Volume 19

DOI: 10.1137/24S1693775

Authors

Bill Chen (Northwestern University)

Project Advisors

Maria Cameron (University of Maryland)

Abstract

Transition path theory is a widely used mathematical framework for quantifying rare noise-driven transitions between two metastable states $A$ and $B$ in systems modeled by stochastic differential equations. Central to this framework is the committor function, the solution to the stationary backward Kolmogorov equation with certain boundary conditions. The committor at a point $x$ is the probability that the process starting at $x$ will first reach metastable state $B$ rather than $A$. In this work, an operator learning approach proposed in Li et al. (2020) is investigated in the context of the committor problem for overdamped Langevin dynamics. The parameters chosen for the numerical tests are relevant to applications in chemical physics, including the temperature which controls the noise amplitude and the parameters which regulate the potential energy landscape. The solution operator to the committor problem with these parameters is represented as a Fourier Neural Operator. This approach yields a family of solutions allowing the user to evaluate the committor at a whole range of parameters values, in contrast to standard numerical methods. Accuracy and efficacy of the approach are demonstrated on a number of benchmark test problems

DOI: 10.1137/25S1821934

Authors

Chanyeong Seo (Georgia Institute of Technology)

Project Advisors

Mohammad Ghomi (Georgia Institute of Technology)

Abstract

This paper provides an algebraic, coordinate-free formula of caustic envelopes with multiple reflections under a two-dimensional convex curve. We define a closed-form envelope parameter, lamda, expressed in the support function invariants, derived by a householder operator and determinants. This simpler caustic calculation formula and algorithm is expected to accelerate visual rendering time in visual design industry. Moreover, the optics industry may get advantage in simulating light rays inside a convex curve optical cavity with any reflectivity.

DOI: 10.1137/24S1722067

Authors

Jamie Mahowald (Corresponding author – University of Texas at Austin)

Project Advisors

Tan Bui-Thanh (University of Texas at Austin)

Abstract

We investigate the generalization capabilities of In-Context Operator Networks (ICONs), a new class of operator networks that build on the principles of in-context learning, for higher-order partial differential equations. We extend previous work by expanding the type and scope of differential equations handled by the foundation model. We demonstrate that while processing complex inputs requires some new computational methods, the underlying machine learning techniques are largely consistent with simpler cases. Our implementation shows that although point-wise accuracy degrades for higher-order problems like the heat equation, the model retains qualitative accuracy in capturing solution dynamics and overall behavior. This demonstrates the model’s ability to extrapolate fundamental solution characteristics to problems outside its training regime.

DOI: 10.1137/25S1833023

Authors

Charles Wang (Corresponding author – Bridgewater-Raritan High School, Bridgewater, NJ)

Project Advisors

Nan Li (City University of New York - NYC College of Technology)

Abstract

Clustering time series is often computationally expensive because most methods require evaluating distances between all pairs of sequences, especially when using measures such as Dynamic Time Warping. This scaling limits the applicability of standard algorithms to large or high-resolution data sets. We propose an efficient framework that represents each time series through polynomial approximation and computes similarity using only the resulting local critical points. This yields compact, structure-preserving summaries that dramatically reduce the cost of pairwise comparisons while retaining essential temporal patterns. The resulting distance measure enhances the extraction of salient dynamical features, suppresses noise, and maintains clustering quality comparable to traditional approaches at a much lower computational cost. Numerical experiments on real data demonstrate that the method offers substantial improvements in efficiency without compromising interpretability or accuracy.

DOI: 10.1137/24S1699371

Authors

Simona Gagarova (Corresponding author – Sofia High School of Mathematics, Bulgaria)

Project Advisors

Japheth Varlack

Abstract

We consider a variant of the Cops and Robbers game on graphs where robbers damage visited vertices. In this version, multiple robbers aim to maximize vertex damage, while a single cop seeks to minimize it. For s robbers against one cop, we completely characterize the graphs on n ≥ 2s + 2 vertices with the smallest possible damage number when s > 2. Furthermore, we determine the s-robber damage number for various families of graphs. We also examine conditions under which the cop can save k vertices from damage against s robbers for k > 3. For triangle-free graphs, we establish that the cop can achieve this objective provided the graph’s maximum degree is at least (s/2) + k − 1. We further present upper and lower bounds on the minimum maximum degree for non-triangle-free graphs, under which the cop can ensure k vertices are saved for k ≤ 2 (s/2)

DOI: 10.1137/25S1790889

Authors

Qinyu Xu (Corresponding author – Duke Kunshan University)

Project Advisors

Dr. Veronica Ciocanel (Duke University)

Abstract

The transport and localization of RNA molecules, crucial for cellular function and development, involve a combination of diffusion and active transport mechanisms. Xenopus laevis oocytes provide a well- established model system for studying RNA transport due to their large size, pronounced spatial organization, and well-characterized RNA localization patterns. Fluorescence Recovery After Photobleaching (FRAP) is an experimental technique that is widely used to investigate the dynamics of molecular movement within cells by observing the recovery of fluorescence intensity in a photobleached region over time. Previous studies have primarily modeled RNA dynamics in FRAP experiments using purely diffusion mechanisms without explicitly incorporating active transport. To advance the understanding of RNA dynamics, we develop a reaction-diffusion-advection partial differential equation (PDE) model integrating both transport and diffusion mechanisms. We propose a pipeline for identifiability analysis to assess the model’s ability to uniquely determine parameter values from observed FRAP data. Based on profile likelihood analysis and reparametrization, we examine the relationship between non-identifiable parameters, which improves the robustness of parameter estimation. We find out that the identifiability of the four parameters of interest is not exactly the same in different regions of the cell. Specifically, transport velocity and diffusion coefficient are identifiable in all regions of the cell, and some combinations of binding rate and unbinding rate are found to be identifiable near the nucleus.

DOI: 10.1137/25S1764323

Authors

Alexander Weiss (Corresponding author – Rensselaer Polytechnic Institute), Paul Bruzzi (Rensselaer Polytechnic Institute)

Project Advisors

Fabian M. Faulstich (Rensselaer Polytechnic Institute)

Abstract

We present a comprehensive and pedagogical study of two core quantum algorithms in quantum linear algebra: the Hadamard test (HT) and quantum phase estimation (QPE). Building on the lecture notes on Quantum Algorithms for Scientific Computing [1] by Lin, we detail their mathematical formulations, circuit implementations, and convergence behavior. Both algorithms are benchmarked on simulators and IBM’s 127-qubit Eagle processor (ibm_rensselaer) using a simple eigenvalue problem. While QPE performs well in simulation, its hardware performance degrades beyond t = 3 ancilla qubits due to circuit depth and two-qubit gate error. In contrast, HT shows stable sampling-based convergence and greater resilience to hardware noise. These results provide a realistic baseline for implementing phase estimation on noisy intermediate-scale quantum (NISQ) era devices.

DOI: 10.1137/25S1768849

Authors

Brooke Tortotelli (Corresponding author – Monmouth University), Ella Gigante (University of Virginia)

Project Advisors

Dr. Joseph Coyle (Monmouth University)

Abstract

Compartmental modeling is frequently used as a model technique when the variables of interest can be grouped into distinct categories or compartments. This is typically the case when simulating the spread and behavior of infectious diseases. When the resulting differential system is coupled in a nonlinear way, numerical techniques are often the only way to approximate the true solutions. Here, we employ a power series approximation demonstrating a way to estimate the radius of convergence as part of an adaptive technique for long term approximations.

DOI: 10.1137/25S1771144

Authors

Yahir Ostos Jimenez (Corresponding author – Universidad Autónoma de Chihuahua)

Project Advisors

M.Sc. Ana Virginia Contreras Garcia (Universidad Autónoma de Chihuahua)

Abstract

In this article, a bijective mapping is presented between the set of natural numbers N and the set Σ∗ of strings generated by a finite alphabet Σ. This bijection assigns a unique position to each string, thereby inducing a shortlex (length-lexicographic) order through the function. The countability of the set Σ∗ is demonstrated, and the importance of this fact is emphasized in applications where an ordered representation of symbol combinations is required. The article also explains how to compute both the absolute position of a string and its relative position within fixed-length subsets, illustrated with concrete examples. These methods provide essential tools for data processing and storage optimization across different areas of application.

DOI: 10.1137/25S1766772

Authors

Casey Holman (Corresponding author – Lawrence University)

Project Advisors

Soumya Mohanty (University of Texas Rio Grande Valley)

Abstract

Gravitational Wave (GW) data is contaminated by many instrumental and environmental artifacts which adversely affect the search sensitivity for astrophysical signals. For the artifacts that cannot be eliminated in hardware, data processing must instead be used to identify and filter them out. Once such artifact is the Wandering Line (WL), a chirp with unpredictable frequency evolution that varies smoothly across a wide frequency band over a sustained period. Due to their large and unpredictable frequency range, WLs are difficult to filter using traditional line removal techniques without risking the removal of astrophysical signals. We propose a novel approach in which the WL is split into smaller segments and Particle Swarm Optimization (PSO) is used to fit a quadratic chirp model that is then subtracted from the data. We test this methodology on simulated WLs generated using random cubic spline frequency functions and added i.i.d. Gaussian white noise. Our results demonstrate that this approach successfully filters simulated WL artifacts with noise that has a standard deviation (s.d.) up to 5 times the WL amplitude. This is more than the noise s.d. to amplitude ratio of 4 we estimate from LIGO data, showing promise for suppressing such artifacts in real GW observations.

DOI: 10.1137/25S178417X

Authors

Ejmen Al-Ubejdij (Corresponding author – Hamad Bin Khalifa University), Elyas Al-Amri (Hamad Bin Khalifa University), Marwan Humaid (Hamad Bin Khalifa University), Hamad Aldous (Hamad Bin Khalifa University), Yousef Abu Dayeh (Carnegie Mellon University in Qatar)

Project Advisors

Dr. Samir Brahim Belhaouari (Hamad Bin Khalifa University)

Abstract

This paper presents a unified combinatorial framework for counting iterations in nested loop structures with generalized constraints. Traditional approaches limit themselves to strictly increasing index sequences, yielding binomial coefficient (n/k). We introduce the ghost element transformation, which enables efficient counting under non-strict inequalities by establishing a bijection between constrained sequences and selections from an augmented set. For the fundamental case with nonstrict inequalities, this transformation yields the formula (n+k−1/k), with extensions to gap constraints and variable offsets. We prove that sequences with minimum spacing d between consecutive indices have count (n−(k−1)(d−1)/k), and provide a general formula for variable offset constraints. Our computational analysis demonstrates that direct enumeration requires O(nk) time, while our formulas evaluate in O(1) time assuming constant-time arithmetic. For typical parameters (n = 100, k = 20), our approach reduces computation time from over 10 hours to under 1 millisecond. The methodology applies broadly to algorithm analysis, combinatorial optimization, and statistical sampling.