SIAM Mini-Symposium Session 2
Organizers
Ziheng Guo (University of Houston), Ming Zhong (University of Houston)
Mini-symposium Abstract:
By combining scientific computing methodologies with machine learning techniques, the new scientific machine learning methods have made significant progress in the science and engineering fields, especially for making scientific discoveries from data. We propose a mini-symposium featuring eight distinguished speakers who will explore the intersection of physics and machine learning, showcasing the latest advancements in this rapidly evolving field. Topics will encompass sparse recovery techniques that enhance data analysis, innovative AI applications in climate prediction, and the role of artificial intelligence in astrophysics. Additionally, we will delve into the use of Physics-Informed Neural Networks (PINNs) for solving partial differential equations, highlighting how these approaches are transforming computational modeling in science and engineering. Each speaker will present cutting-edge research that not only demonstrates the power of machine learning but also emphasizes its foundational ties to physical principles. This mini-symposium aims to foster interdisciplinary dialogue and inspire collaborative efforts in utilizing AI for tackling complex scientific challenges.
Presentation 1
Speaker: Lifan Wang (Texas A&M University)
Co-author: N/A
Title: AI Assistant Inversions in Astrophysical Radiation Transport and Applications to Cosmology
Abstract: Recent advancements in learning-based methods are reshaping image and data analysis, transitioning towards more data-driven approaches. While deep learning techniques have shown considerable promise in various Computer Vision domains, they often require extensive training data and lack performance guarantees. To address these limitations, there is increasing interest in integrating learning-based methods with traditional model-driven approaches, known as scientific machine learning. This hybrid strategy seeks to combine the strengths of both approaches, such as incorporating convolutional neural networks with structural constraints or enhancing classical solvers with data-driven techniques. However, significant theoretical and practical challenges remain in effectively integrating these methods and ensuring reliable performance.
Presentation 2
Speaker: Juntao Huang (Texas Tech University)
Co-author: N/A
Title: Hyperbolic machine learning moment closure models for the radiative transfer equation
Abstract: In this talk, we take a data-driven approach and apply machine learning to the moment closure problem for the radiative transfer equation. Instead of learning the unclosed high order moment, we propose to directly learn the gradient of the high order moment using neural networks, called the gradient-based moment closure. Moreover, we introduce two approaches to enforce the hyperbolicity of our gradient-based machine learning moment closures. A variety of benchmark tests, including the variable scattering problem, the Gaussian source problem and the two material problem, show both good accuracy and generalizability of our machine learning closure model.
Presentation 3
Speaker: Chunyang Liao (UCLA)
Co-author: N/A
Title: Cauchy Random Features for Operator Learning in Sobolev Space
Abstract: Operator learning, which is a well-established field involving learning operators that map one function to another, has become an important tool in scientific machine learning. In this talk, we present a Cauchy random feature method for learning operators between Sobolev spaces. Moreover, we provide an error analysis for our proposed method. Comprehensive numerical comparisons with kernel operator learning method and popular neural net approaches indicate several advantages of random feature based operator learning framework, such as simplicity, comparable test error, and fast implementation.
Presentation 4
Speaker: Dongwei Chen (Clemson University)
Co-author: N/A
Title: Discovering Climate Change via Optimal Transport
Abstract: Climate change is an essential topic in climate science, and the accessibility of accurate, high-resolution datasets in recent years has facilitated the extraction of more insights from big-data resources. Nonetheless, current research predominantly focuses on mean-value changes and largely overlooks changes in the probability distribution. In this study, a novel method called Wasserstein Stability Analysis (WSA) is developed to identify probability density function (PDF) changes, especially the extreme event shift and nonlinear physical value constraint variation in climate change. WSA is applied to the early 21st century and compared with traditional mean-value trend analysis. The results indicate that despite no significant trend, the equatorial eastern Pacific experienced a decline in hot extremes and an increase in cold extremes, indicating a La Niña-like temperature shift. Further analysis at two Arctic locations suggests sea ice severely restricts the hot extremes of surface air temperature. This impact is diminishing as the sea ice melts. By revealing PDF shifts, WSA emerges as a powerful tool to re-examine climate change dynamics, providing enhanced data-driven insights for understanding climate evolution.
Organizers
Kenneth Duru (University of Texas at El Paso), Thomas Hagstrom (Southern Methodist University)
Mini-symposium Abstract:
Robust and high order accurate numerical methods for numerical simulations of PDEs have increasingly become an appealing choice for several modelling applications, owing to their efficiency and scalable performance on modern supercomputers.
Ideally numerical methods should mimic the properties of the continuous system they approximate, but continuous systems have infinitely many invariants of which only a finite number can be preserved by any discretization. To minimize the effects of numerical artifacts from contaminating results of numerical simulations it is desirable that numerical methods preserve some important invariants present in the physical model. For a target modelling application this necessitates choosing a subset of the invariants, based on the particular modelling application, for the discrete model to preserve.
This mini-symposium will cover presentations on recent advances on robust and structure preserving high order numerical methods for PDEs. These will include SBP and DG methods, and mixed and compatible finite element methods.
Presentation 1
Speaker: Kieran Ricardo (Australian National University)
Co-author: David Lee (Bureau of Meteorology Australia), Kenneth Duru (University of Texas at El Paso)
Title: Thermodynamic consistency and structure-preservation in summation by parts methods for the moist compressible Euler equations
Abstract: We present a thermodynamically consistent and structure preserving formulation of the moist compressible Euler equations. When discretized with a summation by parts method, our spatial discretization satisfies the first and second laws of thermodynamics independent of the equation of state used, and additionally conserves tracer variance and energy. These properties are achieved by discretizing a skew symmetric form of the moist Euler equations, using entropy as a prognostic variable, and the summation-by-parts property of discrete operators. We experimentally verify our theoretical results through numerical simulations with a discontinuous Galerkin method.
Presentation 2
Speaker: Drew Anderson (Baylor University)
Co-author: Rob Kirby (Baylor University), Andreas Klöckner (University of Illinois Urbana-Champaign), Marius Mitrea (Baylor University)
Title: An exact nonlocal domain truncation boundary condition for time-harmonic electromagnetic scattering
Abstract: Maxwell’s equations describe the movement of electromagnetic waves and are well-known throughout the scientific world. We seek to solve the exterior scattering problem in 3 dimensions by truncating the domain and numerically solving over a finite domain. Usually, obtaining a numerical solution is achieved using transmission boundary conditions, PML, or the DtN map, but we instead apply an exact but nonlocal boundary condition that has previously been proven to work for the Helmholtz equation. Since this involves using layer potentials and the dyadic Green’s function, we computationally utilize both Pytential and Firedrake to obtain accurate numerical results, converging at the expected rate as our mesh is refined. To prove the quasi-optimal convergence of our solution, we use the theory of collective compactness after converting our variational formulation to a second-kind integral equation.
Organizers
Son-Young Yi (University of Texas at El Paso), Sanghyun Lee (Florida State University), Maria Vasilyeva (Texas A&M-Corpus Christi)
Mini-symposium Abstract:
This mini-symposium aims to bring together leading experts and researchers to discuss the latest advancements in computational methods for subsurface modeling and foster interdisciplinary collaboration. This session will explore innovative numerical methods and multiscale modeling techniques, leveraging either classical or advanced machine-learning approaches and high-performance computing to address challenges and enhance the accuracy and efficiency of complex subsurface simulations. Emphasis will be placed on their applications to modeling critical multi-physics processes in the subsurface, such as groundwater contamination, hydrocarbon reservoir management, and subsurface energy systems, including geothermal or carbon sequestration.
Presentation 1
Speaker: Maria Vasilyeva (Texas A&M University - Corpus Christi)
Co-author: N/A
Title: Multiscale methods for multi-continuum problems in fractured porous media
Abstract: We consider the coupled system of partial differential equations that describe complex processes in fractured porous media. In the flow processes, the permeability of significant differences between each continuum and highly heterogeneous itself. Multiscale multi-continuum approaches are used to describe such complex phenomena and construct an accurate model. To solve the resulting system numerically, we construct a fine grid that resolves heterogeneity and lower dimensional fractures on the grid level and uses the finite element method or finite volume method for approximation by space with an implicit time scheme. The resulting discrete system is computationally expensive to solve due to fine-scale resolution, leading to large system with many unknowns.
To construct an efficient and accurate coarse-scale solver, we introduce two approaches: (1) spectral multiscale space introduced in the Generalized Multiscale Finite Element Method and (2) nonlocal multi-continua approach with carefully designed macroscale parameters. We use a Galerkin coupling to form a coarse-scale system in both approaches. We discuss the extension of the method to include a poroelastic effect, nonlinear flow models, and multiphase flow. Several computational aspects are discussed, including decoupling techniques for multi-continua and multi-physics problems and two-grid preconditioning using a coarse-scale solver.
Presentation 2
Speaker: Buzheng Shan (Texas A&M University)
Co-author: Yalchin Efendiev (Texas A&M University)
Title: Multi-continuum splitting scheme for multiscale flow problems
Abstract: In this work, we propose a multi-continuum splitting scheme for multiscale problems, illustrated through a parabolic equation with high-contrast coefficients. With the framework of multi-continuum homogenization, the solution space is decomposed to separate fast and slow dynamics, allowing for partially explicit time discretization schemes that reduce computational cost. The stability conditions we derive are contrast-independent, assuming appropriate choice of continua. We also discuss methods for optimal decomposition of the solution space. It can relax the stability conditions and enhance computational efficiency. Numerical results demonstrate the accuracy and efficiency of the proposed approach across various coefficient fields and continua.
Presentation 3
Speaker: Nana Mbroh (Texas A&M University - Corpus Christi)
Co-author: N/A
Title: Richardson Extrapolation for Multiscale Approximation of Parabolic Problem in Heterogenous Domain
Abstract: In this talk, we consider a linear parabolic problem. We employ the theta scheme for the time approximation and the generalized multiscale finite element scheme for the spatial approximation. It is well known that the generalized multiscale finite element method yields good results for problems in heterogeneous domains. We aim to understand the effect of Richardson extrapolation on the results after its application. Rigorous analysis for both the time and space discretization is performed, and extensive numerical experiments are conducted to confirm any theoretical findings.
Presentation 4
Speaker: Ludmil Zikatanov (Pennsylvania State University)
Co-author: Yuwen Li (Zhejiang University), Cheng Zuo (Pennsylvania State University)
Title: Reduced Krylov Basis Methods for Parametric Partial Differential Equations
Abstract: We present a user-friendly reduced basis method for solving a family of parametric PDEs by preconditioned Krylov subspace methods including the CG method, GMRes, and BiCGStab. The proposed methods use a preconditioned Krylov subspace method for a high-fidelity discretization of one parameter instance to generate orthogonal basis vectors of the reduced basis subspace. Then the large-scale discrete parameter-dependent problems are approximately solved in the low-dimensional Krylov subspace. As shown in the theory and experiments, only a small number of Krylov subspace iterations are needed to simultaneously generate approximate solutions of a family of high-fidelity and large-scale systems in the reduced basis subspace. This reduces the computational cost dramatically because (1) to construct the reduced basis vectors, we only solve one large-scale problem on the high-fidelity level; and (2) the problems for any value in the parameter set have much smaller dimensions they are restricted to the subspace defined during the Krylov iterations.
Organizers
Collin Victor (Texas A&M University), Ning Ning (Texas A&M University)
Mini-symposium Abstract:
Data assimilation refers to classes of algorithms that integrate observational data with the underlying physical model in order to better predict dynamical systems. This minisymposium aims to explore recent theoretical and practical advancements in data assimilation, focusing on continuous methods for dynamical systems, Bayesian methods for uncertainty management, and machine learning techniques to enhance assimilation accuracy using large amounts of observational data. Key topics include Bayesian methods, ensemble techniques, variational approaches, and methods of continuous data assimilation. Our goals are to foster collaboration and discussion among researchers from diverse methodological backgrounds and to connect practical applications with theoretical insights. Ultimately, we aim to bridge the gap between practice and theory, enhancing our understanding of complex systems.
Presentation 1
Speaker: Amanda Rowley (University of Nebraska-Lincoln)
Co-author: Adam Larios (University of Nebraska-Lincoln), Trenton Franz (University of Nebraska-Lincoln)
Title: Modeling Groundwater Flow with Continuous Data Assimilation
Abstract: Continuous data assimilation (or AOT data assimilation), which was introduced by Azouani, Olsen, and Titi in 2014, is a computationally-efficient algorithm that has been shown analytically and computationally to recover the true solution for a wide variety of regimes exponentially fast in time, in addition to being robust with respect to noisy data, stochastic forcing, and errors in parameters. In this talk, we will apply continuous data assimilation to the Richards equation, a nonlinear system that models fluid flow in unsaturated soil. We will examine convergence of the AOT approach numerically and analytically, assuming unknown initial data and sparse-in-time-and-space observations.
Presentation 2
Speaker: Collin Victor (Texas A&M University)
Co-author: Ning Ning (Texas A&M University)
Title: Assessing the Ensemble Kalman Filter and AOT Algorithms for Continuous Data Assimilation
Abstract: In this talk, we compare the performance of two continuous data assimilation (CDA) algorithms: the Azouani-Olson-Titi (AOT) algorithm and the Ensemble Kalman Filter (EnKF). While EnKF is widely used across a variety of CDA applications, AOT remains relatively underutilized. Through extensive computational experiments, we examine both algorithms' error convergence properties and computational efficiency when applied to the one-dimensional Kuramoto-Sivashinsky equation and the two-dimensional Navier-Stokes equations. Our results indicate that while both methods demonstrate similar error convergence, AOT offers a substantial computational cost advantage.
Presentation 3
Speaker: Nathaniel Morgan (Baylor University)
Co-author: David Kahle (Baylor University), Michael Gallaugher (Baylor University)
Title: ggclassify: Graphics for Bivariate Classification Methods
Abstract: Classification and clustering problems are fundamental in machine learning and applied statistics, but in terms of ready-made software implementations for R, visualizing classifiers remains a clunky do-it-yourself process. To address this problem we created ggclassify, a R package that extends the ggplot2 framework with functions that plot bivariate classifiers rules given data and a method. It does this through implementing new stats and geoms for visualizing the decision spaces and boundaries of common classifiers/clustering rules, e.g. LDA, QDA, random forests, and k means.
Organizers
Qin Sheng (Baylor University), Bruce Wade (University of Louisiana), Julienne Kabre (Nova Southeastern University)
Mini-symposium Abstract:
The development of highly accurate and preservative numerical methods for data approximation and solutions of differential equations is an ongoing quest even after decades of successful approaches. The research is particularly accelerated by recent demands arising from various applications in sciences and engineering. The significance of their numerical strategies has been universally acknowledged and validated through the improvement of discrete methods in diverse branches, including approximation methods, finite difference methods, discrete Galerkin methods, and so on. In recent years, structure preserving methods, also known as geometric numerical integrators, have also emerged as a central topic in computational mathematics. It has been realized that an integrator should be designed to preserve as much as possible intrinsic features of the underlying problems. Preservative algorithms can be effectively utilized for simulations of a variety of theoretical and application problems.
This mini-symposium is dedicated to recent advances in pursuits for high-accuracy and preservative algorithms when data approximation or differential equations are targeted. We intend to accommodate a broad spectrum of investigations.
Presentation 1
Speaker: Yangwen Zhang (Department of Mathematics, University of Louisiana at Lafayette)
Co-author: N/A
Title: Efficient Eigen-Decomposition for Low-Rank Symmetric Matrices in Graph Signal Processing: An Incremental Approach
Abstract: Graph spectral analysis has emerged as an important tool to extract underlying structures among data samples. Central to graph signal processing (GSP) and graph neural networks (GNN), graph spectrum is often derived via eigen-decomposition (ED) of graph representation (adjacency/Laplacian) matrix. Many real-world applications feature dynamic graphs whose representation matrix size varies over time. Such evolving graph usually shares part of the same structures with the previous graphs. We consider efficient ways to estimate the K dominant eigen-vectors of the graph representation matrix. We focus on an iterative ED algorithm for low-rank symmetric matrices to update the top K eigen-pairs of the representation matrix for a graph with increasing node size. To accommodate the growing graph size, we propose two Incremental ED algorithms for Low Rank symmetric matrices (ILRED) algorithms based on an iterative eigen-updating strategy. We also provide analysis on the resulting error performance, computational complexity and memory usage to showcase the efficiency of ILRED. The experimental results in both synthetic and real-world datasets with the context of spectral clustering and graph filtering validate the power of the proposed ILRED algorithms.
Presentation 2
Speaker: Brian E. Moore (Department of Mathematics, University of Central Florida)
Co-author: N/A
Title: Backward Error Analysis for Some Structure-Preserving Discretizations
Abstract: Backward error analysis is a tool that has been useful for understanding the qualitative numerical solution behavior of structure-preserving algorithms for solving differential equations. By constructing a near-by differential equation, called the modified equation, for which the exact solution is the numerical solution of the original system of equations, we may analyze how well an algorithm reproduces properties that it does not completely preserve. For example, symplectic integrators do not preserve energy, but backward error analysis shows that the numerical solution is exponentially close to preserving a near-by energy, which in turn guarantees excellent long-time simulations. In this talk we will consider results from a backward error analysis for discretizations of PDEs and systems with linear damping.
Presentation 3
Speaker: W. Y. Chan (Division of Science and Mathematics, Texas A&M University-Texarkana)
Co-author: N/A
Title: A Numerical Method of Finding Critical Square-Domains of Coupled Quenching Problems
Abstract: In this talk, we introduce a coupled-parabolic nonlinear quenching problem within square domains. The size of the critical domains depends on the existence of a steady-state solution. The integral solution is represented in terms of Green’s function and a conformal mapping is employed to transform the square domain into a circle. Our objective is to numerically determine the critical domains associated with this problem.
Presentation 4
Speaker: Julienne Kabre (Nova Southeastern University)
Co-author: N/A
Title: A Non-uniform Preservative Splitting Numerical Scheme of a Variable Coefficient Quenching Problem
Abstract: In this talk, we will explore the numerical solution of the two-dimensional quenching type nonlinear reaction-diffusion problem via dimensional splitting using non-uniform grid. The differential equation possesses a variable diffusion coefficient and a nonlinear forcing term that leads to a strong quenching singularities. Our current investigations focus on the construction of a finite difference nonuniform spatial grid implementation of a Peaceman-Rachford procedure for solving the aforementioned problem. Temporal adaptation is implemented through arclength estimations of the rate-of-change of the numerical solution. The positivity, monotonocity and localized linear stability of the non-uniform variable step splitting scheme are analyzed. Preliminary computational experiments results are presented to demonstrate the viability and accuracy of the numerical method.
Organizers
Christopher (C.J.) Bott (Texas A&M University), Jordy Lopez Garcia (Texas A&M University), Frank Sottile (Texas A&M University)
Mini-symposium Abstract:
Nonlinear algebra, a field informed and enriched by algebraic geometry, is at the forefront of theoretical and applied mathematics. Recently, this field has been successfully implemented in solving problems involving real algebraic geometry, the geometry of tensors, complexity theory, and nonconvex optimization.
In this mini-symposium, we bring together established and young researchers using methods of nonlinear algebra in their respective fields. We expect the talks to be accessible to graduate students, hence engaging discussions and promoting collaboration between diverse groups of scientists.
Presentation 1
Speaker: Carlos Arreche (The University of Texas at Dallas)
Co-author: Avery Bainbridge (The University of Texas at Dallas), Ben Obert (The University of Texas at Dallas), Alavi Ullah (The University of Texas at Dallas)
Title: Constructing flat algebraic connections whose monodromy groups are complex reflection groups
Abstract: Complex reflection groups comprise a generalization of Weyl groups of semisimple Lie algebras, and even more generally of finite Coxeter groups. They have been heavily studied since their introduction and complete classification in the 1950s by Shephard and Todd, due to their many applications to combinatorics, representation theory, knot theory, and mathematical physics, to name a few examples. For a given complex reflection group G, we explain how to construct connection matrices for an algebraic vector bundle with flat connection whose monodromy group is G. We exhibit these connection matrices explicitly for many (low-rank) irreducible complex reflection groups in the Shephard-Todd classification.
Presentation 2
Speaker: Tomasz Mandziuk (Texas A&M University)
Co-author: N/A
Title: New effective methods in border apolarity
Abstract: One of classical measures of complexity of a tensor is the notion of border rank. Recently border apolarity was developed as a tool for its computation. It was successfully used to obtain border ranks of interesting tensors, however, as implemented, it was unable to solve important problems of interest. In the talk I give an example of a tensor whose border rank cannot be computed using standard border apolarity but can be computed using new methods that I developed.
Presentation 3
Speaker: Weixun Deng (Texas A&M University)
Co-author: Isabella Robinson (University of Houston), J. Maurice Rojas (Texas A&M University), Cordelia Russell (University of New Mexico)
Title: Computing Isotopy Type of Positive Zero Sets of Near-Circuit Polynomials
Abstract: We present initial results on the algorithmic classification of the isotopy types of positive zero sets of polynomials supported on near-circuits. A near-circuit is a set of $n+3$ points in $\mathbb{Z}^n$ that do not lie in any affine hyperplane. Our main contribution is a significant improvement on the upper bound for the number of connected components of these zero sets, showing it to be at most $3$, an advancement over the previous bound of $O(n)$. Furthermore, we provide a criterion for determining when the number of connected components is at most $2$. A key tool in our approach is the computation of cusps on the associated signed $\mathcal{A}$-discriminant contour.
Presentation 4
Speaker: Julia Lindberg (University of Texas-Austin)
Co-author: Jose Rodriguez (University of Wisconsin-Madison)
Title: Invariants of SDP exactness in quadratic programming
Abstract: The Shor relaxation of a quadratic program is a semidefinite program that provides a bound on the objective value of a quadratic program. This relaxation is exact if the optimal solution to the semidefinite program is rank one. In this talk I will study the Shor relaxation by fixing a feasible set and considering the space of objective functions for which this relaxation emits a rank one solution. I will first give conditions under which this region is invariant under the choice of generators defining the feasible set. I will then describe this region when the feasible set is invariant under the action of a subgroup of the general linear group. If time permits, I will conclude by applying these results to quadratic binary programs.
Organizers
Loic Cappanera (University of Houston), Gabriela Jaramillo (University of Houston)
Mini-symposium Abstract:
Nonlocal processes play an important role in many applications including ground-water transport, biology, electro magnetic fluids, and continuum mechanics, to name just a few. These nonlocal processes give rise to integral operators, which present many analytical and numerical challenges. The goal of this mini-symposium is to gather experts in this wide field in order to highlight the different analytical and numerical approaches for studying these systems.
Presentation 1
Speaker: Christian Glusa (Sandia National Laboratories)
Co-author: N/A
Title: Optimal control for fractional order equations
Abstract: We consider adjoint-based optimization for the inference of kernel parameters of a fractional-order state equation. We will discuss optimality conditions, error estimates and techniques to efficiently explore the parameter space and approximate gradients.
Presentation 2
Speaker: Nicole Buczkowski (Worcester Polytechnic Institute)
Co-author: Mikil Foss (University of Nebraska-Lincoln), Michael Parks (Oak Ridge National Laboratory), Petronela Radu (University of Nebraska-Lincoln), Jeremy Trageser (Sandia National Laboratories)
Title: Comparing and Contrasting Features of two Nonlocal Biharmonic Operators
Abstract: Nonlocal operators are used in modeling due to their capability of handling discontinuities and modeling a range of interactions through different choices for kernels. This includes several applications, e.g. peridynamics (fracture mechanics), flocking, and image processing. The biharmonic operator appears in many models, notably in the deformations of beams and plates. We consider two different ways of formulating a nonlocal biharmonic operator: using a fourth difference operator in the integrand or iterating the nonlocal Laplacian. In this talk, we compare and contrast various facets of these operators, including their physical formulations, convergence of each operator to its classical counterpart, and the two associated nonlocal clamped boundary value problems.
Presentation 3
Speaker: Davood Damircheli (Mississippi State University)
Co-author: Robert Lipton (Louisiana State University), Debdeep Bhattacharya (Grinnell College)
Title: Numerical Simulation of particle beds in 3 dimensions using Peridynamics coupled to the Direct Element Method.
Abstract: Traditional numerical approaches to particle simulation for sand and soils use the Direct Element Method (DEM). Using DEM the particles are traditionally modeled as disks or convex shapes with rigid boundaries. The interparticle interactions involve friction and damping. Here we go beyond rigid particles and model the particles (convex or nonconvex) as particles that are elastic and can fracture autonomously using peridynamics. We simulate suspensions of particles settling under gravity for suspensions of up to one thousand particles in 3 dimensions. Here the particles can deform elastically and inelastically. We discuss the scalability of the method for high performance computing.
Organizers
Marco Campos (University of Houston), William Ott (University of Houston)
Mini-symposium Abstract:
Topological Data Analysis (TDA) has emerged as a powerful framework for extracting intrinsic geometric and topological structures hidden within complex datasets. The talks will cover a range of topics, from foundational theories to novel applications, showcasing how TDA can reveal insights in high-dimensional data that traditional methods often overlook. Sessions will include discussions on persistent homology, topological inference, and the integration of TDA with machine learning. The symposium will cover the current state of TDA, future directions in research, and how this cutting-edge methodology is being used to tackle some of the most pressing data analysis challenges across various scientific domains.
Presentation 1
Speaker: Henry Adams (University of Florida)
Co-author: Johnathan Bush (James Madison University), Ziga Virk (University of Ljubljana and Institute IMFM)
Title: The connectivity of Vietoris-Rips complexes of spheres
Abstract: Though Vietoris-Rips complexes are frequently built in applied topology to approximate the "shape" of a dataset, their theoretical properties are poorly understood. Interestingly, the Vietoris-Rips complexes of the circle obtain the homotopy types of the circle, the 3-sphere, the 5-sphere, the 7-sphere, ..., as the scale parameter increases. But little is known about Vietoris-Rips complexes of the n-sphere $S^n$ for $n >= 2$, which we equip with the geodesic metric of diameter pi. We show how to control the homotopy connectivity of Vietoris-Rips complexes of spheres in terms of coverings of spheres and projective spaces. For t > 0, suppose that the first nontrivial homotopy group of the Vietoris-Rips complex of the n-sphere at scale pi-t occurs in dimension k. Then there exist 2k+2 balls of radius t that cover $S^n$, and no set of k balls of radius t/2 cover the projective space $RP^n$.
Presentation 2
Speaker: Dorcas Ofori-Boateng (Portland State University)
Co-author: Segovia Dominguez, Ignacio (University of Texas-Dallas), Akcora, Cuneyt G. (University of Manitoba), Kantarcioglu, Murat (University of Texas-Dallas), Gel, Yulia R. (University of Texas-Dallas)
Title: Topological Anomaly Detection in Dynamic Multilayer Blockchain Networks
Abstract: Motivated by the recent surge of criminal activities with cross-cryptocurrency trades, we introduce a new topological perspective to structural anomaly detection in dynamic multilayer networks. We postulate that anomalies in the underlying blockchain transaction graph that are composed of multiple layers are likely to also be manifested in anomalous patterns of the network shape properties. As such, we invoke the machinery of clique persistent homology on graphs to systematically and efficiently track evolution of the network shape and, as a result, to detect changes in the underlying network topology and geometry. We develop a new persistence summary for multilayer networks, called stacked persistence diagram, and prove its stability under input data perturbations.
We validate our new topological anomaly detection framework in application to dynamic multilayer networks from the Ethereum Blockchain and the Ripple Credit Network, and demonstrate that our stacked PD approach substantially outperforms state-of-art techniques.
Presentation 3
Speaker: Marco Campos (University of Houston)
Co-author: Dan Krofcheck, William Ott (University of Houston)
Title: Zigzag Persistence as a Measure of Topology Preservation in Temporal Link Prediction
Abstract: Temporal link prediction focuses on learning representations of dynamically evolving networks to predict both future and missing interactions. Unfortunately, many of these models tend to be either over-confident or low performing in the presence of heterogeneous relationships, chaotic underlying dynamics, and complex structural elements which can create problems in high consequence settings. Therefore, it is crucial to learn about the geometric and topological structures that these models are generating and how they relate to the data that they represent. Persistent homology has been effectively used to detect structures in high-dimensional spaces by computing connected components, cycles, and generalizations of higher-dimensional holes using a single package. However, this technique has traditionally been limited to static datasets. Although there have been theoretical extensions for some time, recent work has developed practical applications for zigzag persistence, particularly in the analysis of temporally evolving graphs, allowing us to discover the same structures as they change over time. This provides us with an opportunity to evaluate temporal link prediction models such as DySAT and TGN by verifying how well they preserve topological structure through their graph embeddings. We seek to train these models on synthetic and real-world data sets with prevalent features that are crucial to their dynamics and verify whether the embeddings, probability thresholds, attention coefficients and other layer outputs of the model share the same features. In doing so, we will be able to test whether the best performing models effectively recreate the underlying deterministic dynamical system governing the graph creation methods.
Presentation 4
Speaker: Yingying Wu (University of Houston)
Co-author: Roger Fu (University of Houston), Richard Peng (Carnegie Mellon University), Qifeng Chen (Hong Kong University of Science and Technology)
Title: Graph Convolutional Networks for Learning Laplace-Beltrami Operators
Abstract: Recovering a high-level representation of geometric data is a fundamental goal in geometric modeling and computer graphics. In this talk, I will introduce a data-driven approach to compute the spectrum of the Laplace-Beltrami operator of triangle meshes using graph convolutional networks. Specifically, we train graph convolutional networks on a large-scale dataset of synthetically generated triangle meshes, encoded with geometric data consisting of Voronoi areas, normalized edge lengths, and Gauss map, to infer eigenvalues of 3D shapes. We attempt to address the ability of graph neural networks to capture global shape descriptors, including spectral information, which were previously inaccessible using existing methods. Our work exhibits promising indicators suggesting that Laplace-Beltrami eigenvalues on discrete surfaces can be learned, and we will show assuring signs to incorporate topological data analysis methods. Additionally, we perform ablation studies showing the addition of geometric data leads to improved accuracy.
Organizers
Md Joshem Uddin (The University of Texas at Dallas), Sayoni Chakraborty (The University of Texas at Dallas), Baris Coskunuzer (The University of Texas at Dallas)
Mini-symposium Abstract:
As data grows in complexity, encompassing high-dimensional spaces, images, and graph structures, understanding its underlying patterns becomes increasingly challenging. However, these complex datasets often lie on low-dimensional manifolds that can be effectively explored using differential geometry and algebraic topology. Geometric and topological methods are uniquely suited to uncovering the structural nuances of such data, offering insights that might remain hidden to conventional data science approaches. In fields like neuroscience, biology, and network science, the geometric characteristics of data are crucial for direct analysis and for improving optimization and machine learning tasks.
This mini-symposium will cover important topics such as persistent homology, discrete curvature, graph learning, and topological data representations. We will focus on how these topics interact and support each other. Our goal is to bring together researchers working on different aspects of geometric and topological data analysis. By focusing on the intersections with machine learning, network science, optimization, and broader data science domains, we hope to push the boundaries of current methodologies and spark innovative collaborations.
Presentation 1
Speaker: Hengrui Luo (Rice University)
Co-author: N/A
Title: Contrastive Learning in Multi-Scale Shape and Topological Data Analysis
Abstract: In this ongoing work, we delve into contrastive learning within the realms of multi-scale shape and topological data analysis, transitioning from local to global contrastion. We begin by presenting a multi-scale contrastive landmark learning method for planar curves, adept at identifying features across various scales and capturing both local and global shape details. Progressing to global contrastion, we introduce an innovative topological contrastive dimension reduction technique. This method accentuates differences in topological features within reduced dimensional spaces, employing persistence diagrams to highlight multi-scale topological variations. The current work focuses on methodological advancements in multi-scale contrastive learning, aimed at unveiling new perspectives in understanding shape and topological data when there are intrinsic differences between more-than-one datasets.
Presentation 2
Speaker: Md Joshem Uddin (The University of Texas at Dallas)
Co-author: N/A
Title: Wise-GNN: Enhancing GNNs with Wise Embeddings
Abstract: Graph Neural Networks (GNNs) have become the cornerstone of graph representation learning. However, recent studies highlight their struggle to capture information from distant nodes, resulting in poor performance in tasks like node classification, especially in heterophilic settings. To address this, we developed a new model called Wise-GNN. This model directly integrates attribute proximity information into GNN embeddings, improving the understanding of attribute similarities among distant nodes. Additionally, we enhance the robustness of node representations by using the topological signatures of node neighborhoods through persistent homology.
Extensive experiments on various GNN models show that combining Wise embeddings with topological signatures significantly boosts GNN performance on benchmark datasets for node classification tasks. We observed average accuracy increases of up to 15\% in both homophilic and heterophilic contexts. Our work advances the capabilities of GNNs by incorporating attribute awareness, thereby paving the way for more robust and efficient graph representation learning.
Presentation 3
Speaker: Peter F. Stiller (Texas A&M University)
Co-author: N/A
Title: Algebraic Geometry and Computer Vision
Abstract: It's well known that there are no general view invariants that can lend themselves to help recognize 3D features in a 2D image. Using techniques from algebraic geometry, namely the theory of correspondences, we will show how to build so-called object/Image equations in the respective 3D and 2D invariants jointly, for use in object recognition and shape analysis.
Presentation 4
Speaker: Chul Moon (Southern Methodist University)
Co-author: N/A
Title: Statistical Modeling of Topological Features in Medical Imaging: Enhancing Prognostic Precision and Interpretation
Abstract: Tumor shape significantly influences growth and metastasis. We introduce a topological feature obtained by persistent homology to characterize tumor progression in brain tumor patients using radiology images, focusing on its influence on time-to-event data. These topological features, invariant to scale-preserving transformations, capture diverse tumor shape patterns. We introduce a functional spatial Cox proportional hazards model that represents these topological features in a functional space, utilizing them as functional predictors alongside their spatial locations. This model allows for interpretable analysis of the relationship between topological shape features and survival risks.
Organizer
Alexandru Hening (Texas A&M University)
Mini-symposium Abstract:
This mini-symposium will be about showcasing new results in mathematical biology. It will be accessible to a broad audience and is meant to expose people to the rich field of mathematical biology (population dynamics, infections disease dynamics, networks, mathematical neuroscience) as well as the deterministic and stochastic tools used in the field.
Presentation 1
Speaker: Alexandru Hening (Texas A&M University)
Co-author: Dang Nguyen (University of Alabama), Peter Chesson (University of Arizona)
Title: Stochastic population dynamics in discrete time
Abstract: We present a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time stochastic difference equations and we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. Using the general theory, we work out several examples including the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka–Volterra models as well as models of perennial and annual organisms.
Presentation 2
Speaker: Ren Yi Wang (Rice University)
Co-author: N/A
Title: Stochastic dynamics of two-compartment models with regulatory mechanisms for hematopoiesis
Abstract: We present an asymptotic analysis of a stochastic two-compartmental cell proliferation system with regulatory mechanisms. We model the system as a state-dependent birth and death process. Proliferation of hematopoietic stem cells (HSCs) is regulated by density of HSC-derived clones and differentiation of HSC is regulated by density of HSCs. By scaling up the initial population, we show the density of dynamics converges in distribution to the solution of a system of ordinary differential equations. The system of ODE has a unique non-trivial equilibrium that is globally stable. Furthermore, we show the scaled fluctuation of the population converges in law to a linear diffusion. With initial data being Gaussian, the limit is a Gauss-Markov process and we prove the process will stabilize exponentially fast in the 2-Wasserstein metric. We apply our results to analyze and compare two regulatory mechanisms in the hematopoietic system. Simulations are conducted to verify our large-scale and long-time approximation of the dynamics. Biological applications in the context of targeted therapy are discussed.
Presentation 3
Speaker: Tamer Oraby (University of Texas-Rio Grande Valley)
Co-author: N/A
Title: Modeling Parental Decision Making about Immunization against Childhood Vaccine Preventable Diseases
Abstract: In this talk, I will introduce my work on modeling parent's decisions about immunizing their children. The models utilize behavioral game theory supported by health, social and cognitive theories to explain observed vaccination behaviors and ensuing disease incidence. I will present some of those deterministic and stochastic models and discuss their dynamic behavior. I will show how stochastic noise can affect the model's behavior and reflect on its practical implications.
Presentation 4
Speaker: Siddharth Sabharwal (Texas A&M University)
Co-author: N/A
Title: Effects of Dispersal on Population Dynamics under Stochastic Environment
Abstract: We study the dynamics of a species whose population is scattered between n patches, and individuals disperse between these patches. In the discrete case we consider the Beverton-Holt model, with a small random perturbation in the intrinsic growth. For the continuous time setting we consider SDE version of the ODE model given in Grumbach et al JOMB '23, and show when we get extinction, or when the total population converges to nonatomic stationary distribution.
Organizer
Darsh Gandhi (University of Texas at Arlington)
Mini-symposium Abstract:
Due to their excitement and curiosity, undergraduate students provide interesting insights into problems in mathematics and the world. Students who participate in undergraduate research improve writing, speaking, and teamwork skills and are better prepared for both graduate school and careers in business, industry, and government. Undergraduate math students around Texas and Louisiana are working on compelling problems in medical imaging, AI and machine learning, mathematical physics, and more, and in this mini-symposium we showcase a few special projects in various branches of applied mathematics.
Presentation 1
Speaker: Austin Carlson (University of Texas-Arlington)
Co-author: Carli Peterson (University of Texas-Arlington), Darsh Gandhi (University of Texas-Arlington), Aaron Lubkemann (University of Texas-Arlington), Emma Richardson (University of Texas-Arlington), Souvik Roy (University of Texas-Arlington), Christopher Kribs (University of Texas-Arlington), Hristo Kojouharov (University of Texas-Arlington)
Title: A SIMPL Model of Phage-Bacteria Interactions Accounting for Mutation and Competition
Abstract: Pseudomonas aeruginosa is an opportunistically pathogenic bacteria that causes fatal infections and epidemics in hospital environments. Due to the growing prevalence of antibiotic-resistant strains of P. aeruginosa, looking for alternative therapies is crucial. Bacteriophage therapy is emerging as a promising solution; however, it has not been approved for human clinical use and suffers from a lack of understanding of the complex dynamics between bacterial cells and phage virions. Mathematical models provide insight into these dynamics. Through a system of ordinary differential equations, we determined appropriate biological assumptions to effectively capture the complex dynamics observed between susceptible, infected, and mutated bacterial cells with bacteriophage virions in a microwell setting. Data fitting based on this model produced a set of parameter estimates unique to our experimental observations of a specific phage and P. aeruginosa strain. In translating observed optical density readings into bacterial concentrations, we also found that bacterial debris has a significant impact on optical density, with a lysed bacterium contributing roughly 40% as much to optical density readings as a living cell. Future works include extending our model to incorporate various antibiotic resistant bacteria species and their phage counterparts, investigating stochastic and time-delay components of the phage bacteria relationship, and determining optimal, patient-specific synergistic phage and antibiotic treatment regimens.
Presentation 2
Speaker: Leah Rogers (Tarleton State University)
Co-author: N/A
Title: Simulating Left Atrial Arrhythmias With an Interactive N-Body Model
Abstract: Supraventricular Tachycardia (SVT) refers to irregularly fast heart rhythms originating from the atria. While SVT itself rarely directly leads to death, its various forms are significant contributors to strokes and heart attacks worldwide. SVT also plays a substantial role in preventable heart failure cases through tachycardiomyopathy and can trigger myocardial infarction by increasing oxygen demand while reducing cardiac output. Although medication is vital in managing SVT, cardiac ablation has emerged as the most effective long-term solution. During this procedure, an electrophysiologist employs tiny catheters to create a 3D map of the heart's electrical activity. Advanced computer software helps identify the source of the irregular heartbeat, allowing targeted radiofrequency ablation of specific areas to restore the heart's normal rhythm. Despite significant advancements in cardiac ablation over the past two decades, understanding the causes and optimal ablation methods for the most prevalent and problematic cardiac arrhythmias, such as Atrial Fibrillation, remains a modern-day mystery. The underlying mechanisms for all Left atrial arrhythmias are still poorly understood. To address this gap, we conducted an in-depth review of current literature and utilized a computational model of the Left Atrium to explore theories related to the causes and mechanisms behind Left atrial arrhythmias. Through these efforts, our aim is to shed light on these challenging arrhythmias and use our model as a study tool to assist doctors, researchers, and medical students in rapidly and inexpensively testing ideas and observing outcomes.
Presentation 3
Speaker: Darsh Gandhi (University of Texas-Arlington)
Co-author: Alexandria Johnson (North Carolina State University), Emma Slack (University of Texas-Arlington), Isaiah Stevens (North Carolina State University), Zachary Turner (Duke University), Michelle Bartolo (Harvard University), Mette Olufsen (North Carolina State University)
Title: A Computational Framework for the Generation of Patient-Specific Vascular Models
Abstract: One-dimensional (1D) cardiovascular models offer a non-invasive method to answer medical questions, including predictions of wave-reflection, shear stress, fractional flow reserve, vascular resistance, and compliance. This model type can predict patient-specific outcomes by solving 1D fluid dynamics equations in geometric networks extracted from medical images. However, the inherent uncertainty in in-vivo imaging introduces variability in network size and vessel dimensions, affecting hemodynamic predictions. Understanding the influence of variation in image-derived properties is essential to assess the fidelity of model predictions. Numerous programs exist to render three-dimensional surfaces and construct vessel centerlines. Still, there is no exact way to generate vascular trees from the centerlines while accounting for uncertainty in data. This study introduces an innovative framework employing statistical change point analysis to generate labeled trees that encode vessel dimensions and their associated uncertainty from medical images. To test this framework, we explore the impact of uncertainty in 1D hemodynamic predictions in a systemic and pulmonary arterial network. Simulations explore hemodynamic variations resulting from changes in vessel dimensions and segmentation; the latter is achieved by analyzing multiple segmentations of the same images. Results demonstrate the importance of accurately defining vessel radii and lengths when generating high-fidelity patient-specific hemodynamics models.
Organizers
Natasha S. Sharma (The University of Texas at El Paso), Giordano Tierra (University of North Texas)
Mini-symposium Abstract:
This mini-symposium aims to bring together a diverse group of talks highlighting recent developments in modeling, numerical analysis, and computational methods for multi-physics problems formulated using systems of coupled partial differential equations.
Presentation 1
Speaker: Justin Swain (University of North Texas)
Co-author: Giordano Tierra (University of North Texas)
Title: Efficient and Unconditionally Energy Stable Numerical Scheme for a New Ternary Cahn-Hilliard System
Abstract: Phase field modeling of mixtures of three components is important for understanding complex phenomena such as phase separation, pattern formation, and spinodal decomposition in ternary mixtures. The ternary Cahn-Hilliard equation uses a gradient flow of an energy functional to model the dynamics of phase separation which describes the mixture using a concentration order parameter that smoothly transitions between the different phases. In this talk I will introduce new numerical schemes for a modified ternary Cahn-Hilliard model. The key idea is to work with an energy which contains a penalization of the total volume conservation constraint in a way that avoids adding additional nonlinearities to the already nonlinear system. This allows us to develop numerical schemes for representing three component phase separation. In particular, I will present three different schemes and discuss the advantages and disadvantages of each of them. I will compare the effectiveness of each of the numerical schemes by presenting simulations of several popular benchmarks in two and three dimensions.
Presentation 2
Speaker: Arkadz Kirshtein (Texas A&M University-Corpus Christi)
Co-author: N/A
Title: Variational phase-field-micromechanics model of solid-state sintering
Abstract: Sintering, a pivotal technology in additive manufacturing, transforms ceramic and metallic powders into solid objects. To achieve products with customized properties, a deep understanding of microstructure evolution during sintering is crucial. Our approach ensures thermodynamic consistency, deriving the driving force for particle motion from the system's free energy. As a result, our proposed phase-field-micromechanics model guarantees microstructure evolution that minimizes the system's energy. We rigorously validate this model against recent theoretical benchmarks. Subsequently, we employ it to simulate the microstructure evolution of polycrystalline powder particles, shedding light on the mechanisms governing crystallite growth. Additionally, we analyze how grain boundary structure and orientation impact sintering kinetics.
Presentation 3
Speaker: Jia Zhao (Binghamton University)
Co-author: Qi Hong (School of Mathematics, Nanjing University of Aeronautics and Astronautics, China), Yuezheng Gong (School of Mathematics, Nanjing University of Aeronautics and Astronautics, China)
Title: Thermodynamically consistent hydrodynamic phase-field computational modeling for fluid-structure interaction with moving contact lines
Abstract: In this talk, I will present a novel computational modeling approach for investigating fluid-structure interactions with moving contact lines. By applying the generalized Onsager principle, we develop a coupled hydrodynamics and phase-field system in a thermodynamically consistent manner. The resulting partial differential equation (PDE) model consists of the Navier-Stokes equations and a nonlinear Allen-Cahn type equation, with volume conservation enforced through an additional penalty term. We propose a fully discrete, structure-preserving numerical scheme that combines several techniques to solve this coupled PDE system effectively and accurately. Finally, various numerical simulations will be shown to verify the model's capabilities and demonstrate the scheme's effectiveness, accuracy, and stability.
Presentation 4
Speaker: Natasha S. Sharma (The University of Texas at El Paso)
Co-author: Tamas Horvath (Oakland University), Giselle G. Sosa Jones (Oakland University)
Title: An unconditionally stable hybridizable discontinuous Galerkin method for the Phase Field Crystal Equation
Abstract: The Phase Field Crystal (PFC) Equation is a sixth-order nonlinear time-dependent partial differential equation introduced by Elder and Grant (2004) as a continuum model to study atomic-scale crystal growth over diffusive time scales. In this talk, we present a hybridizable discontinuous Galerkin method to solve the PFC equation with temporal discretization realized by the convex splitting scheme. We will discuss key properties of unique solvability and unconditional stability satisfied by the scheme and present numerical experiments to illustrate the performance of our proposed method.
Organizers
Ziad Ghanem (The University of Texas at Dallas), Wieslaw Krawcewicz (The University of Texas at Dallas)
Mini-symposium Abstract:
This mini-symposium brings together experts to explore methods for addressing symmetric bifurcation problems associated with non-linear differential equations. Participants will share their latest findings and approaches from the frontiers of various fields, some novel some classical, fostering a dialogue on the advances and application of symmetric bifurcation theory.
Presentation 1
Speaker: Wieslaw Krawcewicz (The University of Texas at Dallas)
Co-author: N/A
Title: Two Parameter $G$-Equivariant Local and Global Bifurcations
Abstract: We present present a general twisted equivariant degree approach for solving local and global symmetric bifurcation problems of periodic solutions to autonomous systems of non-linear differential equations. The general setting for this method is a bifurcation equation with spatial symmetries represented by a group $\Gamma$ and with two parameters $(\alpha,\beta) \in \mathbb R \times \mathbb R$ where $\alpha$ is the standard bifurcation parameter and $\beta$ corresponds to the a priori unknown frequency of a periodic solution.
Presentation 2
Speaker: Ziad Ghanem (The University of Texas at Dallas)
Co-author: Wieslaw Krawcewicz (The University of Texas at Dallas), Carlos Garcia-Azpetia (Universidad Nacional Autónoma de México)
Title: Global Bifurcation of Non-Stationary Solutions for Symmetric Systems of Nonlinear Wave Equations
Abstract: We demonstrate how the twisted equivariant Leray-Schauder degree can be used to prove a global bifurcation result for the existence of non-stationary branches of solutions to a parameterized family of $\Gamma$-symmetric delayed wave equations with non-trivial damping and non-linear forcing.
Presentation 3
Speaker: Casey Crane (The University of Texas at Dallas)
Co-author: Chaoquan Chen, Huafeng Xiao/School of Electrical Engineering, Southeast University, and Travis Hensley/The University of Texas at Dallas
Title: Global Hopf Bifurcation in Symmetric Configuration of Distributed Delay Differential Equations
Abstract: Motivated by locomotion and swarm formation problems in drones and robotics, we examine systems of coupled oscillators with distributed delays. By transforming the problem into an operator equation in a suitable functional space, we use equivariant degree theory to establish conditions for the existence of symmetric two-parameter global Hopf bifurcations and characterize their symmetries.
Presentation 4
Speaker: Qingwen Hu (The University of Texas Permian Basin)
Co-author: Shi Yu
Title: Vibration-assisted stabilization of turning processes with state-dependent delay
Abstract: We develop vibration-assisted stabilization strategies of turning processes using feed rate and spindle controls, where the feed rate is proportional to the delayed bending position of the cutting tool in the feed direction and the spindle speed is either a constant or proportional to the delayed bending position orthogonal to the feed direction. We show that turning processes which contain intrinsic state- dependent delays can be stabilized by the proposed stabilization strategies and develop analytical descriptions of the stability regions in the parameter space for the system with different stabilization strategies. In contrast to the smoothness of the stability lobes for the vibration-assisted stabilization strategy with constant spindle speed, the stability lobes for the system with the spindle speed proportional to the delayed bending position can be non-smooth at finitely many points, for which we provide algorithms for determining their locations. Numerical simulations of the stability regions are also given to illustrate the general results.
Organizers
Andrei Martinez-Finkelshtein (Baylor University), Brian Simanek (Baylor University)
Mini-symposium Abstract:
The mini-symposium on Orthogonal Polynomials and Special Functions, organized within the annual meeting of the SIAM Texas-Louisiana section, will bring together researchers from both states and the rest of the USA to explore recent advances in the field, including
theory, computational aspects, and applications in approximation theory and mathematical physics.
Presentation 1
Speaker: Alexander Solynin (Texas Tech University)
Co-author: N/A
Title: Capacities of constellations of disks and balls
Abstract: We will discuss a few problems on the logarithmic capacity of configurations consisting of $n\ge 3$ disks in $\mathbb{R}^2$. In particular, we will identify configurations which minimize the logarithmic capacity under certain restrictions and discuss properties of related transcendental functions. Then, we will prove that the linear string maximizes the logarithmic capacity among all strings consisting of $n$ disks and that the circular necklace maximizes the logarithmic capacity over the set of all necklaces consisting of $n$ disks, each of radius one. Similar problems for the Newtonian capacity of constellations of balls in $\mathbb{R}^3$ will be also discussed.
Presentation 2
Speaker: Simon Foucart (Texas A&M University)
Co-author: N/A
Title: Computation of Multivariate Chebyshev Polynomials
Abstract: Chebyshev polynomials of the first kind are conceptually trivial to generalize but highly nontrivial to construct in practice. As a univariate preamble, when the domain is a union of intervals, we show how Riesz-Fejer theorem allows us to compute them exactly by way of semidefinite programming. Moving to the multivariate setting, where uniqueness does not hold anymore, we reveal how the so-called Moment-SOS hierarchy can be exploited to compute best uniform approximants to monomials. Although the workflow technically outputs a candidate for one of the best approximants (and importantly the associated signature), this candidate can be proven to be genuine with enough effort, e.g. for the monomial $x_1^2 x_2^2 x_3$ on the Euclidean ball and for the monomial $x_1^2 x_2 x_3$ on the simplex. We conclude by showing that another interpretation of multivariate Chebyshev polynomials has a closed-form solution on diagonally determined domains.
Presentation 3
Speaker: John Lopez (Tulane University)
Co-author: Victor Moll (Tulane University), Kenneth McLaughlin (Tulane University)
Title: Asymptotics and Zeros of a special family of Jacobi Polynomials
Abstract: Classical Jacobi polynomials $p_n(x)=p_n^{(\alpha, \beta)}(x)$, where $\alpha$ and $\beta$ are real parameters greater than -1 , are well-known for their orthogonality on the interval $[-1,1]$, with respect to the weight function $w(x ; \alpha, \beta)=(1-x)^\alpha(1+x)^\beta$. By analytic continuation on the parameters $\alpha$ and $\beta$, these polynomials can be studied for general complex parameters. However, when $\operatorname{Re}(\alpha) \leq-1$ or $\operatorname{Re}(\beta) \leq-1$, the classical orthogonality property on $[-1,1]$ no longer holds and consequently, the zeros may no longer be real and simple.
In this talk, we will present the Riemann Hilbert analysis of a specific family of Jacobi polynomials with non-classical parameters of the form $\alpha_m=m+1 / 2$ and $\beta_m=$ $-m-1 / 2$. We derive global asymptotics for this family of polynomials using the Riemann Hilbert formulation as in [2]. We use this analysis to study the location of their zeros. Specifically, we prove that the zeros accumulate on the left branch of the curve $\left|1-z^2\right|=1$. Particularly interesting is the behavior near the origin, where local asymptotics and thus the zeros are determined in terms of parabolic cylinder functions.
These polynomials arise in connection with the evaluation of integrals by Boros and Moll [1]. It is worth highlighting that they correspond to a limiting case not considered in the works [3,4,5], thus extending our understanding of Jacobi polynomials in the non-classical parameter regimes.
References
[1] G. Boros, V. H. Moll, An integral hidden in Gradshteyn and Ryzhik, J., Appl. Math. 106 (1999), 361-368.
[2] A.B.J. Kuijlaars, A. Martínez-Finkelshtein, R. Orive, Orthogonality of Jacobi polynomials with general parameters, Elect. Trans. in Numer. Anal 19 (2005), 1-17.
[3] A. Martínez-Finkelshtein, P. Martínez-González, and R. Orive, Zeros of Jacobi polynomials with varying non-classical parameters, Special functions, World Scientific (2000), 98-113.
[4] A.B.J. Kuijlaars, A. Martínez-Finkelshtein, Strong asymptotics for Jacobi polynomials with varying nonstandard parameters, Journal d'analyse Mathématique, Springer 94 (2004),195-234.
[5] A. Martínez-Finkelshtein, R. Orive, Riemann-Hilbert analysis for Jacobi polynomials orthogonal on a single contour, Journal of Approximation Theory, 134-2 (2005), 137-170.
Presentation 4
Speaker: Kenneth McLaughlin (Tulane University)
Co-author: N/A
Title: 20 minutes about the parabolic cylinder parametrix for Riemann-Hilbert problems
Abstract: I will explain how to extract a form of WKB asymptotics for the parabolic cylinder function using a Riemann-Hilbert characterization.