SIAM Mini-Symposium Session 5
Organizers
Andrea Bonito (Texas A&M University), Alan Demlow (Texas A&M University), Maxim Olshanskii (University of Houston)
Mini-symposium Abstract:
Geometric partial differential equations appear in a wide range of applied modeling settings, such as fluid flows, image processing, and optimal transport. As such numerical methods for such PDE have developed into a highly active research area. Talks in this session will focus on numerical methods for partial differential equations on surfaces and other topics related to geometric PDE.
Presentation 1
Speaker: Frederic Marazzato (University of Arizona)
Co-author: N/A
Title: Computation of origami-inspired structures and mechanical metamaterials
Abstract: Origami folds have found a large range of applications in Engineering as solar panels for satellites or to produce inexpensive mechanical metamaterials.
This talk will first focus on the direct problem of computing the deformation of periodic origami surfaces. A homogenization process for origami folds proposed in [Nassar et al, 2017] and then extended in [Xu, Tobasco and Plucinsky, 2023], is first discussed. The talk will then focus on the PDEs describing Miura surfaces and their approximation by a numerical method.
This talk will then focus on the inverse problem of trying to determine an optimal fold set approximating a given target surface. The folding of a thin elastic sheet is modeled as a two-dimensional nonlinear Kirchhoff plate with an isometry constraint. We formulate the problem in the framework of special functions of bounded variation $SBV$, and propose to use a phase-field damage model and a Discontinuous Galerkin method to approximate the solutions of this problem. We prove that the approximation $\Gamma$-converges to the sharp interface model. Finally, some numerical examples are presented.
Presentation 2
Speaker: Jeremy Shahan (Louisiana State University)
Co-author: N/A
Title: Shape Optimization with Unfitted Finite Element Methods
Abstract: We present a formulation of a PDE-constrained shape optimization problem that uses an unfitted finite element method (FEM). The geometry is represented (and optimized) using a level set approach and we consider objective functionals that are defined over bulk domains. For a discrete objective functional (i.e. one defined in the unfitted FEM framework), we derive the exact Fr\'echet, shape derivative in terms of perturbing the level set function directly. In other words, no domain velocity is needed. We also show that the derivative is (essentially) the same as the shape derivative at the continuous level, so is rather easy to compute. In other words, one gains the benefits of both the optimize-then-discretize and discretize-then-optimize approaches.
Presentation 3
Speaker: Vladimir Yushutin (University of Tennessee-Knoxville)
Co-author: N/A
Title: A parabolic inf-sup theory for TraceFEM
Abstract: We consider the surface heat equation and propose a TraceFEM discretization in space in which the time derivative is stabilized as well. This modification allows us to establish necessary and sufficient conditions for inf-sup stability of the scheme on a general sequence of bulk meshes in terms of $H^1$-stability of a stabilized $L^2$-projection and of an inverse inequality constant that accounts for the lack of conformity of TraceFEM. Furthermore, we prove that the latter two quantities are bounded uniformly for a sequence of shape-regular and quasi-uniform bulk meshes. The obtained uniform discrete inf-sup stability implies several quintessential results, namely convergence to minimal regularity solutions, parabolic quasi-best approximation, optimal order-regularity energy and $L^2 L^2$ error estimates.
Presentation 4
Speaker: Alan Demlow (Texas A&M University)
Co-author: N/A
Title: Nodal FEM for the surface Stokes equations
Abstract: Construction of finite element methods for the surface Stokes equations is difficult because the velocity field must be both tangential and H1 conforming. It is not possible to enforce both of these constraints strongly in on the triangulated discrete surfaces on which surface FEM are typically constructed. Most such methods to date weakly enforce either tangentiality or $H^1$ conformity by penalization. We present a method for constructing tangentially conforming/H1 nonconforming elements with sufficient weak continuity properties to avoid the need for penalization. Error analysis is given for MINI elements, and we also discuss extension to Taylor-Hood elements. This is joint work with Michael Neilan.
Organizers
Difeng Cai (Southern Methodist University), Marcin Jurek (Southern Methodist University)
Mini-symposium Abstract:
This session focuses on the recent development in uncertainty quantification, statistical and machine learning, theoretical and computational methods. Relevant topics include Gaussian processes, Bayesian methods, posterior distributions, computational techniques, modelling techniques, etc.
Presentation 1
Speaker: Marcin Jurek (Southern Methodist University)
Co-author: N/A
Title: Bayesian non-parametrics for spatio-temporal data sets
Abstract: Many environmental phenomena which evolve through time are extremely complex and difficult to model adequately. For example, modern remote sensing tools in atmospheric sciences allow us to obtain millions of measurements of certain variables with high frequency. At the same time, representing their temporal evolution requires very complex models and enormous computing power. In addition, since these complicated models often take a very long time to develop theoretically and implement. This allows but a handful of highly specialized researchers to use those tools.
A promising approach to solving these problems is based on Gaussian Process State Space models, a Bayesian Nonparametric method which consists of imposing a Gaussian process prior on the unknown evolution system. However, the existing techniques using GPSSMs were built for low-dimensional systems, and without adjustments, they would be computationally infeasible if the dimension of the system is high. In this talk, we show this approach can be scaled to high-dimensional environmental problems.
Presentation 2
Speaker: Sherry Wang (University of Texas at Arlington)
Co-author: N/A
Title: Variational Bayesian Multimodal Multiple Instance Classification
Abstract: In multiple instance learning (MIL), objects are represented by bags of instances. Each instance shares the same feature set but has unique feature values. MIL aims to train models that predict bag-level outcomes based on these instances, making it a weakly supervised approach due to the lack of instance-level labels. While MIL methods focusing on binary classification are abundant, they often cannot identify which specific instances drive bag labels and have limited or little interpretability. Our research team recently introduced MICProB, a Bayesian multiple instance classification (MIC) algorithm that addresses these issues. However, MICProB is computationally intensive and best suited for unimodal instances. To overcome these limitations, we propose a novel variational Bayesian multimodal MIC (vMMIC) algorithm. vMMIC handles diverse instance types and significantly improves computational efficiency through Bayesian variational inference, combined with data augmentation. We benchmark vMMIC against MICProB and many other MIC approaches on both simulated and real-world data. Results demonstrate vMMIC's superior performance, computational efficiency, and interpretability.
Presentation 3
Speaker: Difeng Cai (Southern Methodist University)
Co-author: N/A
Title: Posterior Covariance Structures in Gaussian Processes
Abstract: Gaussian process (GP) regression is a powerful tool for uncertainty quantification. A central task in GP lies in the computation of the posterior distribution, which can be prohibitively expensive due to the calculation of the posterior covariance matrix. In this talk, we present a comprehensive study on the posterior covariance that helps to circumvent the cost. We offer geometric interpretations that reveal how the posterior covariance is affected by different factors, including observation data, bandwidth parameter in the prior covariance. The new geometric understanding can be used to infer the posterior distribution without direct calculation of the posterior covariance matrix. It can also be used to design efficient preconditioners and approximations for the dense covariance matrix. Extensive experiments demonstrate the theoretical findings as well as practical applications.
Presentation 4
Speaker: Amanda Hering (Baylor University)
Co-author: N/A
Title: Simultaneous Mean and Sparse Covariance Monitoring via Maximum Likelihood Estimation with Positive-Definite Thresholding
Abstract: Multivariate statistical process monitoring is used to detect changes in the distribution of a multivariate process in real time. With modern systems, the number of variables being recorded and the complexity of their interactions are increasing. As the dimension of a process increases, the number of parameters required to estimate the covariance matrix increases quadratically; however, not every pair of monitored variables are related. In such cases, there is no need to estimate their covariance. We propose a new control chart that monitors shifts in the mean and covariance of a multivariate process and employs a sparse covariance estimation technique to reduce the number of parameters that must be estimated. When compared with competitors in a simulation study, this method produces a similar detection rate for shifts in the mean and/or variance and a superior detection rate for shifts in the covariance. Furthermore, this method has a substantially faster execution time, which is advantageous when deployed in real time. These methods are demonstrated on a case study of a fault that was deliberately introduced into a direct potable reuse water treatment system.
Organizers
Sang-Eun Lee (Tulane University), Rubaiyat Islam (Tulane University)
Mini-symposium Abstract:
Fluid dynamics is a fundamental ingredient of many biological and chemical systems, such as arterial blood flow and collective motion of autophoretic particles. Computational and mathematical modeling of these complex systems requires capturing of fluid-structure interactions and moving interfaces. This mini-symposium will be a forum to present advances in numerical methods for such complex systems, as well as successes in modeling in a variety of CFD applications.
Presentation 1
Speaker: Arshia Singhal (Rice University)
Co-author: N/A
Title: Closed Loop Solute Transport in Blood Vessels and Organs
Abstract: Hypoplastic left heart syndrome (HLHS) is a congenital heart disease that accounts for 2-3% of congenital heart diseases in the United States and 40% of all neonatal cardiac deaths. HLHS causes oxygenated blood to mix with deoxygenated blood, resulting in death. This raises a critical need to accurately model the transport of oxygen in blood vessels and organs throughout the human body to improve outcomes in patients with HLHS. Previously, numerical reduced models have been created to solve for blood flow and concentration of one solute. These models reduce the dimensions of the vessels and organs to improve computational efficiency. This work extends the models from open network of blood vessels to closed loops and includes organs such as the heart. Appropriate transmissibility conditions at each vessel junction and organ bed are constructed, that are based on balance laws. The class of interior penalty discontinuous Galerkin methods is used for the discretization of the models.
Presentation 2
Speaker: Yifan Wang (Texas Tech University)
Co-author: N/A
Title: Hyperuniform Micropost Microfluidic Devices for Efficient Circulating Tumor Cell Classification: A Computational and Machine Learning Approach
Abstract: Circulating tumor cells (CTCs) serve as vital biomarkers for early cancer detection, but current techniques for isolating and analyzing them from blood samples often yield heterogeneous populations, requiring expensive, time-consuming processing by trained professionals. In this presentation, we introduce a novel approach utilizing hyperuniform micropost microfluidic devices (MDs) for more efficient CTC classification. By integrating computational fluid-structure interaction modeling in a simulated microfluidic channel with machine learning techniques, we propose a more accessible and effective alternative. Our method utilizes a cell-based framework to model CTC dynamics in erythrocyte-laden plasma flow, generating large datasets of CTC trajectories. These trajectories, accounting for two distinct CTC phenotypes, are then analyzed using Convolutional Neural Networks (CNNs) and Recurrent Neural Networks (RNNs) to classify phenotypes based on trajectory data. The results demonstrate that this hyperuniform micropost MD design, coupled with data-driven classification, offers a promising and cost-efficient tool for distinguishing CTC phenotypes, potentially improving early cancer detection methods.
Presentation 3
Speaker: Sang-Eun Lee (Tulane University)
Co-author: Ricardo Cortez (Tulane University), Lisa Fauci (Tulane University)
Title: Collective Dynamics of Self-avoidant, Secreting Particles
Abstract: Motivated by autophoretic droplet swimmers that move in response to a self-produced chemical gradient, here we examine the collective dynamics of individual motile agents using a reaction-diffusion system. The agents have an unlimited supply of a chemical, secrete it at a given rate, but are anti-chemotactic so move at a given speed in the direction of maximal decrease of this chemical. In both one- and two-dimensional periodic domains, we find intriguing long-time behavior of the system. Depending upon a non-dimensional parameter that involves secretion rate, agent velocity, domain size and diffusion, we find that the position of the agents either relax to regularly spaced arrays, approach these regular arrays with damped oscillation, or exhibit undamped, periodic trajectories. We examine the progression of particles that are initially seeded randomly, and we also examine the stability of the steady and periodic states. In addition, we present results when these agents are embedded in an incompressible fluid, thus adding advection of the chemical field to the dynamics of this complex system.
Presentation 4
Speaker: Sheng Xu (Mathematics Department, Southern Methodist University)
Co-author: N/A
Title: The immersed interface method for flows around moving complex geometries
Abstract: I will present the extension of the immersed interface method to simulate flows around moving complex geometries. The immersed interface method is a framework to solve differential equations with discontinuous solutions across interfaces. When it is employed to simulate a flow around a moving rigid object, the object is modeled as the fluid in the rigid motion enforced by singular force along the interface and body force inside the interface. The flow is then simulated by incorporating necessary jump conditions across the interface into discretization schemes on a fixed grid. In this talk, I will first describe how to implement the jump conditions in the immersed interface method. I will then present how to compute the jump conditions for objects with complex surfaces that are represented by triangular meshes. Lastly, I will show testing results to demonstrate the accuracy, efficiency and robustness of the method.
Organizers
Shanyin Tong (Columbia University), Lu Zhang (Rice University)
Mini-symposium Abstract:
This mini-symposium aims to bring together researchers to discuss the latest advances and applications in computational inversion and reduced order modeling. Computational inversion techniques are crucial for extracting meaningful information from indirect measurements in various scientific and engineering disciplines. At the same time, reduced-order modeling plays an important role in simplifying complex systems, making them more tractable for analysis and real-time simulation. The mini-symposium will cover a wide range of topics, including novel algorithms for inversion, the integration of machine learning with traditional inversion methods, state-of-the-art approaches to model reduction, etc.
Presentation 1
Speaker: Andreas Mang (University of Houston)
Co-author: Jannatul Ferdous Chhoa (University of Houston)
Title: Fast iterative methods for large-scale initial value control problems
Abstract: We present fast iterative methods for PDE-constrained optimization problems with initial value controls. Our applications are in medical imaging. A hyperbolic transport equation and a geodesic equation on the group of diffeomorphisms govern the PDE-constrained optimization problem. The underlying inverse problem is inherently ill-posed and infinite-dimensional in the continuum, resulting in high-dimensional, ill-conditioned inversion operators after discretization. This poses significant mathematical and computational challenges. We discuss effective numerical methods for evaluating forward and adjoint operators, preconditioning the reduced space Hessian, and effective strategies for numerical optimization. We test the performance of the proposed numerical scheme on synthetic and real-world data.
Presentation 2
Speaker: Shari Moskow (Drexel University)
Co-author: Vladimir Druskin (Worcester Polytechnic Institute), Mikhail Zaslavskiy (Southern Methodist University)
Title: Reduced Order Model inversion of monostatic data lifted to full MIMO
Abstract: The Lippmann--Schwinger--Lanczos (LSL) algorithm has recently been shown to provide an efficient tool for imaging and direct inversion of synthetic aperture radar data in multi-scattering environments \cite{DrMoZa3}, where the data set is limited to the monostatic, a.k.a. single input/single output (SISO) measurements. The approach is based on constructing data-driven estimates of internal fields via a reduced-order model (ROM) framework and then plugging them into the Lippmann-Schwinger integral equation. However, the approximations of the internal solutions may have more error due to missing the off diagonal elements of the multiple input/multiple output (MIMO) matrix valued transfer function. This, in turn, may result in multiple echoes in the image. Here we present a ROM-based data completion algorithm to mitigate this problem. First, we apply the LSL algorithm to the SISO data as in \cite{DrMoZa3} to obtain approximate reconstructions as well as the estimate of internal field. Next, we use these estimates to calculate a forward Lippmann-Schwinger integral to populate the missing off-diagonal data (the lifting step). Finally, to update the reconstructions, we solve the Lippmann-Schwinger equation using the original SISO data, where the internal fields are constructed from the lifted MIMO data. The steps of obtaining the approximate reconstructions and internal fields and populating the missing MIMO data entries can be repeated for complex models to improve the images even further. Efficiency of the proposed approach is demonstrated on 2D and 2.5D numerical examples, where we see reconstructions are improved substantially.
Presentation 3
Speaker: Donsub Rim (Washington University in St. Louis)
Co-author: Woojin Cho (Yonsei U.), Kookjin Lee (Arizona State U.), Noseong Park (KAIST), Gerrit Welper (U Central Florida)
Title: Fast-LRNR: Sparse Physics Informed Backpropagation
Abstract: Many recent works are exploring the potential advantages in leveraging deep learning models in solving parametrized partial differential equations (pPDEs). This talk concerns one approach leading to computational speed ups when compared to classical numerical methods. We will introduce specialized neural networks called Low Rank Neural Representations (LRNRs), whose weight and bias parameters are endowed with a singular-value-decomposition-like low rank structure. We show that backpropagation operations that are necessary in physics informed approaches, e.g. the physics informed neural networks (PINNs) framework, can be computed efficiently due to the low rank structure. We dub these efficient operations sparse physics informed backpropagation (SPInProp). SPInProp operations have computational complexity that scales only with a reduced dimension, which is independent of the resolution or width of the neural network approximation. We present computational experiments that demonstrate LRNRs can be adapted to solve pPDEs using only SPInProps within PINNs.
Presentation 4
Speaker: Jingni Xiao (Drexel University)
Co-author: N/A
Title: Regularity of Non-scattering Geometries
Abstract: Non-scattering is a phenomenon where no scattering response is generated when an inhomogeneity is probed by an incident wave. It has recently been proven under certain "admissible conditions" that the shape of inhomogeneities exhibiting non-scattering feature must be regular. We shall discuss some of these results, and also provide examples of irregular non-scattering inhomogeneities that violate the admissible conditions.
Organizer
Robert Kirby (Baylor University)
Mini-symposium Abstract:
With the maturation of robust, high-order, and structure-preserving spatial discretization of partial differential equations, recent research has turned to find equally powerful time-stepping methods. New insights are allowing us to tackle the practical challenges of classical but forgotten methods with optimal theoretical properties. This minisymposium will explore recent contributions on solvers, structure preservation, and high-level software for time stepping partial differential equations.
Presentation 1
Speaker: William Barham (University of Texas at Austin)
Co-author: N/A
Title: A diagnostic tool for symplectic integrators
Abstract: Symplectic integrators are a favorable method for the numerical integration of Hamiltonian ordinary differential equations and are frequently used for the temporal integration of wave-like partial differential equations. These integrators are designed such that the time-advance map is a canonical transformation thus conserving the symplectic form exactly. A consequence of this defining property of symplectic integrators is the conservation of a numerical energy which remains close to the true energy for the duration of the simulation. This energy stability property is frequently cited as a justification for the use of symplectic integrators, however there are non-symplectic, energy-conserving integrators. Another justification for the value of symplectic integration is the exact conservation of the symplectic form. This more fundamental property of symplectic integrators is sometimes overlooked due to its abstractness and difficulty in measuring its impact. However, there are practical benefits: in the context of wave equations, conservation of the symplectic form implies the adiabatic invariance of wave-action. This work proposes a diagnostic tool for directly measuring the conservation of the symplectic form to better distinguish between the properties of symplectic and energy-conserving integrators.
Presentation 2
Speaker: Robert Kirby (Baylor University)
Co-author: Scott MacLachlan (Memorial University of Newfoundland)
Title: Irksome: Software for evolving finite elements with Runge-Kutta methods
Abstract: High-order spatial discretization of the partial differential equations governing fluids and many other application areas has been a highly-active research area for many years.
As these techniques have grown in use, there has been a renewed interest in also achieving high temporal accuracy.
Classical multi-step methods suffer from the Dahlquist Barrier Theorem, not providing A-stability at high order. Among Runge-Kutta schemes, diagonally implicit methods require an implementation quite similar to backward Euler, but they suffer from order reduction in the presence of stiff problems. On the other hand, fully implicit methods provide high stage order and other interesting theoretical properties but are quite complicated to implement and lead to very difficult algebraic problems to solve at each time step.
In this talk, I will introduce the Irksome, which is a Python package built on top of Firedrake. Irksome extends the Unified Form Language to model time-dependent problems, so that users can define a semi-discrete variational form in Python, select the Butcher tableau for a Runge-Kutta method, and generate a performant implementation. In addition to explicit and diagonally implicit methods, Irksome is an enabling technology for fully implicit methods. It not only automating the generation of the algebraic systems but also providing leading solver techniques that can make fully implicit methods even more efficient than alternative schemes.
Presentation 3
Speaker: John Stephens (Baylor University)
Co-author: Robert Kirby (Baylor University), Dan Shapero (University of Washington)
Title: A High-Order Uniformly Bounds-Constrained Finite Element Method via Variational Inequalities
Abstract: The solutions to partial differential equations frequently satisfy bounds constraints. When using finite element or finite difference methods, if one wishes to construct an approximate solution that satisfies these same bounds, great care is required. In a finite element context, one can replace a discrete variational problem with a discrete variational inequality. This allows for the selection of an approximate solution from a set of functions which satisfy the bounds constraints. Solving nonlinear optimization problems, though incurring a practical expense, bypasses known order barriers for linear problems and allows for the possibility of high-order and uniformly bounds-constrained finite element methods.
It is difficult to work with the entire set of bounds-constrained polynomials. However, the polynomials whose coefficients, when represented in the Bernstein basis, satisfy the bounds constraints form a convenient subset with which to work. Selecting an approximation from this set via a variational inequality, one obtains an approximation which is uniformly bounds-constrained, independent of the mesh used. Recent work seeks to extend this approach to collocation-type Runge-Kutta methods. Using a stage-value formulation, the collocating polynomial can be cast in the Bernstein basis to enforce bounds constraints uniformly in time. Examples are shown in which optimal order accuracy is observed.
Presentation 4
Speaker: Jonathan Zhang (University of Texas at Austin)
Co-author: Leszek Demkowicz (University of Texas at Austin)
Title: Nonlinear Elasticity with DPG
Abstract: Various variational formulations are presented for the Cook’s Membrane problem: a 2D tapered cantilever beam accompanied with a hyperelastic strain energy density that is subjected to a load on the right edge. A total of four formulations are presented: (1) the principle of virtual work, (2) the principle of virtual work with DPG, (3) a Galerkin formulation based on the Lagrangian with the definition of det F as the constraint, and (4) a DPG formulation based on the Lagrangian with a relaxed constraint. Each formulation comes with its own considerations, such as several additional trace and field unknowns. The formulations are evaluated based on the strains they support with all other numerical parameters constan.
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 minisymposium will present work done by researchers in the SIAM Section, showcasing both algebraic and analytic approaches to their study.
Presentation 1
Speaker: Jake Fillman (Texas A&M University)
Co-author: N/A
Title: Closed spectral gaps for periodic discrete operators
Abstract: We will discuss some recent results related to the existence and number of closed spectral gaps for discrete periodic Schrödinger operators.
Presentation 2
Speaker: Jonah Robinson (Texas A&M University)
Co-author: Matthew Faust (Michigan State University), Frank Sottile (Texas A&M University)
Title: Invariants of the Dispersion Relation for Discrete Periodic Operators
Abstract: We utilize tools of computational algebraic geometry to study the dispersion relation for discrete Schrödinger operators with generic parameters on various periodic graphs, with the goal of better understanding the geometry of critical points of the Bloch variety. In pursuit of this, we study several invariants of the dispersion relation such as the Newton polytope, singular and degenerate loci, and the singular loci of its facial systems. With the help of Macaulay2 and the Texas A&M Mathematics department's computational cluster, we conduct this investigation for several large families of periodic graphs. In this talk, we present various observed phenomena as well as several conjectures and results arising from our observations. This is joint work with Matthew Faust and Frank Sottile.
Presentation 3
Speaker: Kenneth Beard (Louisiana State University)
Co-author: N/A
Title: The Good the Bad and the Incommensurate: twisted trilayer graphene
Abstract: In contrast to twisted bilayer graphene (TBG), twisted trilayer graphene (TTG) does not maintain periodicity with respect to a moiré Brillouin zone. To obtain observables (e.g. density of states, momentum local density of states, Chern numbers) one must contend with an intractable infinite dimensional eigenvalue problem, even in momentum space. The present literature lacks a convergent algorithm for approximating these observables. We are developing such an algorithm for TTG under a tight-binding model, with hopes of extending to more intricate geometries (e.g. mechanical relaxation, mixed materials, carbon nanotubes).
Presentation 4
Speaker: Yi-Sheng Lim (Texas A&M University)
Co-author: Josip Zubrinic (University of Zagreb)
Title: An operator-asymptotic approach to periodic homogenization
Abstract: We present an operator-asymptotic approach to the problem of homogenization of periodic composite media. Upon writing the solution $u(x)$ to the resolvent problem as a superposition of elementary plane waves with wave vector ("quasimomentum") $\chi$, we seek a uniform-in-$|\chi|$ approximation for $u(x)$, by means of an asymptotic procedure in powers of the inverse wavelength $|\chi|$.
As a consequence, we obtain $L^2 \rightarrow L^2$, $L^2 \rightarrow H^1$, and higher order $L^2 \rightarrow L^2$ norm-resolvent estimates. The correctors for the $L^2 \rightarrow H^1$, and higher order $L^2 \rightarrow L^2$ results are constructed from boundary value problems that arise naturally during the asymptotic procedure, and are shown to coincide with terms of the classical two-scale expansion.
This is joint work with Josip Zubrinic (University of Zagreb).
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: Jimmie Adriazola (Southern Methodist University)
Co-author: Nicholas Bagley (Southern Methodist University), Alejandro Aceves (Southern Methodist University), Wei Zhu (Georgia Tech), Panayotis Kevrekidis (UMass Amherst)
Title: Numerical Discovery of Lax Pairs from Data
Abstract: We will discuss a computational approach for discovering integrability from data. Namely, we use numerical optimization to discover the Lax pair that gives rise to dynamics of interest. We test our approach on the simple harmonic oscillator and the Henon-Heiles system. This approach allows us to thoroughly explore the parameter space of Hamiltonian systems for new integrable cases as well as rule out nonintegrable configurations. We will also discuss how to adapt our approach to Hamiltonian PDE systems such as the Korteweg de-Vries equation.
Presentation 2
Speaker: Austin Marstaller (Southern Methodist University)
Co-author: Alejandro Aceves (Southern Methodist University)
Title: Dynamics of the Ablowitz-Ladik NLS with nonlocal coupling
Abstract: The Ablowitz-Ladik (A-L) model is the famous integrable cousin of the Discrete Nonlinear Schrődinger equation. We consider a variation of this model with a weighted nonlocal pairwise coupling instead of nearest neighbor coupling. Modulation instability of the continuous wave solution with nonzero and zero phase is investigated and these results are related to the classical A-L model. Additionally, a discussion regarding the formation of breathers in a statistical mechanics framework will be given. This is a joint work with Dr. Alejandro Aceves.
Presentation 3
Speaker: Erwin Suazo (University of Texas Rio Grande Valley)
Co-author: N/A
Title: Numerical simulations for fractional differential equations of higher order and a Wright-type transformation
Abstract: In this work, a new relationship is established between the solutions of higher order fractional differential equations and a Wright-type transformation. Solutions could be interpreted as expected values of functions in a random time process. As applications, we solve the fractional beam equation, fractional electric circuits with special functions as external sources, derive d'Alembert's formula and show the existence of explicit solutions for a general fractional wave equation with variable coefficients. Due to this relationship, we present two methods for simulating solutions of fractional differential equations. The two approaches use the interpretation of the Caputo derivative of a function as a Wright-type transformation of the higher derivative of the function. In the first approach, we use the Runge-Kutta method of hybrid orders 4 and 5 to solve ordinary differential equations combined with the Monte Carlo integration to conduct the Wright-type transformation. The second method uses a feedforward neural network to simulate the fractional differential equation.
Presentation 4
Speaker: Changyan Shi (University of Texas Rio Grande Valley)
Co-author: Baofeng Feng (University of Texas Rio Grande Valley), Bingyuan Liu (University of Texas Rio Grande Valley)
Title: Soliton solutions to a coupled Hirota equation under zero and nonzero boundary conditions
Abstract: In this talk, we construct soliton solutions to a coupled Hirota equation under both zero and nonzero boundary conditions. We begin by deriving bright-bright soliton solution starting from three-component KP hierarchy, and obtaining dark-dark soliton solution from a single-component KP-Toda hierarchy with two singular shifts. Finally, we introduce a new tau function along with several new bilinear equations, which help us to derive bright-dark soliton solution. This is a joint work with Bao-Feng Feng and Bingyuan Liu.
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: Scott Cook (Tarleton State University)
Co-author: N/A
Title: Political Redistricting, Markov Chains, and the Texas County Line Rule
Abstract: Every 10 years, US states redraw voting district boundaries to reflect the preceding decennial census. This highly contentious political process profoundly impacts the next 10 years of election results. Mathematicians have developed Markov Chain Monte Carlo (MCMC) methods to produce large ensembles of “fair†electoral maps to inform both political redistricting and legal challenges. We present advances to two popular MCMC methods (single-flip and recombination) that incorporate the “County Line Rule†from the Texas Constitution and analyze its effect on recent elections.
Presentation 2
Speaker: Amber Bozer (Tarleton State University)
Co-author: N/A
Title: Investigating the cortical representation of the sensory-discriminative and affective-motivational dimensions of pain using electroencephalography
Abstract: Pain is complex and consists of three dimensions: sensory-discriminative, affective-motivational, and cognitive-evaluative. The pain literature lacks a comprehensive and methodologically robust profile of the EEG cortical activity that underlies cognitive processing of the multidimensional pain experience. The primary objective of this research is to provide a methodologically robust, cortex-wide, full-frequency band profile of cortical EEG activity associated with the cognitive processing of the sensory-discriminative and affective-motivational dimensions of pain. We recorded cortical activity using EEG in participants without and with pain at both baseline resting-state and during experimentally-induced pain (cold pressor task) with simultaneous presentation of previously validated sensory dimension and affective dimension questions. Sensory and affective dimension stimuli were presented while neural oscillations were recorded in a double-walled/insulated sound attenuating chamber (Whisper Room) with LED lights. Oscillations were recorded using an Advanced Brain Monitoring B-Alert 24 electrode EEG (referenced to mastoids) and iMotions software. Matlab and Cartool were used to filter (.05-50Hz) data, assign electrode locations, reject artifacts and apply fast fourier transforms to obtain frequency band data (delta .05-3 Hz; theta 4-7 Hz; alpha 8-13 Hz; beta 14-30 Hz; and gamma 31-50 Hz). ANOVAs were computed for each electrode and frequency band with participant group (chronic pain vs. no pain) as the between-subjects variable, time point (baseline, baseline with stimuli, cold pressor task, and cold pressor task with stimuli) and dimension (sensory vs. affective dimension questions) as within-subjects variables. These studies reveal differential processing of pain dimension stimuli at rest and during acute pain.
Presentation 3
Speaker: Bryant Wyatt (Tarleton State University)
Co-author: N/A
Title: Modeling Supraventricular Tachycardia Using Dynamic Computer-Generated Left Atrium
Abstract: Supraventricular Tachycardia (SVT) occurs when the heart's atria beat rapidly or irregularly compared to the ventricles. Although not immediately fatal, this disharmony contributes to strokes, heart attacks, and heart failure. Catheter ablation is the primary treatment, wherein an electrophysiologist creates a 3D heart map, guiding a catheter to burn aberrant tissue with RF energy. Despite advances, gaps persist in understanding SVT triggers and optimal ablation sites, especially in cases like atrial fibrillation (AF). To address these gaps, our team has created a model of the left atrium that beats in near real-time and is adjustable down to the level of individual muscles. Users can implement ablation strategies on our digital twin to quickly gain insights outside of the operating room. Patient data can be imported directly from a CT scan and electro-cardial mapping. This approach accelerates SVT comprehension without endangering lives. Our work holds life-saving potential that could revolutionize cardiac care.
Presentation 4
Speaker: Tahsin Khajah (The University of Texas at Tyler)
Co-author: N/A
Title: Efficient High frequency multiple scattering analyses in Reduced Spatial Dimensions
Abstract: Volume discretization methods face multiple challenges when used for exterior scattering analyses including high computational cost, need for artificial domain truncation and accurate boundary representation. These challenges are more pronounced in multiple scattering analyses where the wave moves back and forth between multiple obstacles. Conventionally such problems are solved by meshing the space between obstacles and truncating the domain with a fictitious boundary enclosing them. This makes mesh generation even more challenging and considerably limits the flexibility to move, rotate, and reshape the scatterers which are required for design and optimization of devices relying on wave propagation phenomena. Hence, spatial dimension reduction is highly desirable. Methods relying on boundary integral equations allow spatial dimension reduction, however, these methods introduce new challenges such as the need to treat singular integral kernels and lead to dense and possibly ill-conditioned matrices limiting them to low- to mid-frequency analyses. Like boundary integral methods, On Surface Radiation Conditions (OSRC) require boundary discretization only and lead to integral equations with smooth kernels. We demonstrate possibility of enhancing the accuracy OSRC for single and multiple scattering analyses and adopt it to develop a well-conditioned method based on fundamental solution. These developments reduce the spatial dimensions by one as the interaction between scatterers are captured analytically; the need for treating singularities is eliminated; and reliable single and multiple scattering analyses can be performed for arbitrarily shaped scatteres in mid- to high-frequency regimes.
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-syposium 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: Samuel Van Fleet (Rice University)
Co-author: James Rossmanith (Iowa State University)
Title: Maximum-Taylor Discontinuous Galerkin Schemes for Linear Hyperbolic Systems
Abstract: In this work we develop the maximum Taylor discontinuous Galerkin (MTDG) method for solving linear systems of hyperbolic partial differential equations (PDEs). The proposed method is a variant of the Lax-Wendroff discontinuous Galerkin (LxW-DG) method from the literature. The process by which the Lax-Wendroff DG method is obtained can be summarized as follows:
1. Compute a truncated Taylor series in time that relates the solution that is being sought to the known solution at the previous time-step.
2. Replace all the temporal derivatives in this Taylor expansion by spatial derivatives by repeatedly invoking the underlying PDE.
3. Multiply this expansion by appropriate test functions, integrate over a finite element, and perform a single integration-by-parts that places a derivative on the test functions as well as introducing boundary terms.
4. Replace the boundary terms by appropriate numerical fluxes. The key innovation in the newly proposed method is that we replace the single integration-by-parts step by an approach that moves all spatial derivatives onto the test functions; this process introduces many new terms that are not present in the Lax-Wendroff DG approach. The regions of stability various MTDG methods are compared to the LxW-DG stability regions. It is shown that compared to the Lax-Wendroff DG method, the maximum Taylor DG method has a larger region of stability and has improved accuracy. These properties are demonstrated by applying MTDG to several numerical test cases.
Presentation 2
Speaker: Matthias Maier (Texas A&M University)
Co-author: Jordan Hoffart (Texas A&M University), Ignacio Tomas (Texas Tech University)
Title: Structure-preserving finite-element schemes for the Euler-Maxwell and Euler-Poisson equations
Abstract: We discuss structure-preserving numerical discretization techniques for the Euler-Poisson and Euler-Maxwell equations that find applications in the modeling and simulation of fluid plasma, self gravitation, and nanoscale optical device modeling. A key feature of the methods presented is that they maintain a discrete energy law, as well as hyperbolic invariant domain properties, such as positivity of the density and a minimum principle of the specific entropy, on a fully discrete level.
We first introduce and discuss the underlying algebraic discretization technique based on collocation, convex limiting, and a high-order IMEX splitting technique and then discuss how the method is applied to the coupled Euler-Poisson and Euler-Maxwell system. We demonstrate how a careful choice of continuous and discontinuous finite element spaces for the PDE subsystems combined with a block elimination of the source subsystem leads to an energy stable formulation with robust linear algebra. Most crucially, the scheme is able to cope with the inherent multiple time scales of the coupled PDE system. A detailed discussion of algorithmic details is given, as well as proofs of the claimed properties.
Presentation 3
Speaker: Raven S. Johnson (Rice University)
Co-author: Jesse Chan (Rice University)
Title: High Order Entropy Stable Methods for Blood Flow Simulations
Abstract: The blood flow equations are a 1D nonlinear hyperbolic system which serves as a reduced model of arterial blood flow. We are interested in constructing entropy stable discontinuous Galerkin (DG) methods for this system, which enforce a semi-discrete cell entropy inequality while retaining high order accuracy. An entropy stable method for the blood flow equations was introduced by Burger, Valbuena, and Vega [(2023). Numer. Methods Partial Differ. Eq.. 39, 2491-2509] using primitive variables. We propose an entropy stable DG method for the blood flow equations based on conservative variables, based on a new condition for entropy stability introduced in Chan, Shukla, et al. [(2024). Journal of Computational Physics, 112876] that eases the analysis of non-conservative terms in the 1D blood flow equations.
Presentation 4
Speaker: Ignacio Tomas (Texas Tech University)
Co-author: N/A
Title: Structure preserving numerical method for the ideal compressible MHD system
Abstract: We present a new method in order to solve the compressible ideal MHD equations capable of preserving total energy, positivity of density and internal energy, and divergence-free conditions of the magnetic field. The scheme uses the source formulation of the system rather than the usual divergence form. We solve a variety of ideal MHD problems, showing that the method is capable of delivering second-order accuracy in space for smooth problems, while also offering unconditional robustness in the shock hydrodynamics regime as well.
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: Robert Viator (Denison University)
Co-author: N/A
Title: Bloch Waves in High-Contrast Photonic Crystals
Abstract: Three-dimensional photonic crystals have been demonstrated numerically and experimentally to exhibit stop-bands in their frequency spectra since the early 2000's. Nonetheless, a mathematical proof of the existence of band gaps for three-dimensional high-contrast photonic crystals remains an open problem. We will review some techniques that have had success for the band-gap problem in two dimensions, discuss their limitations in the context of Maxwell's equations, and then review some recent advances towards understanding the Bloch bands associated with 3D photonic crystals in the high-contrast limit.
Presentation 2
Speaker: Jordan Hoffart (Texas A&M University)
Co-author: N/A
Title: A comparison of finite element spaces for the discontinuous Galerkin approximation of the Maxwell eigenvalue problem in first order form
Abstract: In 2023, Ern and Guermond presented a discontinuous Galerkin method to approximate the Maxwell eigenvalue problem in first order form. They prove that, on affine simplicial meshes and with vector valued polynomials of degree at most $k$, the method gives a spectrally correct approximation, even when the eigenspaces have low Sobolev regularity, the domain has a nontrivial topology, or the material coefficients are discontinuous. In this talk, we consider their method, but now on general quadrilateral or hexahedral meshes. In this setting, we present a few different finite element spaces and show which ones do or do not give a spectrally correct approximation.
Presentation 3
Speaker: Yuliia Yershova (Texas A&M University)
Co-author: N/A
Title: Negative group velocity in doubly-porous media via sharp norm-resolvent estimates
Abstract: I will discuss the possibility for doubly-porous media to exhibit negative group velocities, or, in other words, to support metamaterial regimes of wave propagation. I will show that the next-order term of the resolvent asymptotics, if taken into account, leads to negative properties of the effective material.
The model I will consider in detail will be based on a periodic graph, which is itself seen as a norm-resolvent limit of a PDE on a thin network. In this talk, I will demonstrate a class of periodic thin networks in the so-called resonant case in which the negative group velocity and, by implication, the negative refraction can be obtained quantitatively in explicitly computed frequency bands. Alongside analytic results, some numerics for that model will be shown.
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: Aynur Bulut (Louisiana State University)
Co-author: N/A
Title: TBD
Abstract: TBD
Presentation 2
Speaker: Luan Thach Hoang (Texas Tech University)
Co-author: N/A
Title: On Galerkin approximations of the Navier-Stokes equations with large Grashof numbers
Abstract: We examine how stationary solutions to Galerkin approximations of the Navier-Stokes equations behave in the limit as the Grashof number tends to infinity. An appropriate scaling is used to place the Grashof number as a new coefficient of the nonlinear term while the body force is fixed. A new type of asymptotic expansion for a family of solutions is introduced. Relations among the terms in the expansion are obtained by following a procedure that compares and totally orders positive sequences generated by the expansion. The same methodology applies to the case of perturbed body forces and similar results are obtained. We demonstrate with a class of forces and solutions that have convergent asymptotic expansions.
Presentation 3
Speaker: Collin Victor (Texas A&M University)
Co-author: N/A
Title: Intertwinement in 2D Turbulence: Linking Synchronization and Determining Modes
Abstract: In this talk, we investigate the interplay between determining modes and synchronization phenomena in the context of two-dimensional (2D) turbulence governed by the Navier-Stokes equations (NSE). We introduce the concept of self-synchronous intertwinement to describe a new relationship that connects the classical determining modes property with the synchronization of two continuous data assimilation (CDA) filters, namely, the synchronization filter and the nudging filter. This concept extends the idea of asymptotic enslavement to a stronger form, demonstrating that the existence of finitely many determining modes leads to robust synchronization of these filters. Our results provide a novel perspective on how different scales of turbulence interact and synchronize, shedding light on the mechanisms by which large-scale structures influence the convergence behavior in data assimilation processes. Numerical experiments corroborate our theoretical findings, offering new insights into the multi-scale dynamics of turbulent flows and their effective modeling through data assimilation techniques.
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 mini-symposium 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: Elżbieta Polak (University of Texas at Austin)
Co-author: Joe Kileel (University of Texas at Austin), Yong Sheng Soh (National University of Singapore)
Title: New Approaches in SO(2) Optimization
Abstract: In this talk I will describe the SO(2) alignment problem. I will present new approaches to optimization in this context, utilizing methods of algebra as well as numerical and Fourier analysis. Based on joint work with Joe Kileel and Yong Sheng Soh.
Presentation 2
Speaker: Gabriel Brown (University of Texas at Austin)
Co-author: N/A
Title: Rank Geometry for Small Sized Tensors: Singularities and Nearest Point Problems
Abstract: Even small real tensors, like 2x2x2, are sufficient to demonstrate interesting phenomena like rank jumping and ill-posedness which do not plague matrices. By viewing the best low-rank approximation problem as a nearest point problem, we develop a clearer picture of the relevant geometry and can provide a new direct proof for the fact that no 2x2x2 rank 3 tensor has a best rank 2 approximation. A critical component in this new geometric understanding is the singular locus of the hyperdeterminant, which is further supported by new and extensive visualizations. Some new and old results for larger tensor sizes are also included.
Presentation 3
Speaker: Yifan Zhang (University of Texas at Austin)
Co-author: N/A
Title: Scalable Interpolative Decomposition for Structured Tensor Data
Abstract: In this talk, we present new scalable algorithms for finding interpolative decompositions of higher order tensors - an extension of matrix CUR decomposition to higher dimensional arrays. We develop efficient sampling-based algorithms for two classes of structured tensor data: sparse tensors and tensors in CP format. Numerical illustrations as well as theoretical guarantees will be provided to justify the algorithm.
Presentation 4
Speaker: Bobby Shi (University of Texas at Austin)
Co-author: N/A
Title: A Look at the Complexity of Decomposition of Generic Low-Rank Tensors
Abstract: We study the computational complexity of an algebraic algorithm for complex symmetric tensor decomposition of Brachat et al. We give conditions for when the algorithm may be performed in polynomial time; through our analysis we reveal implications for geometry and identifiability that may be of independent interest. We provide examples and numerics to illustrate our points.