Final Doctoral Defenses

Date: Thursday, May 23, 2024

Time: 1:00- 2:30 pm


Advisor(s): Dr. Yanzhi Zhang

Title: Nonlocal Modeling and Computation: Numerical Methods and Data-Driven Approaches

Abstract: In recent decades, nonlocal models have garnered significant attention for describing complex systems. These models excel in capturing long-range interactions, multiscale properties, and memory effects more accurately than local models. Among these nonlocal operators, the fractional Laplacian $(-\Delta)^{\alpha/2}$ stands out as one of the most popular for describing anomalous phenomena in homogeneous media, and its variable-order generalization $(-\Delta)^{\alpha({\bf x})/2}$ for heterogeneous media. While nonlocal operators enable more accurate modeling, they also introduce new numerical challenges. Currently, numerical studies for these nonlocal models still remain limited, especially for the variable-order cases.

The objective of this dissertation is to develop accurate and efficient methods for solving nonlocal problems, with the application in seismic wave propagation in heterogeneous media. In the first part, we present a spectral method for the fractional Laplacian, which can be viewed as an exact discrete analogue of the fractional Laplacian. Furthermore, we introduce the first finite difference and pseudospectral methods for the variable-order fractional Laplacian, addressing the challenges brought by the spatial dependency in the power $\alpha({\bf x})$. In the second part, we propose a data-driven surrogate solver for time-dependent nonlocal problems. This surrogate solver utilizes the convolutional neural network (CNN) and works independently of the nonlocal PDE. Leveraging limited time-series observed data, it can learn the underlying dynamics and predict long-term future solutions accurately. Furthermore, the trained surrogate solver demonstrates robustness to initial conditions beyond the observed data.

The proposed finite difference and pseudospectral methods as well as the CNN surrogate solver offer accurate and efficient approaches for solving nonlocal problems. These methods could effectively address the numerical challenges arising from the nonlocality and spatial dependency inherent in fractional Laplacian, so as to advance the applications of nonlocal models in other fields.

Date: Wednesday, May 29, 2024

Time: 10:00 - 11:30 am


Advisor(s): Dr. Cihan Dagli

Title: Deep Learning Architecture Design for Nano-Satellite Image Super-Resolution

Abstract: Increasing threats to U.S. national security satellite constellations have resulted in an increased interest in constellation resilience and satellite redundancy. NanoSats have contributed to commercial, scientific and government applications in remote sensing, communications, navigation and research and have the potential to enhance satellite constellation resilience. However, the inherent size, weight and power limitations of NanoSats enforce constraints on imaging hardware; the small lenses and short focal lengths result in imagery with low spatial resolution, which limits the utility of CubeSat images for military planning purposes and national intelligence applications.

This research proposes a deep learning architecture capable of enhancing low-resolution NanoSat images to produce high-resolution imagery suitable for defense mission planning and intelligence operations. The ability to make more effective use of NanoSat technology expands the variety of imagery sources available to the Department of Defense (DoD) and contributes to improved satellite redundancy and constellation resilience.

Enhancing NanoSat imagery has potential benefit to a range of military and national intelligence missions by providing access to inexpensive, low-resolution, commercial imagery with the ability to improve the resolution for more detailed planning purposes. Military operations planners and geospatial engineers could benefit from an additional source of high-resolution imagery for determining terrain conditions, identifying potential adversary courses of action, and preparing for humanitarian assistance and disaster relief missions. Geospatial analysts would have access to an alternative intelligence source with varying revisit rates and alternative look angles to develop a more complete intelligence picture of their area of interest.