Volume 18

DOI: 10.1137/24S1710498

Authors

Yutong Bu (Corresponding author – Emory University)

Project Advisors

Julianne Chung (Emory University)

Abstract

Solving large-scale Bayesian inverse problems presents significant challenges, particularly when the exact (discretized) forward operator is unavailable. These challenges often arise in image processing tasks due to unknown defects in the forward process that may result in varying degrees of inexactness in the forward model. Moreover, for many large-scale problems, computing the square root or inverse of the prior covariance matrix is infeasible such as when the covariance kernel is defined on irregular grids or is accessible only through matrix-vector products. This paper introduces an efficient approach by developing an inexact generalized Golub-Kahan decomposition that can incorporate varying degrees of inexactness in the forward model to solve large-scale generalized Tikhonov regularized problems. Further, a hybrid iterative projection scheme is developed to automatically select Tikhonov regularization parameters. Numerical experiments on simulated tomography reconstructions demonstrate the stability and effectiveness of this novel hybrid approach.

DOI: 10.1137/24S1686000

Authors

Riley Link (Corresponding author – Creighton University) , Ethan Ebbighausen (University of North Carolina Chapel Hill)

Project Advisors

Alen Alexanderian (North Carolina State University), Joseph Hart (Sandia National Laboratories)

Abstract

We consider trajectory optimal control problems in which parameter uncertainty limits the applicability of control trajectories computed prior to travel. Hence, efficient trajectory adjustment is needed to ensure successful travel. However, it is often prohibitive or impossible to recalculate the optimal control in-transit due to strict time constraints or limited onboard computing resources. Thus, we propose a framework for quick and accurate trajectory approximations by using postoptimality sensitivity information. This allows the reduction of uncertain parameter space and an instantaneous approximation of the new optimal controller while using sensitivity data computed and stored pretransit.

DOI: 10.1137/24S1710917

Authors

Warin Watson (Corresponding author – Colorado Mesa University), Cash Cherry (Colorado School of Mines), Rachelle Lang (University of Wisconsin – Madison)

Project Advisors

Lars Ruthotto (Emory University)

Abstract

We demonstrate that automatic differentiation (AD), which has become commonly available in machine learning frameworks, is an efficient way to explore ideas that lead to algorithmic improvement in multi-scale affine image registration and affine super-resolution problems. In our first experiment on multi-scale registration, we implement an ODE predictor-corrector method involving a derivative with respect to the scale parameter and the Hessian of an image registration objective function, both of which would be difficult to compute without AD. Our findings indicate that exact Hessians are necessary for the method to provide any benefits over a traditional multi-scale method; a Gauss-Newton Hessian approximation fails to provide such benefits. In our second experiment, we implement a variable projected Gauss-Newton method for super-resolution and use AD to differentiate through the iteratively computed projection, a method previously unaddressed in the literature. We show that Jacobians obtained without differentiating through the projection are poor approximations to the true Jacobians of the variable projected forward map and explore the performance of other approximations in the problem of super-resolution. By addressing these problems, this work contributes to the application of AD in image registration and sets a precedent for further use of machine learning tools in this field.

DOI: 10.1137/24S1703392

Authors

Jimmy Vineyard (Corresponding author – University of Wisconsin - Madison), Peizhe Li (University of Wisconsin – Madison), Seungyeon Oh (University of Wisconsin – Madison), Jack Maloney (University of Wisconsin – Madison)

Project Advisors

Haley Colgate Kottler (University of Wisconsin - Madison), Amy Cochran (University of Wisconsin - Madison)

Abstract

The two-stage Markov task, widely-used for measuring model-based relative to model-free learning in humans, has faced skepticism regarding its effectiveness. We suggest a modification to better distinguish the two learning approaches. Our revised task incorporates an additional phase for learning local contingencies, mirroring a strategy from machine learning for separating model-free from model-based algorithms. We evaluated the effectiveness of our revised task through simulations, employing model-free and model-based strategies.

DOI: 10.1137/24S1717993

Authors

Justin Le (Corresponding author – Arizona State University)

Project Advisors

Sebastien Motsch (Arizona State University), Johannes Brust (Arizona State University)

Abstract

We provide an overview of the diffusion model as a method to generate new samples. Generative models have been recently adopted for tasks such as art generation (Stable Diffusion, Dall-E) and text generation (ChatGPT). Diffusion models in particular apply noise to sample data and then “reverse” this noising process to generate new samples. We will formally define these noising and denoising processes, then present algorithms to train and generate with a diffusion model. Afterward, we will explore a potential application of diffusion models in improving classifier performance on imbalanced

DOI: 10.1137/25S172954X

Authors

Aiden Rohrbach (Corresponding author – University of Rochester)

Project Advisors

Xuwen Chen (University of Rochester)

Abstract

We address a Landau temperature fluctuation problem by applying Bernstein polynomials and the Weierstrass approximation theorem. We consider the problem detailed in [X. Wang and Q.H. Liu, Temperature fluctuations for a finite system of classical spin-1/2 particles, Annals of Physics 322 (2007), 2168-2178] in which temperature fluctuations diverge as the temperature of a particle system tends to 0. An application of Bernstein polynomials allows us to solve this issue by proving that the m’th moment of the configurational temperature of a finite particle system tends to the temperature of the surroundings, implying the fluctuations in the temperature must tend to 0.

DOI: 10.1137/24S1692241

Authors

Xiaoxing Yu (Corresponding author – Pomona College)

Project Advisors

Christina J. Edholm (Scripps College)

Abstract

Mathematical models of language change promise to provide insight on how community-wide changes emerge from individual speaker interactions. We develop an agent-based model to test the validity of the gravity theory of linguistic diffusion. Our model uses an exemplar model and the functional-teleological explanation of chain shifts to build up a picture of how individual agents change their vowel systems, and studies the emergent patterns across language communities. We find that the predictions of the gravity model hold true in a subset of cases determined at least in part by patterns of interaction and the particularities of how vowels are produced and perceived. To be specific, the gravity model’s predictions will hold true in situations where two interacting communities have categories within each other’s tolerance thresholds.

Supplementary Materials #S169224R

DOI: 10.1137/24S1675503

Authors

Murphy John (Corresponding author – University of New Mexico), Samantha Mackley (University of Missouri)

Project Advisors

Dr. Mihhail Berezovski (Emory-Riddle Aeronautical University)

Abstract

Accurately predicting the demand for aviation is a complex problem that is essential for the success of private aviation providers. Factors such as seasonality and location affect the demand for private flights, but high-demand events and holidays introduce additional and often unexpected influences on these services. In European destinations, travel is heavily characterized by high-demand events and holidays. This research utilizes detailed characterization data centered in Europe containing over 1.1 million private flights between 2,016 locations from 2018 and 2019. Leveraging advanced data analysis techniques, this project constructs a spatio-temporal forecasting model to accurately predict the demand for private jet travel during high-demand events and holidays in European destinations. This research delivers valuable insights to providers of private aviation, enabling them to proactively respond to market fluctuations and optimize their operational strategies.

DOI: 10.1137/24S168324X

Authors

Alexander Le (Corresponding author – United States Air Force Academy)

Project Advisors

Dr. Maila Hallare (United States Air Force Academy), Maj. A.J. Wallerstein (United States Air Force Academy)

Abstract

Knowing baseplate kinematics during mortar fire allows for more accurate shelling during military operations. During mortar fire, ignition causes a rapid increase in barrel pressure, exerting a recoil force onto the baseplate and causing it to sink into the ground. This sinkage causes the barrel to move and fire shells slightly off trajectory. Accordingly, shells may miss their targets or hit unintentional ones. Using second-order linear differential equations, mortar kinematics was analyzed as an externally-forced spring-mass-damper system. Soil properties accounted for springiness and damping in this model, with recoil as the external forcing function. Simulations of the differential equations show that mortar movement was significantly minimized with larger baseplate area, and that launch angle did not significantly affect vertical sinkage. This project recommends that mortar baseplate area should be maximized to allow for more accurate artillery strikes.

DOI: 10.1137/24S1681069

Authors

Aidan Tillman (Corresponding author – Northeastern University), Yuzhi Liu (University of Michigan Ann Arbor)

Project Advisors

Stuart Brorson (Northeastern University)

Abstract

We define algorithms for computing the eigenvalue counting functions (ECFs) of the Laplacian with Dirichlet and Neumann boundary conditions for a variety of convex 2D and 3D domains using floating point precision in Python. These domains include the square, disk, ball, cube, and cylinder. We use these computed ECFs to validate Polya’s conjecture on these domains. The conjecture has not been proven in completeness on the ball, so this work presents numerical evidence to the positive. We also use these calculations to study the potential geometric information in the lower-order terms of the asymptotics by comparing ECFs between different shapes. We do this by subtracting the like terms in the asymptotics of the ECFs to draw out the lower-order information. In the 3D case, a difference can be seen in the lower-order asymptotics of a cube and cylinder with the same volume and surface area. In particular, the cube’s o(k2) term has a larger variance than the cylinder’s term, and in both cases, the term is mostly positive.

Supplementary Material

DOI: 10.1137/24S1694586

Authors

Jacob Vogelpohl (Corresponding author – Morehead State University)

Project Advisors

Rus May (Morehead State University)

Abstract

We make connections between the well-known gambler’s ruin problem and the enumeration of bounded Motzkin and Dyck paths. We start with a basic recurrence relation for a variation of the gambler’s ruin that permits ties, derive explicit formulas for the corresponding probability generating functions, explain the correspondence between this ruin variation and Motzkin paths, and obtain algebraic and rational expressions for the generating functions that enumerate height-bounded Motzkin and Dyck paths.

DOI: 10.1137/24S1637283

Authors

Lauren Eriksen (Corresponding author – Purdue University), Julian Bennett (North Carolina State University)

Project Advisors

Xingjie Li (University of North Carolina at Charlotte)

Abstract

In this paper, we develop a series of modified Susceptible-Infected- Removed (SIR) models to study the COVID-19 pandemic, assess the effectiveness of government protocols, and investigate the rationale behind them in Florida from March to June 2020. Our research involves the application of multiple regressions to pinpoint and identify Covid-19 lockdown periods, followed by a series of modifications to the SIR model and parameters fitting from the data. This involved creating new model approximations, such as the time-delayed SIR model and the reinfected SIR model, in order to take into account factors such as incubation and reinfection, and get the lowest error discrepancy possible for our infection rate. We were able to conclude that as we included more factors, our error rate became lower and our simulation results became more accurate compared to the actual data. We could also identify outlier metropolitan areas and draw certain conclusions on the level of government performance. We then moved on to find the correlations, if any, between infection rates and external factors. We looked at demographic and weather data to demonstrate whether correlations existed. We found that there are a few factors with high correlations, including graduate education and low temperatures.

DOI: 10.1137/24S1636472

Authors

Shuaicheng Tong (Corresponding author – University of California, Los Angeles), Linghai Liu (Brown University), Lisa Zhao (University of California, Berkeley)

Project Advisors

Samy Wu Fung (Colorado School of Mines)

Abstract

Recent efforts in applying implicit networks to solve inverse problems in imaging have achieved competitive or even superior results when compared to feedforward networks. These implicit networks require only constant memory during backpropagation, regardless of the number of layers. However, they are not necessarily easy to train because they require backpropagating through a fixed point, which leads to computationally expensive gradient calculations. In particular, this process requires solving a large linear system whose size is determined by the number of features in the fixed point iteration. To circumvent such calculations, Jacobian-free Backpropagation (JFB) was recently proposed. This paper explores its application in image deblurring problems. Our results show that JFB is comparable to the classical TV method, Plug-n-Play, and Deep Equilibrium at a reduced computational cost.

DOI: 10.1137/24S1666331

Authors

Haozhi Hong (Corresponding author – Queen’s University), Zoey Drassinower (Queen’s University), Ari Fialkov (Queen’s University), Daniel Forestell (Queen’s University), Emily Hunter (Queen’s University)

Project Advisors

Catherine Pfaff (Queen’s University)

Abstract

This paper focuses on how a soccer team can progress the ball up the field from the defensive third to the attacking third. We define a “safe configuration” of soccer players as one in which the ball possessor is part of a collection of players on the team that are pairwise connected by open passing lanes. We then study how teams can shift between these configurations in progressing the ball up the field. We provide some evidence of the benefits of “safe configurations.”

DOI: 10.1137/24S1691600

Authors

Elias Hohl (Corresponding author – ETH Zürich)

Project Advisors

Giovanni Felder (ETH Zürich)

Abstract

We find analytical solutions of the Laplace equation on a rectangular domain with Dirichlet-Neumann boundary condition type transition occurring on a side of the rectangle. The problem can be tackled either by cutting the domain in two parts and solving the obtained coupled Laplace equations using a Fourier series approach, or by creating a conformal mapping based on Christoffel-Schwarz and Mobius transformations between the original domain and another rectangular domain where the solution of the problem is less challenging. Apart from solving for the temperature field, we also compute a coefficient, dependent only on the geometry of the domain, from which the thermal resistance of a system with the given cross-section can be computed. We conduct an experiment to confirm the validity of our model. Using our results from the Christoffel-Schwarz method, we derive regularity results for the Fourier coefficients of the coupling method. We not only determine the order of decay, but also find formulas for explicit upper bounds of the absolute value of the coefficients. These results can be used to obtain maximum error bounds for a numerical computation considering only a finite number of Fourier coefficients. Furthermore, we prove that the solutions obtained via the coupling method converge to the actual solution.

DOI: 10.1137/23S1553479

Authors

Noah Dahle (Corresponding author – University of Tennessee, Knoxville), Kristina Wilson (University of Tennessee, Knoxville)

Project Advisors

Dr. Olivia Prosper (University of Tennessee, Knoxville)

Abstract

It is common for a mapping platform to relay information to the user about the traffic in the area. Traffic conditions have different degrees of traffic intensity that are represented by colors. We have studied Google Maps traffic data through a simulation to model these conditions. Our methodology explores the prospect of how different traffic condition colors will imply different travel times. This modeling was a means to develop an user interface to determine the fastest path between two points in a particular area of Knoxville, Tennessee. We accomplished this by using Dijkstra’s Algorithm. The final product is a program that accepts and corrects user input for day of week, time, and two points, and the output is the path the user should take, along with the travel time according to the Google Maps traffic conditions at that day and time. This model can be applied to different cities to study their traffic patterns or in city planning.

Supplementary Materials