SIAM Mini-Symposium Session 3

1. MS 14 (SDRICH Rm 207) - Recent Advances in Physics Inspired Machine Learning, Part 2/2

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: Tran Hoang (Oak Ridge National Laboratory)

Co-author: Tran Hoang (Oak Ridge National Laboratory)

Title: Nonconvex regularization for joint sparse recovery

Abstract: Sparse regularization has found applications in signal processing, data analytics, dynamical system identification and neural network compression. The most prominent approach to sparse recovery is using the convex l1-type norms. However, it has been shown that these regularizations are less robust and less effective at enforcing sparsity compared to non-convex norms. Designing efficient optimization schemes for non-convex regularizations is challenging, while a general, rigorous theory verifying their advantages over convex regularizations remains largely elusive. In this talk, we will discuss our recent theoretical advances and numerical methods for the problem, as well as the prospect of the approach for energy-efficient, large-scale machine learning.

Presentation 2

Speaker: Changhong Mou (University of Wisconsin-Madison)

Co-author: N/A

Title: Combining stochastic parameterized reduced order models with machine learning for data assimilation and uncertainty quantification with partial observations

Abstract: A hybrid data assimilation algorithm is developed for complex dynamical systems with partial observations. The method starts with applying a spectral decomposition to the entire spatiotemporal fields, followed by creating a machine learning model that builds a nonlinear map between the coefficients of observed and unobserved state variables for each spectral mode. A cheap low-order nonlinear stochastic parameterized extended Kalman filter (SPEKF) model is employed as the forecast model in the ensemble Kalman filter to deal with each mode associated with the observed variables. The resulting ensemble members are then fed into the machine learning model to create an ensemble of the corresponding unobserved variables. In addition to the ensemble spread, the training residual in the machine learning-induced nonlinear map is further incorporated into the state estimation, advancing the diagnostic quantification of the posterior uncertainty. The hybrid data assimilation algorithm is applied to a precipitating quasi-geostrophic (PQG) model, which includes the effects of water vapor, clouds, and rainfall beyond the classical two-level QG model. The complicated nonlinearities in the PQG equations prevent traditional methods from building simple and accurate reduced-order forecast models. In contrast, the SPEKF forecast model is skillful in recovering the intermittent observed states, and the machine learning model effectively estimates the chaotic unobserved signals. Utilizing the calibrated SPEKF and machine learning models under a moderate cloud fraction, the resulting hybrid data assimilation remains reasonably accurate when applied to other geophysical scenarios with nearly clear skies or relatively heavy rainfall, implying the robustness of the algorithm for extrapolation.

Presentation 3

Speaker: Xiong Wang (John Hopkins University)

Co-author: N/A

Title: Interacting Particle Systems on Networks: joint inference of the network and the interaction kernel

Abstract: Modeling multi-agent systems on networks is a fundamental challenge in a wide variety of disciplines. We jointly infer the weight matrix of the network and the interaction kernel, which determine respectively which agents interact with which others and the rules of such interactions from data consisting of multiple trajectories. The estimator we propose leads naturally to a non-convex optimization problem, and we investigate two approaches for its solution: one is based on the alternating least squares (ALS) algorithm; another is based on a new algorithm named operator regression with alternating least squares (ORALS). Both algorithms are scalable to large ensembles of data trajectories. We establish coercivity conditions guaranteeing identifiability and well-posedness. The ALS algorithm appears statistically efficient and robust even in the small data regime but lacks performance and convergence guarantees. The ORALS estimator is consistent and asymptotically normal under a coercivity condition. We conduct several numerical experiments ranging from Kuramoto particle systems on networks to opinion dynamics in leader-follower models.

Presentation 4

Speaker: Ziheng Guo (University of Houston)

Co-author: N/A

Title: Noise Guided Learning from Observation of Stochastic Dynamics

Abstract: We develop an innovative learning framework that incorporate the noise structure to infer the governing equations from observation of trajectory data generated by stochastic dynamics. Our approach can proficiently captures both the noise and the drift terms. Furthermore, it can also accommodate a wide range of noise types, including correlated and state-dependent variations. Moreover, our method demonstrates scalability to high-dimensional systems when combined with popular deep learning approaches. Through extensive numerical experiments, we showcase the exceptional performance of our learning algorithm in accurately reconstructing the underlying stochastic dynamics.

2. MS 20 (SDRICH Rm 208) - Data and model-driven approaches for inverse problems, Part 1/2

Organizers

Daniel Onofrei (University of Houston), Andreas Mang (University of Houston)

Mini-symposium Abstract:

This mini-symposium will focus on recent developments in data- and model-driven approaches for inverse problems in challenging applications. Despite significant progress in the mathematical sciences, there are still notable challenges. In inference, the hidden parameters are often linked to indirect and noisy measurements through complex systems modeled as solutions of partial or ordinary differential equations. As a result, the parameter-to-observable maps can be highly nonlinear, leading to a non-convex optimization landscape with ill-conditioned inversion operators. Additionally, the observed data are typically noisy, and the unknown parameters of interest may be infinite-dimensional in the continuum.

The aim of this mini-symposium is to attract researchers at the forefront of inverse problems and data science to present their latest work on designing and analyzing mathematical methods for inverse problems.

Presentation 1

Speaker: Richard Albanese (Albanese Defense and Energy Development LLC)

Co-author: Myoung An (Prometheus Inc), Richard Medina (Albanese Defense and Energy Development LLC)

Title: Electromagnetic Fields and Subsurface Objects

Abstract: At sufficiently long wavelengths electromagnetic radiation can penetrate soils and aqueous media for practically significant depths. In this report, we will review penetration depth data for dry soils and for water such as that of lakes and sea. The emphasis will be on penetration depth as a function of incident frequency. The mathematical difficulty for communication to or detection of subsurface objects is not related to computing penetration depth in various media but is associated with the long wavelengths that are needed for practically useful penetration. We will introduce a timed array that is specifically designed to broadcast very long wavelengths. Each radiating element in the antenna array radiates a basis function in a series representation of the desired far field signal. While different basis sets have been studied and will be touched on it this talk, the best signal representation we have found thus far is based on the Weyl-Heisenberg system, for example, the Gabor expansion. We will discuss the properties and utility of this representation in an electromagnetic context and indicate how the representation is inferred, itself a kind of inverse problem, from the signal to be broadcast. Last, we will discuss the long wavelength object detection challenge.

Presentation 2

Speaker: Traian Dogaru (U.S. Army DEVCOM Army Research Laboratory)

Co-author: Fauzia Ahmad (Temple University)

Title: Inverse Problems in Radar Imaging

Abstract: In this paper, we frame radar imaging in the context of general inverse problems. Creating the scene's reflectivity map is the first stage of any radar system's signal processing chain. This procedure is firmly rooted in the electric field integral equation describing electromagnetic wave scattering. However, unlike in general electromagnetic non-linear inverse problems, the usual approach to radar imaging consists of linearizing the integral equation (via the Born approximation) to obtain the classic linear system Ax = b upon discretization. We discuss the prevalent methods of solving this system within the radar engineering community. By far the most popular approach is the matched filter, which can be written as x = AHb and approximates the L2-norm minimization of the residual error term. More recently developed techniques regularize this optimization problem via a sparsity condition imposed on the scattering scene, which is related to the L1-norm of the latter. Both implementations based on antenna arrays and synthetic aperture radar (SAR) are discussed, together with traditional and advanced algorithms for radar reflectivity map formation. One of the current research efforts focuses on sparse, multi-static array implementation in the near field, with applications ranging from body scanners at security checkpoints to ground penetrating radar (GPR). Examples of radar imaging based on both computer simulations and experimental data are presented.

Presentation 3

Speaker: Taoufik Meklachi (Penn State Harrisburg)

Co-author: N/A

Title: An Asymptotic Approach to Scattering Resonances in High-Contrast Nonlinear Optical Media with Kerr Effect

Abstract: In this work, we present the development of an asymptotic method for calculating the scattering resonances of small-volume, high-contrast scatterers in the context of nonlinear waves. This problem is particularly relevant in the field of nonlinear optics, where the scattering behavior of metallic nanoparticles plays a crucial role. The study of such scattering phenomena is not only of fundamental scientific interest but also holds significant practical implications for various applications, such as in the design of optical devices and sensors. Moreover, we extend our analysis to consider double-layer high-contrast media, with at least one of the layers exhibiting the Kerr effect - a nonlinear optical phenomenon where the refractive index of the material changes in response to the intensity of the incident light.

Presentation 4

Speaker: Daniel Onofrei (Department of Mathematics, University of Houston)

Co-author: N/A

Title: On the problem of active control of fields through the lenses of integral operators, inverse sources and optimization

Abstract: In this talk we will present first a theoretical introduction for the general problem of active control of fields and then focus on the particular case of electromagnetic fields. We will then show how by employing techniques from inverse source analysis and through suitable layer potential operators one can prove the existence of a class of active control sources achieving the desired effects. We then present how these controls can be further studied through the equivalent formulation of the underlying problem as a constrained minimization problem and in particular focus on the implementation of a practical strategy for far field control of electromagnetic fields by using a swarm of flying drones with applications to communications. Numerical support will be presented for several interesting applied scenarios.

3. MS 15 (SDRICH Rm 210) - Recent Advancements in Numerical Methods for Subsurface Simulations, Part 2/3

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: Todd Arbogast (University of Texas at Austin)

Co-author: Chieh-Sen Huang (National Sun Yat-sen University), Chenyu Tian (University of Texas at Austin)

Title: A higher order finite volume multilevel WENO scheme for advection-diffusion equations

Abstract: We present a multi-level weighted essentially non oscillatory (ML-WENO) reconstruction on general computational meshes for solving partial differential equations exhibiting advection and degenerate diffusion. The reconstruction combines stencil polynomial approximations of various degrees, including constants, defined on any set of stencils that need not be arranged hierarchically. The nonlinear weighting biases the reconstruction away from both inaccurate oscillatory polynomials of high degree (i.e., those crossing a shock or steep front) and smooth polynomials of low degree, thereby selecting the smooth polynomial(s) of maximal degree of approximation. We also give a general result for determining when a stencil polynomial approximation is accurate. The reconstruction leads to a very general finite volume scheme that can handle shocks or steep fronts in the solution. We apply our new ML-WENO scheme to Richards equation modeling flow in porous media.

Presentation 2

Speaker: Young Ju Lee (Texas State University)

Co-author: N/A

Title: Positivity and maximum principle preserving discontinuous Galerkin finite element schemes for a coupled flow and transport

Abstract: We introduce a new concept of the local flux conservation and investigate its role in the coupled flow and transports. We demonstrate how the proposed concept of the locally conservative flux can play a crucial role in obtaining the L2 norm stability of the discontinuous Galerkin finite element scheme for the transport in the coupled system with flow. In particular, the lowest order discontinuous Galerkin finite element for the transport is shown to inherit the positivity and maximum principle when the locally conservative flux is used, which has been elusive for many years in literature. The theoretical results established in this paper are based on the equivalence between Lesaint-Raviart discontinuous Galerkin scheme and Brezzi-Marini-Sűli discontinuous Galerkin scheme for the linear hyperbolic system as well as the relationship between the Lesaint-Raviart discontinuous Galerkin scheme and the characteristic method along the streamline. Sample numerical experiments have then been performed to justify our theoretical findings.

Presentation 3

Speaker: Yiran Wang (University of Alabama)

Co-author: N/A

Title: Physics-preserving IMPES based multiscale methods for immiscible two-phase flow in highly heterogeneous porous media

Abstract: In this talk, I will introduce a novel physics-preserving multiscale approach to tackle the challenge of immiscible two-phase flow problems. These are typically described as a coupled system comprising Darcy's law and mass conservation equations. Physics-preserving IMplicit Pressure Explicit Saturation (P-IMPES) scheme, is designed to uphold local mass conservation for both phases while remaining unbiased. Notably, when the time step is kept below a certain threshold, the P-IMPES scheme ensures bounds-preserving saturation for both phases. For velocity updates, we employ the Mixed Generalized Multiscale Finite Element Method (MGMsFEM), a highly efficient solver that operates on a coarse grid to compute unknowns. We adopt an operation splitting technique to manage the complexities of two-phase flow, utilizing an upwind strategy for explicit saturation iteration, while employing the MGMsFEM to compute velocity via a decoupled system on a coarse mesh. To validate the effectiveness and robustness of our proposed method, we conduct a series of comprehensive experiments. Additionally, we provide a rigorous analysis to establish the theoretical underpinnings of the method, which are corroborated by our numerical findings. Both simulations and analysis demonstrate that our approach strikes a favorable balance between accuracy and computational efficiency.

Presentation 4

Speaker: Seonghee Jeong (Florida State University)

Co-author: Sanghyun Lee (Florida State University)

Title: Optimal control problem for coupled stationary flow and transport equations

Abstract: In this work, we present an optimal control problem constrained by coupled flow and transport equations. We explore two different cases: one with no pointwise constraints and the other with pointwise control constraints. The optimality conditions and convergence analysis are obtained for both cases, and they are numerically solved using a finite element method. The primal-dual active set algorithm is used for the pointwise control constraint case. Numerical examples are provided to validate the theoretical results.

4. MS 21 (SDRICH Rm 212C) - Recent Developments and Applications of High-fidelity Numerical Simulations, Part 1/2

Organizers

Yong Yang (West Texas A&M University), Yonghua Yan (Jackson State University), Caixia Chen (Jackson State University)

Mini-symposium Abstract:

This mini-symposium will explore the latest advancements in numerical simulation techniques across a wide array of scientific disciplines, including physics, chemistry, biology, and engineering. This session will feature a series of talks highlighting the transformative role of high-fidelity simulations in solving complex, real-world problems. Topics will span from fluid dynamics and high-speed flow simulations to molecular modeling in chemistry, the simulation of biological processes at the cellular and molecular levels, and the integration of machine learning with numerical methods. By bringing together researchers from diverse fields, the mini-symposium aims to foster interdisciplinary collaboration and share innovative strategies for leveraging high-fidelity simulations to advance scientific understanding and technological development. Attendees will gain a comprehensive overview of how state-of-the-art numerical techniques are being applied to push the boundaries of research across multiple domains.

Presentation 1

Speaker: Yonghua Yan (Jackson State University)

Co-author: Yong Yang (West Texas A&M University), Caixia Chen (Jackson State University)

Title: A compact-based high order correction method for computational fluid dynamics

Abstract: A novel high order correction method is proposed to enhance numerical simulation accuracy for high-speed flows by refining the Weighted Essentially Non-Oscillatory (WENO) flux through compact-based higher order corrections. Numerical experiments demonstrate improved sharpness in capturing shock waves and precision in resolving small scale structures in complex conditions. Despite the significantly improved accuracy, the extra computational cost brought by the new methods only marginally increases compared to original WENO. The method is further investigated on lower order schemes and still shows its ability to improve both accuracy and robustness.

Presentation 2

Speaker: Yuzhong Huang (Jackson State University)

Co-author: Yonghua Yan (Jackson State University)

Title: Predicting High-Speed Complex Fluid Dynamics Using Machine Learning

Abstract: While machine learning has made significant inroads across scientific research domains, applying it directly to predict 3D complex flows remains challenging. In this study, we address this difficulty by leveraging high-fidelity numerical solutions obtained from previous simulations. Our approach involves a machine learning-based prediction method specifically tailored for the 3D evolution of coherent vortex structures in high-speed flows. To achieve this, we employ advanced vortex identification techniques alongside Principal Component Analysis (PCA) and Proper Orthogonal Decomposition (POD) analysis. These methods allow us to extract common spatial modes associated with the vortex structures and their corresponding time coefficients. Additionally, deep learning-based time series models come into play for predicting these time coefficients. Through this process, we reconstruct the temporal development of major coherent structures within complex fluid flows. Remarkably, our predicted results exhibit strong consistency with the outcomes obtained from numerical simulations, demonstrating the efficacy of our approach in capturing the intricate dynamics of 3D flow phenomena.

Presentation 3

Speaker: Shiming Yuan (Jackson State University)

Co-author: Paris Smith (Jackson State University), Caixia Chen (Jackson State University), Yuzhong Huang (Jackson State University)

Title: Asymmetric Structure Evolution in Late Transitional Boundary Layers: A DNS Study

Abstract: This study aims to contribute to the understanding of coherent structure evolution in late transitional boundary layers. Utilizing DNS of Mach 0.5 flow over a flat plate, we examine the nonlinear growth of large vortex structures and the development of asymmetric coherent structures. Our methodology employs only TS waves as inlet input. Preliminary results suggest a potential correlation between these asymmetric structures and the gradient of vortex magnitude. While further research is necessary to confirm these findings, this work offers insights into the complex dynamics of transitional boundary layers in compressible flows.

Presentation 4

Speaker: Jagdish Gnawali (Texas Tech University)

Co-author: W. Brent Lindquist (Texas Tech University), S. T. Rachev (Texas Tech University)

Title: Hedging in Trinomial Option Pricing with Perpetual Derivative

Abstract: We introduce a fairly general, recombining trinomial tree model in the natural world. Market completeness is ensured through the addition of a perpetual derivative. Using a replicating portfolio, we develop prices for European call and put options and generate the unique relationships between the risk-neutral and real-world parameters of the model. We discuss calibration of the model to empirical data in the cases in which the risky asset returns are treated as either arithmetic or logarithmic. From historical data for selected large cap stocks, we develop implied parameter surfaces for all real-world parameters in the model.

5. MS 5 (SDRICH Rm 213) - Recent trends in highly accurate and preservative numerical methods for data approximation and differential equations, Part 3/3

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: Henry Han (School of Computer Science & Engineering, Baylor University)

Co-author: N/A

Title: KAN can be more likely to encounter overfitting than MLP?

Abstract: KAN (Kolmogorov-Arnold Network) emerges as a promising alternative to the MLP (Multi-Layer Perceptron) due to its powerful function approximation capabilities. KAN leverages spline functions with learnable parameters to enable more customized learning. However, it remains unclear whether this model is more robust to overfitting, as MLPs are known to be sensitive to it. In this study, we examine overfitting in both KAN and MLP. Our findings show that KAN with default B-spline activation functions exhibits higher Rademacher complexity than MLP during learning, indicating a greater capacity for overfitting. However, a KAN using wavelet activation functions demonstrates lower Rademacher complexity compared to both a KAN with default spline functions and an MLP with the same network architecture. This suggests that the choice of activation function in KAN plays a critical role in its susceptibility to overfitting. Using wavelet activation functions introduces better control over model complexity and promotes sparsity. These results indicate that, with the appropriate activation function, KAN can more effectively balance model capacity and generalization.

Presentation 2

Speaker: Eduardo Servin Torres (Department of Mathematics and CASPER, Baylor University)

Co-author: Qin Sheng (Department of Mathematics and CASPER, Baylor University)

Title: A preliminary report of the numerical analysis for approximating solutions of $n$-dimensional Kawarada equations

Abstract: Kawarada equations are well suited to model multi-physical phenomena found in energy and biomedical fields. Their strong singularities are due to the non-linear terms, which force the solution's time derivative to become unbounded while the solutions themselves remain bounded. In traditional simulation approaches, the non-linear terms are locally frozen, that is, no changes are to be allowed within each temporal step advancement. Based on such an approach the numerical stability is analyzed. In this talk, we will focus on a new novel analysis without freezing the non-linear source terms. Careful numerical stability and convergence results will be established. Both pointwise and averaged convergence orders will be provided utilizing the Milne device. Non-uniform temporal meshes will be used throughout. At last, 3D simulations will be shown to illustrate our theoretic results and more general approaches in $n$-dimensional cases.

Presentation 3

Speaker: Jonathan Hu (School of Engineering and Computer Science, Baylor University)

Co-author: Joshua T. Young (Baylor University), Curtis Menyuk (University of Maryland - Baltimore County)

Title: Modeling Transverse Mode Instability in High-Power Amplifiers

Abstract: High-energy lasers using fiber amplifiers have been the subject of significant interest due to their capability to produce high output powers. One of the most prominent nonlinear effects limiting output power in fiber amplifiers is the transverse mode instability (TMI) [1]. TMI occurs due to quantum defect heating, which creates a thermally-induced refractive index grating that facilitates coupling between different optical transverse modes. To better understand and mitigate TMI, efficient computational models are necessary. Coupled-mode equations have proven effective in modeling TMI [2], but they are computationally demanding. In this talk, we describe a model for analyzing TMI in fiber amplifiers that we developed [3] and led to a 100× speedup relative to the earlier model [2] with no loss of accuracy. This phase-matched model disregards the rapidly oscillating terms in the coupled-mode equations, which do not significantly impact the coupling between the fundamental mode and higher-order modes [3]. Instead, this model focuses solely on the terms that play a crucial role in mode coupling. Consequently, it allows for a much coarser longitudinal discretization along the fiber. We compare the full model [2] and the phase-matched model [3] for TMI over a realistic fiber length of 10 meters. The computational speedup of the phase-matched model enables the integration of both the TMI and Brillouin instability [4] simulations within a single framework for the first time. These two nonlinear optical effects both limit the power of fiber amplifiers and had previously been considered separately. Combining both instabilities in a single simulation makes it possible to optimize the fiber design to minimize both effects simultaneously and yield the highest power threshold [5], facilitating the development of powerful optimization tools for fiber amplifiers.

[1] C. Jauregui, C. Stihler, and J. Limpert, “Transverse mode instability,” Adv. Opt. Photonics 12(2), 429–484 (2021).

[2] S. Naderi, I. Dajani, T. Madden, et al., “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21(13), 16111–16129 (2013).

[3] C. R. Menyuk, J. T. Young, J. Hu, et al., “Accurate and Efficient Modeling of the Transverse Mode Instability in High Energy Laser Amplifiers,” Opt. Express 29(12), 17746–17757 (2021).

[4] J. T. Young, C. R. Menyuk, and J. Hu, “SBS suppression using PRBS phase modulation with different orders,” Opt. Express 31, 18497–18508 (2023).

[5] J. T. Young, A. J. Goers, D. M. Brown, et al., “Tradeoff between the Brillouin and transverse mode instabilities in Yb-doped fiber amplifiers,” Opt. Express 30, 40691–40703 (2022)

Presentation 4

Speaker: Bo Deng (Department of Mathematics, University of Nebraska-Lincoln)

Co-author: N/A

Title: Error-free Training for Artificial Neural Networks

Abstract: We define one important aspect of intelligence to not make the same mistakes twice. To achieve this artificial intelligence a system must be able to learn from its mistakes every time. For artificial neural networks models, this entails to train them error-free without any training data left behind. In this talk, I will discuss a newly discovered method that can train ANN models to perfect precision. I will outline the ideas from Dynamical Systems that guarantee the convergence of the error-free training algorithm, and show simulations on the most popular benchmark data for training algorithms in the field.

6. MS 6 (SDRICH Rm 214) - Nonlinear Algebra in Applications, Part 3/3

Organizer

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: Daniel Bernstein (Tulane University)

Co-author: N/A

Title: Matroid lifts and representability

Abstract: Given matroids $M$ and $N$ on the same ground set, one says that $M$ is a lift of $N$ if every flat of $N$ is a flat of $M$. This happens when $M$ is the matroid on the column-set of a matrix, and $N$ is the matroid on the columns of a row-submatrix. In this talk, I will discuss how matroid lifts can be used to derive conditions necessary for a matroid to be representable. Time permitting, I will discuss how similar results might be obtained to derive conditions necessary for a matroid to be algebraic.

Presentation 2

Speaker: Yingying Wu (University of Houston)

Co-author: N/A

Title: Comparison Theorems of Moduli Spaces

Abstract: The moduli space of an object contains rich information about that object and consequently provides information to discover or construct the object being parameterized by the moduli space. The moduli space of trees and the moduli space of networks are homeomorphic to algebraic fans spanned by root subsystems of type D that arise in the moduli space of smooth marked del Pezzo surfaces. This correspondence sheds a promising light on the study of mathematical biology in terms of understanding and discovering evolutionary spaces. I will also briefly introduce how they are tied to the moduli space of algebraic curves. Then, I will extend to the moduli space of super curves, which are algebraic curves with additional supersymmetric or supergeometric structures. I will conclude my talk with an exposition on the construction of dual graphs of SUSY curves with Neveu-Schwarz and Ramond punctures and describe how the moduli space of genus 0 SUSY graphs coincides with the aforementioned moduli space of trees.

Presentation 3

Speaker: John Cobb (Auburn University)

Co-author: Matthew Faust (Michigan State University), David Barnhill (Naval Postgraduate School)

Title: Likelihood Correspondence of Statistical Models

Abstract: Maximum likelihood estimation (MLE) is a fundamental problem in statistics. Characteristics of the MLE problem for algebraic statistical models are reflected in the geometry of the likelihood correspondence, a variety that ties together data and their maximum likelihood estimators. I'll discuss how to construct the likelihood correspondence for the large class of toric models and find a Grőbner basis in the case of complete and joint independence models. These results provide insight into their properties and offer faster computational strategies for solving the MLE problem.

Presentation 4

Speaker: Shixuan Zhang (Texas A&M University)

Co-author: N/A

Title: Spurious minima in nonconvex sum-of-square optimization via syzygies

Abstract: We study spurious local minima in a nonconvex low-rank formulation of sum-of-squares optimization on a real variety X. We reformulate the problem of finding a spurious local minimum or stationary points in terms of syzygies of the underlying linear series, and also bring in topological tools to study this problem. When the variety X is of minimal degree, there exist spurious stationary points if and only if both the dimension and the codimension of the variety are greater than one, answering a question by Legat, Yuan, and Parrilo. Moreover, for surfaces of minimal degree, we provide sufficient conditions to exclude points from being spurious local minima. In particular, all spurious local minima on the Veronese surface, corresponding to ternary quartics, lie on the boundary and can be written as a binary quartic, up to a linear change of coordinates, complementing work by Scheiderer on decompositions of ternary quartics as a sum of three squares. For general varieties of higher degree, we give examples and characterizations of spurious local minima in the interior, and provide numerical experiment results demonstrating the effectiveness of the low-rank formulation.

7. MS 7 (SDRICH Rm 216 - Accurate, robust, and efficient methods for computational fluid dynamics, Part 2/2

Organizers

Jesse Chan (Rice University), Matthias Maier (Texas A&M University)

Mini-symposium Abstract:

Robust and reliable numerical methods for fluid dynamics are increasingly necessary for high-performance computing, where manual interventions and problem-dependent parameter tuning become untenable. This mini-symposium targets theoretical and practical aspects of such methods, especially for the nonlinear partial differential equations governing complex fluid flows.

Presentation 1

Speaker: David Pecoraro (Texas A&M University)

Co-author: N/A

Title: Greedy Viscosity for the Compressible Euler Equations in the Low Mach Regime

Abstract: It is known that viscosity based methods for the compressible Euler equations become inefficient and inaccurate in the low Mach regime. This lack of performance stems from a sound speed based artificial viscosity which results in a restrictive CFL condition and excessive diffusion of the numerical solution. In this talk a first order, invariant domain preserving, graph based viscosity scheme is examined in the low Mach regime and a method to alleviate the restrictive CFL condition via reduction of the graph viscosity is presented.

Presentation 2

Speaker: Lander Besabe (University of Houston)

Co-author: Lander Besabe (University of Houston), Michele Girfoglio (SISSA), Annalisa Quaini (University of Houston), and Gianluigi Rozza (SISSA)

Title: A data-driven reduced order model for two-layer quasi-geostrophic oceans

Abstract: The simulation of ocean flows is extremely challenging for several reasons. The first is of course the scale of the problem: the area of an ocean basin is of the order of millions of Km$^{2}$. The second reason is the nature of the flow itself, which requires very fine computational meshes to resolve all the eddy scales. The third reason is connected to the first two: high resolution meshes over very large domains lead to a prohibitive computational cost with nowadays computational resources. In this talk, we present a data-driven reduced order model (ROM) for the two-layer quasi-geostrophic equations, a simplified, yet nontrivial, model for ocean dynamics. The main building blocks of our ROM are Proper Orthogonal Decomposition (POD) and a Long Short Term Memory (LSTM) architecture. Using an extension of the classical double-gyre wind forcing test, we assess the accuracy of the POD-LSTM ROM both in the reconstruction and prediction of the flow and quantify the drastic reduction in the computational time allowed by the method.

Presentation 3

Speaker: Samuel Kwan (Rice University)

Co-author: N/A

Title: A first order meshfree method for time-dependent nonlinear conservation laws

Abstract: We introduce a robust first order accurate mesh-free method to numerically solve time-dependent nonlinear conservation laws. The main ingredient behind the method are first order consistent summation by parts derivative operators which can be efficiently constructed. We study the performance of such derivative operators, and then combine these operators with a numerical flux-based formulation to approximate the solution of nonlinear conservation laws. We observe numerically that, while the resulting meshfree differentiation operators are only $O(h^\frac{1}{2})$ accurate in the $L^2$ norm, the error when approximating solutions to PDEs is O(h).

Presentation 4

Speaker: Darsh Nathawani (Louisiana State University)

Co-author: Darsh Nathawani (Louisiana State University), Chris Kees (Louisiana State University)

Title: A robust and accurate approach to simulate free surface flows

Abstract: Droplet breakup from a free surface flow comprises complex dynamics of the interfaces. Droplet pinch-off happens in finite time due to surface tension. The singularity region yields a self-similar solution and preserves a topological symmetry. Therefore, an asymptotic model with a front-tracking approach can accurately represent the interface and precisely predict the droplet volume. We aim to develop a robust and accurate approach to simulate the droplet pinch-off process and compare to more generic free surface and multiphase flow approaches that can simulate pinch-off and other complex flows with topological changes, such as level set methods. Challenges in these problems include accurately modeling the jump conditions across the interface, obtaining high-order convergence, and maintaining conservation properties, particularly mass and volume in incompressible flows. Recent advancements in the Immersed Finite Element Methods (IFEM) provide strongly enforcing jump conditions by modifying the basis functions in the cut cells, thereby explicitly enforcing jumps in scalar variables gradients. The locally constructed element basis results in a non-conforming finite element method. Adding a penalty term for the cell boundary yields a symmetric non-conforming method with optimal convergence. An alternative approach is using Nitsche's approach to enforce the jump condition weakly along the immersed boundary, as was recently demonstrated for fluid-solid flows using level set methods for the fluid-solid interface. We compare the front-tracking and level set methods in this work.

8. MS 22 (SDRICH Rm 326) - Advances in Algebra, Topology and Geometry with Applications to Data Analysis

Organizer

Bradley Vigil (Texas Tech University)

Mini-symposium Abstract:

In the rapidly evolving landscape of data analysis, the interplay between algebra, topology, and geometry has opened new frontiers for extracting meaningful insights from complex data. This minisymposium focuses on recent research advances that combine one or more of algebra, topology and geometry to address both current and emerging problems in data science; these include the recognition of patterns and structures in high-dimensional data sets, alongside effective methods to interpret the results of a data analysis to motivate novel insights and discoveries. Contributions of these recent research developments to robust and scalable machine learning, network analysis, and network dynamics will be emphasized.

Presentation 1

Speaker: Bradley Vigil (Texas Tech University)

Co-author: Travis Thompson (Texas Tech University)

Title: Making a Complex Choice

Abstract: From tumors to trending news, nature is filled with complex processes; sometimes bursting and sometimes flowing, complex processes generate complex data. Topological data analysis has evolved, over the last two decades, into a framework that couples theory, from algebra and topology, to large-scale computations with the aim of providing quantitative insights into data that we could only previously describe qualitatively. Since its introduction, a number of authors have proposed various sequential choices of simplicial complexes to describe a set of data. In this talk, I will describe my development of a new perspective on making complex choices; I will demonstrate the flexibility of the theory by defining the first filtrations for studying propagations on networks that generalize the ideas of the Vietoris-Rips complex to networks without the use of any metric embedding.

Presentation 2

Speaker: Brighton Nuwagira (University of Texas at Dallas)

Co-author: Li Qiwei (University of Texas at Dallas), Baris Coskunuzer (University of Texas at Dallas)

Title: Topo-CNN: Breast Cancer Detection with Topological Deep Learning

Abstract: We investigate the application of topological deep-learning techniques in medical diagnosis, particularly in breast cancer screening. Despite the effectiveness of convolutional neural networks (CNNs) in analyzing medical images, challenges such as interpretability and the need for extensive labeled data persist. Topological machine learning offers a unique approach to uncovering shape patterns within complex datasets, potentially addressing these challenges. Additionally, while CNNs excel at capturing detailed local spatial relationships within images, topological data analysis (TDA) methods, in particular persistent homology, provide a holistic perspective by analyzing data globally, offering complementary insights to CNN features. We propose the integration of these features into our Topo-CNN model, specifically tailored for breast cancer screening, the most prevalent cancer in women. Experimental results on benchmark ultrasound datasets consistently demonstrate that incorporating topological features statistically significantly enhances the performance of CNNs across various architectures. Furthermore, we prove a generalization of Alexander duality theorem for cubical persistence, implying that persistent homology output remains the same for sublevel and superlevel filtrations for image data, in contrast to the graph setting.

Presentation 3

Speaker: Peter Stiller (Texas A&M University)

Co-author: N/A

Title: Clustering Theory and Functoriality

Abstract: A well-known result of Kleinberg shows that no clustering algorithm exists that satisfies three eminently reasonable and desirable properties. Later work of Carlsson and Memoli adopted "functoriality" as the guiding property for the design of mathematically analyzable clustering algorithms. We will discuss alternative criteria which preserve functoriality, but which offer a broader array of functorial clustering schemes.

Presentation 4

Speaker: Henry Adams (University of Florida)

Co-author: Joshua Mirth, Yanqin Zhai, Johnathan Bush (University of Florida), Enrique Alvarado (UC Davis), Howie Jordan (University of Colorado Boulder), Mark Heim, Bala Krishnamoorthy (Washington State University), Markus Pflaum (University of Colorado Boulder), Aurora Clark (University of Utah), Yang Zhang

Title: Representations of Energy Landscapes by Sublevelset Persistent Homology

Abstract: Encoding the complex features of an energy landscape is a challenging task, and often chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e. 2- or 3-dimensions. Even though disconnectivity graphs or merge trees summarize the connectivity of the local minima of an energy landscape via the lowest-barrier pathways, there is more information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane C_m H_{2m+2} potential energy landscapes, for all m, and in all homological dimensions. We further compare both the analytical configurational potential energy landscapes and sampled data from molecular dynamics simulation, using the united and all-atom descriptions of the intramolecular interactions.

9. MS 9 (SDRICH Rm 225) - Geometric and Topological Approaches in Data Science and Machine Learning, part 3/3

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: Keaton Hamm (University of Texas at Arlington)

Co-author: N/A

Title: Manifold Learning in Wasserstein Space

Abstract: We will discuss the problem of manifold learning on data that is a collection of probability measures. The notion of geometry will arise from the Wasserstein distance between probability measures. Analogues of multidimensional scaling and isomap will be discussed, and we show how one can make the computation of the embedding feasible by linearization. Experiments on synthetic data and toy experiments on MNIST exhibit the accuracy of the embeddings.

Presentation 2

Speaker: Weihua Geng (Southern Methodist University)

Co-author: N/A

Title: A Biophysics DNN Model with Topological and Electrostatic Features

Abstract: In this project, we provide a deep-learning neural network (DNN) based biophysics model to predict protein properties. The model uses multi-scale and uniform topological and electrostatic features generated with protein structural information and force field, which governs the molecular mechanics. The topological features are generated using the element specified persistent homology (ESPH) while the electrostatic features are fast computed using a Cartesian treecode. These features are uniform in number for proteins with various sizes thus the broadly available protein structure database can be used in training the network. These features are also multi-scale thus the resolution and computational cost can be balanced by the users. The machine learning simulation on over 4000 protein structures shows the efficiency and fidelity of these features in representing the protein structure and force field for the predication of their biophysical properties such as electrostatic solvation energy. Tests on topological or electrostatic features alone and the combination of both showed the optimal performance when both features are used. This model shows its potential as a general tool in assisting biophysical properties and function prediction for the broad biomolecules using data from both theoretical computing and experiments.

Presentation 3

Speaker: Astrit Tola (The University of Texas at Dallas)

Co-author: N/A

Title: TopER: Topological Embeddings in Graph Representation Learning

Abstract: Graph embeddings serve as the cornerstone for graph representation learning, facilitating the exploration of graphs by machine learning methods. However, prevalent deep learning techniques rely on black-box, high-dimensional graph embeddings. There is a pressing need for an interpretable, low-dimensional embedding approach to empower efficient graph visualization and provide practical tools to study graph datasets effectively.

In this paper, we present a novel low-dimensional graph embedding method called Topological Evolution Rate (TopER), which simplifies a key concept of topological data analysis known as filtration. TopER calculates the evolution rate of graph substructures induced by a filtration function on nodes or edges, resulting in interpretable 2D visualizations of graph datasets. Our experiments demonstrate that this new embedding method achieves highly competitive performance com- pared to the latest deep learning models in graph classification tasks on benchmark datasets. We further provide theoretical stability guarantees for TopER.

Presentation 4

Speaker: Qixing Huang (University of Texas at Austin)

Co-author: N/A

Title: 4DRecons: 4D Neural Implicit Deformable Objects Reconstruction from a single RGB-D Camera with Geometrical and Topological Regularizations

Abstract: I will present novel approach 4DRecons that takes a single camera RGB-D sequence of a dynamic subject as input and outputs a complete textured deforming 3D model over time. 4DRecons encodes the output as a 4D neural implicit surface and presents an optimization procedure that combines a data term and two regularization terms. The data term fits the 4D implicit surface to the input partial observations. We address fundamental challenges in fitting a complete implicit surface to partial observations. The first regularization term enforces that the deformation among adjacent frames is as rigid as possible (ARAP). To this end, we introduce a novel approach to compute correspondences between adjacent textured implicit surfaces, which are used to define the ARAP regularization term. The second regularization term enforces that the topology of the underlying object remains fixed over time. This regularization is critical for avoiding self-intersections that are typical in implicit-based reconstructions. We have evaluated the performance of 4DRecons on a variety of datasets. Experimental results show that 4DRecons can handle large deformations and complex inter-part interactions and outperform state-of-the-art approaches considerably

10. MS 23 (SDRICH Rm 224) - Novel Perspectives on Infectious Diseases Modeling

Organizers

Tamer Oraby (University of Texas Rio Grande Valley), Mohammad Mihrab Uddin Chowdhury (Clemson University), Md Rafiul Islam (University of the Incarnate Word)

Mini-symposium Abstract:

Mathematical modeling and computational approaches are crucial in understanding the dynamics of infectious diseases, particularly through the lens of human and pathogen factors. As life sciences increasingly integrate quantitative approaches, these tools have become indispensable in the shaping of effective control and prevention strategies. This mini-symposium will explore how advanced mathematical and computational models can predict disease progression and inform interventions. By convening researchers, mathematicians, and public health professionals, the symposium aims to foster collaboration and knowledge exchange, highlighting how new perspectives can enhance disease modeling and lead to more effective evidence-based strategies to combat infectious diseases.

Presentation 1

Speaker: Thoa Thieu (School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley)

Co-author: William Holmes (Indiana University)

Title: Modeling Interactions Between Microtubule Cytoskeleton and Insulin Granules in pancreatic $\beta$ cell

Abstract: Glucose-stimulated insulin secretion (GSIS) in pancreatic $\beta$ cells is vital to metabolic homeostasis. Such GSIS dynamics are affected by numerous factors and the cell's microtubule (MT) cytoskeleton plays a critical role in regulating insulin secretion. However, that critical role is still not fully understood. Recent results have shown two mechanisms by which the microtubule cytoskeleton negatively regulates insulin secretion: 1) by limiting the amount of insulin near the plasma membrane and 2) by inhibiting or breaking the interactions between the plasma membrane and the remaining nearby insulin granules. In this work, we construct a computational model of the MT cytoskeleton in insulin-producing cells in 3D. Using this model, we investigate how the structure of the MT cytoskeleton influences the transportation of insulin within the cell, with a specific focus on how the porosity of the filament network influences the ability of insulin to reach the plasma membrane.

Presentation 2

Speaker: Tamer Oraby (The University of Texas Rio Grande Valley)

Co-author: Kamal Jnawali (State University of New York at Oswego), Michael Tyshenko (Risk Sciences International)

Title: Mitigating Economic and Health Externalities in Cross-Border Disease Transmission: A Two-Country Model

Abstract: In this talk, I will introduce my work on modeling economic and health externality of diseases. Healthy populations are needed for productive workforce to contribute to the nation's economic growth. Using a two-country example, I studied the externality of a disease of poverty with its effect on economy and showed the importance of its mitigation through swift health aid. The model was parameterized using real-world data and calibrated to a disease of poverty. Finally, I will discuss the model's several components and present its practical recommendations.

Presentation 3

Speaker: Andras Balogh (University of Texas Rio Grande Valley)

Co-author: Tamer Oraby (University of Texas Rio Grande Valley)

Title: Leveraging High-Performance Computing to Model Disease Spread and Vaccination Decision-Making in Stochastic Networks

Abstract: In this study, we present a high-performance parallel computational framework designed to analyze two interconnected stochastic networks. These networks simulate parental vaccination decisions within a disease-spreading contact network. Our bilayer network model employs either Erdős-Renyi (random) or Barabasi-Albert (scale-free) structures. We incorporate a Bayesian aggregation rule for observational social learning and extend our model to include other decision-making frameworks, such as voting and DeGroot models. Utilizing a novel sparse storage format for adjacency matrices and the Python library CuPy, accelerated with NVIDIA CUDA on GPUs, we conduct statistical analyses on hundreds of random networks, each comprising hundreds of thousands of nodes representing households.

Presentation 4

Speaker: Md Rafiul Islam (University of the Incarnate Word)

Co-author: Claus Kadelka (Iowa State University), Audrey McCombs (Sandia National Laboratories), Jake Alston (Iowa State University), Noah Morton (Iowa State University)

Title: Impact of Ethnic Homophily on Disease Transmission and Vaccination Strategies

Abstract: In this talk, we will explore the intersection of disease transmission modeling and vaccination strategies, with a particular focus on the impact of ethnic homophily. Ethnic homophily, the tendency for individuals to form social connections with those of the same ethnicity, is a significant yet often overlooked factor in public health initiatives aimed at curbing the spread of infectious diseases. This study delves into the intricate relationship between ethnic homophily and disease transmission dynamics, emphasizing the critical role it plays in shaping effective vaccination strategies. We will discuss how understanding these social patterns can enhance the efficacy of vaccination programs and ultimately contribute to more equitable health outcomes across diverse communities.

11. MS 10 (SDRICH Rm 324) - Integral and Meshfree Methods in Fluid Mechanics and Electrokinetics, Part 2/2

Organizer

Vladimir Ajaev (Southern Methodist University), Johannes Tausch (Southern Methodist University)

Mini-symposium Abstract:

The focus of the minisymposium is on the numerical solution of partial differential equations for fluid flow and electrical charge transport in situations when meshfree and boundary integral methods offer a promising alternative to the standard finite element approaches. The minisymposium will address recent advances in the theory, computational techniques, and applications of these methods. In the context of fluid mechanics, the linearity of Stokes flow can be used to reduce problems in terms of boundary integral equations or expand the solutions in infinite series. To that end, efficient numerical methods will be discussed that address the non-locality and singularities of the integral operator as well as the convergence of series expansions. Similar modeling and computational issues arise in more complicated situations, which may have structured geometries or geometries that evolve in time. The invited speakers will consider two-phase flow of a viscous drop immersed in a surrounding viscous fluid which contains a soluble surfactant. Other talks will address electrical charge transport in electrolyte solutions. The approaches discussed in the mini-symposium are important for a wide range of practical applications such as impedance spectroscopy, energy storage, electrochemistry, and microfluidics.

Presentation 1

Speaker: Johannes Tausch (Southern Methodist University)

Co-author: N/A

Title: Boundary element method for transient Stokes flow

Abstract: Flow problems with high viscosities or small length scales are often well approximated by a vanishing Reynolds number. This is known as Stokes flow. For the steady-state case boundary integral equation methods are by now widely used. However, for transient Stokes the corresponding boundary integral formulations are much less understood. In the time dependent case, the Green's tensor can be expressed in terms of incomplete gamma functions. This reveals the nature of the space-time singularities of the corresponding boundary integral operators. The Galerkin discretization with piecewise polynomial ansatz functions then leads to a block lower triangular system. We will discuss convergence with respect to mesh refinements and will discuss how the adaptive cross approximation (ACA) method can be used for its efficient solution.

Presentation 2

Speaker: Svetlana Tlupova (Farmingdale State College)

Co-author: J. T. Beale (Duke University)

Title: Extrapolated regularization of nearly singular integrals on surfaces

Abstract: We will discuss recent work on the numerical evaluation of single and double layer integrals for harmonic potentials or Stokes flow, at points near the surface. The singular kernel is first regularized using a length scale $\delta$ in order to control the discretization error. Local analysis is used to express the smoothing error having terms with unknown coefficients multiplied by known quantities. The extrapolation strategy is to compute the regularized integral for three choices of the smoothing parameter $\delta$ and solve the 3x3 linear system for the extrapolated value of the integral. This improves the smoothing error to $O(\delta^5)$, uniformly for target points on or near the surface. The extrapolation method can be extended to $O(\delta^7)$, with smaller errors, although the order of accuracy is less predictable in practice. We will present some numerical examples and discuss the choice of the regularization parameter $\delta$.

Presentation 3

Speaker: Joar Bagge (University of Texas Austin)

Co-author: George Biros (University of Texas Austin), Dhwanit Agarwal (University of Texas Austin)

Title: Simulation of deformable capsules moving through a pipe in periodic Stokes flow

Abstract: The flow of deformable capsules in either an unconfined or pipe-confined fluid is of interest e.g. in the study of suspensions of biological cells moving through microfluidic devices. Due to the small length scales involved, the flow is described by the Stokes equations, and boundary integral methods can be used to solve the flow equations. We present a GPU-accelerated framework for fully three-dimensional simulations of deformable capsules. We observe a speedup of around 10.5 when running on a single NVIDIA A100 GPU, compared to running on a 128-core AMD EPYC 7763 CPU.

An important part of our algorithm is the FFT-based fast Ewald summation method, which computes periodic interactions in O(N log N) time for N grid points. The capsule is represented using a partition of unity discretization, with numerical integration using the trapezoidal rule and regularization of the singular layer potential kernels.

A point of interest is to characterize the long-term dynamics of the capsule, under a wide range of flow and material parameters. This requires a large number of runs that can be made in parallel. The simulations are challenging due to the deforming capsule surface and the difficulty of maintaining a surface mesh of high quality over a long period of time. We will comment on these difficulties and the overall time-stability of the scheme.

Presentation 4

Speaker: Thomas Anderson (Rice University)

Co-author: Marc Bonnet (ENSTA Paris), Luiz M. Faria (ENSTA Paris), Carlos Perez-Arancibia (University of Twente)

Title: Fast, provably high-order accurate methods for volume integral operators

Abstract: Volume integral operators (VIOs) are fundamental tools for solving volumetric problems with integral equation methods, including wave scattering problems in inhomogeneous media. We will discuss a treatment using Green's third identity and a local polynomial interpolant of the density function that transforms a VIO into regularized VIOs (as well as layer potentials, and a polynomial PDE solution): the (globally!) regularized operators can be evaluated to provably high-order accuracy using entirely generic volumetric quadrature and can make efficient use of fast algorithms. Detailed error and stability analysis is provided in two dimensions for this regularizing effect, including connections to quadrature of functions of limited regularity at isolated points, and the guarantees will provide a point of contrast with the locally-corrected (near-)singular correction methods in widespread use.

12. MS 11 (SDRICH Rm 325) - Numerical methods for multi-physics problems, Part 3/3

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: Xiaoming He (Missouri University of Science and Technology)

Co-author: Xuejian Li, Wei Gong, and Craig Douglas

Title: Variational data assimilation with finite element discretization for second order parabolic interface equation

Abstract: In this talk, we present a finite element method of variational data assimilation for a second order parabolic interface equation on a two-dimensional bounded domain. The Tikhonov regularization plays a key role in translating the data assimilation problem into an optimization problem. Then the existence, uniqueness, and stability are analyzed for the solution of the optimization problem. We utilize the finite element method for spatial discretization and backward Euler method for the temporal discretization. Then based on the Lagrange multiplier idea, we derive the optimality systems for both the continuous and the discrete data assimilation problems for the second order parabolic interface equation. The convergence and the optimal error estimate are proved with the recovery of Galerkin orthogonality. Moreover, three iterative methods, which decouple the optimality system and significantly save computational cost, are developed to solve the discrete time evolution optimality system. Finally, numerical results are provided to validate the proposed method.

Presentation 2

Speaker: Peimeng Yin (The University of Texas at El Paso)

Co-author: Eirik Endeve, Cory D. Hauck and Stefan R. Schnake (Oak Ridge National Laboratory)

Title: Dynamical low-rank approximation for neutrino kinetic equations

Abstract: Dynamical low-rank approximation (DLRA) is an emerging tool for reducing computational costs and provides memory savings when solving high-dimensional problems. In this work, we propose and analyze a semi-implicit dynamical low-rank discontinuous Galerkin (DLR-DG) method for the space homogeneous kinetic equation with a relaxation operator, modeling the emission and absorption of particles by a background medium. Both DLRA and the discontinuous Galerkin (DG) scheme can be formulated as Galerkin equations. To ensure their consistency, a weighted DLRA is introduced so that the resulting DLR-DG solution is a solution to the fully discrete DG scheme in a subspace of the standard DG solution space. Similar to the standard DG method, we show that the proposed DLR-DG method is well-posed. We also identify conditions such that the DLR-DG solution converges to the equilibrium. Numerical results are presented to demonstrate the theoretical findings.

Presentation 3

Speaker: Matthew Dallas (University of Dallas)

Co-author: Sara Pollock (University of Florida), Leo G. Rebholz (Clemson University)

Title: Improving Newton's Method Near Bifurcation Points with Anderson Acceleration

Abstract: We present an adaptive safeguarding scheme with a tunable parameter, which we call adaptive gamma-safeguarding, that one can use in tandem with Anderson acceleration to improve the performance of Newton's method when solving problems at or near singular points. Newton-Anderson with adaptive gamma-safeguard converges locally for singular problems; it detects nonsingular problems automatically and responds by scaling the iterates towards standard Newton asymptotically. The result is a flexible algorithm that performs well for singular and nonsingular problems and can recover convergence from both standard Newton and Newton-Anderson with the right parameter choice. We will discuss three strategies one can use when implementing Newton-Anderson and gamma-safeguarded Newton-Anderson to solve parameter-dependent problems near singular points. These strategies will be demonstrated by applying them to two incompressible flow problems near bifurcation points.

Presentation 4

Speaker: Mallikarjunaiah S. Muddamallappa (Texas A&M University-Corpus Christi)

Co-author: N/A

Title: On Wave Propagation in New Class of Elastic Bodies

Abstract: In this talk, I will introduce a model for wave propagation in elastic materials, where the material's moduli are dependent on density. I will derive a new formulation based on implicit theories of elasticity, incorporating internal variables dependent on pointwise density. Analytical solutions for a one-dimensional elastodynamics problem will be detailed, specifically for a certain category of constitutive law. Additionally, I will develop a continuous Galerkin-type finite element method using Picard's type linearization. I will present some interesting results obtained from simulating important bench problems.

13. MS 19 (SDRICH Rm 344) - Bifurcation and Symmetry in Nonlinear Differential Equations, Part 2/2

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: Jingzhou Liu (The University of Texas at Dallas)

Co-author: Carlos García-Azpetia (Universidad Nacional Autónoma de México), Anna Gołebiewska (Nicolaus Copernicus University), Wieslaw Krawcewicz (The University of Texas at Dallas)

Title: Global Bifurcation In Four-Component Bose-Einstein Condensates In Space

Abstract: We investigate a system of coupled Bose-Einstein condensates in the domain of a unitary ball in $R^3$. The coupling is due to atom-to-atom interactions that occur between different components. The multi-component Bose-Einstein condensate is described by a system of Gross-Pitaevskii equations, which has an explicit trivial branch of constant solutions. Our main theorem establishes that this trivial branch undergoes multiple global bifurcations at simple critical values with a kernels of dimensions 3(2k + 1), for $k \in \mathbb{N}+$. Handling these high dimension kernels poses a challenge from the perspective of bifurcation theory. Our methodology, which relies on the G-equivariant gradient degree, effectively manages these complexities and, in the case k = 1, we establishes the existence of at least 13 different branches bifurcating with different symmetries.

Presentation 2

Speaker: Oleg Makarenkov (The University of Texas at Dallas)

Co-author: N/A

Title: Bifurcation of attracting limit cycles from equilibrium in switched models of passive walkers

Abstract: The simplest model of a passive biped walking down a slope is given by switched coupled pendula. The linear equations of small angle approximation can be solved in closed form and the existence of a family of cycles (i.e. potential walking cycles) can be computed in closed form. As observed by Garcia et al. [J. Biomech. Eng. 120 (1998)], the family of cycles disappears when the angle increases and only isolated asymptotically stable cycles (walking cycles) persist. The talk presents a proof [Proc. A. 476 (2020)] of this statement using a suitable perturbation theorem for maps.

Presentation 3

Speaker: Tomoki Ohsawa (The University of Texas at Dallas)

Co-author: N/A

Title: Relative Dynamics and Stability of Point Vortices

Abstract: I will talk about a Hamiltonian formulation of the relative dynamics of the planar $N$-vortex problem as well as a stability condition for its relative equilibria. The relative dynamics is the dynamics of the "shape" formed by the point vortices: For example, if $N = 3$, it is the dynamics of the shape of the triangle formed by three vortices, regardless of their position and orientation of the triangle. A relative equilibrium is a solution in which the vortices move in such a manner that the shape formed by the vortices does not change in time. The main result is a sufficient condition for the stability of relative equilibria using the Hamiltonian formulation of the relative dynamics.

14. MS 24 (SDRICH Rm 340) - Spectral methods in disordered systems, Part 1/2

Organizers

Wencai Liu (Texas A&M University), Matthew Powell (Georgia Institute of Technology), Xueyin Wang (Texas A&M University)

Mini-symposium Abstract:

This mini-symposium will cover many topics under the umbrella of spectral theory and its applications to studying various disordered systems.

Presentation 1

Speaker: Stephen Shipman (Louisiana State University)

Co-author: N/A

Title: Overview of algebra in periodic graph operators

Abstract: I will attempt to give an overview of commutative algebra in spectral theory of periodic graph operators. I will clarify the role of analysis vs the role of algebra and the contributions of algebraic techniques to questions that seem analytic in nature. The most salient algebraic object is the Bloch (Fermi) variety in energy-momentum (momentum) space. I will present some new algebraic questions arising from tight-binding models of bilayer graphene.

Presentation 2

Speaker: Iris Emilsdottir (Rice University)

Co-author: N/A

Title: Johnson-Schwartzman Gap Labeling for Shifts

Abstract: Johnson-Schwartzman type theorems are a key tool in the field of gap labeling. Using them, we can compute groups that contain all spectral gap labels for several classes of dynamically defined operators. The Schwartzman group, however, does not tell us which labels can occur for which sampling function. Our focus will be on operators defined by shifts on compact alphabets. We will discuss recent work with Damanik and Fillman where we find the Schwartzman groups for several families of shifts, give upper bounds for the Schwartzman group of quasi-Sturmian subshifts, and construct cases where the spectral gap labels are bona-fide subsets of the Schwartzman group.

Presentation 3

Speaker: Xin Liu (Texas A&M University)

Co-author: N/A

Title: Singular limit in compressible flows

Abstract: Depending on the physical scale of the flows, different models are used to study the dynamics. To justify these models rigorously, singular limits are often involved. The famous fast oscillation theory from Schochet makes use of the structure of eigenvalue-eigenfunction pairs for the corresponding operator to identify the slow waves, the fast waves, and the resonances (wave interaction) in the nonlinear problems. In this talk, I will use our recent study of the incompressible limit for hydrostatic flow as an example to explain the key ingredient in singular limit in compressible flows.

15. MS 25 (SDRICH Rm 226) - Spectral theory with applications to mathematical physics

Organizer

John Zweck (University of Texas at Dallas)

Mini-symposium Abstract:

We highlight recent work in spectral theory with applications to twisted bilayer graphene, Schrődinger operators with periodic potentials, and the stability of nonlinear waves and short pulse lasers.

Presentation 1

Speaker: Christoph Fischbacher (Baylor University)

Co-author: N/A

Title: Slow propagation velocities of discrete Schrődinger operators in large periodic potential

Abstract: I will present some recent joint work with Abdul-Rahman, Darras, and Stolz (https://arxiv.org/abs/2401.11508). While periodic Schrődinger operators have purely ac spectrum and exhibit ballistic transport, I will show that if the potential is large enough, it is possible to make the velocity of this transport arbitrarily small. I will discuss the special case of period 2, where things can be computed explicitly and then talk about the case of general period p.

Presentation 2

Speaker: Tal Malinovitch (Rice University)

Co-author: N/A

Title: Twisted Bilayer Graphene in Commensurate Angles

Abstract: Graphene is an exciting new two-dimensional material. Though it was considered theoretical for a long time, it was isolated about 20 years ago. Since then, it has drawn much attention due to its numerous exciting properties. More recently, it was discovered that when twisting two layers of graphene with respect to each other, at certain angles called "magic angles", exotic transport properties emerge. The primary tool for studying this thus far is the famous Bistritzer-MacDonald model, which relies on several approximations.

This work aims to build the first steps in studying magic angles without using this model. Thus, we study a model for TBG without the approximations mentioned above in the continuum setting, using two copies of potential with the symmetries of graphene, sharing a common origin and twisted with respect to each other (so-called TBG in AA stacking). We describe the angles for which the two twisted lattices are commensurate and prove the existence of Dirac cones for such angles. Furthermore, we show that for small potentials, the slope of the Dirac cones is small for commensurate angles that are close to incommensurate angles. This work is the first in a series of works to build a more fundamental understanding of the phenomenon of magic angles.

In this talk, I will introduce the main phenomena of twisted bilayer graphene and state our main results.

Presentation 3

Speaker: Yongming Li (Texas A&M University)

Co-author: Jonas Lűhrmann (Texas A&M University)

Title: Asymptotic stability of solitary waves for the 1D cubic NLS under even perturbations

Abstract: I will present an overview for our proof of the asymptotic stability of solitary waves for the 1D cubic NLS under even perturbations, which is based on a combination of modulation techniques and a space-time resonances approach. The main challenges are the threshold resonances of the linearized operator and the resulting slow local decay of the Schrődinger waves. Remarkable null structures in the evolution equation for the radiation term as well as in the modulation equations play an important role in the proof. This is joint work with Jonas Lűhrmann (Texas A&M University).

Presentation 4

Speaker: John Zweck (University of Texas at Dallas)

Co-author: Erika Gallo (University of Texas at Dallas), Yuri Latushkin (University of Missouri)

Title: Spectral stability via the Fredholm determinant of a trace class Birman-Schwinger operator

Abstract: We propose a novel computational method to determine the spectral stability of stationary pulse solutions of nonlinear wave equations such as the cubic-quintic complex Ginzburg-Landau equation in one spatial dimension. Specifically, we show that the point spectrum of the linearization of the equation about a pulse is given by the zero set of the regular Fredholm determinant of a trace-class Birman-Schwinger operator. This operator is defined in terms of a Green's kernel for the linearized equation. We adapt a method of Bornemann to numerically approximate the Fredholm determinant by a matrix determinant, and we quantify the error in this approximation. Numerical results show excellent agreement with existing methods. The new method avoids the computational challenge of solving a stiff system for the Jost solutions that is inherent in the standard approach based on computation of the Evans function. The motivation for developing the method was for future extensions to determine the spectral stability of time-periodic pulses, for which an Evans function most likely cannot be defined.