Numerical Analysis and Computation Commons^™

A Collection Of Fast Algorithms For Scalar And Vector-Valued Data On Irregular Domains: Spherical Harmonic Analysis, Divergence-Free/Curl-Free Radial Basis Functions, And Implicit Surface Reconstruction, Kathryn Primrose Drake

Boise State University Theses and Dissertations

This dissertation addresses problems that arise in a diverse group of fields including cosmology, electromagnetism, and graphic design. While these topics may seem disparate, they share a commonality in their need for fast and accurate algorithms which can handle large datasets collected on irregular domains. An important issue in cosmology is the calculation of the angular power spectrum of the cosmic microwave background (CMB) radiation. CMB photons offer a direct insight into the early stages of the universe's development and give the strongest evidence for the Big Bang theory to date. The Hierarchical Equal Area isoLatitude Pixelation (HEALPix) grid is …

Joint Inversion Of Gpr And Er Data, Diego Domenzain

Boise State University Theses and Dissertations

Imaging the subsurface can shed knowledge on important processes needed in a modern day human's life such as ground-water exploration, water resource monitoring, contaminant and hazard mitigation, geothermal energy exploration and carbon dioxide storage. As computing power expands, it is becoming ever more feasible to increase the physical complexity of Earth's exploration methods, and hence enhance our understanding of the subsurface.

We use non-invasive geophysical active source methods that rely on electromagnetic fields to probe the depths of the Earth. In particular, we use Ground penetrating radar (GPR) and Electrical resistivity (ER). Both methods are sensitive to electrical conductivity while …

Radial Basis Function Finite Difference Approximations Of The Laplace-Beltrami Operator, Sage Byron Shaw

Boise State University Theses and Dissertations

Partial differential equations (PDEs) are used throughout science and engineering for modeling various phenomena. Solutions to PDEs cannot generally be represented analytically, and therefore must be approximated using numerical techniques; this is especially true for geometrically complex domains. Radial basis function generated finite differences (RBF-FD) is a recently developed mesh-free method for numerically solving PDEs that is robust, accurate, computationally efficient, and geometrically flexible. In the past seven years, RBF-FD methods have been developed for solving PDEs on surfaces, which have applications in biology, chemistry, geophysics, and computer graphics. These methods are advantageous, as they are mesh-free, operate on arbitrary …

Joint Inversion Of Compact Operators, James Ford

Boise State University Theses and Dissertations

The first mention of joint inversion came in [22], where the authors used the singular value decomposition to determine the degree of ill-conditioning in inverse problems. The authors demonstrated in several examples that combining two models in a joint inversion, and effectively stacking discrete linear models, improved the conditioning of the problem. This thesis extends the notion of using the singular value decomposition to determine the conditioning of discrete joint inversion to using the singular value expansion to determine the well-posedness of joint linear operators. We focus on compact linear operators related to geophysical, electromagnetic subsurface imaging.

The operators are …

Numerical Computing With Functions On The Sphere And Disk, Heather Denise Wilber

Boise State University Theses and Dissertations

A new low rank approximation method for computing with functions in polar and spherical geometries is developed. By synthesizing a classic procedure known as the double Fourier sphere (DFS) method with a structure-preserving variant of Gaussian elimination, approximants to functions on the sphere and disk can be constructed that (1) preserve the bi-periodicity of the sphere, (2) are smooth over the poles of the sphere (and origin of the disk), (3) allow for the use of FFT-based algorithms, and (4) are near-optimal in their underlying discretizations. This method is used to develop a suite of fast, scalable algorithms that exploit …

Stability And Convergence For Nonlinear Partial Differential Equations, Oday Mohammed Waheeb

Boise State University Theses and Dissertations

If used cautiously, numerical methods can be powerful tools to produce solutions to partial differential equations with or without known analytic solutions. The resulting numerical solutions may, with luck, produce stable and accurate solutions to the problem in question, or may produce solutions with no resemblance to the problem in question at all. More such numerical computations give no hope of solving this troublesome feature and one needs to resort to investing time in a theoretical approach. This thesis is devoted not solely to computations, but also to a theoretical analysis of the numerical methods used to generate computationally the …