Date of Award
2023
Document Type
Open Access Dissertation
Degree Name
Doctor of Philosophy in Mathematical Sciences (PhD)
Administrative Home Department
Department of Mathematical Sciences
Advisor 1
Cécile Piret
Committee Member 1
Allan Struthers
Committee Member 2
Argyrios Petras
Committee Member 3
Jiguang Sun
Abstract
The bulk of this dissertation is mainly composed of four chapters, which are organized as follows: Chapter 1 provides an introduction to the Radial Basis Functions (RBF) method by briefly outlining its historical developments and reviewing the RBF interpolation and the RBF-Finite Difference (FD) methodologies, and their advantages/disadvantages. Chapter 2 describes the Orthogonal Gradients (OGr) method and the Fast OGr method and how these can be used to compute differential operators restricted to hypersurfaces and space curves ($\Gamma$) embedded in R3. We will highlight a challenge of pairing Fast OGr with RBF-FD on nearly flat local clusters and how to overcome it. Numerical results of computing the surface (curve) Laplace-Beltrami operator $\Delta_{\Gamma}$, will be presented. Chapter 3 presents a novel local parameterization method with RBF-FD for computing arbitrary differential operators restricted to arbitrary manifolds. Numerical results will be presented for computing the Laplace-Beltrami operator of curves embedded in R2. Additionally, results will be presented for a stable, novel methodology of evolving curves via two examples. Chapter 4 is a study of how to couple the spatial differentiation matrices of the RBF-FD method with the Parareal Algorithm, a parallel-in-time integration method. The details of the Parareal algorithm will be reviewed along with potential theoretical speed-up for multi-processor computations. This will be followed by numerical results demonstrating the successful loose and tight couplings of the methodologies. For certain time-dependent partial differential equations the differentiation matrix produced by the RBF-FD method sometimes contains spurious eigenvalues which have a positive real part. We examine how the Parareal algorithm provides an opportunity for mitigating the effects of the spurious eigenvalues in the spectrum.
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 License.
Recommended Citation
Blazejewski, Jacob James, "NOVEL APPROACHES TO COMPUTE MANIFOLD OPERATORS WITH THE RADIAL BASIS FUNCTIONS METHOD", Open Access Dissertation, Michigan Technological University, 2023.