SIAM Mini-Symposium Session 1
Organizer
Lisa M. Kuhn (Southeastern Louisiana University)
Mini-symposium Abstract:
Models driven by partial differential equations (PDEs) and PDE systems are renowned for their complexity and high computational demands. Enhancing the accuracy and efficiency of solutions to PDEs is becoming increasingly crucial. This minisymposium will highlight recent advancements in efficient numerical algorithms, including parallel computing and machine learning methods, for solving PDEs. It will provide a forum for discussing modeling techniques, algorithmic innovations, and computational tools used to simulate PDE solutions.
Presentation 1
Speaker: Lisa M. Kuhn (Southeastern Louisiana University)
Co-author: Allen Mire (Southeastern Louisiana University)
Title: Neural Finite Element Control
Abstract: Artificial neural networks are increasingly being employed to solve partial differential equations, but issues of controlling these equations using such methods remain unresolved. This talk will present recent results that blend a neural finite element approach with a partial differential equation control technique using a one-dimensional heat equation. Linear quadratic control is utilized on the neural element model. Training is applied on the associated algebraic Riccati equations. The results show that the control objective is achieved, indicating the neural element method shows promise for future partial differential equation control techniques.
Presentation 2
Speaker: Justin Biggs (Louisiana Tech University)
Co-author: Jonathan Walters (Louisiana Tech University)
Title: Simulating multiple component beam structures with linear control design and heave flight dynamics
Abstract: This work focuses on analyzing a beam-mass-beam system representing a micro aerial vehicle. The aircraft is simulated using two Euler-Bernoulli beams connected to a rigid mass with free end conditions. Heave flight dynamics and linear control design are also applied to the system. A finite element method, employing cubic B-spline basis vectors, is used to solve the system.
Presentation 3
Speaker: Arash Goligerdian (University of Houston)
Co-author: William Fitzgibbon (University of Houston)
Title: Environmentally Sustained Pathogen Models
Abstract: A mathematical model is developed to describe the spread of a waterborne disease within a spatially distributed host population. The population is assumed to inhabit a bounded domain, denoted by Ω, situated in the upper half-plane, with part of its boundary coincident with the real axis. The host population becomes infected through a pathogen introduced into Ω via a waterway that flows along the real axis.
Presentation 4
Speaker: Davood Damircheli (Louisiana State University)
Co-author: N/A
Title: A C0 Interior Penalty Method for a Nonlinear Time-fractional Fourth-order Partial Differential Equation
Abstract: We derive and study a C0 interior penalty method for a nonlinear time fractional fourth-order partial differential equation (PDE). The method employs poly-fractonomials as the basis for both test and trial spaces. Error estimates and convergence rates are computed through numerical procedures to demonstrate the effectiveness of the proposed method.
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: Lu Zhang (Rice University)
Co-author: N/A
Title: A fully discrete energy-based discontinuous Galerkin method for variable-order time fractional wave equations
Abstract: This paper presents and analyzes a fully discrete scheme for solving variable-order time fractional wave equations. The method begins by transforming the original equation into a system that introduces an additional velocity variable. An energy-based discontinuous Galerkin scheme is then used for spatial discretization. To accurately approximate the Caputo-type variable-order time fractional derivative, a specific point within each time interval is identified, ensuring at least second-order accuracy. The proposed scheme is shown to be unconditionally stable, and it also achieves optimal error estimation in an energy norm with certain numerical fluxes. Numerical experiments show optimal convergence in the $L^2$ norm for both spatial and temporal discretization, achieved through the use of carefully selected numerical fluxes and tailored approximation spaces.
Presentation 2
Speaker: Jesse Chan (Rice University)
Co-author: N/A
Title: Enforcing cell entropy inequalities using knapsack limiting on simplicial elements
Abstract: Algebraic flux correction methods blend high and low order methods together to ensure that the solution satisfies physical constraints (such as positivity) while maintaining conservation and accuracy. However, such strategies do not provably satisfy a semi-discrete cell entropy inequality. To remedy this, we enforce a semi-discrete cell entropy inequality by formulating limiting factors as solutions to an optimization problem with an efficient solution algorithm. Numerical experiments suggest that, in contrast to stricter versions of an entropy inequality, enforcement of a cell entropy inequality preserves high-order accuracy while still avoiding convergence to non-entropic solutions. We conclude by discussing how to extend these strategies to simplicial meshes.
Presentation 3
Speaker: Tom Hagstrom (Southern Methodist University)
Co-author: N/A
Title: Discontinuous Galerkin discretization of regularly hyperbolic systems
Abstract: Regularly hyperbolic systems are a natural analogue of Friedrichs systems for second order formulations of wave propagation problems. We present a general recipe for constructing provably-stable discontinuous Galerkin discretization of such systems. As a particular example we will demonstrate our approach with applications to linearized systems arising in general relativity.
Presentation 4
Speaker: Kenneth Duru (University of Texas at El Paso)
Co-author: Kerian Ricardo (Australian National University) and David Lee (Bureau of Meteorology Australia)
Title: On entropy stable and mimetic discontinuous Galerkin finite element methods for the rotating thermal shallow water equations in complex geometries
Abstract: The rotating thermal shallow water equations (RTSW) have recently gained attention as a test bed for atmospheric models owing to the similarity in their mathematical structure to the full compressible Euler equations of atmospheric motion. These equations extend the rotating shallow water equations to include a temperature-like quantity, known as buoyancy, that is transported by the flow and modulates the pressure gradient forcing. However, the development of robust numerical methods for the RTSW on curvilinear meshes pose a significant challenge because the energy functional is no longer a convex function of the prognostic variables and will not ensure numerical stability when preserved. We will derive an entropy functional which is convex and which must be preserved in order to preserve model stability at the discrete level. We present a novel discontinuous Galerkin finite element method for numerical simulations of the RTSW in complex geometries using curvilinear meshes, with arbitrary accuracy. We will prove entropy stability and conservation of mass, buoyancy, vorticity, and energy. This is achieved by using novel numerical fluxes, summation-by-parts principle, and splitting the pressure and convection operators so that we can circumvent the chain rule at the discrete level. We will present numerical simulations on a cubed sphere mesh and verify the theoretical results. The numerical experiments demonstrate the robustness of the method for a regime of well developed turbulence, where it can be run stably without any dissipation. The entropy stable fluxes are sufficient to control the grid scale noise generated by geostrophic turbulence, eliminating the need for artificial stabilization.
Organizers
Ron Morgan (Baylor University), Hayden Henson (Baylor University)
Mini-symposium Abstract:
Krylov iterative methods are fundamental tools for solving large systems of linear equations. Use of polynomials can significantly enhance Krylov methods, but also, some polynomials can stand on their own. We will look at polynomial preconditioned GMRES, including application to indefinite matrices. Also, it will be shown that high-degree polynomials can accurately approximate the inverse of a large matrix. Applications include solution of linear equations with multiple right-hand sides and a Multilevel Monte Carlo method for trace estimation. Finally, interpreting the conjugate gradient method as a polynomial method leads to a streamlined approach for solving multiple right-hand side systems with a symmetric matrix.
Presentation 1
Speaker: Hayden Henson (Baylor University)
Co-author: Ron Morgan (Baylor University)
Title: Polynomial Preconditioning for Indefinite Linear Equations & Interior Eigenvalues
Abstract: Indefinite spectra occur for linear equations in many applications. Examples include Helmholtz equations such as the wave equation and in quantum chromodynamics. Indefinite problems can be very difficult for Krylov iterative methods. We investigate adding polynomial preconditioning for such problems and show it can give tremendous improvement. Several difficulties can arise in polynomial preconditioning for indefinite matrices. These are mentioned along with some algorithmic solutions. Eigenvalue problems will also be explored.
Presentation 2
Speaker: Ron Morgan (Baylor University)
Co-author: N/A
Title: Polynomial Approximation of the Inverse of a Large Matrix
Abstract: A polynomial of A can give a good approximation to the inverse of A. For sparse matrices, this in some sense gives a sparse version of the inverse. For ill-conditioned matrices, a high degree polynomial may be needed. GMRES and Lanczos iterations can be used to develop this polynomial. Application is given to linear equations with multiple right-hand sides. Once a polynomial is developed, it can be applied to each right-hand side, and eigenvalue deflation can be included.
Presentation 3
Speaker: Shayan Nadeem (Baylor University)
Co-author: N/A
Title: A Multi-polynomial Monte Carlo Method for QCD Trace Calculations
Abstract: Estimating the trace of the inverse of a large matrix is an important problem in lattice quantum chromodynamics. A multilevel Monte Carlo method is proposed for this problem that uses different degree polynomials for the levels. The polynomials give approximations to the inverse of the matrix. They are developed from the GMRES algorithm. Deflation of eigenvalues is used to reduce the variance. The new Multi-polynomial Monte Carlo method can significantly improve the trace computation for matrices that have a difficult spectrum due to small eigenvalues.
Presentation 4
Speaker: Abby Williams (Baylor University)
Co-author: Ishara Saparamadu (Baylor University)
Title: Simple CG and BiCG for Multiple Right-hand Sides
Abstract: A very simple approach to solving multiple right-hand side systems is proposed. For symmetric problems, the conjugate gradient method is a very efficient way to solve linear equations. We will use the same parameters from solving the first system for other systems. This corresponds to applying a polynomial approximation of the inverse of the matrix, and it requires no dot products. Deflation of eigenvalues using seeding can be included. The natural stability control of symmetric Lanczos will be discussed along with the possibility of additional stability control. For nonsymmetric problems, a similarly simple version of BiCG can be used, and it is naturally more stable than BiCGStab in this context.
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 mini-symposium 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: Ning Ning (Texas A&M University)
Co-author: Edward Ionides (University of Michigan, Ann Arbor)
Title: Iterated Block Particle Filter for High-dimensional Parameter Learning: Beating the Curse of Dimensionality
Abstract: Parameter learning for high-dimensional, partially observed, and nonlinear stochastic processes is a methodological challenge. Spatiotemporal disease transmission systems provide examples of such processes giving rise to open inference problems. We propose the iterated block particle filter (IBPF) algorithm for learning high-dimensional parameters over graphical state space models with general state spaces, measures, transition densities and graph structure. Theoretical performance guarantees are obtained on beating the curse of dimensionality (COD), algorithm convergence, and likelihood maximization. Experiments on a highly nonlinear and non-Gaussian spatiotemporal model for measles transmission reveal that the iterated ensemble Kalman filter algorithm (Li et al., 2020) is ineffective and the iterated filtering algorithm (Ionides et al., 2015) suffers from the COD, while our IBPF algorithm beats COD consistently across various experiments with different metrics.
Presentation 2
Speaker: Joshua Hudson (University of Arkansas)
Co-author: Animikh Biswas (University of Maryland), Adam Larios (University of Nebraska-Lincoln), Elizabeth Carlson (CalTech)
Title: Uniquely recovering viscosity in 2D NSE from finitely many determining modes
Abstract: It is a classical result that, for the Navier--Stokes equations in 2D (with periodic boundary conditions), trajectories on the attractor are determined by finitely many modes (called determining modes). In this talk, we will discuss the case where the viscosity is unknown. We will show that trajectories on the attractor are still determined by finitely many modes, and discuss the problem of recovering the viscosity from the determining modes. We will present theoretical results guaranteeing the solution of this inverse problem, as well as some numerical results that motivated this line of research.
Presentation 3
Speaker: Zezhong Zhang (Oak Ridge National Laboratory)
Co-author: Feng Bao (Florida State University), Guannan Zhang (Oak Ridge National Laboratory)
Title: An Ensemble Score Filter for Tracking High-Dimensional Nonlinear Dynamical Systems
Abstract: We propose an ensemble score filter (EnSF) for solving high-dimensional nonlinear filtering problems with superior accuracy. A major drawback of existing filtering methods, e.g., particle filters or ensemble Kalman filters, is the low accuracy in handling high-dimensional and highly nonlinear problems. EnSF attacks this challenge by exploiting the score-based diffusion model, defined in a pseudo-temporal domain, to characterizing the evolution of the filtering density. EnSF stores the information of the recursively updated filtering density function in the score function, instead of storing the information in a set of finite Monte Carlo samples (used in particle filters and ensemble Kalman filters). Unlike existing diffusion models that train neural networks to approximate the score function, we develop a training-free score estimation that uses a mini-batch-based Monte Carlo estimator to directly approximate the score function at any pseudo-spatial-temporal location, which provides sufficient accuracy in solving high-dimensional nonlinear problems as well as saves a tremendous amount of time spent on training neural networks. High-dimensional Lorenz-96 systems are used to demonstrate the performance of our method. EnSF provides surprising performance, compared with the state-of-the-art Local Ensemble Transform Kalman Filter method, in reliably and efficiently tracking extremely high-dimensional Lorenz systems (up to 1,000,000 dimensions) with highly nonlinear observation processes.
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: Goong Chen (Department of Mathematics and Institute for Quantum Science Engineering, Texas A&M University)
Co-author: N/A
Title: Animal Shapes and Their Modal Analysis
Abstract: Eigenfunctions and eigenvalues of physical systems and engineering structures can reveal many of the system’s fundamental features and, therefore, become a basis for the study of inverse problems. In this series of papers, we take a reverse, direct problem point of view; namely, given the shapes of animals, can we see the patterns of their motions or behaviors through their eigenmode analysis? This modal analysis, we believe, has never been done for living animals. Our modal analysis is based on the model of continuum mechanics of solids, simplified and idealized by isotropy, homogeneity and linearity, leading to the 3D system of linear elastodynamic equations. We will compute and illustrate those animal shapes' eigenmodes. The dynamical aspects will be emphasized, which is achieved by visualization through video animation by incorporating the time-harmonic dependence of the eigenmodes. Several types of animals, including human beings, horses, camels, ducks, geese, the eagle, and the T-Rex dinosaur, are modeled and computed. Certain physical interpretations of the motion patterns from modal analysis are made. In addition, by visualization, one can see that symmetry plays an important role in motion patterns. One of our main conclusions is that shapes alone can usefully reflect or explain some animal’s behavior or motion patterns.
Presentation 2
Speaker: Goong Chen (Department of Mathematics and Institute for Quantum Science Engineering, Texas A&M University)
Co-author: N/A
Title: Applications of Modal Analysis to Animated-Film Making, Including the Role of Artificial Intelligence
Abstract: The making or production of high-quality animated films/videos is feasible for the interested non-experts these days through the aid of many movie-making software packages. A case in point is the making of dinosaur films such as the blockbuster Jurassic Park series made by the famous film producer Steven Spielberg. Such films are rather entertaining and fun to watch. However, the motions of those prehistoric animals such as dinosaurs do not satisfy the fundamental laws of physics such as the conservations of momentum, angular momentum and energy.
In this talk, the speaker will first give a quick review of the film-making techniques based on the powerful open-source software Blender. From there, we introduce a different technique based upon the linear combination of time-dependent eigenmodes of a T-Rex dinosaur. Such (time-dependent) eigenmodes of the linear elastodynamic equations have been proven to satisfy those conservation laws cited above and, therefore, make the motions of the dinosaur look physically more reasonable. A short video is made by us as a demonstration-of-principle for this idea. When perturbation of shapes of the T-Rex dinosaur or a dinosaur of a different species occur, we demonstrate how to apply AI techniques of machine learning and others to mimic the motions of the T-Rex dinosaur. Many other relevant videos will also be shown to the audience at the presentation if time permits.
Presentation 3
Speaker: Zhaosheng Feng (School of Mathematical and Statistical Sciences, University of Texas-RGV)
Co-author: N/A
Title: Parabolic System of Aggregation Formation in Bacterial Colonies
Abstract: The goal of this talk is to study a fourth-order nonlinear parabolic system with dispersion for describing bacterial aggregation. Analytical solution of traveling wave is found by taking into account the dispersion coefficient. Numerically, we demonstrate that the initial concentration of bacteria in the form of a random distribution over time transforms into a periodic wave, followed by a transition to a stationary solitary wave without dispersion.
Presentation 4
Speaker: Qin Sheng (Department of Mathematics and CASPER, Baylor University)
Co-author: Yin-Tzer Shih (National Chung Hsing University, Taiwan) and Eduardo Servin Torres (Baylor University)
Title: Remarks on Local and Global Errors of Splitting Methods in Connection with ADI and LOD Strategies
Abstract: This presentation delves into the local and global error estimates for important splitting methods such as the alternating-direction implicit (ADI) and locally one-dimensional (LOD) algorithms in data managements and approximations. It is found that the commutativity of underlying matrices plays intensive roles in the analysis and applications. It is shown that the global errors behave as polynomial functions in splitting procedure, and decay quickly as the discretization steps tend to zero. It is also noticed that global error analysis is in general for more applicable and superior as compared with traditional local error analysis when relatively large Courant numbers are used in numerical partial differential equation solution procedures. Simulated examples based on multidimensional Kawarada equations will be provided.
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
Void
Presentation 2
Speaker: Frank Sottile (Texas A&M University)
Co-author: N/A
Title: Periodic Operators for Algebraic Geometers
Abstract: Understanding the spectrum of the Schr\"odinger operator on a periodic medium is a fundamental problem in mathematical physics that is best approached using methods from analysis. The discrete version of this concerns operators on periodic graphs. In this discrete version, the primary objects are real algebraic varieties, and thus algebraic geometry becomes relevant for the study of discrete periodic operators. The purpose of this talk will be to explain this to algebraic geometers, and emphasize some of the computational and combinatorial aspects of this study.
Presentation 3
Speaker: Paul Dessauer (Texas A&M University)
Co-author: N/A
Title: Identifiability of Directed-Cycle Linear Compartmental Models
Abstract: A parameter of a mathematical model is identifiable if it can be determined from experimental data, with a model being called identifiable if each of its parameters are identifiable. We examine the identifiability of an important class of linear compartmental models called directed-cycle models. Our main result is a complete characterization of the directed-cycle models for which every parameter is (generically, locally) identifiable.
Presentation 4
Speaker: Christopher (C.J.) Bott (Texas A&M University)
Co-author: Frank Sottile (Texas A&M University)
Title: Computation of Schubert Galois Groups
Abstract: While Galois Theory may be familiar in the contexts of solving equations and studying field extensions, many do not know that historically Galois Theory was deeply connected to enumerative geometry. We review Galois groups geometrically in this way, and describe our study of computing Galois groups in the setting of Schubert Calculus.
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 minisymposium targets theoretical and practical aspects of such methods, especially for the nonlinear partial differential equations governing complex fluid flows.
Presentation 1
Speaker: Jean-Luc Guermond (Texas A&M University)
Co-author: N/A
Title: Third-order A-stable alternating implicit Runge--Kutta schemes
Abstract: We design pairs of six-stage, third-order, alternating implicit Runge--Kutta (RK) schemes that can be used to integrate in time two stiff operators by an operator-split technique. We also design for each pair a companion explicit RK scheme to be used for a third, nonstiff operator in an IMEX fashion. The main application we have in mind are (non)linear parabolic problems, where the two stiff operators represent diffusion processes (for instance, in two spatial directions) and the nonstiff operator represents (non)linear transport. We identify necessary conditions for linear A(α)-stability by considering a scalar ODE with two (complex) eigenvalues lying in some fixed cone of the half-complex plane with nonpositive real part. We show numerically that it is possible to achieve A(0)-stability when combining two operators with negative eigenvalues, irrespective of their relative magnitude. Finally, we show by numerical examples including two-dimensional nonlinear transport problems discretized in space using finite elements that the proposed schemes behave well.
Presentation 2
Speaker: Juntao Huang (Texas Tech University)
Co-author: N/A
Title: The Runge-Kutta sparse grid discontinuous Galerkin method with stage-dependent mesh for transport equations
Abstract: In this talk, we present a class of Runge-Kutta (RK) sparse grid discontinuous Galerkin (DG) methods with stage-dependent mesh for transport equations. The new method extends beyond the traditional method of lines framework and utilizes stage-dependent sparse grid DG finite element spaces for the spatial discretization operators. It features fewer floating-point operations and may achieve larger time step sizes. Numerical tests for transport problems in high dimensions are provided to demonstrate the performance of the new method.
Presentation 3
Speaker: Christina Taylor (University of Texas at Austin)
Co-author: N/A
Title: A High-Order Entropy Stable Discontinuous Galerkin Method for Cut Meshes
Abstract: Cut meshes allow the complex domain geometries to be captured well on relatively coarse, easy to use unfitted background meshes. However, the arbitrary shape of cut elements can make it difficult to produce accurate quadrature rules and traditional summation-by-parts operators, both of which are key components of high-order accurate entropy stable schemes. Using the skew-hybridized summation-by-parts formulation of Chan ["Skew-Symmetric Entropy Stable...", 2019], we reduce producing a generic, high-order accurate entropy stable method on cut meshes to the ability to produce sufficiently accurate quadrature rules on cut elements. We demonstrate a means to produce accurate quadrature rules using Caratheodory pruning and present numerical results for our method for the shallow water and compressible Euler equations.
Presentation 4
Speaker: Madison Sheridan (Texas A&M University)
Co-author: N/A
Title: An invariant-domain preserving approximation technique for Lagrangian hydrodynamics
Abstract: In this talk we will present an explicit approximation technique for the Lagrangian hydrodynamics equations equipped with an arbitrary equation of state. The approximation of the state variable is done with piecewise constant finite elements and the approximation of the mesh motion is done with higher-order continuous finite elements. For the calculation of the mesh velocities we employ an implicit method that is locally mass conservative. The method is invariant-domain preserving and can be used in combination with higher-order methods to make them invariant domain preserving as well. Numerical examples will be presented.
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: Burak Aksoylu (Texas A&M University-San Antonio)
Co-author: N/A
Title: Four Mutual Properties of Classical and Nonlocal Wave Equations
Abstract: The main advantage that our nonlocal (NL) operators provide is the ability to enforce local boundary condition (BC) through the use of a forcing function only on the local boundary, not in the interior of the domain. The ability to incorporate such a widely accepted BC type into NL formulations is quite valuable.
We provide a comparative study on classical and NL wave equations. The NL operators employ local BCs, and this is why a comparison to the classical wave equation is relevant. We find out that the two equations are qualitatively identical in terms of the balance of linear momentum (BLM), conservation of energy, and the resonance and beating phenomena. For both equations, the BLM is satisfied for the Neumann and periodic BCs and fails for Dirichlet and antiperiodic BCs.
We also reveal a close connection between classical and NL wave equations. In d'Alembert's formula on a bounded domain, the BC is encoded in the solution using the extension artifice known as the method of images. Whereas in our integral formulation, since the only degree of freedom is the kernel function, it is encoded in the kernel of the operator. This is a striking difference from the local formulation. What is even more striking is the following similarity: we discovered that the same combination of the function piece (even or odd) and extension type (antiperiodic or periodic) is used in structuring the kernel function. We were able to discover such suitable kernel structures thanks to functional calculus.
Presentation 2
Speaker: Cristian Meraz (University of Houston)
Co-author: Gabriela Jaramillo (University of Houston)
Title: Existence of Weak Solutions to the Nonlocal Klausmeier Model
Abstract: We establish the existence and uniqueness of weak solutions for a nonlocal Klausmeier model within a small-time interval $[0, T)$. The Klausmeier model comprises coupled, nonlinear partial differential equations governing plant biomass and water dynamics in semiarid regions. Unlike the original model, which posits classical diffusion for plant biomass spread, we opt for a nonlocal diffusive operator in alignment with ecological field data that validates long-range dispersive behaviors of plants and seeds. The equations, defined on a finite interval in $\mathbb{R}$, feature homogeneous Dirichlet boundary conditions for the water equation and nonlocal Dirichlet volume constraints for the plant biomass equation. We describe the nonlocal operator using a symmetric and spatially extended convolution kernel possessing mild integrability and regularity properties. We employ the Galerkin method with a nontraditional approach to establish existence and uniqueness. The key challenge comes from the nonlocal operator; we define it on a subspace of $L^{2}$ instead of $H^{1}$, precluding the use of Aubin's compactness theorem for weak convergence of nonlinear terms. To overcome this, we introduce two new equations for the spatial derivatives of plant biomass and water. This procedure allows us to recover enough regularity to establish compactness and complete the proof of weak convergence for the approximate solutions within some small-time interval $[0, T)$.
Presentation 3
Speaker: Joshua Siktar (Texas A&M University)
Co-author: Tadele Mengesha (University of Tennessee-Knoxville), Abner Salgado (University of Tennessee-Knoxville)
Title: Computational Aspects of Nonlocal Optimal Conductivity Problems
Abstract: In this problem we will describe a nonlocal optimal design problem where the objective functional is of a compliance type, the constraint is of a truncated fractional type, and the admissible design class consists of positive functions with pointwise bounds. While theoretical results on the analysis and discretization of such problems will be mentioned, the main focus of this talk will be on the implementation of a projected gradient descent algorithm for this problem. Leveraging the compliance structure of the cost functional alongside parallelization greatly reduces the amount of time needed to compute the descent direction during each iteration of the algorithm.
Presentation 4
Speaker: Gabriela Jaramillo (University of Houston)
Co-author: N/A
Title: Spiral waves and spiral chimeras in nonlocal oscillatory media
Abstract: Systems that can be classified as oscillatory media consist of small oscillating elements that interact with each other via some form of coupling. Interest in these systems stems from their ability to generate beautiful structures, including target patterns and spiral waves. When coupling between oscillators occurs over long spatial scales the set of unstable wavenumbers that generate these patterns is no longer constrained to a narrow band. This leads to interesting new structures like spiral chimeras. In this talk I will explain an approach based on the method of multiple-scales that can be used to understand the emergence of spiral waves and spiral chimeras.
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: Wang Li (The University of Texas at Arlington)
Co-author: N/A
Title: Large-Scale Semi-supervised Learning via Graph Structure Learning over High-dense Points
Abstract: We focus on developing a novel scalable graph-based semi-supervised learning (SSL) method for a small number of labeled data and a large amount of unlabeled data. Due to the lack of labeled data and the availability of large-scale unlabeled data, existing SSL methods usually encounter either suboptimal performance because of an improper graph or the high computational complexity of the large-scale optimization problem. In this paper, we propose to address both challenging problems by constructing a proper graph for graph-based SSL methods. Different from existing approaches, we simultaneously learn a small set of vertexes to characterize the high-dense regions of the input data and a graph to depict the relationships among these vertexes. A novel approach is then proposed to construct the graph of the input data from the learned graph of a small number of vertexes with some preferred properties. Without explicitly calculating the constructed graph of inputs, two transductive graph-based SSL approaches are presented with the computational complexity in linear with the number of input data. Extensive experiments on synthetic data and real datasets of varied sizes demonstrate that the proposed method is not only scalable for large-scale data, but also achieve good classification performance, especially for extremely small number of labels.
Presentation 2
Speaker: Difeng Cai (Southern Methodist University)
Co-author: N/A
Title: Geometric approaches for kernel compression in machine learning
Abstract: The kernel method is a powerful tool for learning nonlinear relationships in machine learning. It is a foundational component in support vector machine, kernel k-means, kernel ridge regression, etc. Computationally, since the kernel matrix that represents the inner product of the nonlinear features is dense, the kernel method becomes impractical for large data sets. To resolve this issue, Nyström method and its variants are developed to construct a small subset of landmark points and obtain a low-rank surrogate of the original kernel matrix. However, the cost of constructing the landmark points can be high and a random choice can lead to poor approximation accuracy. In this talk, we derive a geometric characterization of good landmark points and propose a linear complexity algorithm for low-rank construction. The subset selection based method is valid for rectangular kernel matrices and high dimensional data. We also show that the approach can be used to construct hierarchical matrix representations. Experiments are presented to demonstrate the efficiency, and the robustness to complex geometry, indefinite kernel, kernel parameters, by comparing to other methods such as k-means, random sampling, as well as adaptive algebraic compressions.
Presentation 3
Speaker: Nan Wu (The University of Texas at Dallas)
Co-author: N/A
Title: Inferring manifolds from noisy data using Gaussian processes
Abstract: We focus on the study of a noisy data set sampled around an unknown Riemannian submanifold of a high-dimensional space. Most existing manifold learning algorithms replace the original data with lower dimensional coordinates without providing an estimate of the manifold in the observation space or using the manifold to denoise the original data. We propose a Manifold reconstruction via Gaussian processes (MrGap) algorithm for addressing these problems, allowing interpolation of the estimated manifold between fitted data points. The proposed approach is motivated by novel theoretical properties of local covariance matrices constructed from noisy samples on a manifold. Our results enable us to turn a global manifold reconstruction problem into a local regression problem, allowing the application of Gaussian processes for probabilistic manifold reconstruction. Simulated and real data examples will be provided to illustrate the performance. This talk is based on the joint work with David Dunson.
Presentation 4
Speaker: Liyan Chen (University of Texas at Austin)
Co-author: N/A
Title: PDGen - A Topology Model for Generative Implicit Shape Models
Abstract: A fundamental problem in learning 3D shapes generative models is that when the generative model is simply fitted to the training data, the resulting synthetic 3D models can present various artifacts. Many of these artifacts are topological in nature, e.g., broken legs, unrealistic thin structures, and small holes. In this talk, we introduce a principled approach that utilizes topological regularization losses to rectify topological artifacts. The objectives are two-fold. The first is to align the persistent diagram (PD) distribution of the training shapes with that of synthetic shapes. The second ensures that the PDs are smooth among adjacent synthetic shapes. We show how to achieve these two objectives using two simple but effective formulations. Specifically, distribution alignment is achieved by learning a generative model of PDs and aligning this PD generator with PDs of synthetic shapes.
Moreover, we enforce the smoothness of the PDs using a smoothness loss on the PD generator, which further improves the behavior of PD distribution alignment. Experimental results on ShapeNet show that our approach leads to much better generalization behavior than state-of-the-art implicit shape generators.
To conclude my talk, I will share ongoing research and several open questions to invite further exploration. Potential topics may include: Parallelizing Persistent Homology for modern GPUs; score distillation of geometries from PD priors; and possibilities of Topology-to-Geometry models through Helmholtz-Hodge decomposition, offering opportunities for collaborations and sparking insightful discussions among attendees.
Organizers
Vladimir Ajaev (Southern Methodist University), Johannes Tausch (Southern Methodist University)
Mini-symposium Abstract:
The focus of the mini-symposium 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 mini-symposium 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: Vladimir Ajaev (Southern Methodist University)
Co-author: Kenneth K. Yamamoto (Southern Methodist University)
Title: Integral equations for models of charge transport in liquid electrolytes: analytical and numerical approaches
Abstract: We consider dynamics of charge transport near a circular electrode in contact with a large volume of high conductivity electrolyte solution and subject to AC voltage. By linearizing the full Poisson-Nernst-Planck system of equations and applying Hankel transform, we reduce the problem to a system of integral equations for a single unknown, the transformed amplitude of the electric charge density. The multiscale nature of the problem allows us to identify small parameters, leading to an asymptotic method for solving the system. The complex impedance can then be represented by a simple formula which is shown to be in very good agreement with the experimental measurements of impedance in high-conductivity salt solutions over a wide range of frequencies. We also develop a numerical solution of the system of integral equations by expanding in a series of suitably chosen Bessel functions and show that it agrees well with the asymptotic results in the parameter region where the latter are valid. The characteristic time scale for charge transport identified in our model leads to a self-similar representation of experimental measurements of the complex impedance for different electrode sizes valid over many orders of magnitude in frequency.
Presentation 2
Speaker: Michael Booty (New Jersey Institute of Technology)
Co-author: Samantha Evans (New Jersey Institute of Technology), Johannes Tausch (Southern Methodist University), Michael Siegel (New Jersey Institute of Technology)
Title: A mesh-free algorithm for soluble surfactant dynamics at large Peclet number
Abstract: We present a mesh-free method for the numerical solution of a boundary value problem that occurs in the dynamics of soluble surfactant near sharp interfaces in two-phase flow at large bulk Peclet number. The governing PDE is an advection-diffusion equation and its solution, which describes bulk-interface surfactant exchange, can be written as a convolution integral that has a non-standard time-dependent kernel. The algorithm for computationally efficient, accurate evaluation of the integral is the focus of the talk. The problem context and practical examples together with comparison to other solution techniques will also be discussed.
Presentation 3
Speaker: Sharon Yang (UT Southwestern)
Co-author: Elyssa Sliheet (Southern Methodist University), Reece Iriye (Southern Methodist University), Daniel Reynolds (Southern Methodist University), Weihua Geng (Southern Methodist University)
Title: Optimized parallelization of boundary integral Poisson-Boltzmann solvers
Abstract: The Poisson-Boltzmann (PB) model describes the electrostatics of solvated biomolecules, including potential, field, energy, and force, offering valuable insights into protein properties, functions, and dynamics. Leveraging advancements in algorithms and hardware, we focus on parallelizing the treecode-accelerated boundary integral (TABI) PB solver using MPI on CPUs and the direct-sum boundary integral (DSBI) PB solver with KOKKOS on GPUs. We provide guidance on optimizing performance, recommending the DSBI solver on GPUs for smaller problems and the TABI solver on CPUs for larger ones. When the number of unknowns is below a certain threshold, the GPU-accelerated DSBI solver converges quickly, enabling its use in PB model-based molecular dynamics or Monte Carlo simulations. In practical applications, our parallelized boundary integral PB solvers are employed to analyze the electrostatics of proteins involved in the spread, treatment, and prevention of COVID-19. These simulations yield global electrostatic solvation energy and local electrostatic surface potential details for each protein.
Presentation 4
Speaker: Jacob E. Davis (Southern Methodist University)
Co-author: Vladimir Ajaev (Southern Methodist University)
Title: A coupled numerical and analytical solution for modeling droplet levitation
Abstract: We develop a mathematical model of heat and mass transfer in a configuration which involves a spherical droplet levitating near a flat liquid layer heated from below. Analytical solutions for vapor concentration in air and the temperature distributions both inside the droplet and in moist air around it are obtained using separation of variables in bipolar coordinates and coupled to the numerical solution for heat transfer in the liquid layer. A finite difference scheme is used to find the grid point temperature values in the liquid layer. In the limit of weak evaporation, the liquid layer surface is cooled locally due to the presence of the droplet, while the effect is reversed for strong evaporation. The latter case is also characterized by a possibility of strong temperature gradients in the droplet itself, an unexpected conclusion given the high liquid-to-air thermal conductivity ratio. A critical value for the scaled latent heat is found where the droplet's affect on temperature balances the affects from phase change, leading to no change in the liquid layer temperature profile. The observations are explained in terms of interplay between geometric and thermal effects of the presence of the droplet. The calculation of the evaporation rate leads to determination of the moist air flow around the droplet, treated in the Stokes flow approximation and using bipolar coordinates, and thus the levitation height. The latter is reduced as a result of heat transfer effects in the liquid layer.
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: Giordano Tierra (University of North Texas)
Co-author: Francisco Guillen-Gonzalez (Universidad de Sevilla, Spain)
Title: Energy-stable numerical schemes preserving boundedness for the Cahn-Hilliard equation with degenerate mobility
Abstract: The study of interfacial dynamics has become a key component to understand the behavior of a great variety of systems, in scientific, engineering and industrial applications. A very effective approach for representing interface problems is the diffuse interface/phase field approach, which describes the interfaces by layers of small thickness and whose structure is determined by a balance of molecular forces, in such a way that the tendencies for mixing and de-mixing compete through a non-local mixing energy. In particular, the Cahn-Hilliard equation was originally introduced to model the thermodynamic forces driving phase separation, arriving to a system with a gradient flow structure. During this talk I will present two new numerical schemes to approximate the Cahn-Hilliard equation with degenerate mobility, by using two different approximations of the mobility. These schemes are both energy stable and preserve the maximum principle approximately, i.e., the amount of the solution being outside of the interval [0,1] goes to zero in terms of a truncation parameter. Additionally, I will present numerical results in order to illustrate the accuracy and the well behavior of the proposed schemes.
Presentation 2
Speaker: An Vu (University of St. Thomas)
Co-author: Loic Cappanera (University of Houston), Caroline Nore (University Paris Saclay)
Title: Finite Element Methods for Multiphase Incompressible Flows with Thermal Convection
Abstract: We introduce a semi-implicit time-stepping thermal solver for the incompressible Navier-Stokes equations coupled with temperature equation. This solver handles systems with variable fluid properties such as density and viscosity, as well as thermal properties like heat diffusivity and thermal conductivity. We use a projection method to enforce the incompressibility, treating momentum (product of density and velocity) and internal energy (product heat capacity, density, and temperature) as primary variables. We show that the scheme is stable and provide the first-order error estimates. Additionally, we present a fully discretized algorithm using finite elements and verify its effectiveness through numerical simulations, including cases with large density differences and fluid interface reversals.
Presentation 3
Speaker: Xu Zhang (Oklahoma State University)
Co-author: Yuan Chen (Ohio State University)
Title: High-Order Immersed C0 Interior Penalty Methods for Biharmonic Interface Problems
Abstract: In this talk, we introduce a group of high-order immersed C0 interior penalty methods for solving biharmonic equations with discontinuous coefficients. High order immersed finite element (IFE) spaces are constructed to incorporate biharmonic interface conditions in the least-squares sense. Basic properties of these new IFE spaces such as unisolvence and partition of unity are established. A modified C0 interior penalty scheme utilizing this new IFE space is developed to solve the biharmonic boundary value problems with interfaces. We prove the well-posedness of the numerical scheme. Extensive numerical experiments indicate optimal convergence in related Sobolev norms. This project is collaborated with Yuan Chen from The Ohio State University.
Presentation 4
Speaker: Ram Manohar (Texas A&M University Corpus Christi)
Co-author: S.M. Mallikarjunaiah (Texas A&M University Corpus Christi)
Title: An hp-Discontinuous Galerkin Discretization of a Strain-Limiting Elasticity Model
Abstract: In this presentation, I will explore an hp-discontinuous Galerkin (hpDG) approximation of a nonlinear strain-limiting elasticity model. The mathematical model describes the stress-strain state in an elastic body characterized by a specific class of algebraically nonlinear constitutive relations. I will delve into the solvability and stability of the discrete scheme for a nonconvex problem and provide a detailed derivation of a priori error estimates in the energy and L^2 norm. Lastly, I will showcase some compelling numerical experiments for the nonconvex domain, such as a single-edge crack under anti-plane shear-type loading.
Organizers
Aidan Gettemy (The University of Texas at Dallas), Susan Minkoff (The University of Texas at Dallas)
Mini-symposium Abstract:
Digital twins model the behavior of their physical counterparts through the bidirectional flow of information. Data from the physical system continuously updates the numerical model, while the model's predictions inform the control of the physical system. By integrating real-time data from sensors and predictive algorithms, digital twins provide a dynamic, up-to-date representation. However, challenges such as improving predictive accuracy, reducing computational costs, and quantifying uncertainties limit broader applications. To address these, the symposium explores topics like integrating remote monitoring with surrogate modeling, optimizing computational efficiency through sensitivity-driven model refinement, improving uncertainty quantification with hyper-reduced order models, and leveraging model reduction for large-scale nonlinear problems. These approaches aim to make digital twins more robust and practical for real-time applications across various industries.
Presentation 1
Speaker: Aidan Gettemy (University of Texas at Dallas)
Co-author: Susan Minkoff (Brookhaven National Laboratory), John Zweck (University of Texas at Dallas), Todd Griffith (University of Texas at Dallas)
Title: Integrating Remote Monitoring and Surrogate Modeling for Wind Turbine Blade Erosion Detection
Abstract: Leading edge erosion (LEE), gradual damage to the front edge of wind turbine blades due to factors like rain, hail, and dust, has become a growing concern for the wind industry. LEE degrades aerodynamic performance, significantly reduces energy output, and shortens the lifespan of turbines. Remote monitoring offers a solution to detect damage early. However, leveraging remote monitoring data requires integrating rapid modeling and accurate analysis. The digital twin, combining physics-based modeling with real-time data, is key to overcoming these challenges by enabling detection and precise prediction of erosion. To adapt existing models to create a digital twin, we develop a surrogate model based on the Gaussian process emulator to predict the effects of leading-edge erosion on wind turbine blades. The surrogate model rapidly and accurately predicts the outputs of the simulator across a range of conditions and can train an effective erosion classification model.
Presentation 2
Speaker: Jonathan Cangelosi (Rice University)
Co-author: Matthias Heinkenschloss (Rice University)
Title: Sensitivity-Driven Surrogate Model Refinement for Efficient Computation of Quantities of Interest in Dynamical Systems
Abstract: In many applications one wishes to compute a quantity of interest (QoI) that depends on the solution of a trajectory simulation or optimization problem where some components of the dynamics depend on a computationally expensive high-fidelity model. Using a computationally inexpensive surrogate model in place of the high-fidelity model allows one to compute the QoI much more cheaply, but at the cost of accuracy. The QoI error may be reduced as needed by incorporating additional high-fidelity data into the surrogate to obtain a more accurate trajectory, but the selection of inputs to the high-fidelity model is important to ensure sufficient decrease of the QoI error at minimal computational cost. In this talk, I propose a model refinement procedure for trajectory simulation problems that selects the best locations to evaluate the high-fidelity model using sensitivity information for a QoI and error bounds for surrogates constructed via interpolation in reproducing kernel Hilbert spaces. I then use this approach to reduce the error in the computed downrange of a hypersonic vehicle whose lift and drag coefficients are expensive to compute.
Presentation 3
Speaker: Suparno Bhattacharyya (Texas A&M University)
Co-author: Jian Tao (Texas A&M University), Eduardo Gildin (Texas A&M University), Jean Ragusa (Texas A&M University)
Title: Uncertainty quantification with hyper-reduced order models
Abstract: Data-driven Reduced Order Models (ROMs) are crucial to Digital Twins, enabling efficient real-time simulations of complex systems. However, optimal performance requires a deep understanding of the associated uncertainties. Uncertainty quantification (UQ) is increasingly critical across industries for system analysis, predictive maintenance, and real-time decision-making. For example, Digital Twins in nuclear reactors must account for uncertainties in temperature and material properties, while aircraft engine Digital Twins assess uncertainties in sensor data and environmental conditions.
In this presentation, we introduce a specific class of reduced-order models: hyper-reduced order models (hyper-ROMs). Unlike traditional ROMs, hyper-ROMs are designed to maximize computational efficiency, especially in nonlinear scenarios, and have shown significant potential for integration with Digital Twins. UQ typically involves many-query computations, which can be computationally expensive, particularly with nonlinear systems. We show that hyper-ROMs make UQ more efficient and practical, for complex, high-dimensional problems.
Presentation 4
Speaker: Dane Grundvig (Rice University)
Co-author: Matthias Heinkenschloss (Rice University)
Title: Line-Search Based Optimization with Online Model Reduction
Abstract: When applying model reduction to optimization, an online only approach can help to alleviate the expense of model construction by leveraging locally accurate models. We propose a line-search algorithm that uses objective function models with tunable accuracy to solve smooth optimization problems with general nonlinear equality constraints. This algorithm specifies how objective function models can be used to generate new iterates in the context of line-search methods, and specifies approximation properties these models have to satisfy.
Moreover, the algorithm assumes that a bound for the model error is available and uses this bound to explore regions where the model is sufficiently accurate. The algorithm has the same first-order global convergence properties as standard line-search methods. However, this algorithm uses only the models and the model error bounds, but never directly accesses the original objective function. Examples include problems where the evaluation of the objective requires the solution of a large-scale system of nonlinear equations. The models are constructed from reduced order models of this system. Numerical results for partial differential equation constrained optimization problems show the benefits of the proposed algorithm. Extensions to constrained optimization are presented.
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: Markus Hunziker (Baylor University)
Co-author: Fritz Gesztesy (Baylor University)
Title: The generalized eigenvalue problem for the classical Euler differential equation and Meijer's $G$-function
Abstract: We present a fundamental system of solutions of the generalized eigenvalue problem for the classical (higher order) Euler differential equation. In the course of deriving the solution, we review some of the basics of generalized hypergeometric functions and Meijer's $G$-functions and some of its special cases where the underlying Mellin-type integrand exhibits higher-order poles.
Our work on the generalized eigenvalue problem for the Euler differential equation was motivated by our study of the essential self-adjointness in $L^2(\mathbb{R}^n;d^nx)$, $n \in \mathbb{N}$, of the differential operator
(\Delta^2 +c|x|^{-4}) |_{C_0^{\infty}(\mathbb{R}^n \backslash \{0\})}, c \in \mathbb{R},
which, upon angular momentum decomposition, leads to special cases of generalized eigenvalue problems for 4-th order Euler differential equations.
Presentation 2
Speaker: Andrei Prokhorov (University of Michigan, Ann Arbor)
Co-author: N/A
Title: Low temperature asymptotic of partition function for logarithmic gas on the torus
Abstract: Classical potential theory can be extended to the Riemann surfaces as described in Bertola, Groot, Kuijlaars (2022). Consider the bipolar Green's function $G(p,q)$ on the torus with periods $1$ and $i\tau$. Given the potential $V(p)$ we can consider the logarithmic energy $E_V[\mu]=\intop_0^1\intop_0^1G(p,q)d\mu(p)d\mu(q)+\intop_0^1V(p)d\mu(p)$ evaluated on probability measures $\mu$ on the interval $[0,1]$. We would like to evaluate it on the empirical measures $\mu_\lambda=\frac{1}{N}\sum_{j=1}^N\delta_{\lambda_j}$. We consider the partition function of the form $Z_N=\intop_{0}^1\ldots \intop_0^1e^{-\beta E_V[\mu_\lambda] }d\lambda_1\ldots d\lambda_N$. We restrict ourself to the low temperature choice $\beta=\alpha N$ and to the choice of the potential $V(x)=Px$. In that case $Z_N=\int_{[0,1]^N} \prod_{1\leq i<j\leq N} \vert \theta_{1}(x_i - x_j) \vert^{-\frac{2\alpha}{N}} \prod_{i=1}^N \theta_{1}(x_i)^{2\alpha} e^{\alpha P x_i} \prod_{i=1}^N dx_i$ and the partition function is related to the conformal blocks of Liouville theory on the torus. Using the probabilistic tools such as derivative GMC measure and martingale estimates we managed to compute large $N$ asymptotics $\log(Z_N)\simeq \phi N$, $N\to\infty$. Here $\phi$ is related to the accessory parameter of Lam\'e equation and admits explicit expression. This is the joint work with Harini Desiraju and Promit Ghosal. (arXiv:2407.05839).
Presentation 3
Speaker: Daniel Perales (Texas A&M University)
Co-author: Octavio Arizmendi (Centro de Investigación en Matemáticas), Katsunori Fujie (Kyoto University) and Yuki Ueda (Hokkaido University of Education)
Title: S-transform in Finite free probability
Abstract: We show a simple way to obtain the limiting spectral distribution of a sequence of polynomials (with increasing degree) directly using their coefficients. Specifically, we relate the asymptotic behavior of the ratio of consecutive coefficients to Voiculescu's S-transform of the limiting measure.
The intuition comes from finite free probability. In this framework, the ratios of coefficients can be understood as a new notion of finite S-transform, which satisfies several analogous properties to those of the S-transform in free probability.
We will mention some of the main ingredients of the proof that include topics of independent interest, such as a partial order in the set of polynomials, and a better understanding of the behavior under differentiation.
As one of many applications, we can compute the limiting S-transform of hypergeometric polynomials, a large class that contains many of the important families of polynomials that naturally appear in finite free probability, such as Laguerre, Hermite and Jacobi.
Joint work with Octavio Arizmendi, Katsunori Fujie and Yuki Ueda (arXiv:2408.09337).
Presentation 4
Speaker: Christoph Fischbacher (Baylor University)
Co-author: N/A
Title: Dissipative quantum systems with non-local point interactions
Abstract: In this talk, I will discuss dissipative operators of the form
$i\frac{d}{dx}+V$ and $-\frac{d^2}{dx^2}+V$, where $V$ is a bounded dissipative potential. Besides $V$, there are two additional sources contributing to the dissipativity of the system: (i) dissipative boundary conditions and (ii) so-called non-local point interactions.
Mechanism (ii) is less standard and leads to interesting new problems, even in the first-order case.
I will discuss necessary and sufficient conditions for the operators to be maximally dissipative, the spectrum of the first-order operators, and the possibility of choosing the non-local point interaction in such a way that it generates a real eigenvalue even if is very dissipative.
Based on previous and ongoing collaborations with Matthias Hofmann, Andres Lopez Patio, Sergey Naboko, Danie Paraiso, Chloe Povey-Rowe, Monika Winklmeier, Ian Wood, and Brady Zimmerman.