Keywords

Finite element method, mean curvature flow, energy penalized minimization algorithm, stochastic partial differential equations, MDI

Abstract

This thesis explores advancements in numerical methods, focusing on their applications to some classes of PDEs, Stochastic PDEs (SPDEs), and integral evaluations. The first part of this proposal provides algorithms for computing the mean curvature flow, including cases with topological changes. We propose energy-penalized and multilevel minimization algorithms. Benchmark problems with random initial conditions reveal that small changes in interaction length led to different solution patterns under topological changes. Our methods demonstrate robustness and efficiency, even with poor initial guesses, making them suitable for a wide range of scenarios.

The second part of this proposal introduces a new approach for time and spatial discretization of semi-linear SPDEs with multiplicative noise, under minimal assumptions on the drift and diffusion terms. Using a Milstein-based time discretization and an interpolation-based finite element method for space, we establish strong convergence of nearly order 1 and various stability results. New Hölder continuity estimates, and energy methods underpin the analysis, ensuring nearly optimal convergence in time and space. Numerical experiments validate the theoretical findings and demonstrate the effectiveness of the proposed scheme.

The final part of this proposal presents adjusted algorithms for the Multilevel Dimension Iteration (MDI) method. High-dimensional integration often suffers from the curse of dimensionality. The recent MDI method has demonstrated significant success in overcoming this issue for smooth, well-behaved functions with simple structures. However, its performance declines when applied to functions with non-smooth features, singularities, or complex structures. This limitation restricts its use in real-world applications. To address these challenges, we propose modifications to the MDI method. These adjustments aim to enhance its ability to handle complex functions and broaden its applicability to practical problems.

Completion Date

2025

Semester

Spring

Committee Chair

Yukun, Li

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Identifier

DP0029416

Document Type

Dissertation/Thesis

Campus Location

Orlando (Main) Campus

Download

Share

COinS