SIAM Mini-Symposium Session 4
Organizer
Lizy John (Electrical and Computer Engineering, University of Texas Austin)
Mini-symposium Abstract:
This mini-symposium will focus on Weightless Neural Networks (WNNs), a class of neural networks that rely on look up tables (LUTs). Hashing techniques to reduce aliasing and hashing costs in memory nodes for weightless neural networks will be presented. H3 hashes were found to be very useful for efficiently realizing WNNs. Soon filter, a recently introduced technique for improving accuracy and efficiency of WNNs will be discussed. Traditionally WNNs are not differentiable, but recent work devised an extended finite difference method to render WNNs differentiable. Training of DWNs as enabled by a novel Extended Finite Difference technique for approximate differentiation of binary values will be discussed. The mathematical foundations of differentiation of look up tables will be discussed. We will also discuss Learnable Mapping, Learnable Reduction, and Spectral Regularization to further improve the accuracy and efficiency of these models.
Presentation 1
Speaker: Felipe M. G. Franca (Federal University of Rio de Jeneiro (UFRJ), Brazil)
Co-author: Lizy K. John
Title: Biological Plausibility of dendritic trees performing decoding, WiSARD, overview of past applications and recent results
Abstract: Mainstream artificial neural network models are based on weighted-sum-and-threshold artificial neurons, as the pioneering Threshold Logic Unit, of McCullogh and Pitts. In this classic model, an important simplification happens in the way inputs to neurons are modeled: all synaptic connections terminate directly at the neuron's soma. In fact, the vast majority of synapses in the central nervous system terminate at the neuron's dendritic tree, the most noticeable morphological structure of the neuron cell. Mimicking biological neurons by focusing on the excitatory/inhibitory decoding performed by the dendritic trees is a different and attractive alternative to the integrate-and-fire McCullogh-Pitts neuron stylisation. It is now more than 60 years that Random Access Memory (RAM) nodes can play the role of artificial neurons that are addressed by Boolean inputs and produce Boolean outputs. An overview on recent theoretical advances, implementations, and applications of weightless neural systems, in classification, clustering, regression etc. will be visited.
Presentation 2
Speaker: Shashank Nag (University of Texas at Austin)
Co-author: N/A
Title: Hashing and Filtering Techniques for Weightless Neural Networks
Abstract: This talk covers advanced hashing and filtering techniques to improve the efficiency of Weightless Neural Networks (WNNs) for edge inference. We will delve into the BTHOWeN and ULEEN architectures, which leverage counting Bloom filters and H3 hash functions to reduce memory use, enhance accuracy, and cut energy consumption. We will also present SoonFilter, which boosts WNN performance through gradient-driven architecture adjustments. These innovations result in significant gains in latency and power efficiency, positioning WNNs as a strong choice for edge computing.
Presentation 3
Speaker: Alan T. L. Bacellar (University of Texas at Austin)
Co-author: N/A
Title: Approximate Differentiation Techniques for Weightless Neural Networks
Abstract: This talk will present recent advancements in rendering Weightless Neural Networks (WNNs) differentiable through novel techniques such as Extended Finite Differences (EFD). We will discuss how EFD enables backpropagation through lookup tables (LUTs) to support gradient-based optimization. Additionally, Learnable Mapping will be introduced, allowing WNNs to dynamically optimize connections between LUTs during training, further improving model performance. Learnable Reduction, a method that reduces model size by replacing popcounts with decreasing pyramidal LUT layers, and Spectral Regularization, a specialized regularization technique for LUTs, will also be explored. These innovations collectively enhance WNN accuracy and efficiency, making them highly competitive for edge inference and tabular data applications.
Presentation 4
Speaker: Mugdha P. Jadhao (University of Texas at Austin)
Co-author: N/A
Title: ConvWNN: Combination of Convolutional and Weightless Neural Networks for Efficient Edge Inference
Abstract: Weightless Neural Networks (WNNs) use table lookups and minimal computations, making them efficient for low-power edge applications. However, WNNs lack positional invariance, limiting their accuracy compared to Deep Neural Networks (DNNs). To address this, we propose ConvWNN, a hybrid model combining quantized convolutional DNNs and WNNs, which aims to improve accuracy while reducing hardware resource use compared to iso-accurate WNNs and quantized DNNs. We evaluate ConvWNN against Binary Neural Networks (BNNs) from Xilinx's FINN on image classification datasets, demonstrating its potential for extreme edge environments.
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: Bart van Bloemen Waanders (Sandia National Laboratories)
Co-author: Joseph Hart (Sandia National Laboratories)
Title: Hyper-Differential Sensitivity Analysis for Large Scale Inverse Problems
Abstract: Inverse problems constrained by partial differential equations (PDEs) play a critical role in model development and calibration. In many applications, there are multiple uncertain parameters in a model that must be estimated. However, high dimensionality of the parameters and computational complexity of the PDE solves make such problems challenging. A common approach is to reduce the dimension by fixing some parameters (which we will call auxiliary parameters) to a best estimate and use techniques from PDE-constrained optimization to estimate the other parameters. In this presentation, hyper-differential sensitivity analysis (HDSA) is used to assess the sensitivity of the solution of the PDE-constrained optimization problem to changes in the auxiliary parameters. We introduce novel theoretical and computational approaches to justify and enable HDSA for ill-posed inverse problems by projecting the sensitivities on likelihood informed subspaces and defining a posteriori updates. Our proposed framework is demonstrated on land-ice dynamics for Greenland.
Presentation 2
Speaker: Gaik Ambartsoumian (University of Texas at Arlington)
Co-author: Divyansh Agrawal, Venkateswaran P. Krishnan, Nisha Singhal (Centre for Applicable Mathematics, Tata Institute of Fundamental Research, India)
Title: A simple range characterization for spherical mean transform in odd dimensions and its applications
Abstract: The spherical mean transform (SMT), sometimes also called the spherical Radon transform, maps a function to its integrals over hyperspheres. The study of this operator has a long history due to its relations to certain PDEs (wave equation, Euler-Poisson-Darboux equation), approximation theory and functional analysis. More recently, SMT and its inversion have been analyzed in connection with applications in tomography.
The talk describes a novel and simple range description for SMT of functions supported in the unit ball of an odd dimensional Euclidean space. The new description comprises a set of symmetry relations between the values of certain differential operators acting on the coefficients of the spherical harmonics expansion of the function in the range of the transform. As a central part of the proof of our main result, we derive a remarkable cross product identity for the spherical Bessel functions of the first and second kind, which may be of independent interest in the theory of special functions. Finally, we present several applications of our new range characterization. The list includes an explicit counterexample proving that unique continuation type results cannot hold for SMT in odd dimensional spaces, a characterization of the null space of the associated backprojection operator, and negation of a conjecture relating SMT, its backprojection operator and the Riesz potential.
Presentation 3
Speaker: Shanyin Tong (Columbia University)
Co-author: Kui Ren (Columbia University), Nathan Soedjak (Columbia University)
Title: A policy iteration method for inverse mean field games
Abstract: Mean-field games study the Nash Equilibrium in a non-cooperative game with infinitely many agents, they have a wide range of applications in engineering, economics, and finance. The coupling structure of the forward-backward equations of MFG raises specific difficulties in finding solutions to MFG, and makes it even more challenging to solve its inverse problem. In this talk, we will present a policy iteration method to solve an inverse problem for a mean-field game model, specifically to reconstruct the obstacle function in the game given partial observation data of value functions. The proposed approach decouples this complex inverse problem, which is an optimization problem constrained by a coupled nonlinear forward and backward PDE system in the MFG, into several iterations of solving linear PDEs and linear inverse problems. It substantially simplifies the problems and improves computational efficiency. We will use several numerical examples to demonstrate the accuracy and efficiency of the proposed method, and also discuss its convergence.
Presentation 4
Speaker: Akwum Onwunta (Lehigh University)
Co-author: N/A
Title: A Deep Neural Network Approach for Parameterized PDEs and Bayesian Inverse Problems
Abstract: We consider the simulation of Bayesian statistical inverse problems governed by large-scale linear and nonlinear partial differential equations (PDEs). Markov chain Monte Carlo (MCMC) algorithms are standard techniques to solve such problems. However, MCMC techniques are computationally challenging as they require a prohibitive number of forward PDE solves. The goal of this work is to introduce a fractional deep neural network (fDNN) based approach for the forward solves within an MCMC routine. We illustrate the efficiency of fDNN on inverse problems governed by nonlinear elliptic partial differential equations and the unsteady Navier-Stokes equations. Two examples are discussed in the former case, depending on several parameters, with significant observed savings. The unsteady Navier-Stokes example illustrates that fDNN can outperform existing DNNs, doing a better job of capturing essential features such as vortex shedding.
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 multiphysics processes in the subsurface, such as groundwater contamination, hydrocarbon reservoir management, and subsurface energy systems, including geothermal or carbon sequestration.
Presentation 1
Speaker: Malgorzata Peszynska (Oregon State University)
Co-author: N/A
Title: Modeling coupled processes in permafrost soils and snow cover
Abstract: We overview our progress on TpHM models in permafrost soils across the scales which was recently coupled to a snow cover model. This multi-physics multiscale problem features several computational and modeling challenges including the choice of operator splitting and solvers, as well as the coupling conditions on the soil-snow interface. For realistic results, it also requires good field and atmospheric data. However, the model is also amenable to surprising simplifications as well as to reduced order modeling. This is joint work with Naren Vohra, Zachary Hilliard, Matt Evans, Praveeni Mathangadeera, and Madison Phelps.
Presentation 2
Speaker: Jihoon Kim (Texas A&M University)
Co-author: N/A
Title: Thermodynamically consistent modeling approach on coupled capillary hysteresis and poromechanics for subsurface hydrogen storage and co2 sequestration
Abstract: We develop a robust numerical algorithm for coupled flow and geomechanics simulation of geological hydrogen storage as well as CO2 sequestration that can account for capillary hysteresis and elastoplasticity simultaneously. For example, the hydrogen storage can cause repeated drainage and imbibition processes induced by periodic injection and production, which can exhibit capillary hysteresis and/or material failure such as Mohr-Coulomb failure. The algorithm developed in this study is fundamentally based on the general elastoplasticity theory that can keep thermodynamic consistency, which provides numerical stability. Also, we take the fixed stress method for simulation of coupled flow and geomechanics, which can yield numerical stability of sequential modeling between flow and geomechanics. Then, from the numerical tests, we find that the results are still stable even though many repeated drainage and imbibition processes combined with several loading-unloading conditions occur.
Presentation 3
Speaker: Min Wang (University of Houston)
Co-author: N/A
Title: The interplay between deep learning and model reduction
Abstract: The integration of Reduced Order Modeling (ROM) and Deep Learning (DL) presents a promising avenue for enhancing computational efficiency and predictive accuracy. ROM techniques reduce complex systems into low-dimensional representations, preserving essential dynamics, while DL methodologies excel at learning intricate patterns from raw data. By com- bining ROM with DL, we aim to develop approaches that inherit merits from both disciplines. On one hand, we explore leveraging ROM as a preprocessing step to train DNNs with limited labeled data, addressing data scarcity. On the other hand, we intend to utilize DL to expedite ROM construction and learn ROMs directly from observational data. In this talk, we will elaborate on specific efforts made along this line.
Presentation 4
Speaker: Ali Pakzad (California State University, Northridge)
Co-author: N/A
Title: A Data-Driven Approach to Large Eddy Simulation of Turbulence
Abstract: One of the major challenges in accurately simulating turbulent flows is determining the initial state of the flow. Similar to weather prediction models, which require the complete present state of the atmosphere as input, fluid dynamics models also rely on accurate initial data. However, in practice, these initial data are often incomplete because measurements are only available at discrete points, such as weather stations or satellites. Data assimilation addresses this issue by integrating spatially coarse observational data into mathematical models, thereby enhancing the accuracy of forecasts for physical systems. This technique, initially developed for 2D fluid dynamics, has recently been extended to 3D models. In this talk, we will discuss recent rigorous results that support this method for a family of 3D globally well-posed modi ed Navier-Stokes equations, which aim to capture large-scale turbulent structures. Numerical computations will also be presented to demonstrate the efficacy of the algorithm.
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: Yong Yang (West Texas A&M University)
Co-author: Yuzhong Huang (Jackson State University)
Title: Flow Structure Analysis in MVG-Controlled Hypersonic Boundary Layers
Abstract: In this study, we delve into the evolution of flow structures within hypersonic boundary layers controlled by micro vortex generators (MVGs). To achieve this, we employ a powerful combination of techniques: the proper orthogonal decomposition (POD) method and a newly developed, high-fidelity vortex identification approach known as Liutex. This synergy allows for a detailed exploration of the flow structure, enabling us to assess energy contributions from each mode based on vortex intensity. Moreover, we compare the POD results from the hypersonic boundary layer with those observed in supersonic boundary layers at Mach 2.5 and 3.5. Our findings reveal a distinct impact on the relative intensity of spanwise vortices at hypersonic speeds. In this regime, streamwise vortex structures emerge as dominant players within the boundary layer.
Presentation 2
Speaker: Caixia Chen (Jackson State University)
Co-author: Yong Yang (West Texas A&M University), Paris Smith (Jackson State University)
Title: Exploring Flow Profile Nonlinearities with Artificial Neural Networks
Abstract: This study investigates the use of artificial neural networks (ANNs) to tackle the nonlinear dynamics inherent in complex fluid flows. By analyzing flow profiles generated from high-fidelity simulations, we demonstrate that ANNs effectively capture intricate features such as discontinuities and subtle wave characteristics at small scales. Even in challenging three-dimensional scenarios, ANNs accurately predict averaged velocity profiles with minimal input data during training. These findings highlight the immense potential of neural networks in advancing our understanding of complex fluid dynamics and their applicability across diverse flow scenarios.
Organizer
Suparno Bhattacharyya (Texas A&M University)
Mini-symposium Abstract:
Understanding the mechanical behavior and response of organs under various conditions is critical for advancing both biomedical research and clinical applications. However, the complexity of these systems often results in computationally intensive simulations. Data-driven surrogate modeling techniques provide a valuable approach by enabling efficient and accurate simulations of organ-level biomechanics.
This mini-symposium focuses on the latest developments in data-driven modeling techniques, specifically, on the mechanical behavior and response of organs. Topics include: Data-Driven surrogate modeling: Utilizing machine learning to predict organ responses under various conditions, based on large-scale simulation data and clinical observations. Multiscale Applications: Linking tissue-level and organ-level models to provide a comprehensive understanding of biomechanical behavior. Clinical Applications: Real-world case studies demonstrating the use of ROM in simulating organ responses for personalized medicine, including surgical planning and disease treatment.
This mini symposium aims to provide a platform for interdisciplinary collaboration, bringing together researchers from applied mathematics, biomedical engineering, and clinical fields.
Presentation 1
Speaker: Hernal Santos (Department of Visualization, Texas A&M University)
Co-author: Jian Tao (Department of Visualization, Texas A&M University)
Title: Comparative Study of DMD and ANN for Inverse Dynamics in Musculoskeletal Modeling
Abstract: Recent advancements in computational modeling have significantly improved simulations of human musculoskeletal systems. This study compares Dynamic Mode Decomposition (DMD), and Artificial Neural Networks (ANN) to develop high-fidelity models of musculoskeletal dynamics. The primary goal is to perform inverse dynamics, analyzing forces and torques exerted during specific maintenance tasks.
DMD decomposes joint motions into principal components and extracts key spatiotemporal patterns from motion capture data. These reduced-order representations, along with ANN models, are evaluated on their ability to predict muscle forces and joint torques from motion capture data. The study assesses each method's accuracy in hazardous scenarios, addressing limitations of traditional physics-based models.
This comparative approach aims to develop Metahuman Models that can reduce human exposure to dangerous conditions during offshore wind turbine operations and maintenance, ultimately enhancing worker safety.
Presentation 2
Speaker: Shinjiro Sueda (Computer Science & Engineering, Texas A&M University)
Co-author: N/A
Title: Data-Driven Inertia and Collision Handling for Biomechanics and Graphics
Abstract: Data-driven approaches are valuable tools in computational modeling, offering the potential to simplify or increase the efficiency of simulations. However, replacing the entire computational pipeline with data-driven models is not always feasible or practical. In this talk, we will present two examples of using neural networks to address small yet computationally intensive portions of the simulation pipeline for biomechanics and graphics. First, we will demonstrate the significance of accounting for the inertial effects of musculotendons as they move around bones and joints, and how these effects can be efficiently modeled with a neural network. Second, we will illustrate how a neural network can be used to model collision responses between contacting triangles. In both cases, these networks can be seamlessly integrated into the simulation pipeline with minimal modifications.
Organizers
Stephen Shipman (Louisiana State University), Frank Sottile (Texas A&M University)
Mini-symposium Abstract:
Schrödinger and related operators on a periodic medium may be modeled as operators on periodic graphs, which enables a mixture of analysis and algebraic geometry to be brought to bear on their study. This mini-symposium will present work done by researchers in the SIAM Section, showcasing both algebraic and analytic approaches to their study.
Presentation 1
Speaker: Wencai Liu (Texas A&M University)
Co-author: Rodrigo Matos (PUC-Rio) and John Treuer (University of California San Diego)
Title: Embedded eigenvalues of perturbed periodic Schrödinger operators
Abstract: The occurrence of embedded eigenvalues in periodic Schrödinger operators under perturbation is closely related to the irreducibility of the Fermi variety, as explored in the works of Kuchment-Vainberg and Shipman. Building on the recent work of Liu, who resolved the irreducibility conjecture for the Fermi varieties of periodic Schrödinger operators, we investigate uniqueness results for perturbed periodic Schrödinger operators on $\mathbb{Z}^d$. Key applications of this research include demonstrating the absence of embedded eigenvalues for operators with impurities that decay faster than any exponential function, as well as determining sharp decay rates for eigenfunctions. This is a joint work with R. Matos and J. Treuer.
Presentation 2
Speaker: Jordy Lopez (Texas A&M University)
Co-author: Matthew Faust (Michigan State University), Stephen Shipman (Louisiana State University), Frank Sottile (Texas A&M University)
Title: Toric compactifications of periodic graph operators
Abstract: A periodic graph operator is a weighted Laplacian plus potential acting on functions on the vertices of a periodic graph. It is well-known that the spectrum of a periodic graph operator is the projection of an algebraic variety known as the Bloch variety. Motivated by work of Bättig, we compactify the Bloch variety of a periodic graph operator inside the toric variety associated to its Newton polytope. For a wider family of periodic graphs, we extend this operator to the toric variety by expressing the compactification as the support of a kernel sheaf. We outline a few spectral-theoretic consequences of this compactification. This is joint work with Matt Faust, Stephen Shipman, and Frank Sottile.
Presentation 3
Speaker: Fan Yang (Louisiana State University)
Co-author: Moises Gomez Solis (Louisiana State University), Dylan Spedale (Louisiana State University)
Title: On the spectrum of magnetic Laplacian on the Lieb lattice
Abstract: We study the magnetic Laplacian on the Lieb lattice, and prove Cantor spectrum for arbitrary irrational magnetic flux. We also provide a complete spectral analysis for the reduced one-dimensional Hamiltonian, proving Cantor spectra for all irrational frequencies, and sharp arithmetic phase transitions. Part of our analysis reveals a novel coexistence phenomenon of point spectrum and absolutely/singular continuous spectrum. This is a joint work with Moises Gomez Solis and Dylan Spedale.
Presentation 4
Speaker: Rui Han (Louisiana State University)
Co-author: N/A
Title: Non-perturbative localization for quasi-periodic Jacobi matrices
Abstract: We talk about non-perturbative Anderson localization for quasi-periodic Jacobi block matrix operators assuming non-vanishing of all Lyapunov exponents. The base dynamics on tori is assumed to be a Diophantine rotation. We will also talk about some applications on bilayer graphene models.
Organizers
Jimmie Adriazola (Southern Methodist University), Baofeng Feng (University of Texas Rio Grande Valley), Alejandro Aceves (Southern Methodist University)
Mini-symposium Abstract:
Integrable and nearly-integrable systems have a long history dating back to the advent of Newtonian mechanics. Despite being an old subject, these systems still remain a fruitful area of research both for theory and applications. Moreover, modern data science methods are proving to be useful for studying integrable systems numerically.
This session will focus on analytical and numerical methods that are advancing our understanding of lattice and continuum systems with an underlying integrable or nearly-integrable structure. Topics will range from inverse-scattering, bilinear, and data-driven methods with applications to fractional and non-local models.
Presentation 1
Speaker: Baofeng Feng (University of Texas Rio Grande Valley)
Co-author: Guo-Fu Yu (Shanghai Jiaotong University), Han-Han Sheng (Shanghai Jiaotong University)
Title: A Generalized Sine-Gordon Equation: Reductions and Integrable Discretizations
Abstract: In this talk, we construct integrable lattice models of a generalized sine-Gordon (gsG) equation. To this end, we firstly link the gSG equation to a set of blinear equations in the KP-Toda hierarchy which can be derived from the discrete KP equation. Based on this finding, we derive semi-discrete and fully discrete analogues of the gsG equation. By introducing a parameter c, we demonstrate that the gsG equation can reduce to the sine-Gordon equation and the short pulse at the levels of continuous, semi-discrete and fully discrete cases. The limiting forms of the N-soliton solutions to the gsG equation in each level also correspond to those of the sine-Gordon equation and the short pulse equation. This is a joint work with Guo-Fu Yu and Han-Han Sheng at Shanghai Jiaotong University, China.
Presentation 2
Speaker: Andrew Comech (Texas A&M University)
Co-author: Nabile Boussaid (Universite Franche-Comte, Besancon, France), Niranjana Kulkarni (Texas A&M University)
Title: Linear stability of bi-frequency solitary waves in the nonlinear Dirac equation and the Dirac--Klein--Gordon system
Abstract: As it has been known for some time, the nonlinear Dirac equation (the Soler model) and the Dirac--Klein--Gordon system, besides usual, one-frequency solitary waves, also admit bi-frequency solitary waves. Moreover, it turns out that the one-frequency solitary waves -- some of them proved to be linearly stable -- can never be asymptotically stable: a small perturbation of a one-frequency solitary wave may always be chosen so that it becomes a bi-frequency solitary wave; being an exact solution, it would never relax to a one-frequency solitary wave. (This explains why there have been no asymptotic stability results for nonlinear Dirac equation, except under certain restrictions on the perturbation to effectively eliminate bi-frequency solitary waves altogether.)
To prove asymptotic stability of bi-frequency solitary waves, we are back to square one: we first need to prove their linear stability. This is the main result of this talk: we develop the approach to the linear stability of bi-frequency solitary waves in the Soler model (or Dirac--Klein--Gordon system) in three spatial dimensions and show that some of these solitary waves are linearly stable.
Presentation 3
Speaker: Juan Carlos Lopez Vieyra (Instituto de Ciencias Nucleares UNAM)
Co-author: N/A
Title: N-body choreographic motion on the Lemniscate: polynomial integrals of motion, superintegrability
Abstract: For 3-body and 5-body planar choreographic motions on a common algebraic (Bernoulli) lemniscate we found explicitly a maximal possible set of (particular) Liouville integrals, 7 and 15, respectively, (including the total angular momentum), which Poisson commute with a corresponding Hamiltonian along the trajectory. Thus, these choreographies are particularly maximally superintegrable. The particular integrals are polynomial in coordinates and velocities. In general, it is conjectured that the total number of (particular) Liouville integrals is maximal possible for any odd number of bodies moving choreographically along the lemniscate and that the corresponding trajectory is particularly, maximally superintegrable. Some of these Liouville integrals are presented explicitly.
Presentation 4
Speaker: Mohammad Hassan Murad (University of Texas at Dallas)
Co-author: Vladimir Dragovic (University of Texas at Dallas)
Title: Circumscribed about a Unit Circle and Inscribed in a Conic from a Given Confocal Pencil
Abstract: We study n-Poncelet polygons in the plane inscribed in a given family of confocal conics C(t) and circumscribed about a unit circle D. In this talk we focus on case n = 3. Cayley's theorem for the existence of 3-Poncelet polygons gives a monic quartic polynomial in t. By using the root classification of a quartic polynomial, we describe a subset of the plane for which we show that if the center of the circle lies there, then there exists at least one and at most four nondegenerate conics from the confocal family, so that a triangle is inscribed in the conic and circumscribed about the circle. We also show that there exist boundary conics of 3-Poncelet polygons corresponding to a double zero of the quartic with a fixed D and they can be achieved as a limit of two boundaries of 3-Poncelet polygons of the same type, while keeping D fixed. We show that the boundary conics corresponding to a triple zero of the quartic with the fixed D exist, but they, however, cannot be achieved as a limit of boundaries of three 3-Poncelet polygons of the same type. We further proved that both boundary conics corresponding to the simple and triple zeros of the quartic are always ellipse. The presentation is based on a joint work with V. Dragovic.
Organizers
Christin Bibby (Louisiana State University), Galen Dorpalen-Barry (Texas A&M University)
Mini-symposium Abstract:
Both hyperplane arrangements and polytopes arise naturally in many disparate areas of mathematics: discrete geometry, topology, algebraic geometry, linear algebra, coding theory, optimization, and more. While every hyperplane arrangement defines a dual zonotope and every polytope as an associated defining hyperplane arrangement, the communities studying these two areas are often disconnected. In this mini-symposium, we plan to bring together experts in hyperplane arrangements and polytopes, in order to foster collaboration between the two areas.
Presentation 1
Speaker: Nathan Williams (University of Texas at Dallas)
Co-author: Colin Defant (Harvard University)
Title: Pop, Crackle, Snap (and Pow): Some Facets of Shards
Abstract: Reading cut the hyperplanes in a real central arrangement H into pieces called shards, which reflect order-theoretic properties of the arrangement. We show that shards have a natural interpretation as certain generators of the fundamental group of the complement of the complexification of H. Taking only positive expressions in these generators yields a new poset that we call the pure shard monoid. When H is simplicial, its poset of regions is a lattice, so it comes equipped with a pop-stack sorting operator Pop. In this case, we use Pop to define an embedding Crackle of Reading's shard intersection order into the pure shard monoid. When H is the reflection arrangement of a finite Coxeter group, we also define a poset embedding Snap of the shard intersection order into the positive braid monoid; in this case, our three maps are related by Snap = Crackle Pop. This is joint work with Colin Defant.
Presentation 2
Speaker: Dan Cohen (Louisiana State University)
Co-author: Christin Bibby (Louisiana State University), Emanuele Delucchi (University of Applied Sciences and Arts of Southern Switzerland)
Title: Supersolvable toric arrangements
Abstract: A toric arrangement is a finite collection of codimension one subtori in a complex torus. If the intersection pattern of these subtori satisfy the combinatorial condition of supersolvabiilty, the complement of the toric arrangement sits atop a tower of fiber bundles. We investigate various topological features of the complement in this context. This is joint work with Christin Bibby (LSU) and Emanuele Delucchi (SUPSI).
Presentation 3
Speaker: Trevor Karn (University of Minnesota)
Co-author: Robert Angarone, Patricia Commins, Satoshi Murai, and Brendon Rhoades
Title: Superspace coinvariants and hyperplane arrangements
Abstract: The superspace ring of polynomial-valued differential forms on n-space admits an action of the symmetric group. The superspace coinvariant ring is the quotient by the ideal of symmetric forms with no constant term. We give the first explicit basis of this quotient, proving a conjecture of Sagan and Swanson. Our techniques use the theory of hyperplane arrangements. We relate the quotient to instances of the Solomon-Terao algebras of Abe-Maeno-Murai-Numata and use exact sequences relating the derivation modules of certain 'southwest closed' arrangements to obtain the desired basis. (This work is joint with Robert Angarone, Patricia Commins, Satoshi Murai, and Brendon Rhoades.)
Presentation 4
Speaker: Theo Douvropoulos (Brandeis University)
Co-author: N/A
Title: Deformations of restricted reflection arrangements
Abstract:
The m-Fuss ideal deformations of the reflection arrangements of Weyl groups W include the Shi and Catalan cases and have a remarkable
property: if h is the Coxeter number of W, their characteristic polynomials are product formulas in m*h with integer roots. This is a rare phenomenon when considering deformations of arbitrary arrangements, which Athanasiadis and Yoshinaga related to the Ehrhart polynomiality of certain associated polytopes.
We construct deformations for all restrictions of W-reflection arrangements, prove that they are free, and that their characteristic
polynomials are given via product formulas that generalize the previous cases. Remarkably, the parallel copies of hyperplanes in our free
deformations are not always equally spaced and their multiplicities are not constant, but given through root-theoretic data of W.
Organizer
Oleg Makarenkov (The University of Texas at Dallas)
Mini-symposium Abstract:
The mini-symposium discusses (i) mathematical methods that are used for the analysis of the dynamics of differential equations with non-smooth right-hand-sides and jumps, (ii) non-smooth models from applied sciences where recent mathematical theory helped to predict and control the dynamics.
Presentation 1
Speaker: Soufiane Abbadi
Co-author: Adannah Duruoha (The University of Texas at Dallas), Collins Boateng (The University of Texas at Dallas), Matthew Williams (The University of Texas at Dallas), Oleg Makarenkov (The University of Texas at Dallas)
Title: Existence and stability of a limit cycle in the model of a nonuniform rimless wheel rolling down a slope
Abstract: The rimless wheel model appears in engineering literature as a zero dynamics reduction of the model of a biped walking down a slope. Nonuniformity in the rimless wheel (i.e. unequal lengths of spokes and unequal angles between successive spokes) can be viewed as nonuniformity of the walking terrain. Existence of a limit cycle for nonuniform rimless wheel model was established in the earlier literature but stability was addressed just briefly. The present talk discusses conditions for asymptotic stability of the limit cycle.
Presentation 2
Speaker: Egor Makarenkov (Allen High School)
Co-author: Sakshi Malhotra (The University of Texas at Dallas)
Title: Design of a benchmark elastoplastic lattice-spring model with minimal fabrication cost and required multi-functional properties
Abstract: We consider an elastoplastic lattice-spring model whose springs are allowed to conduct current. Denoting elastic limit of spring i by c_i, we take the electric resistance of each spring as R_i=1/c_i. We then compute the maximal response force F that such an elasto-plastic model can achieve (in response to gradually increasing displacement-controlled loading). A combination of R_i and F allow us to introduce a functional that measures multi-functional performance of the model. The fabrication cost of the model is taken as summation over all c_i. By going over all possible topologies on 4 springs, we develop analytic computations to determine the topology that minimizes the manufacturing cost while keeping the multi-functional performance above a chosen constant.
Presentation 3
Speaker: Sakshi Malhotra (The University of Texas at Dallas)
Co-author: Yang Jiao (Arizona State University), Oleg Makarenkov (The University of Texas at Dallas)
Title: Optimization of a lattice spring model with elastoplastic conducting springs: A case study
Abstract: We consider a simple lattice spring model in which every spring is elastoplastic and is capable to conduct current. The elasticity bounds of spring i are taken as [-c_i,c_i] and the resistance of spring i is taken as 1/c_i, which allows us to compute the resistance of the system. The model is further subjected to a gradual stretching and due to plasticity, the response force increases until a certain terminal value. We demonstrate that the recently developed sweeping process theory can be used to optimize the interplay between the terminal response force and the resistance on a physical domain of parameters c_i.
Presentation 4
Speaker: Oleg Makarenkov (The University of Texas at Dallas)
Co-author: Josean Albelo-Cortes (Southern Methodist University)
Title: Topological properties of network spring models that determine terminal distributions of plastic deformations
Abstract: A recent result by Gudoshnikov et al [SIAM J. Control Optim. 2022] ensures finite-time convergence of the stress-vector of elastoplastic network spring models under assumption that the vector g'(t) of the applied displacement controlled-loading lies strictly inside the normal cone to the associated polyhedral set (that depends on mechanical parameters of the springs). Determination of the terminal distribution of stresses has been thereby linked to a problem of spotting a face on the boundary of the polyhedral set where the normal cone contains vector g'(t). In this talk the above-mentioned problem of spotting an eligible face is converted into a search for an eligible set of springs that have a certain topological property with respect to the graph of the network.
Organizers
Bryant Wyatt (Tarleton State University), Christopher Mitchell (Tarleton State University)
Mini-symposium Abstract:
The mini symposium "Computational Modeling in the Sciences," at the upcoming SIAM conference, will explore the application of computational techniques across various scientific disciplines. Presentations will cover topics such as the use of advanced methods for analyzing complex systems, modeling wave propagation, and applying statistical approaches to societal issues. The symposium will also feature discussions on innovative simulations in plasma physics and biomedical sciences, highlighting the role of computational models in understanding intricate physical and biological phenomena. This session will provide valuable insights into how computational modeling drives advancements and addresses challenges in diverse areas of science.
Presentation 1
Speaker: Christopher Mitchell (Tarleton State University)
Co-author: N/A
Title: Expanding Classical ODE Models with Bayesian Inference for Improved Disease Outbreak Management
Abstract: Classical models of disease outbreaks, based on nonlinear ordinary differential equations (ODEs), have been instrumental in controlling the spread of infectious diseases and are credited with saving millions of lives. These models rely on accurately estimating parameters that are often poorly understood and challenging to infer, particularly when the available data is weak, noisy, or limited to specific subpopulations. Traditional point-estimation methods can be fragile under these conditions, leading to unreliable predictions. Bayesian inference offers a more resilient approach by replacing point-estimates with posterior distributions, enabling the generation of an ensemble of forecasts that provide a broader understanding of potential outbreak scenarios. This project focuses on developing ODE models for tuberculosis and avian influenza using amortized Bayesian inference within the BayesFlow Python library. By combining these new tools with classical equilibrium analysis techniques, particularly the basic reproductive number, we explore the ODE parameter space more efficiently and robustly. This approach will yield valuable insights into interventions such as treatment, vaccination, and culling, ultimately helping to prevent or mitigate future outbreaks.
Presentation 2
Speaker: Sebastian Acosta (Department of Pediatrics, Baylor College of Medicine and Texas Children's Hospital)
Co-author: N/A
Title: Pseudo-differential Models for Ultrasound Waves
Abstract: To strike a balance between modeling accuracy and computational efficiency for simulations of ultrasound waves in soft tissues, we derive a pseudo-differential factorization of the wave operator. This factorization allows us to approximately solve the wave equation via one-way (transmission) or two-way (transmission and reflection) sweeping schemes tailored to high-frequency wave fields. We also provide a proof-of-concept numerical implementation of the proposed method for ultrasound wave propagation.
Presentation 3
Speaker: Hristo Kojouharov (University of Texas at Arlington)
Co-author: N/A
Title: Theoretical and Numerical Study of Stem Cell Therapy for Post-Myocardial Infarction Left Ventricular Remodeling
Abstract: The human heart is a vital organ that has limited regeneration and repair capabilities. Following a myocardial infarction (MI), permanent cell death and a reduced ability of the heart to recover are the leading causes of morbidity and mortality worldwide. This work presents a novel mathematical model for investigating stem cell therapy possibilities for left ventricular remodeling after myocardial infarction. The model not only successfully predicts the interactions between cardiac cells and the immune system, but it also accurately reproduces post-MI cardiomyocyte regeneration after a stem cell therapy with oxygen restoration. The performance and functionality of the new model are illustrated through a numerical study of the resulting system of nonlinear ordinary differential equations (ODE). The optimal time to infuse stem cells for different types of oxygen restorations is identified. The proposed nonlinear ODE model has the potential to provide researchers with a predictive computational tool for better understanding MI pathogenesis and developing new cell-based therapies.
Presentation 4
Speaker: Benny Rodriguez Saenz (Center for Astrophysics, Space Physics, and Engineering Research at Baylor University)
Co-author: N/A
Title: Studying Dust Behavior in Weakly Ionized Plasmas with Magnetic Fields Via the DRIAD Code
Abstract: Complex plasmas are composed of micron-sized dust particles that are suspended in a gas with low ionization levels. The charging of these dust particles occurs as a result of interactions with electrons and ions, and is influenced by factors such as the temperature of the plasma, the density of the gas, and the strength of electric fields. Magnetic fields also have an impact on the charging of dust particles and their subsequent behavior but it is not well understood. The effect is contingent upon the levels of magnetization exhibited by various charged species present in the complex plasma. Despite the fact that current theories mainly concentrate on dust particles that are spherical in shape, practical situations encountered, for instance, in experiments related to fusion and in astrophysical settings often entail dust particles with irregular shapes. In order to bridge this knowledge gap, an examination into the charging mechanism of dust aggregates is undertaken. More specifically, we compare how aggregates become charged in scenarios where no magnetic field is present (B = 0 T) to situations where a magnetic field is present (0 T < B < 3.5 T). Our analysis takes into account the variation in the flow of electrons and ions towards specific points on the surface of the aggregate. The manner in which charge is distributed across the aggregate's surface results in conflicting torques, ultimately influencing the orientation and movement of dust particles within the plasma medium.
Organizers
Yunhui He (University of Houston), Loic Cappanera (University of Houston)
Mini-symposium Abstract:
Nonlinear Partial Differential Equations (PDEs) play an important role in our real-world and arise in many areas such as geoscience, materials science, and energy technologies, which often model complex coupled problems, such as Navier-Stokes problems, poroelasticity, and Magnetohydrodynamics. However, it remains a challenge to derive and analysis numerical methods of such models due to, for example, physical parameters and unstructured domain. There has been an increasing interest in the development and analysis of fast and efficient solvers for these problems. The goal of this mini-symposium is to bring researchers from practical use and theoretical aspects to share and discuss recent achievements and challenges on the development and analysis of numerical solutions for nonlinear PDEs with finite elements methods. Specifically, the topics of this mini-symposium focus on different type of nonlinear models, finite element methods, preconditioning skills and multilevel techniques.
Presentation 1
Speaker: Amnon J. Meir (Southern Methodist University)
Co-author: N/A
Title: On the Equations of Poroelasticity and Electroporoelasticity
Abstract: Complex physical systems and phenomena frequently involve multiple components, complex or coupled domains, complex physics or multi-physics, as well as multiple spatial and temporal scales. Such phenomena and systems are often modeled by systems of coupled partial differential equations, or integro-partial differential equations, often nonlinear. Such phenomenon of interest include poroelasticity and electroporoelasticity.
After introducing the equations of poroelasticity (the equations of flow through porous media coupled to the elasticity equations) and electroporoelasticity (the equations of poroelasticity coupled to Maxwell's equations) which naturally arise in geoscience, hydrology, and petroleum exploration, as well as various areas of science and technology, I will describe some recent results (well posedness), the numerical analysis of a finite-element based method for approximating solutions, and some interesting challenges.
Presentation 2
Speaker: Mark Simmons (University of Houston)
Co-author: Loic Cappanera (University of Houston), Giselle Sosa Jones (Oakland University)
Title: Discontinuous Galerkin Method for Three-Phase Flows in Porous Media
Abstract: In the oil and gas industry, to enhance recovery techniques and improve clean-up strategies for subsurface contamination leftover from carbon dioxide buildup, there is an increasing need for efficient, robust, and accurate numerical methods. In this talk, we give a brief overview of the discontinuous Galerkin finite element method derived to solve three-phase flow problems in a highly heterogeneous porous medium. This method involves solving for three unknowns: liquid pressure, aqueous, and vapor saturation, all of which are solved sequentially. We will see the validation of our scheme via manufactured solutions and the quarter of a 5-spot problem as well as applications to viscous fingering generation. Then, we present a new stability version of our algorithm method that introduces a stability term in our formulation, which results in time-independent stiffness matrices for our saturation variables. Numerical investigations show how this new method yields similar accuracy results and increases the computational efficiency of our solver.
Presentation 3
Speaker: Melvin Creff (Texas A&M University)
Co-author: Jean-Luc Guermond (Texas A&M University)
Title: Preconditioning of the generalized Stokes problem arising from the approximation of the time-dependent Navier-Stokes equations.
Abstract: We consider standard iterative methods for solving the generalized Stokes problem arising from the time and space approximation of the time-dependent incompressible Navier-Stokes equations. Various preconditioning techniques are considered (Cahouet&Chabard and augmented Lagrangian), and one investigates whether these methods can compete with traditional pressure-correction and velocity-correction methods in terms of CPU time per degree of freedom and per time step.
Numerical tests on fine unstructured meshes (68 millions degrees of freedom) demonstrate convergence rates that are independent of the mesh size and improve with the Reynolds number.
Three conclusions are drawn:
(1) Although very good parallel scalability is observed for the augmented Lagrangian method, thorough tests on large problems reveal that the overall CPU time per degree of freedom and per time step is best for the standard Cahouet&Chabard preconditioner.
(2) Whether solving the pressure Schur complement problem or solving the full couple system at once does not make any significant difference in terms of CPU time per degree of freedom and per time step.
(3) All the methods tested, whether matrix-free or not, are on average 30 times slower than traditional pressure-correction and velocity-correction methods.
Hence, although all these methods are very efficient for solving steady state problems, they are not yet competitive for solving time-dependent problems.
Presentation 4
Speaker: Yunhui He (University of Houston)
Co-author: Maxim Olshanskii (University of Houston)
Title: A preconditioner for the grad-div stabilized equal-order finite elements discretizations of the Oseen problem
Abstract: The work considers grad-div stabilized equal-order finite elements (FE) methods for the linearized Navier-Stokes equations. A block triangular preconditioner for the resulting system of algebraic equations is proposed which is closely related to the Augmented Lagrangian (AL) preconditioner. A field-of-values analysis of a preconditioned Krylov subspace method shows convergence bounds that are independent of the mesh parameter variation. Numerical studies support the theory and demonstrate the robustness of the approach also with respect to the viscosity parameter variation, as is typical for AL preconditioners when applied to inf-sup stable FE pairs. The numerical experiments also address the accuracy of grad-div stabilized equal-order FE method for the steady state Navier-Stokes equations.
Organizers
Manaswinee Bezbaruah (Texas A&M University), Jordan Hoffart (Texas A&M University)
Mini-symposium Abstract:
The control of electromagnetic properties of different materials has led to the development of several novel devices and techniques such as: optical materials with tunable frequency, cloaking, nano-antennas with extremely short wavelength resonance, wireless nano communications, optical holography, and waveguides. Recent developments in numerical methods have significantly advanced our ability to simulate and analyze these electromagnetic phenomena. We will explore the forefront of these advancements, focusing on innovative techniques that enhance the accuracy, efficiency, and applicability of numerical simulations in electromagnetism. Key topics include finite element methods tailored to electromagnetic problems, spectral methods for high-frequency applications, and robust algorithms for handling nonlinearities and multi-physics coupling.
Presentation 1
Speaker: Loic Cappanera (University of Houston)
Co-author: Caroline Nore (University Paris-Saclay) Sabrina Benard (University Paris-Saclay) Wietze Herreman (University Paris-Saclay)
Title: Magnetic field based discretization for problems with discontinuous electric potential. Applications to liquid metal batteries.
Abstract: We introduce a magnetic field based finite element formulation to approximate magneto-static problems with discontinuous electric potential. We compare it with a classical formulation based on the electrical scalar potential that has the disadvantages of requiring the use Biot-Savart law to reconstruct the magnetic field, needed to compute the Lorentz force for magnetohydrodynamics problems. After validating the proposed method using various setups with manufactured solutions, we extend the method to time-dependent magnetohydrodynamics problems where the Maxwell and Navier-Stokes equations are coupled.
The resulting scheme is applied to the study of Liquid Metal batteries, a promising grid-scale energy storage device, where the alloy composition affects the electrical potential distribution at the liquids interface and so can hinder the battery operation and efficiency to store/release energy.
Presentation 2
Speaker: Daniel Massatt (Louisiana State University)
Co-author: N/A
Title: Momentum Space and Continuum Models of Incommensurate Bilayer 2D Materials
Abstract: Electronic structure of incommensurate 2D materials is well approximated by Wannierized tight-binding models, but these models do not immediately realize the critical relationship between momenta and energy vital for applications such as the construction of a many-body basis. Continuum models have become excellent tools for describing the low energy physics such as the Bistritzer-MacDonald model, but much is yet to be understood about the accuracy of these models.
Here we discuss the momentum space model, and provide a procedure for producing continuum models with controlled accuracy. We show the momentum space model is a direct transformation of the tight binding model, and is exponentially accurate with respect to truncation. We then construct a method for producing continuum models realized as Taylor expansions of the momentum space model. As a case study, we show the Bistritzer-MacDonald model is a Taylor expansion of the Wannier tight-binding model for twisted bilayer graphene, and illustrate how to improve accuracy of the model with additional terms. The procedure can be applied to a large class of tight-binding models such as TMDC bilayers, and error can be tuned by parameter choices.
Presentation 3
Speaker: Liet Vo (University of Texas Rio Grande Valley)
Co-author: N/A
Title: A high-order perturbation of envelopes (hope) method for vector electromagnetic scattering by periodic inhomogeneous media
Abstract: The scattering of electromagnetic waves by three-dimensional periodic structures is important for many problems of crucial scientific and engineering interest. Due to the complexity and three-dimensional nature of these waves, fast, accurate, and reliable numerical simulation of these are indispensable for engineers and scientists alike. For this, High-Order Spectral methods are frequently employed and in this talk, I present the HOPE method which is in this class. Our approach is perturbative in nature where we view the deviation of the permittivity from a constant value as the deformation and we pursue regular perturbation theory. More specifically, we expand the three-dimensional, vector-valued electric field in a Taylor series in this small deformation parameter, derive recursions that each term in this series must satisfy, invoke a novel elliptic theory to establish bounds on the size of each correction, and thereby show that the purported Taylor series does, in fact, converge. Beyond this, we show that each of these terms in the Taylor series is jointly analytic in all three spatial variables by estimating solutions of governing equations for derivatives of these terms. Numerical experiments are also provided to validate the theoretical results.
Presentation 4
Speaker: Mansi Bezbaruah (Texas A&M University)
Co-author: Matthias Maier (Texas A&M University), Winnifred Wollner (University of Hamburg)
Title: Shape and Eigenvalue Optimization of Microstructures Governed by Maxwell's Equations
Abstract: This talk is concerned with a class of shape optimization problems involving optical metamaterial comprised of periodic nanoscale inclusions. We will first summarize the underlying microscale model, the corresponding homogenization theory, and the eigenvalue representation that will serve as a basis for the shape optimization problems. We will then introduce a deformation field on the cell problem and the eigenvalue problem. Finally, we will formulate and solve the shape optimization problems using an adjoint approach.
Organizer
Xin Liu (Texas A&M University)
Mini-symposium Abstract:
The proposed session on "The many scales of mathematical analysis of fluid" will bring together researchers from a variety of disciplines to discuss the latest advances in this field. The session will cover a wide range of topics, including: 1. The mathematical analysis of incompressible fluid flows, 2. The mathematical analysis of fluid flows in geophysics, 3. The mathematical analysis of multiphase fluid flows, etc.
Presentation 1
Speaker: Alex Vasseur (University of Texas at Austin)
Co-author: Geng Chen (University of Kansas), Moon-Jin Kang (Korea Advanced Institute of Science and Technology)
Title: From Navier-Stokes to discontinuous solutions of compressible Euler
Abstract: The compressible Euler equation can lead to the emergence of shock discontinuities in finite time, notably observed behind supersonic planes. A very natural way to justify these singularities involves studying solutions as inviscid limits of Navier-Stokes solutions with evanescent viscosities. The mathematical study of this problem is however very difficult because of the destabilization effect of the viscosities.
Bianchini and Bressan proved the inviscid limit to small BV solutions using the so-called artificial viscosities in 2004. However, until very recently, achieving this limit with physical viscosities remained an open question.
In this presentation, we will provide the basic ideas of classical mathematical theories to compressible fluid mechanics and introduce the recent method of a-contraction with shifts. This method is employed to describe the physical inviscid limit in the context of the barotropic Euler equation, and to solve the Bianchini and Bressan conjecture in this special case. This is a joint work with Geng Chen and Moon-Jin Kang.
Presentation 2
Speaker: Jiayun Meng (University of Texas at Austin)
Co-author: Alex Vasseur (University of Texas at Austin), Young-Sam Kwon (Dong-A University, South Korea)
Title: a-contraction theory for general viscous systems of conservation laws
Abstract: The a-contraction theory is a recent energy-based method developed to tackle the stability and asymptotic limits for general solutions to conservation laws. In this talk, I will focus on the time asymptotic stability of composite waves of viscous shock and rarefaction for viscous conservation laws. I will describe how the a-contraction theory, first developed in this context to the compressible Navier-Stokes equation, can be extended to such general situations. This is a joint work with Young-Sam Kwon and Alexis Vasseur.
Presentation 3
Speaker: Paul Blochas (University of Texas at Austin)
Co-author: N/A
Title: Uniform Asymptotic Stability for Convection-Reaction-Diffusion Equations in the Inviscid Limit Towards Riemann Shocks
Abstract: In this talk, I will present a result about the study of the stability in time of a family of viscous shocks approximating a given Riemann shock. The goal of that work is to show some uniform asymptotic orbital stability property of such waves in the vanishing viscosity limit.
Even at the linear level, to ensure uniformity, the propagator of the linearization is decomposed into a decreasing part and a phase modulation in a highly non-standard way.
Furthermore, we introduce a multi-scale norm depending on the viscous parameter. To avoid the use of arguments based on parabolic regularization that would preclude a result uniform in the viscosity parameter, the nonlinear estimates on this norm are closed through some suitable maximum principle.
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: Long Li (Rice University)
Co-author: N/A
Title: Almost periodicity in time of the nonlinear defocusing Schrődinger operators
Abstract: KdV equations and nonlinear Schrődinger equations are considered as the most important examples of completely integrable systems. Akhiezer first realized that integrating such systems can be reduced to a Jacobian inversion problem. After decades of intense activities of Novikov, Levitan, Dubrovin, Its, Matveev, Widom, Baker, Lax, Marchenko, Mackean, Trubowitz, Zakharov, Shaba, Hasumi and many others, the theory has been largely completed. In particular, the almost periodicity of the solitary solution is understood as a nice property of the generalized Abel map by Sodin-Yuditskii. It was then realized by Volberg-Yuditskii that almost periodicity is a consequence of the dichotomy of Direct Cauchy Theorem of a Widom domain. The theory has been implemented by Binder-Damanik-Lukic-Yuditskii for KdV equations. They proved that under suitable conditions of the initial data, the KdV equation admits a unique solution that is almost periodic in both space and time variables. However, the counterpart of NLS is missing. We fill this gap by providing fundamental results in both direct and inverse spectral aspects. We noticed a different behavior of the translation flow compared to the Schrődinger case, which is crucial to understand the gap edges situation and the almost periodicity.
Presentation 2
Speaker: Jonathan Stanfill (Ohio State University)
Co-author: N/A
Title: Exotic structures of spectral $\zeta$-functions for Sturm-Liouville operators
Abstract: We discuss the structure of the spectral $\zeta$-function associated with Sturm-Liouville operators. In particular, we will point out what is a "standard" behavior in physical models. We will then give examples of more exotic behavior and show that varying the self-adjoint extensions of a given problem can drastically change the structure of the spectral $\zeta$-function. This includes a very unusual and remarkable structure consisting of a series of branch points located at every nonpositive integer value of $s$. By comparing the Pőschl--Teller potential to the classic Bessel potential, we further illustrate that perturbing a given potential by even a smooth potential on a finite interval can greatly affect the meromorphic structure of the spectral $\zeta$-function in surprising ways. Based on joint work with Guglielmo Fucci and Mateusz Piorkowski.
Presentation 3
Speaker: Qiaochu Ma (Texas A&M University)
Co-author: N/A
Title: Mixed quantization and quantum ergodicity
Abstract: Quantum Ergodicity (QE) is a classical topic in quantum chaos, it states that on a compact Riemannian manifold whose geodesic flow is ergodic, the Laplacian has a density-one subsequence of eigenfunctions that tends to be equidistributed. We present a uniform version of QE (UQE) for a certain series of unitary flat vector bundles. The holonomy of flat bundles leads to beautiful geometrical phenomena. The key technique involves the combination of Weyl quantization and Berlin-Toeplitz quantization. This is joint work with Minghui Ma.
Organizers
Julia Lindberg (University of Texas Austin), Joe Kileel (University of Texas Austin)
Mini-symposium Abstract:
Many problems in data science have algebraic structure, which algorithms can potentially exploit for computational gain. This minisymposium focuses on polynomial optimization and tensor decomposition methods for data analysis. As motivating applications, talks will touch on statistical modeling, data streaming, and neural network training. Some speakers will present algebro-geometric theory behind polynomial and tensor based methods, which comprise part of their appeal, while other speakers will demonstrate that these computational approaches can be applied at scale (at least in some cases). Our sessions aim to bridge the gap between these abstract algebraic theories and practical computational methods and bring together researchers interested in both areas.
Presentation 1
Speaker: Suhan Zhong (Texas A&M University)
Co-author: N/A
Title: Polynomial Approach for Bilevel Optimization
Abstract: Bilevel optimization is a historically challenging problem due to its hierarchical structure. Efficient computational method is highly wanted to find the global optimality of such problems without convex assumptions. In this talk, I'll introduce a novel method to solve bilevel optimization defined by polynomials. The method is based on Lagrange multiplier expressions and polynomial extensions. Under proper assumptions, this method can solve bilevel polynomial optimization globally with Moment-Sum-Of-Squares relaxations.
Presentation 2
Speaker: David Kahle (Baylor University)
Co-author: N/A
Title: Experiments with Variety Distributions and Variety Regression
Abstract: Recent work in computational algebraic statistics has revealed practical numerical strategies to generate points on or near the real variety of a given collection of polynomials. These algorithms have since been implemented in the R language, providing a convenient Monte Carlo environment in which to experiment with various problems of interest in real algebraic geometry. After reviewing the basic concept of variety distributions-and in particular the variety normal distribution-and the sampler implementation, in this talk we present a number of such experiments, including the inverse problem of variety regression.
Presentation 3
Speaker: Sean Plummer (University of Arkansas)
Co-author: N/A
Title: Asymptotic Expansion of the Evidence Lower Bound for Transformation Based VI in Singular Models
Abstract: Recent work of Bhattacharya et al. (2024) on variational model selection in singular models shows that the evidence lower bound (ELBO) corresponding the mean-field approximation of the standard form of a singular model asymptotically recovers the leading order behavior of the log-marginal likelihood. Unfortunately, the standard form of a singular model cannot be computed for most statistical models as it requires full knowledge of the resolution map. This naturally suggests using a transformation based variational family constructed using normalizing flows to implicitly learn an approximation to the resolution. We delineate the conditions under which the normalizing flow is capable of approximating the resolution map for the singular model and the corresponding evidence lower bound (ELBO) to this normalizing flow based variational families recovers either the correct leading order or the full behavior asymptotic behavior of the log-marginal likelihood. Additionally, aiming to leverage statistical-computational trade-offs in mean-field VI, we introduce our two-step estimation procedure in which we first learn the approximate resolution mapping, then analytically compute the mean-field approximation using the CAVI algorithm. We justify the use of this two-step procedure by showing that the CAVI algorithm corresponding to this two-step procedure is guaranteed to converge and under the same conditions on the normalizing flows, the ELBO corresponding to the two-step procedure recovers the correct leading order behavior of the log-marginal likelihood. We numerically demonstrate this two-step approach is able to achieve better estimates for the RLCT of the model while simultaneously improving computational speed through the use of smaller normalizing flow architectures.
Presentation 4
Speaker: Zehua Lai (University of Texas at Austin)
Co-author: N/A
Title: ReLU Transformers and Piecewise Polynomials
Abstract: We highlight a perhaps important but hitherto unobserved insight: The attention module in a ReLU-transformer is a cubic spline. Viewed in this manner, this mysterious but critical component of a transformer becomes a natural development of an old notion deeply entrenched in classical approximation theory. Conversely, if we assume the Pierce--Birkhoff conjecture, then every spline is also an encoder.
Organizers
William Ott (University of Houston), Amanda Alexander (University of Houston)
Mini-symposium Abstract:
Because a variety of biological phenomena are best modeled stochastically, probability theory has become an indispensable tool for solving many biological problems. Talks in this session will highlight recent advances in mathematical biology that involve probabilistic methods.
Presentation 1
Speaker: William Ott (University of Houston)
Co-author: Sean Campbell (University of Houston), Jae Kyoung Kim (Korea Advanced Institute of Science and Technology), LieJune Shiau (University of Houston Clear Lake), Yun Min Song (Korea Advanced Institute of Science and Technology)
Title: Noisy delay denoises biochemical oscillators
Abstract: Genetic oscillations are generated by delayed transcriptional negative feedback loops, wherein repressor proteins inhibit their own synthesis after a temporal production delay. This delay is distributed because it arises from a sequence of noisy processes, including transcription, translation, folding, and translocation. Because the delay determines repression timing and therefore oscillation period, it has been commonly believed that delay noise weakens oscillatory dynamics. Here, we demonstrate that noisy delay can surprisingly denoise genetic oscillators. Specifically, moderate delay noise improves the signal-to-noise ratio and sharpens oscillation peaks, all without impacting period and amplitude. We show that this denoising phenomenon occurs in a variety of well-studied genetic oscillators and we use queueing theory to uncover the universal mechanisms that produce it.
Presentation 2
Speaker: Andrea Barreiro (Southern Methodist University)
Co-author: Antonio J. Fontanele (University of Arkansas), Prashant C. Raju (University of Arkansas), Shree Hari Gautam (University of Arkansas), Woodrow L. Shew (University of Arkansas), Cheng Ly (Virginia Commonwealth University)
Title: Sensory input to cortex encoded on low-dimensional periphery-correlated subspaces
Abstract: As information about the world is conveyed from the sensory periphery to central neural circuits, it mixes with complex ongoing cortical activity. How do neural populations keep track of sensory signals, separating them from noisy ongoing activity? We demonstrate that sensory signals are encoded more reliably in low-dimensional subspaces defined by correlations between neural activity in primary sensory cortex and upstream sensory brain regions. We analytically show that these correlation-based coding subspaces can reach optimal limits as noise correlations between cortex and upstream regions are reduced, and that this principle generalizes across diverse sensory stimuli in the olfactory system and the visual system of awake mice.
Presentation 3
Speaker: Alexandru Hening (Texas A&M University)
Co-author: N/A
Title: Long-term behavior of stochastic SIQRS epidemic models
Abstract: We analyze and classify the dynamics of SIQRS epidemiological models with susceptible, infected, quarantined, and recovered classes, where the recovered individuals can become reinfected. We include two important types of random fluctuations. The first type is due to small fluctuations of the various model parameters and leads to white noise terms. The second type of noise is due to significant environmental regime shifts that can happen at random. The environment switches randomly between a finite number of environmental states, each with a possibly different disease dynamic. We prove that the long-term fate of the disease is fully determined by a real-valued threshold \lambda. When \lambda<0 the disease goes extinct asymptotically at an exponential rate. On the other hand, if \lambda>0 the disease will persist indefinitely. We end our analysis by looking at some important examples where \lambda can be computed explicitly, and by showcasing some simulation results that shed light on real-world situations.