http://hdl.handle.net/11122/13039 2026-04-14T02:12:31Z 2026-04-14T02:12:31Z Fochesatto, Stefano http://hdl.handle.net/11122/16333 2026-04-11T01:02:42Z 2024-11-01T00:00:00Z

Adaptive mesh refinement for variational inequalities Fochesatto, Stefano Variational inequalities play a pivotal role in a wide array of scientific and engineering applications. This project presents two techniques for adaptive mesh refinement (AMR) in the context of variational inequalities, with a specific focus on the classical obstacle problem. We propose two distinct AMR strategies: Variable Coefficient Elliptic Smoothing (VCES) and Unstructured Dilation Operator (UDO). VCES uses a nodal active set indicator function as the initial iterate to a time-dependent heat equation problem. Solving a single step of this problem has the effect of smoothing the indicator about the free boundary. We threshold this smoothed indicator function to identify elements near the free boundary. Key parameters such as timestep and threshold values significantly influence the efficacy of this method. The second strategy, UDO, focuses on the discrete identification of elements adjacent to the free boundary, employing a graph-based approach to mark neighboring elements for refinement. This technique resembles the dilation morphological operation in image processing, but tailored for unstructured meshes. We also examine the theory of variational inequalities, the convergence behavior of finite element solutions, and implementation in the Firedrake finite element library. Convergence analysis reveals that accurate free boundary estimation is pivotal for solver performance. Numerical experiments demonstrate the effectiveness of the proposed methods in dynamically enhancing mesh resolution around free boundaries, thereby improving the convergence rates and computational efficiency of variational inequality solvers. Our approach integrates seamlessly with existing Firedrake numerical solvers, and it is promising for solving more complex free boundary problems. Master's Project (M.S.) University of Alaska Fairbanks, 2024.

2024-11-01T00:00:00Z Spehlmann, Michael http://hdl.handle.net/11122/16323 2026-04-10T01:02:51Z 2025-05-01T00:00:00Z

Bayesian spacial process convolutions for geostatistical modelling Spehlmann, Michael The declining rate of new metal discoveries, coupled with surging demand for critical minerals, underscores the need for improved geostatistical tools in early-stage exploration. Traditional fre- quentist methods like kriging are ill-suited for target generation due to their reliance on dense data, stationarity assumptions, and tendency to smooth anomalies. Meanwhile, exploration workflows remain largely qualitative, relying on geological intuition rather than probabilistic frameworks. A Bayesian spatial process convolution (SPC) model offers a promising alternative, leveraging hierarchical Bayesian formulations to model spatial processes with uncertainty quantification. This paper presents a Bayesian SPC model designed for geostatistical modeling, particularly in early-stage mineral exploration. The model is evaluated through simulations that mimic real-world geological complexity, including categorical boundaries and spatially coherent processes. The SPC model demonstrates its ability to recover smooth spatial structures and quantify uncertainty. Master's Project (M.S.) University of Alaska Fairbanks, 2025.

2025-05-01T00:00:00Z Masterman, Everett http://hdl.handle.net/11122/16319 2026-04-10T01:02:26Z 2025-05-01T00:00:00Z

Comparison of student success between asynchronous online and in-person sections of calculus 1 using multiple statistical methods Masterman, Everett This observational study compares outcomes for students taking the in-person and asynchronous online sections of Calculus 1 at the University of Alaska Fairbanks. Propensity score covariate models, propensity score stratification, propensity score matching, and multiple logistic regression were used to predict student pass rates in the course and on the final exam after accounting for demographic variables, student engagement, and student preparation. Propensity score methods showed no difference between asynchronous and in-person students. Multiple logistic regression showed that students in the in-person modality performed better after accounting for covariates. Master's Project (M.S.) University of Alaska Fairbanks, 2025.

2025-05-01T00:00:00Z Sus, Olha http://hdl.handle.net/11122/16274 2025-11-21T01:02:17Z 2025-08-01T00:00:00Z

Optimal control and inverse problems for partial differential equations and variational inequalities Sus, Olha This dissertation addresses optimal control problems for nonlinear evolutionary variational inequalities involving Volterra-type operators and inverse problems for the Dirac system on finite metric graphs. The first part presents the historical background, novelty, and motivation behind the research studies. In the second part, we focus on solving the initial value problem for nonlinear evolutionary variational inequalities with Volterra-type operators, proving the existence of a unique solution using the Banach fixed-point theorem. The third part explores an optimal control problem for these inequalities, establishing the existence of a solution under specific assumptions on the given data. In the last part of the dissertation, we examine the inverse dynamic problem for the Dirac system on finite metric tree graphs, as well as a graph with a single cycle (a ring with two attached edges). First, we solve the forward problem for this system on general graphs using a novel dynamic algorithm and then address the inverse problem for the same system on finite metric tree graphs. We recover unknown data such as the topology (connectivity) of a tree, edge lengths, and matrix potential functions associated with each edge. This is achieved using the dynamic response operator as the inverse data and the leaf peeling method. We also determine the minimum time required to uniquely identify the unknown data. Finally, we demonstrate the solution to the inverse problem for the Dirac system on a ring with two attached edges, establishing the minimum time needed to uniquely determine the unknown parameters for this graph. Dissertation (Ph.D.) University of Alaska Fairbanks, 2025

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