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Full-Text Articles in Other Mathematics
Preconditioned Nesterov’S Accelerated Gradient Descent Method And Its Applications To Nonlinear Pde, Jea Hyun Park
Preconditioned Nesterov’S Accelerated Gradient Descent Method And Its Applications To Nonlinear Pde, Jea Hyun Park
Doctoral Dissertations
We develop a theoretical foundation for the application of Nesterov’s accelerated gradient descent method (AGD) to the approximation of solutions of a wide class of partial differential equations (PDEs). This is achieved by proving the existence of an invariant set and exponential convergence rates when its preconditioned version (PAGD) is applied to minimize locally Lipschitz smooth, strongly convex objective functionals. We introduce a second-order ordinary differential equation (ODE) with a preconditioner built-in and show that PAGD is an explicit time-discretization of this ODE, which requires a natural time step restriction for energy stability. At the continuous time level, we show …
Generalized Finite-Difference Time-Domain Schemes For Solving Nonlinear Schrödinger Equations, Frederick Ira Moxley Iii
Generalized Finite-Difference Time-Domain Schemes For Solving Nonlinear Schrödinger Equations, Frederick Ira Moxley Iii
Doctoral Dissertations
The nonlinear Schrödinger equation (NLSE) is one of the most widely applicable equations in physical science, and characterizes nonlinear dispersive waves, optics, water waves, and the dynamics of molecules. The NLSE satisfies many mathematical conservation laws. Moreover, due to the nonlinearity, the NLSE often requires a numerical solution, which also satisfies the conservation laws. Some of the more popular numerical methods for solving the NLSE include the finite difference, finite element, and spectral methods such as the pseudospectral, split-step with Fourier transform, and integrating factor coupled with a Fourier transform. With regard to the finite difference and finite element methods, …
Results In Lattices, Ortholattices, And Graphs, Jianning Su
Results In Lattices, Ortholattices, And Graphs, Jianning Su
Doctoral Dissertations
This dissertation contains two parts: lattice theory and graph theory. In the lattice theory part, we have two main subjects. First, the class of all distributive lattices is one of the most familiar classes of lattices. We introduce "π-versions" of five familiar equivalent conditions for distributivity by applying the various conditions to 3-element antichains only. We prove that they are inequivalent concepts, and characterize them via exclusion systems. A lattice L satisfies D0π, if a ✶ (b ✶ c) ≤ (a ✶ b) ✶ c for all 3-element antichains { a, b, c}. We consider …
Spatiotemporal Dynamics In A Lower Montane Tropical Rainforest, Robert Michael Lawton
Spatiotemporal Dynamics In A Lower Montane Tropical Rainforest, Robert Michael Lawton
Doctoral Dissertations
Disturbance in a forest’s canopy, whether caused by treefall, limbfall, landslide, or fire determines not only the distribution of well-lit patches at any given time, but also the ways in which the forest changes over time. In this dissertation, I use a 25 year record of treefall gap formation find a novel and highly patterned process of forest disturbance and regeneration, providing a local mechanism by examining the factors that influence the likelihood of treefall. I then develop a stochastic cellular automaton for disturbance and regeneration based on the analysis of this long term data set and illustrate the potential …