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Articles 1 - 14 of 14
Full-Text Articles in Numerical Analysis and Computation
Boundary Integral Equation Methods For Superhydrophobic Flow And Integrated Photonics, Kosuke Sugita
Boundary Integral Equation Methods For Superhydrophobic Flow And Integrated Photonics, Kosuke Sugita
Dissertations
This dissertation presents fast integral equation methods (FIEMs) for solving two important problems encountered in practical engineering applications.
The first problem involves the mixed boundary value problem in two-dimensional Stokes flow, which appears commonly in computational fluid mechanics. This problem is particularly relevant to the design of microfluidic devices, especially those involving superhydrophobic (SH) flows over surfaces made of composite solid materials with alternating solid portions, grooves, or air pockets, leading to enhanced slip.
The second problem addresses waveguide devices in two dimensions, governed by the Helmholtz equation with Dirichlet conditions imposed on the boundary. This problem serves as a …
Ensemble Data Fitting For Bathymetric Models Informed By Nominal Data, Samantha Zambo
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 Component-Wise Approach To Smooth Extension Embedding Methods, Vivian Montiforte
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
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 …
Variable Compact Multi-Point Upscaling Schemes For Anisotropic Diffusion Problems In Three-Dimensions, James Quinlan
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. …
Recover Data In Sparse Expansion Forms Modeled By Special Basis Functions, Abdulmtalb Mohamed Hussen
Recover Data In Sparse Expansion Forms Modeled By Special Basis Functions, Abdulmtalb Mohamed Hussen
Dissertations
In data analysis and signal processing, the recovery of structured functions (in terms of frequencies and coefficients) with respect to certain basis functions from the given sampling values is a fundamental problem. The original Prony method is the main tool to solve this problem, which requires the equispaced sampling values.
In this dissertation, we use the equispaced sampling values in the frequency domain after the short time Fourier transform in order to reconstruct some signal expansions, such as the exponential expansions and the cosine expansions. In particular, we consider the case that the phase of the cosine expansion is quadratic. …
Enhancement Of Krylov Subspace Spectral Methods Through The Use Of The Residual, Haley Dozier
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. …
Adaptive Meshfree Methods For Partial Differential Equations, Jaeyoun Oh
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
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 …
Radial Basis Function Differential Quadrature Method For The Numerical Solution Of Partial Differential Equations, Daniel Watson
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 …
Numerical Solution Of Partial Differential Equations Using Polynomial Particular Solutions, Thir R. Dangal
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 …
Fast Method Of Particular Solutions For Solving Partial Differential Equations, Anup Raja Lamichhane
Fast Method Of Particular Solutions For Solving Partial Differential Equations, Anup Raja Lamichhane
Dissertations
Method of particular solutions (MPS) has been implemented in many science and engineering problems but obtaining the closed-form particular solutions, the selection of the good shape parameter for various radial basis functions (RBFs) and simulation of the large-scale problems are some of the challenges which need to overcome. In this dissertation, we have used several techniques to overcome such challenges.
The closed-form particular solutions for the Matérn and Gaussian RBFs were not known yet. With the help of the symbolic computational tools, we have derived the closed-form particular solutions of the Matérn and Gaussian RBFs for the Laplace and biharmonic …
On The Selection Of A Good Shape Parameter For Rbf Approximation And Its Application For Solving Pdes, Lei-Hsin Kuo
On The Selection Of A Good Shape Parameter For Rbf Approximation And Its Application For Solving Pdes, Lei-Hsin Kuo
Dissertations
Meshless methods utilizing Radial Basis Functions~(RBFs) are a numerical method that require no mesh connections within the computational domain. They are useful for solving numerous real-world engineering problems. Over the past decades, after the 1970s, several RBFs have been developed and successfully applied to recover unknown functions and to solve Partial Differential Equations (PDEs).
However, some RBFs, such as Multiquadratic (MQ), Gaussian (GA), and Matern functions, contain a free variable, the shape parameter, c. Because c exerts a strong influence on the accuracy of numerical solutions, much effort has been devoted to developing methods for determining shape parameters which provide …
Monte Carlo Simulation Of Electron-Induced Air Fluorescence Utilizing Mobile Agents: A New Paradigm For Collaborative Scientific Simulation, Christopher Daniel Walker
Monte Carlo Simulation Of Electron-Induced Air Fluorescence Utilizing Mobile Agents: A New Paradigm For Collaborative Scientific Simulation, Christopher Daniel Walker
Dissertations
A new paradigm for utilization of mobile agents in a modular architecture for scientific simulation is demonstrated through a case study involving Monte Carlo simulation of low energy electron interactions with molecular nitrogen gas. Design and development of Monte Carlo simulations for physical systems of moderate complexity can present a seemingly overwhelming endeavor. The researcher must possess or otherwise develop a thorough understanding the physical system, create mathematical and computational models of the physical system’s components, and forge a simulation utilizing those models. While there is no single route between a collection of physical concepts and a Monte Carlo simulation …