment. This project studies the atmospheres of four Super- Earths, planets that have a mass greater than the Earth but smaller than Neptune. This is accomplished by processing the raw spectroscopic data from the Hubble Space Telescope and modeling the atmosphere using the program 𝝉-Rex3. This survey found clear results from two of the selected …

Delta-Shaped Approximation Based Homotopy Analysis Method For Nonlinear Poisson-Type Partial Differential Equations, Cyril Ocloo

Dissertations

This research aims to solve nonlinear Poisson-type partial differential equations (PDEs) by the approach of the homotopy analysis method (HAM) incorporated with approximate particular solutions (APS) using Delta-shaped basis (DSB) approximations.

With the inclusion of the h auxiliary parameters, we tackle nonlinear problems by studying the mathematical characteristics of the h curve. This is to ensure the numerical convergence of the HAM.

In the solution process, we use the homotopy analysis method to convert a nonlinear PDE into linear inhomogeneous PDEs, which are solved using the method of approximate particular solutions with DSB.

A proper value of the h is …

Scalable Solution Of Time-Dependent Pdes Through Component-Wise Exponential Integrator, Chelsea Drum

Dissertations

Exponential integrators, such as exponential Runge-Kutta or Rosenbrock methods, are designed specifically for the time integration of stiff systems of ordinary differential equations (ODEs) and allow the use of larger time steps than other general-purpose ODE solvers. However, these methods rely on computing matrix function-vector products that are traditionally computed using a Krylov projection, such as Lanczos or Arnoldi iteration, that involves substantial computational expense at high spatial resolution. Krylov Subspace Spectral (KSS) methods' frequency-dependent approach, designed to circumvent stiffness in linear problems, computes these products with greater scalability. We propose the combination of such KSS methods with exponential integrators …

A Tale Of Two Diagonalizations: Methods To Diagonalize A 1-D Piecewise Constant Indefinite Schrödinger Operator, Sarah Wright

Dissertations

We present two numerical methods for computing the solution of a partial differential equation (PDE) for modeling acoustic pressure, known as an extra-wide angle parabolic equation, that features the square root of a differential operator. The differential operator is the negative of an indefinite Schrödinger operator with a piecewise constant potential. This work primarily deals with the 3-piece case; however, a generalization is made to the case of an arbitrary number of pieces. In the first method, the Rayleigh-Secant Method, through restriction to a judiciously chosen lower-dimensional subspace, approximate eigenfunctions are used to obtain estimates for the eigenvalues of the …

Solving The Cable Equation, A Second-Order Time Dependent Pde For Non-Ideal Cables With Action Potentials In The Mammalian Brain Using Kss Methods, Nirmohi Charbe

Master's Theses

In this thesis we shall perform the comparisons of a Krylov Subspace Spectral method with Forward Euler, Backward Euler and Crank-Nicolson to solve the Cable Equation. The Cable Equation measures action potentials in axons in a mammalian brain treated as an ideal cable in the first part of the study. We shall subject this problem to the further assumption of a non-ideal cable. Assume a non-uniform cross section area along the longitudinal axis. At the present time, the effects of torsion, curvature and material capacitance are ignored. There is particular interest to generalize the application of the PDEs including and …

Spectral Multistep Methods For The Scalable Simulation Of Time-Dependent Phenomena, Bailey Rester

Dissertations

Krylov subspace spectral (KSS) methods are high-order accurate, one-step explicit time-stepping methods for partial differential equations (PDEs) that also possess stability characteristic of implicit methods. Unlike other time-stepping approaches, KSS methods compute each Fourier coefficient of the solution from an individualized approximation of the solution operator of the PDE, using techniques developed by Golub and Meurant for approximating bilinear forms involving matrix functions. As a result, KSS methods scale effectively to higher spatial resolution.

This dissertation will present spectral multistep methods, explicit and implicit, designed through the combination of KSS methods and Adams methods. This combination allows spectral multistep methods …

Multi-Valued Solutions For The Equation Of Motion, Darcy-Jordan Model, As A Cauchy Problem: A Shocking Event, Chandler Shimp

Master's Theses

Shocks are physical phenomenon that occur quite often around us. In this thesis we examine the occurrence of shocks in finite amplitude acoustic waves from a numerical perspective. These waves, or jump discontinuities, yield ill-behaved solutions when solved numerically. This study takes on the challenge of finding both single- and multi-valued solutions.

The previously unsolved problem in this study is the representation of the Equation of Motion (EoM) in the form of the Darcy-Jordan model (DJM) and expressed as a dimensionless IVP Cauchy problem. Prior attempts to solve have resulted only in implicit solutions or explicit solutions with certain initial …

Ensemble Data Fitting For Bathymetric Models Informed By Nominal Data, Samantha Zambo

Dissertations

Due to the difficulty and expense of collecting bathymetric data, modeling is the primary tool to produce detailed maps of the ocean floor. Current modeling practices typically utilize only one interpolator; the industry standard is splines-in-tension.

In this dissertation we introduce a new nominal-informed ensemble interpolator designed to improve modeling accuracy in regions of sparse data. The method is guided by a priori domain knowledge provided by artificially intelligent classifiers. We recast such geomorphological classifications, such as ‘seamount’ or ‘ridge’, as nominal data which we utilize as foundational shapes in an expanded ordinary least squares regression-based algorithm. To our knowledge …

A Modified Preconditioned Conjugate Gradient Method For Approximating The Scattering Amplitude, Samson Ayo

Master's Theses

In this thesis, we look at an iterative method for approximating the scattering amplitude that involves solving two linear systems: a forward system Ax=b and an adjoint system A^Ty=g. Once these two systems are solved, the scattering amplitude, defined by g^Tx=y^Tb is easily obtained.

We derive a conjugate gradient-like iteration for a nonsymmetric saddle point matrix that is constructed to have a real positive spectrum. We investigate the use of Schur Complement preconditioners with block-diagonal factorization to speed up the convergence of our method and compare …

A Component-Wise Approach To Smooth Extension Embedding Methods, Vivian Montiforte

Dissertations

Krylov Subspace Spectral (KSS) Methods have demonstrated to be highly scalable methods for PDEs. However, a current limitation of these methods is the requirement of a rectangular or box-shaped domain. Smooth Extension Embedding Methods (SEEM) use fictitious domain methods to extend a general domain to a simple, rectangular or box-shaped domain. This dissertation describes how these methods can be combined to extend the applicability of KSS methods, while also providing a component-wise approach for solving the systems of equations produced with SEEM.

Efficient Denoising Of High Resolution Color Digital Images Utilizing Krylov Subspace Spectral Methods, Eva Lynn Greenman

Dissertations

The solution to a parabolic nonlinear diffusion equation using a Krylov Subspace Spectral method is applied to high resolution color digital images with parallel processing for efficient denoising. The evolution of digital image technology, processing power, and numerical methods must evolve to increase efficiency in order to meet current usage requirements. Much work has been done to perfect the edge detector in Perona-Malik equation variants, while minimizing the effects of artifacts. It is demonstrated that this implementation of a regularized partial differential equation model controls backward diffusion, achieves strong denoising, and minimizes blurring and other ancillary effects. By adaptively tuning …

Stability Analysis Of Krylov Subspace Spectral Methods For The 1-D Wave Equation In Inhomogeneous Media, Bailey Rester

Master's Theses

Krylov subspace spectral (KSS) methods are high-order accurate, explicit time-stepping methods for partial differential equations (PDEs) that also possess the stability characteristic of implicit methods. Unlike other time-stepping approaches, KSS methods compute each Fourier coefficient of the solution from an individualized approximation of the solution operator of the PDE. As a result, KSS methods scale effectively to higher spatial resolution. This thesis will present a stability analysis of a first-order KSS method applied to the wave equation in inhomogeneous media.

Diagonalization Of 1-D Schrodinger Operators With Piecewise Constant Potentials, Sarah Wright

Master's Theses

In today's world our lives are very layered. My research is meant to adapt current inefficient numerical methods to more accurately model the complex situations we encounter. This project focuses on a specific equation that is used to model sound speed in the ocean. As depth increases, the sound speed changes. This means the variable related to the sound speed is not constant. We will modify this variable so that it is piecewise constant. The specific operator in this equation also makes current time-stepping methods not practical. The method used here will apply an eigenfunction expansion technique used in previous …

Variable Compact Multi-Point Upscaling Schemes For Anisotropic Diffusion Problems In Three-Dimensions, James Quinlan

Dissertations

Simulation is a useful tool to mitigate risk and uncertainty in subsurface flow models that contain geometrically complex features and in which the permeability field is highly heterogeneous. However, due to the level of detail in the underlying geocellular description, an upscaling procedure is needed to generate a coarsened model that is computationally feasible to perform simulations. These procedures require additional attention when coefficients in the system exhibit full-tensor anisotropy due to heterogeneity or not aligned with the computational grid. In this thesis, we generalize a multi-point finite volume scheme in several ways and benchmark it against the industry-standard routines. …

An Adaptive Approach To Gibbs’ Phenomenon, Jannatul Ferdous Chhoa

Master's Theses

Gibbs’ Phenomenon, an unusual behavior of functions with sharp jumps, is encountered while applying the Fourier Transform on them. The resulting reconstructions have high frequency oscillations near the jumps making the reconstructions far from being accurate. To get rid of the unwanted oscillations, we used the Lanczos sigma factor to adjust the Fourier series and we came across three cases. Out of the three, two of them failed to give us the right reconstructions because either it was removing the oscillations partially but not entirely or it was completely removing them but smoothing out the jumps a little too much. …

A Time Integration Method Of Approximate Particular Solutions For Nonlinear Ordinary Differential Equations, Cyril Ocloo

Master's Theses

We consider a time-dependent method which is coupled with the method of approximate particular solutions (MAPS) of Delta-shaped basis functions and the method of fundamental solutions (MFS) to solve nonlinear ordinary differential equations. Firstly, we convert a nonlinear problem into a sequence of time-dependent non-homogeneous boundary value problems through a fictitious time integration method. The superposition principle is applied to split the numerical solution at each time step into an approximate particular solution and a homogeneous solution. Delta-shaped basis functions are used to provide an approximation of the source function at each time step. The purpose of this is to …

Automatic Numerical Methods For Enhancement Of Blurred Text-Images Via Optimization And Nonlinear Diffusion, Aaditya Kharel

Honors Theses

In this paper, we propose an automatic numerical method for solving a nonlinear partialdifferential- equation (PDE) based image-processing model. The Perona-Malik diffusion equation (PME) accounts for both forward and backward diffusion regimes so as to perform simultaneous denoising and deblurring depending on the value of the gradient. One of the limitations of this equation is that a large value of the gradient for backward diffusion can lead to singularity formation or staircasing. Guidotti-Kim-Lambers (GKL) came up with a bound for backward diffusion to prevent staircasing, where the backward diffusion is only limited to a specific range beyond which backward diffusion …

Multi-Point Flux Approximations Via The O-Method, Christen Leggett

Master's Theses

When an oil refining company is drilling for oil, much of the oil gets left behind after the first drilling. Enhanced oil recovery techniques can be used to recover more of that oil, but these methods are quite expensive. When a company is deciding if it is worth their time and money to use enhanced oil recovery methods, simulations can be used to model oil flow, showing the behavior and location of the oil. While methods do exist to model this flow, these methods are often very slow and inaccurate due to a large domain and wide variance in coefficients. …

Krylov Subspace Spectral Methods With Non-Homogenous Boundary Conditions, Abbie Hendley

Master's Theses

For this thesis, Krylov Subspace Spectral (KSS) methods, developed by Dr. James Lambers, will be used to solve a one-dimensional, heat equation with non-homogenous boundary conditions. While current methods such as Finite Difference are able to carry out these computations efficiently, their accuracy and scalability can be improved. We will solve the heat equation in one-dimension with two cases to observe the behaviors of the errors using KSS methods. The first case will implement KSS methods with trigonometric initial conditions, then another case where the initial conditions are polynomial functions. We will also look at both the time-independent and time-dependent …

Enhancement Of Krylov Subspace Spectral Methods Through The Use Of The Residual, Haley Dozier

Dissertations

Depending on the type of equation, finding the solution of a time-dependent partial differential equation can be quite challenging. Although modern time-stepping methods for solving these equations have become more accurate for a small number of grid points, in a lot of cases the scalability of those methods leaves much to be desired. That is, unless the timestep is chosen to be sufficiently small, the computed solutions might exhibit unreasonable behavior with large input sizes. Therefore, to improve accuracy as the number of grid points increases, the time-steps must be chosen to be even smaller to reach a reasonable solution. …

Comparative Error Analysis Of The Black-Scholes Equation, Chuan Chen

Honors Theses

Finance is a rapidly growing area in our banking world today. With this ever-increasing development come more complex derivative products than simple buy-and-sell trades. Financial derivatives such as futures and options have been developed stemming from the traditional stock, bond, currency, and commodity markets. Consequently, the need for more sophisticated mathematical modeling is also rising. The Black-Scholes equation is a partial differential equation that determines the price of a financial option under the Black-Scholes model. The idea behind the equation is that there is a perfect and risk-free way for one to hedge the options by buying and selling the …

Adaptive Meshfree Methods For Partial Differential Equations, Jaeyoun Oh

Dissertations

There are many types of adaptive methods that have been developed with different algorithm schemes and definitions for solving Partial Differential Equations (PDE). Adaptive methods have been developed in mesh-based methods, and in recent years, they have been extended by using meshfree methods, such as the Radial Basis Function (RBF) collocation method and the Method of Fundamental Solutions (MFS). The purpose of this dissertation is to introduce an adaptive algorithm with a residual type of error estimator which has not been found in the literature for the adaptive MFS. Some modifications have been made in developing the algorithm schemes depending …

Rapid Generation Of Jacobi Matrices For Measures Modified By Rational Factors, Amber Sumner

Dissertations

Orthogonal polynomials are important throughout the fields of numerical analysis and numerical linear algebra. The Jacobi matrix J for a family of n orthogonal polynomials is an n x n tridiagonal symmetric matrix constructed from the recursion coefficients for the three-term recurrence satisfied by the family. Every family of polynomials orthogonal with respect to a measure on a real interval [a,b] satisfies such a recurrence. Given a measure that is modified by multiplying by a rational weight function r(t), an important problem is to compute the modified Jacobi matrix Jmod corresponding to the new measure from knowledge of J. There …

Automatic Construction Of Scalable Time-Stepping Methods For Stiff Pdes, Vivian Ashley Montiforte

Master's Theses

Krylov Subspace Spectral (KSS) Methods have been demonstrated to be highly scalable time-stepping methods for stiff nonlinear PDEs. However, ensuring this scalability requires analytic computation of frequency-dependent quadrature nodes from the coefficients of the spatial differential operator. This thesis describes how this process can be automated for various classes of differential operators to facilitate public-domain software implementation.

Radial Basis Function Differential Quadrature Method For The Numerical Solution Of Partial Differential Equations, Daniel Watson

Dissertations

In the numerical solution of partial differential equations (PDEs), there is a need for solving large scale problems. The Radial Basis Function Differential Quadrature (RBFDQ) method and local RBF-DQ method are applied for the solutions of boundary value problems in annular domains governed by the Poisson equation, inhomogeneous biharmonic equation, and the inhomogeneous Cauchy-Navier equations of elasticity. By choosing the collocation points properly, linear systems can be obtained so that the coefficient matrices have block circulant structures. The resulting systems can be efficiently solved using matrix decomposition algorithms (MDAs) and fast Fourier transforms (FFTs). For the local RBFDQ method, the …

Eignefunctions For Partial Differential Equations On Two-Dimensional Domains With Piecewise Constant Coefficients, Abdullah Muheel Momit Aurko

Master's Theses

In this thesis, we develop a highly accurate and efficient algorithm for computing the solution of a partial differential equation defined on a two-dimensional domain with discontinuous coefficients. An example of such a problem is for modeling the diffusion of heat energy in two space dimensions, in the case where the spatial domain represents a medium consisting of two different but homogeneous materials, with periodic boundary conditions.

Since diffusivity changes based on the material, it will be represented using a piecewise constant function, and this results in the formation of a complicated mathematical model. Such a model is impossible to …

Numerical Solution Of Partial Differential Equations Using Polynomial Particular Solutions, Thir R. Dangal

Dissertations

Polynomial particular solutions have been obtained for certain types of partial differential operators without convection terms. In this dissertation, a closed-form particular solution for more general partial differential operators with constant coefficients has been derived for polynomial basis functions. The newly derived particular solutions are further coupled with the method of particular solutions (MPS) for numerically solving a large class of elliptic partial differential equations. In contrast to the use of Chebyshev polynomial basis functions, the proposed approach is more flexible in selecting the collocation points inside the domain. Polynomial basis functions are well-known for yielding ill-conditioned systems when their …