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SEARCDE 2024

Titles and Abstracts

The 42nd Southeastern-Atlantic Regional Conference on Differential Equations

View source content in SEARCDE 2024 Titles and Abstracts (Google Sheet)

 
 
B1 Dhruba Adhikari Kennesaw State University Nontrivial Solutions of Inclusions Involving Perturbations of Positively Homogeneous Maximal Monotone Operators Let $X$ be a real reflexive Banach space and $X^*$ be its dual space. Let $G_1$ and $G_2$ be open subsets of $X$ such that $\overline G_2\subset G_1$, $0\in G_2$, and $G_1$ is bounded. Let $L: X\supset D(L)\to X^*$ be a densely defined linear maximal monotone operator, $A:X\supset D(A)\to 2^{X^*}$ be a maximal monotone and positively homogeneous operator of degree $\gamma>0$, and $C:X\supset D(C)\to X^*$ be a bounded demicontinuous operator of type $(S_+)$ w.r.t. $D(L).$ We assume that $A$ is bounded and establish the existence of nonzero solutions of $Lx+Ax+Cx\ni 0$ in $G_1\setminus G_2$ and remove the restriction $\gamma= 1$ from the existing results. We also present applications to parabolic partial differential equations in general divergence form satisfying Dirichlet initial-boundary conditions.
C5 Hamza Adjerid Virginia Tech Nonlinear feedback control for Stokes-type DAEs We study optimal feedback control for a class of differential algebraic equations (DAEs) with quadratic nonlinearity. Systems of this specific form would arise in the discretization of control problems associated with the Navier-Stokes equations, but here we focus our attention on finite dimensional systems that share this form. Our approach is to use the strangeness framework for DAEs to appropriately recast the control problem as a system of differential equations with polynomial drift terms. At this stage it is appropriate to develop standard approximations to the feedback control laws. We also describe an implementation that exploits sparse matrices in the original system rather than the dense matrices that result from the index reduction process. We separately consider the cases when the control term does and does not appear in the algebraic constraints since the former is significantly easier to derive.
C2 Michael Aguadze Norfolk State University Using Machine Learning to Measure the Impact of Treatment as a Control for Marijuana Use Marijuana use is a serious health concern in the U.S. Epidemiological models such as SEU (S-Susceptible, E-Latent, U-Users) have been studied by several authors. However, these models do not fit the data very accurately when treatment is used as a control. We implement a new approach using a neural network-augmented SEU model to quantify the effectiveness of treatment control in mitigating marijuana use. The basic reproduction number for the SEU model is evaluated, as well as the reproduction number for the model incorporating treatment control.
C5 Sanwar Ahmad Virginia State University On accelerating iterative gradient type methods for solving nonlinear optimization problem: application to Electrical Impedance Tomography problems In scientific inquiry, an inverse problem involves deducing the underlying causes behind a set of observations. Typically, inverse problems are severely challenging due to their ill-posed nature. Generally, three methodologies are employed to tackle such ill-posed problems: i) direct approaches, ii) iterative techniques, and iii) statistical methods. Iterative methods, formulated within nonlinear optimization frameworks, are robust to noise and limited data; however, they come with high computational demands due to the computation involved in the forward operator's Jacobian matrix. In this presentation, we propose an alternative strategy to streamline this process by updating the Jacobian at each iteration that significantly reducing computational costs and hastening convergence and presented some numerical examples to demonstrate the efficiency of the proposed method.
A6 Tahmineh Azizi Washington University An application of the Grünwald-Letinkov fractional derivative to a study of drug diffusion in pharmocokinetic compartmental models Pharmacokinetic compartmental models are crucial for analyzing and understanding the rates of chemical or biochemical reactions of drugs and their distribution throughout the body. Typically, these models are represented as systems of differential equations. In this paper, we explore fractional calculus (FC) and its applications in biomedicine. We present three distinct integer-order pharmacokinetic models with two and three compartments and provide their numerical solutions to illustrate the absorption, distribution, metabolism, and excretion (ADME) of drugs or nanoparticles (NPs) in various compartments.

To derive the fractional-order versions of these pharmacokinetic models, we first apply the nonstandard finite difference (NSFD) method to the two- and three-compartment models, as NSFD often yields more accurate results compared to traditional standard finite difference (SFD) methods. We then discretize these pharmacokinetic models using the Grünwald-Letnikov discretization technique. Various numerical methods are employed to solve the fractional-order two- and three-compartment pharmacokinetic models for oral drug administration, demonstrating their effectiveness in capturing the dynamics of drug distribution and metabolism.
A1 Shalmali Bandyopadhyay The University of Tennessee at Martin Positive Solutions to Singular Second Order BVP on Time Scales We study singular second order BVPs with nonlinear boundary conditions on general time scales. We prove existence of a positive solution using sub and super solution methods, fixed point theory and perturbation methods used in approximating regular problems.
A7 Isabel Barrio Sanchez University of Pittsburgh Long-term H¹-Stability of Cauchy's Method for the Navier-Stokes Equations We study the stability of Cauchy's theta method for all time for the Navier-Stokes equations. We focus solely on the time discretization and first prove L^2 stability for all θ in (1/2,1]. We then use that bound to prove H^1 stability.
A6 Jeff Borggaard Virginia Tech Polynomial Feedback Control of Navier-Stokes Equations In control of fluid dynamics, the fluidic pinball problem presents a significant benchmark problem. This problem seeks to control the vortex shedding behind three cylinders using cylinder rotation as the actuation mechanism and drive it to a desired steady-state solution. This talk describes a novel model-based approach that combines the Quadratic-Quadratic Regulator (QQR) [Borggaard and Zietsman, 2020] feedback control methodology with interpolatory model order reduction to address this challenge. The QQR is based on finding polynomial approximations to the Hamilton-Jacobi-Bellman equations, that can serve as a Lyapunov function and be used to develop polynomial feedback control laws. This approach is demonstrated for two different Reynolds numbers, $Re_D = 30$ and $Re_D = 50$. In the case of $Re_D = 30$, the QQR controller is able to control the system and achieves the desired performance criteria $40.1\%$ faster than the linear controller. In the case of $Re_D = 50$, only the QQR controller successfully controls the flow, while the linear controller fails. This is joint work with Ali Bouland at Virginia Tech.
B4 Balázs Boros University of Wisconsin-Madison The smallest bimolecular mass-action systems admitting Andronov-Hopf bifurcation We systematically address the question of which small, bimolecular reaction networks endowed with mass-action kinetics are capable of Andronov-Hopf bifurcation. It is easily shown that any such network must have at least three species and at least four irreversible reactions. We are able to fully classify three-species, four-reaction, bimolecular networks: with the extensive help of computer algebra, we divide these networks into those which forbid Andronov-Hopf bifurcation and those which admit Andronov-Hopf bifurcation.

Beginning with 14670 three-species, four-reaction, bimolecular networks which admit positive equilibria, we show that the great majority of these are incapable of Andronov-Hopf bifurcation. At the end of this process, we are left 138 networks with the potential for Andronov-Hopf bifurcation. These fall into 87 distinct classes, up to a natural equivalence. Out of the 87 classes we find that 86 admit nondegenerate Andronov-Hopf bifurcation. The remaining exceptional network robustly admits a degenerate Andronov-Hopf bifurcation.
C4 Matthew Broussard North Carolina State University Analysis and Comparison of Interface Conditions for a Coupling of Poroleastic Equations and Lumped Hydraulic Circuit For the coupled system between the poroelasticy PDE equations and an LHC ODE we investigate the wellposedness of this multiscale model for two different interface conditions and compare the results of numerical simulations for these differing conditions.
Plenary Speaker 3 Jeff Calder University of Minnesota PDEs and graph-based semi-supervised learning Graph-based semi-supervised learning is a field within machine learning that uses both labeled and unlabeled data with an underlying graph structure for classification and regression tasks. In problems where very little labeled data is available, the classical Laplacian regularization gives very poor results. This can be explained through its PDE continuum limit, which is an ill-posed elliptic equation. Much work recently has been focused on designing graph-based learning methods with well-posed continuum limits, including the p-Laplacian, higher order Laplacians, re-weighted Laplacians, and Poisson equations.

In this talk, we will survey this literature, and present our recent work on using Poisson equations for semi-supervised learning. We will present theoretical results which establish that learning with Poisson equations is provably well-posed at arbitrarily low label rates, and experimental results showing that it outperforms existing graph-based semi-supervised learning methods on challenging data sets. We will also present some recent work on applications of Poisson learning to graph-based active learning, where the goal is to select a training set with the most informative examples, often in a sequential online setting starting at extremely low label rates.
B7 Qinying Chen University of Delaware Evaporation-driven tear film thinning and breakup in two space dimensions Evaporation profiles have a strong effect on tear film thinning and breakup (TBU), a key factor in dry eye disease (DED). In experiments, TBU is typically seen to occur in patterns that locally can be circular (spot), linear (streak), or intermediate . We investigate a two-dimensional (2D) model of localized TBU using a Fourier spectral collocation method to observe how the evaporation distribution affects the resulting dynamics of tear film thickness and osmolarity, among other variables. We find that the dynamics are not simply an addition of individual 1D solutions of independent TBU events, and we show how the TBU quantities of interest vary continuously from spots to streaks for the shape of the evaporation distribution. We also find a significant speedup by using a proper orthogonal decomposition to reduce the dimension of the numerical system. The speedup will be especially useful for future applications of the model to inverse problems, allowing the clinical observation at scale of quantities that are thought to be important to DED but not directly measurable in vivo within TBU locales.
A1 Ming Chen University of Pittsburgh Bifurcation for hollow vortex desingularization A hollow vortex is a region of constant pressure bounded by a vortex sheet and suspended inside a perfect fluid; we can think of it as a spinning bubble of air in water. In this talk, we present a general method for desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices. The machinery simultaneously treats the translating, rotating, and stationary regimes. Through global bifurcation theory, we further obtain maximal curves of solutions that continue until the onset of a singularity. As specific examples, we obtain the first existence theory for co-rotating hollow vortex pairs and stationary hollow vortex tripoles, as well as a new construction of Pocklington’s classical co-translating hollow vortex pairs. All of these families extend into the non-perturbative regime, and we obtain a rather complete characterization of the limiting behavior along the global bifurcation curve. This is a joint work with Samuel Walsh (Missouri) and Miles Wheeler (Bath).
C1 Vani Cheruvu The University of Toledo Haar wavelet based Quasilinearization Method Quasi-linearization technique can be viewed as a generalization of Newton-Raphson method (functional equations) to differential equations. The two important aspects of the quasi-linearization technique are (i). linearizes the non-linear equation and (ii) provides a sequence of functions which in general converges rather rapidly to the solution of the original non-linear equation. In this talk, we will discuss the Quasi-linearization method and its application to couple of differential equations. I will conclude the talk with Haar wavelet based quasi-linearization method.
A2 Maya Chhetri University of North Carolina at Greensboro An interpolation approach to $L^{\infty}$ a priori estimates for elliptic problems with nonlinearity on the boundary We establish an explicit $L^{\infty}(\Omega)$ a priori estimate for weak solutions to subcritical elliptic problems with nonlinearity on the boundary, in terms of the powers of their $H^1(\Omega)$ norms. To prove our result, we combine in a novel way Moser type estimates together with elliptic regularity and Gagliardo–Nirenberg interpolation inequality.
Plenary Speaker 2 Gheorghe Craciun University of Wisconsin-Madison Polynomial dynamical systems and reaction networks: persistence and global attractors The mathematical analysis of global properties of dynamical systems with polynomial right-hand side can be very challenging (for example: Hilbert’s 16th problem about polynomial dynamical systems in 2D, or the analysis of chaotic dynamics in the Lorenz system). On the other hand, any polynomial dynamical system can essentially be regarded as a model of a reaction network. Key properties of reaction systems are closely related to fundamental results about global stability in classical thermodynamics. For example, the Global Attractor Conjecture can be regarded as a finite dimensional version of Boltzmann’s H-theorem. We will discuss some of these connections, as well as the introduction of toric differential inclusions as a tool for proving the Global Attractor Conjecture.
We will also discuss some implications for the more general Persistence Conjecture (which says that solutions of weakly reversible systems cannot “go extinct”), as well as some applications to biochemical mechanisms that implement cellular homeostasis.
A3 Tom Cuchta Marshall University Periodic functions with nonuniform domains In the real numbers and integers, periodic functions are naturally defined by $f(t+\omega)=f(t)$, but if the domain is not closed under addition, then this definition does not apply. In this talk, we will introduce and motivate the a new definition for periodic functions introduced in 2022 within the theory of dynamic equations on time scales. Time permitting, some new results about the existence of periodic solutions of systems homogeneous and nonhomogeneous dynamic equations with nonuniform domains will be presented.
B7 Kubilay Dagtoros Norfolk State University Direct and Indirect Simulation Techniques In this talk, we will go over some direct and indirect simulation techniques. Principal integral transform allows us to simulate from a population with a cost of solving an integral equation. Often times, finding inverse of a cdf comes with its own challenges. Another approach is to use some indirect methods such as the Accept/Reject Algorithm. We will compare these methods and review their perfromances as well as their advantages and disadvantages. Moreover, we will go over a thorough analysis of family of axuillary functions to find the best performing functions in the indirect method.
A4 Kushani De Silva University of North Carolina at Greensboro Exploring the Dynamics and Stability of Dengue Transmission: The Influence of Vector-Pathogen Interactions and Climatic Factors Dengue fever has long posed a significant threat to human health, and a definitive cure is still out of reach. Much of the research in this area focuses on understanding the transmission dynamics between the mosquitoes (vectors) and the humans (hosts). Although the former relationship is crucial for comprehending disease transmission, we argue that delving into vector-specific factors is essential for disease eradication. As of today, the only solution to this disease is successful vector control methods including genetic modifications. To this end, this study focuses on understanding how vector genetics contribute to infection according to different functional responses and explores the complex relationship between the vector population and pathogen loads. Specifically, the vector competence of Aedes mosquitoes plays a crucial role in determining the intensity of the infection and the persistence of the disease. We estimated this genetic parameter in vectors outside of laboratory settings, and based on these findings, we discussed disease persistence criteria using a dynamical system model that considers the relationship between vectors and pathogen loads. Stability analysis of the dynamical system provides valuable insights into vector competence and disease transmission intensity, helping to predict future disease trajectories. Additionally, these findings are further refined by incorporating climate data into the dynamical system model to account for climate change impacts.
A2 Prerona Dutta Xavier University of Louisiana Well-posedness and continuity properties of the two-component Fornberg-Whitham system in Besov spaces The two-component Fornberg-Whitham (FW) system is a model for analyzing surface waves in an incompressible fluid. Fan et al. proposed this system in 2011 by generalizing the FW equation that was derived in 1978 to study wave breaking. Establishing well-posedness of the FW system and studying continuity properties of its data-to-solution map in various spaces is a challenging problem. Among well-known function spaces, Besov spaces are of growing interest in the study of nonlinear PDEs, as they are based on Sobolev spaces and more effective at measuring regularity properties of functions. In this talk, we present recent results on the well-posedness and non-uniform dependence on periodic initial data for the two-component FW system in Besov spaces.
C1 Mohamed El-Houssieny Detroit Public Schools Community District Comparison of Adomian Decomposition and Laplace Adomian Decomposition Methods Adomian decomposition method (ADM) has a unique feature that it directly deals with the nonlinear problem avoiding any linearization or discretization. This is a semi-analytic method and assumes that the solution is decomposed into a rapidly convergent series and the nonlinear term as a series of Adomian Polynomials. This results in the reduction of any differential equation into a set of recursive relations for the Adomian solution series. We present ADM and Laplace ADM and compare their results with a couple of PDEs. This is a joint work with Dr. Vani Cheruvu from the University of Toledo.
B1 Junyuan Fang University of Tennessee, Knoxville Harnack inequality for degenerate parabolic equations in non-divergence form In this talk, I will present some recent results on regularity of a class of linear parabolic equations in non-divergence form in which the leading coefficients are measurable and they can be singular or degenerate as a weight belonging to the $A_{1+\frac{1}{n}}$ class of Muckenhoupt weights. Krylov-Safonov type Harnack inequality for solutions is proved under some smallness assumption on a weighted mean oscillation of the weight. I will also give an introduction of a class of generic weighted parabolic cylinders in which several growth lemmas are proved. As a corollary, Hölder regularity estimates of solutions with respect to a quasi-distance is also obtained.

This talk is based on work joint with Sungwon Cho and Tuoc Phan.
B5 Rui Fang University of Pittsburgh Adaptive Parameter Selection in Nudging Based Data Assimilation Data assimilation combines (imperfect) knowledge of a flow's physical laws with (noisy, time-lagged, and otherwise imperfect) observations to produce a more accurate prediction of flow statistics. Assimilation by nudging (from 1964), while non-optimal, is easy to implement and its analysis is clear and well-established. Nudging's uniform in time accuracy has even been established under conditions on the nudging parameter χ and the density of observational locations, H, Larios, Rebholz, and Zerfas, 2019. One remaining issue is that nudging requires the user to select a key parameter. The conditions required for this parameter, derived through á priori (worst case) analysis are severe (Section 2.1 herein) and far beyond those found to be effective in computational experience. One resolution, developed herein, is self-adaptive parameter selection. This report develops, analyzes, tests, and compares two methods of self-adaptation of nudging parameters. One combines analysis and response to local flow behavior. The other is based only on response to flow behavior. The comparison finds both are easily implemented and yields effective values of the nudging parameter much smaller than those of á priori analysis.
B2 Pierre Aime Feulefack University of Pennsylvania Bifurcation results and multiple solutions for the fractional $(p,q)$-Laplace operator We investigate a nonlinear nonlocal eigenvalue problem involving the sum of fractional (p,q)-Laplace operators and subject to Dirichlet boundary conditions in an open bounded domain. We prove bifurcation results from trivial solutions and from infinity for the considered nonlinear nonlocal eigenvalue problem. We also show the existence of multiple solutions of the nonlinear nonlocal problem using variational methods.
B3 Evangelia Ftaka North Carolina State University Piecewise Regular Solutions to Scalar Balance Laws with Singular Nonlocal Sources We will present a local well-posed result for piecewise regular solutions with a single shock of scalar balance laws, with singular integral of convolution type kernels. In a neighborhood of the shock curve, a detailed description of the solution is provided for a general class of initial data.
A5 Davis Funk West Virginia University Solving the Wave Equation on Discrete Time Scales We present a solution to an initial value problem for the 1-dimensional wave equation on time scales through the application of a Fourier transform and its inverse via contour integrals. The time scale of the spatial dimension is set to the integers and a broader class of discrete time scales, while the time dimension is set to the positive real numbers.
A7 John Gemmer Wake Forest University Tipping in a low-dimensional model of a tropical cyclone A presumed impact of global climate change is the increase in frequency and intensity of tropical cyclones. Due to the possible destruction that occurs when tropical cyclones make landfall, understanding their formation should be of mass interest. In 2017, Kerry Emanuel modeled tropical cyclone formation by developing a low-dimensional dynamical system which couples tangential wind speed of the eye-wall with the inner-core moisture. For physically relevant parameters, this dynamical system always contains three fixed points: a stable fixed point at the origin corresponding to a non-storm state, an additional asymptotically stable fixed point corresponding to a stable storm state, and a saddle corresponding to an unstable storm state. We present a case study of both rate and noise-induced tipping between the stable states, relating to the destabilization or formation of a tropical cyclone. While the stochastic system exhibits transitions both to and from the non-storm state, noise-induced tipping is more likely to form a storm, whereas rate-induced tipping is more likely to cause the storm to destabilize. When rate-induced tipping causes the storm to destabilize, a striking result is that both wind shear and maximal potential velocity have to increase at a substantial rate in order to affect tipping away from the active hurricane state. For storm formation through noise-induced tipping, we identify a specific direction along which the non-storm state is most likely to get activated.
A7 Shohreh Gholizadeh Siahmazgi Wake Forest University Mean Exit Times for Perturbed Gradient Systems Using the Freidlin-Wentzell theory of large deviations, we study noise-induced transitions for stochastic differential equations with additive noise. Specifically, we focus on a system in which the drift consists of the negative gradient of a potential weakly perturbed by a non-gradient perturbation. For gradient systems, the well-known Eyring-Kramers' law states that the mean exit time depends exponentially on the difference in potential evaluated at the equilibrium points with a prefactor that depends on the curvatures of the potential. In this talk, we present our work on extending the Eyring-Kramers' law to the perturbed gradient case. In this case, the problem of the mean exit time reduces to solving perturbed matrix Riccati equations.
C2 Gleb Gribovskii University of North Carolina at Greensboro A Game-Theoretic Model of Optimal Condom Usage to Prevent Chlamydia Infections Chlamydia trachomatis is one of the most common sexually transmitted infections worldwide, often asymptomatic and leading to severe reproductive health complications if untreated. In this work, we construct a game-theoretic model to examine individual decisions regarding condom usage as a preventive measure against chlamydia transmission. We consider two behavioral models including non-discriminating and discriminating populations. In the first scenario, condom usage rate is independent of chlamydia symptoms. In the second scenario, everyone will use condoms when they recognize chlamydia symptoms. Our findings indicate that the optimal population condom usage falls short of the herd immunity level in both scenarios if the cost of condom usage is not negligible. The optimal condom usage is closer to herd immunity level in non-discriminating population if the cost of condom usage relative to the cost of infection is close to zero. These results highlight the significance of reducing the relative (perceived) cost of condom usage to prevent chlamydia infections.
C5 Kanan Gupta University of Pittsburgh Nesterov acceleration despite very noisy gradients Momentum-based gradient descent methods use information gained along the trajectory, in
addition to the local information from the gradient, in order to achieve an accelerated rate of convergence. These methods have been well-studied for convex optimization. Computing the gradient is often too expensive and it is approximated using stochastic gradient estimates in practice. However, the analyses of accelerated methods fail in the setting of stochastic gradient descent, even for the simple case of convex functions. We address this gap with our algorithm, AGNES, which provably achieves acceleration for smooth convex minimization tasks with noisy gradient estimates if the noise intensity is proportional to the magnitude of the gradient. Nesterov’s accelerated gradient descent does not converge under this noise model if the constant of proportionality exceeds one. AGNES fixes this deficiency and provably achieves an accelerated convergence rate no matter how small the signal to noise ratio in the gradient estimate. Empirically, we demonstrate that this is an appropriate model for mini-batch gradients in overparameterized deep learning. Finally, we show that AGNES outperforms stochastic gradient descent with momentum and Nesterov’s method in the training of CNNs.
B7 Andrew Hicks Carnegie Mellon University Modeling and Simulation of the Cholesteric Landau-de Gennes Model This talk gives an analysis of modeling and numerical issues in the Landau-de Gennes (LdG) model of nematic liquid crystals (LCs) with cholesteric effects. We derive various time-step restrictions for a (weighted) $L^2$ gradient flow scheme to be energy decreasing. Furthermore, we prove a mesh size restriction, for finite element discretizations, that is critical to avoid spurious numerical artifacts in discrete minimizers that is not well-known in the LC literature, particularly when simulating cholesteric LCs that exhibit ``twist''. Furthermore, we perform a computational exploration of the model and present several numerical simulations in 3-D, on both slab geometries and spherical shells, using a fully-implicit gradient flow scheme applied to a finite element discretization of the model. The simulations are consistent with experiments, illustrate the richness of the cholesteric model, and demonstrate the importance of the mesh size restriction.
A3 John Holmes The Ohio State University The Fokas transform method for Burgers' equation We investigate the well-posedness of the viscous Burgers’ equation on the half-line using Fokas’ novel unified transform method. We show that the Cauchy problem is well-posed for initial data in Sobolev spaces $H^s$, and boundary data in $H^{(2s+1)/4}$, for $s>-1/2$. We establish this result by finding appropriate subsets of these Sobolev spaces for which we can prove the necessary linear and bi-linear estimates. Then, we show a fixed point of the operator constructed by the Fokas transform, via Picard iteration.
A6 Robert Ireri Marshall University A Numerical Method for Coefficient Reconstruction of a Periodic Inverse Source Problem This research focuses on reconstructing a periodic array of sources from measurements of waves emitted from those sources. This array can have infinitely many sources along one direction in space, thus is unbounded. A practical application of this problem is in Non-Destructive Testing, where the goal is to assess integrity of the sources without causing them any damage. To solve this problem, we employ the Quasi-Reversibility Method (QRM) which has been successfully applied to inverse source problems in the bounded setting. Both theoretical and numerical studies are investigated. In this talk, I will present the results that I have obtained so far: the derivation of a finite-difference approximation of the problem, and the implementation of a corresponding numerical scheme in Python. This is joint work with Dr. Trung Truong.
C2 Rahnuma Islam University of Pittsburgh Stochastic Immunology model and its analysis We present a stochastic model in immunology describing the evolution of influenza A disease in the human organism. Our analysis includes two types of viral production: bursting or budding. Euler-Maruyama simulation algorithm and jump-diffusion process are used to analyze the stochastic model. Numerical simulations explore the dynamic behavior of the reproduction system in vitro in the absence of any immune component. There are two absorbing states for the dynamics of our system: extinction of virus cells and extinction of host cells. The simulation also investigates the mean times of absorption and probabilities of absorption depending on the parameters of the system.
B4 Jiaxin Jin University of Louisiana at Lafayette Infinitesimal Homeostasis in Mass-Action Systems Homeostasis occurs in a biological system when a chosen output variable remains approximately constant despite changes in an input variable. In this work we specifically focus on biological systems which may be represented as chemical reaction networks and consider their infinitesimal homeostasis, where the derivative of the input-output function is zero. The specific challenge of chemical reaction networks is that they often obey various conservation laws complicating the standard input-output analysis. We derive several results that allow to verify the existence of infinitesimal homeostasis points both in the absence of conservation and under conservation laws where conserved quantities serve as input parameters. In particular, we introduce the notion of infinitesimal concentration robustness, where the output variable remains nearly constant despite fluctuations in the conserved quantities. We provide several examples of chemical networks which illustrate our results both in deterministic and stochastic settings.
A3 Md Ibrahim Kholil Norfolk State University A Uniqueness Theorem for Inverse Problems in Quasilinear Anisotropic Media We study the question of whether one can uniquely determine a scalar quasilinear conductivity in an anisotropic medium by making voltage and current measurements at the boundary. We prove a global uniqueness in the C infinity, by showing that the C infinity quasilinear conductivity in an anisotropic medium can be uniquely determined by the voltage and current measurements at the boundary, i.e., by the Dirichlet to Neumann map, assuming that an anisotropic linear conductivity can be identified by its Dirichlet to Neumann map up to a diffeomorphism that fixes the boundary.
B5 Valentin Kunz The Ohio State University Several Complex Variables and the Quarter-Plane problem The 'quarter-plane problem' refers to the boundary value problem which models the interaction of a monochromatic plane-wave, e.g., an acoustic pressure field, with the tip of an infinitely thin corner in three spatial dimensions, e.g., the tip of a turbofan blade. In this talk, we will outline how the theory of several complex variables can help us gain a better understanding of such physical phenomena. We are particularly interested in finding an (analytic) asymptotic formula for the field observed at a great distance from the quarter-plane, which, in turn, requires knowledge of the singularity structure of some two-complex-variable spectral functions.
B1 Van Le University of Tennessee, Knoxville Existence and uniqueness of solutions to stationary Navier-Stokes equations in the upper-half plane We study the incompressible stationary Navier-Stokes equations in the upper-half plane with homogeneous Dirichlet boundary condition and non-zero external forcing terms. Existence of weak solutions is proved under a suitable condition on the external forces by using stream functions and density lemmas with suitable weights. Weak-strong uniqueness criteria based on various growth conditions at the infinity of weak solutions are also given by employing an energy estimate and Hardy’s inequality.
A6 Tom Lewis University of North Carolina at Greensboro Convergent methods for approximating sublinear semipositone reaction diffusion equations In this talk we discuss some of the analytic issues for proving a numerical approximation method reliably approximates the solutions to sublinear semipositone boundary values problems. We will highlight new analytic techniques for proving admissibility, stability, and convergence of simple finite difference methods. The admissibility and stability results will be based on adapting the method of sub- and supersolutions. The convergence analysis shows that all limit points for the approximating sequences are indeed solutions to the boundary value problem. The new tools serve as a foundation for approximating semipositone boundary value problems in higher dimensions and on more general domains.
C1 Boya Liu North Dakota State University Recovery of time-dependent coefficients in hyperbolic equations on Riemannian manifolds from partial data In this talk we discuss inverse problems of determining time-dependent coefficients appearing in the wave equation in a compact Riemannian manifold of dimension three or higher. More specifically, we are concerned with the case of conformally transversally anisotropic manifolds, or in other words, compact Riemannian manifolds with boundary conformally embedded in a product of the Euclidean line and a transversal manifold. With an additional assumption of the attenuated geodesic ray transform being injective on the transversal manifold, we prove that the knowledge of a certain partial Cauchy data set determines time-dependent coefficients of the wave equation uniquely in a space-time cylinder. We shall discuss two problems: (1) Recovery of a potential appearing in the wave equation, when the Dirichlet and Neumann values are measured on opposite parts of the lateral boundary of the space-time cylinder. (2) Recovery of both a damping coefficient and a potential appearing in the wave equation, when the Dirichlet values are measured on the whole lateral boundary and the Neumann data is collected on roughly half of the boundary. This talk is based on joint works with Teemu Saksala (NC State University) and Lili Yan (University of Minnesota).
Plenary Speaker 4 Anna Mazzucato Penn State University Direct and inverse problems for elastic dislocations in geophysics Elastic dislocations have been used to model creeping faults in the Earth’s crust in between seismic events. The forward problem amounts to solving a non-standard transmission problem for a system of linear PDES in elastostatics, knowing the fault and how much the rock has slipped at the fault. The inverse problem consists in determining the geometry of the fault and the slip at the fault from surface measurements, which can be obtained from GPS and satellite data. While the direct problem is well posed, the inverse problem is generally ill-posed unless assumptions are made on the fault.
I will present a uniqueness result for the inverse problem and an iterative reconstruction algorithm based on a distributed shape derivative, which measures the change in the rock displacement under infinitesimal movements of the fault and the slip. I will close with some simple numerical tests from synthetic data.
If time permits, I will also discuss non-linear and non-local viscoelastic models for the fault dynamics.
This is joint work with Andrea Aspri (University of Milan), Elena Beretta (NYU-Abu Dhabi), Maarten de Hoop (Rice University), and PhD student Arum Lee.
C4 Shixu Meng Virginia Tech Exploring Low Rank Structures in Inverse Problems and PDEs In high dimensional PDEs and inverse problems, numerical methods may suffer from the curse of dimensionality where the computational cost scales exponentially with the number of dimensions of the problem. In addition, inverse problems are even more challenging since they can be intrinsically non linear and ill-posed, and the measurement data are inevitably corrupted by noise, potentially large-scale or limited, across different application areas. The underlying low rank structures are pivotal to develop both theoretical stability estimates and efficient numerical algorithms. This talk is concerned with a comprehensive framework to develop low rank structures in the context of inverse scattering problem, as well as a low rank method based on preconditioning low rank Generalized Minimal Residual Method for solving matrix differential equations.
B4 Maya Mincheva Northern Illinois University Efficient computation of Hopf bifurcation points for mass action systems A method for identifying Hopf bifurcation points in mass action ordinary differential equations (ODE) systems is presented. The method is based on a Hopf bifurcation theorem for parametric systems, algebraic geometry, majorization theory and convex analysis. Selecting parameter values such that the next to last Hurwitz determinant $\det H_{n-1}$ is zero is required for the Hopf bifurcation theorem.
B5 Rachel Morris North Carolina State University Uniform convergence guarantees for adversarially robust learning Recent work has shown that neural networks trained to maximize classification accuracy can be tricked by well-targeted adversarial attacks – even though their effects may be visually imperceptible to the human eye. This has sparked many new approaches to classification which include an adversary in the training process: such an adversary can improve robustness and generalization properties, at the cost of decreased accuracy and increased training time. In this presentation, I will discuss a range of adversarial training models, with a special focus on the underlying connection between adversarial training problems and nonlocal perimeter minimization. In particular, these problems can be cast as modified isoperimetric problems, and hence we expect that many of the ideas from geometric measure theory should apply in the context of adversarially robust learning. The theoretical results discussed in the presentation provide a rigorous comparison with the standard Bayes’ classification problem – an important step towards understanding the regularity of minimizers of the adversarial training problem.
B5 Ryan Murray North Carolina State University Regularization via Dirichlet energies for active learning Recently there has been significant effort to extend tools from the calculus of variations in order to understand problems from machine learning and data science. This has been especially successful in the context of supervised learning, namely regression and classification. On the other hand, in many realistic contexts there is a disparity between labeled and unlabeled data. One such setting is in active learning, wherein practitioners seek to select unlabeled points at which to obtain data labels. I will discuss recent work, with Kevin Miller, which identifies a new Bayesian-inspired variational model for active learning. This talk will discuss analytical and computational results relating to these PDE-based models. The talk will not assume special knowledge about data science.
B3 Timothy Myers Howard University A Constructive Solution to The Ornstein-Uhlenbech Operator Equation on a Separable Banach Space The paper $\it{Constructive\space Analysis\space on\space Banach\space Spaces,\rm (Real\space Analysis\space Exchange, \bf{44}\rm\space (2019)\ 1-36)}$ presents a proof that every separable infinite-dimensional Banach space, denoted $\mathcal{B}$, has an isomorphic, isometric embedding in $\mathbb{R}^{\infty}=\mathbb{R}\times\mathbb{R}\times\cdots$. The authors used this result and a method due to Yamasaki $\rm(\it{Measures\space on\space Infinite\space Dimensional\space Spaces},\rm\ World\space Scientific,\space (1985))$ to construct a sigma-finite Lebesgue measure $\lambda_{\mathcal{B}}$ for $\mathcal{B}$ and defined the associated integral $\int_{\mathcal{B}}\cdot\ d\lambda_{\mathcal{B}}$ in a way that equals a limit of finite-dimensional Lebesgue integrals. \begin{align*}\end{align*}The objective of this talk is to apply this theory to developing a constructive solution to the Ornstein-Uhlenbech operator equation : \begin{equation} \tag{1}\label{1}\frac{\partial u(x,t)}{\partial t}=\Delta u(x,t)-x\cdot \nabla u(x,t)\ ,\hspace{.1in} u(x,0)=\varphi(x) \end{equation}where $x\in\mathcal{B}$, $\varphi\in\mathcal{C}^2_0[\mathcal{B}]$, and $t\in[0,\infty)$. Our approach is constructive in the sense that the solution $u(x,t)$ of equation \eqref{1} is expressible as an integral $\int_{\mathcal{B}}\cdot\ d\lambda_{\mathcal{B}}$ which, by the aforementioned definition, equals a limit of Lebesgue integrals on Euclidean space as the dimension $n\to\infty$. Thus with this theory we may evaluate infinite-dimensional quantities, such as the solution $u(x,t)$, by means of finite-dimensional approximation.
C4 Loc Nguyen University of Carolina at Charlotte The Carleman-contraction mapping approach for the inverse scattering problem This talk addresses the inverse scattering problem in a domain Ω. The input data, measured outside Ω, involve the waves generated by the interaction of plane waves with various directions and unknown scatterers fully occluded inside Ω. The output of this problem is the spatially dielectric constant of these scatterers. Our approach to solving this problem consists of two primary stages. Initially, we eliminate the unknown dielectric constant from the governing equation, resulting in a system of partial differential equations. Subsequently, we develop the Carleman contraction mapping method to effectively tackle this system. It is noteworthy to highlight this method’s robustness. It does not request a precise initial guess of the true solution, and its computational cost is not expensive. Some numerical examples are presented.
A2 Tien Khai Nguyen North Carolina State University Generic properties of solutions to Hamilton-Jacobi equations The talk presents a result on generic properties of solutions to first order Hamilton-Jacobi equations associated with the classical calculus of variations in $\mathbb{d}^n$ . For generic initial data, the set of conjugate points is contained within the image of a $d-2$ dimensional manifold, and has locally bounded $(d-2)$- dimensional Hausdorff measure.
B6 Rebecca Oduro Marshall University First-order Nabla Riemann--Liouville fractional difference equations This talk presents the first-order homogeneous Riemann--Liouville fractional difference equation. We will explore the known case when the coefficient is constant and an unknown case when the coefficient is time varying. When the coefficient is constant, it is well-known that the solution is expressible in terms of the discrete Mittag-Leffler function. While a closed-form solution remains elusive for the time varying coefficient case, solving it numerically provides insights to the system's behavior. The results of this investigation have implications for the broader understanding of fractional dynamics, particularly the Nabla Gompertz equation.
A5 Kayode Oluwasegun Drexel University Investigation of oceanic wave solutions to a modified (2 + 1)-dimensional coupled nonlinear Schrodinger system Oceanic wave characteristics can be investigated by a modified integrable generalized
(2 + 1)-dimensional nonlinear Schrodinger (NLS) system of equations with variable coefficients. Variable coefficients enhance the modeling capability of the NLS equation, making it a more powerful tool for understanding and predicting wave behaviors in complex oceanic and other physical systems. In this work, two newly modified methods, specifically the improved nonlinear Riccati equation method and the improved sub-equation method, have been proposed to investigate the aforementioned nonlinear system. Through the utilization of these methods, we successfully obtain traveling and solitary wave solutions for this nonlinear system. We emphasize several constraint conditions that serve to guarantee the existence of these solutions. More comprehensive information about the physical dynamical representation of some of the solutions presented is illustrated through graphical depictions. The Mathematica software package is employed to produce both three-dimensional and their corresponding contour plots, thereby improving the visualization and comprehension of the solutions. This work illustrates that the two proposed approaches provide straightforward and efficient means of acquiring various types of solitons, rational, trigonometric, hyperbolic, and exponential solutions. Moreover, they present a more potent mathematical tool for addressing a variety of other nonlinear partial differential equations that hold significance in the field of applied science and engineering.
C2 Sujan Pant Norfolk State University Understanding the Obesity Epidemic: A Mathematical Model
for the Dynamics between Insulin and Glucose		 In this talk, we describe a mathematical model for the dynamics between insulin and glucose, using parameter values according to the American diet. The main interest is to create awareness of the obesity epidemic. Regular high consumption of sugar and carbohydrates in the American diet affects the body with insulin resistance; more than half of the population in the states suffer from this condition. Some epidemiologists describe insulin resistance as a hidden epidemic; if not treated on time, an individual can develop type 2 diabetes, Alzheimer's disease, heart disease, metabolic syndrome, fat liver, high blood pressure, some types of cancer, and obesity, among other
diseases. The majority of these diseases affect predominantly the most vulnerable populations: minority communities, the same communities that use social services such as "food and health" assistance programs; this cycle cannot be broken without solid regulations from the top government institutions.
A4 Lorand Parajdi "Babeș-Bolyai" University Lower and Upper Solution Method for Control Problems: Application to an Allogeneic Bone Marrow Transplantation Model This presentation focuses on the lower and upper solution method for control problems related to abstract operator equations of the fixed-point type. Specifically, I will introduce an algorithm designed to approximate the solutions of general control problems. The algorithm primarily involves a bisection method applied to the control variable within an interval defined by a pair of lower and upper solutions. I will outline the method's general framework and demonstrate its application to a control problem concerning cell evolution after allogeneic bone marrow transplantation in patients with acute myeloid leukemia. This work is based on recent collaborations with Prof. Radu Precup, Ph.D., from "Babeș-Bolyai" University and MD Ioan Ștefan Haplea from "Iuliu Hațieganu" University of Medicine and Pharmacy, Cluj-Napoca, Romania.
B6 Md Mashud Parvez Old Dominion University A Strict Physicality-Preserving Scheme for a 2D Q-Tensor Flow with a Singular Potential We introduce a numerical scheme for a two-dimensional (2D) dynamic Q-tensor model describing nematic liquid crystals. This model is formulated as an \(L^2\)-gradient flow driven by the liquid crystal free energy. In this work, we construct a numerical scheme, prove its strict physicality, and demonstrate its convergence. Our convergence analysis further establishes the well-posedness of the original PDE system for the 2D Q-tensor model. Several numerical experiments are provided to validate the efficiency and accuracy of the proposed scheme.
C4 Jesse Paul University of North Carolina at Greensboro The Monge Ampere Equation and Prescribed Gaussian Curvature: Numerical Methods This talk will discuss the Monge Ampere equation and a novel approach to computing solutions using the narrow stencil method developed by Feng and Lewis.
C3 Joseph Paullet Penn State Behrend Generalized Boundary-Layer Flow Due to a Shrinking Permeable Sheet In this talk we analyze a differential equation model, recently proposed by Merkin and Pop ($\it{J. Eng. Math.,}$ $\bf{121}$ (2020), pp. 1-17.) of boundary-layer flow generated by a shrinking permeable sheet.  The model is governed by two physical parameters, the transpiration rate, $S$, and the coupling between the fluid and the moving surface, $c$.  We prove several results regarding the existence or nonexistence of solutions in the $c$-$S$ parameter space. In regions where solutions are shown to exist, we prove that there is a continuum of infinitely many solutions.  This is in contrast to previous numerical studies, which reported either a unique solution or dual solutions.     
Plenary Speaker 1 Bob Pego Carnegie Mellon University Rigidly breaking potential flows and a countable Alexandrov theorem for polytopes Variational relaxation of the least-action principle for free-boundary incompressible flow yields pressureless Euler flow following Wasserstein geodesics. Action-minimizing incompressible flows are locally rigid, determined by convex and locally affine velocity potentials for initially convex bodies. An alternative characterization of these flows involves absence of mass concentration in Monge-Ampere measures associated with certain Hamilton-Jacobi equations. There is a close relation to geometric theorems of Minkowski and Alexandrov for convex polytopes, and to the adhesion model in cosmology, a multi-D model which reduces to sticky particle flow in 1D.
C3 Wenlong Pei The Ohio State University The variable time-stepping DLN method for fluid models Dahlquist, Liniger, and Nevanlinna proposed a two-step time-stepping scheme for systems of ordinary differential equations (ODEs) in 1983. The little-explored variable time-stepping scheme has advantages in numerical simulations for its fine properties such as unconditional G-stability and second-order accuracy. We simplify its implementation through time filters (pre-filter and post-filter) on a certain first-order implicit method. The adaptivity algorithm for this variable time-stepping scheme, highly reducing computation cost as well as keeping time accuracy, has been applied to systems of ODEs and fluid models. Moreover this time-stepping method can be combined with other efficient algorithms for robust simulations.
C3 Adam Pickarski North Carolina State University Large data limits and scaling laws for tSNE This work considers large-data asymptotics for t-distributed stochastic neighbor embedding (tSNE), a widely-used non-linear dimension reduction algorithm. We identify an appropriate continuum limit of the tSNE objective function, which can be viewed as a combination of a kernel-based repulsion and an asymptotically-vanishing Laplacian-type regularizer. We prove then prove that embeddings of the original tSNE algorithm cannot have any consistent limit as $n \to \infty$. We propose a rescaled model which mitigates the asymptotic decay of the attractive energy, and which does have a consistent limit.
B2 Antonio Pierrottet Clemson University Recovering all coefficients in the Schrödinger equation by finite sets of measurements We consider the inverse problem of recovering all spatial dependent coefficients, in the Schr\"odinger equation defined on an open bounded domain with smooth enough boundary. We show that we can directly achieve Lipschitz stability, thus recovering all coefficients from the corresponding boundary measurements of their solutions by appropriately selecting a finite number of initial conditions.
B6 Jocelyn Quaintance MCIT Online, University of Pennsylvania Parabolic Compactification: Construction and Critical Points, Finite and at Infinity A parabolic compactification is a homeomorphism (continuous bijection) between $\mathbb{R}^n \cup \{\infty U\}_{U\in S^{n-1}}$, where $S^{n-1} = \{x\in\mathbb{R}^n\mid \|x\| = 1\}$, and a bowl on the paraboloid $P := \{x =(x_1,x_2,\dots ,x_n,\sum_{i=1}^nx_i^2)\}\subset \mathbb{R}^{n+1}$. We will present the geometric intuition behind the parabolic compactification, and in the process, define the meaning of $\infty U$ and $UE\mathbb{R}^n$. We will also derive detailed algebraic formulas for the homeomorphism and show that such formulas encapsulated by a ``master equation'' of the form $w(t) = \theta^{-1}(t)z(t)$, where $w(t)\in \mathbb{R}^n \cup \{\infty U\}_{U\in S^{n-1}}$ and
$z(t)\in\{x=(x_1,x_2,\dots, x_n)\in \mathbb{R}^{n}\mid \|x\|^2 \leq \gamma\}$. We then use this master equation to discuss correspondence between finite critical points of $w'(t)=F(w(t))$ and $z'(t)=H(z(t))$. We will also utilize the master equation to determine critical points at ``infinity'' of the system
\[
\frac{dw_1}{dt} = \frac{k_1w_1w_2}{k_2+w_1} - k_3w_1^2,\qquad
\frac{dw_2}{dt} = -\frac{k_1w_1w_2}{k_2+w_1} + k_3w_1^2,
\]
which appears in the work of P. Yu and G. Craciun.
This is joint work with Harry Gingold of WVU.
B2 Madhumita Roy North Carolina State University Existence of global attractors for a semilinear wave equation with
nonlinear boundary dissipation and nonlinear interior and boundary sources with critical exponents		 In this talk we study a long-time dynamics for a nonlinearly forced 3-D wave equation subject to nonlinear boundary dissipation. Non-linear forces are of critical exponent and supported both in the interior and on the boundary. While global attractors for wave equations with geometrically
restricted damping and critical interior forcing have been known, a treatment of critical nonlinearity supported on the boundary, within the context of theory of global attractors, was an open question. The method devised for solution to the problem relies on “hidden regularity” established for Neumann hyperbolic problems and new tangential-boundary estimates which are also of indepen-
dent interest.
A2 Teemu Saksala North Carolina State University Inverse Problem for Hyperbolic Partial Differential Operators on Riemannian Manifolds Without Boundary In this talk we consider an inverse problem for a hyperbolic partial differential operator on a Riemannian manifold without boundary. Such a manifold can be either compact, like a sphere or torus, or unbounded, like Euclidean or Hyperbolic spaces. The hyperbolic partial differential operator we are studying is a self-adjoint first order perturbation of the Riemannian wave operator. In particular, this operator has time-independent lower order terms which can be written in the form of a vector field and a function which models the magnetic and electric potentials respectively. Our goal is to recover the speed of sound as well as these lower order terms by sending lots of waves from some open set of the manifold and measuring these waves on the same open set. This is called the local source-to-solution map. In this talk I will introduce the natural obstruction for recovering the aforementioned quantities from such measurements. This is the gauge of the problem. I will outline a proof that shows that modulo this gauge we can recover the wave speed, together with the magnetic and electric potentials from the local source-to-solution map. Our proof is based on a variation of the celebrated boundary control method (BC-method) which was developed by Belishev and Kurylev, and used to solve Gel'fand's inverse boundary spectral problem: "Can you recover a Riemannian manifold with boundary from the spectral data of its Dirichlet-Laplacian?" The BC-method reduces the PDE-based inverse problem to a geometric inverse problem of recovering a Riemannian manifold from a family of distance functions.

This talk is based on my earlier work with Tapio Helin, Matti Lassas and Lauri Oksanen and on an ongoing project with Andrew Shedlock.
C3 Jaffar Ali Shahul Hameed Florida Gulf Coast University Positive Solutions for a Derivative Dependent $p$-Laplacian Equation with Integral Boundary Conditions In this talk, we will discuss the existence of two non-trivial positive solutions to a class of boundary value problems (BVP), involving a $p$-Laplacian, of the form:
\begin{align*}
(\Phi_p(x^{'}))^{'} + g(t)f(t,x,x^{\prime}) & = 0, \quad t \in (0,1),\\
x(0)-ax^{'}(0) & = \alpha[x],\\
x(1)+bx^{'}(1) & = \beta[x],
\end{align*}
where $\Phi_{p}(x) = |x|^{p-2}x$ is a one dimensional $p$-Laplacian operator with $p>1, a,b$ are real constants. Here $\alpha,\beta$ are given by Riemann-Stieltjes integrals
\[ \alpha[x] = \int \limits_{0}^{1} x(t)dA(t), \quad \beta[x] = \int \limits_{0}^{1} x(t)dB(t),\]
where $A$ and $B$ are functions of bounded variations. We will use the fixed point index theory to establish our results.
B3 Andrew Shedlock North Carolina State University Lipschitz Stability of Travel Time Data The travel time map will map any point in a domain to the distance function of that point to any other point in some fixed measurement set. The travel time map allows for the domain to be embedded into a set in the space of continuous functions on this measurement set called the Travel Time Data for the domain. The celebrated boundary control method shows that if two sets of travel time data are equal, then the original domains are equal up to isometry. In particular, the boundary control method is of central importance for Inverse Problems for Hyperbolic PDEs. Building off of the uniqueness results developed by the boundary control method, we establish a stability result for a broad class of domains where the Gromov-Hausdorff Distance between the original domains is bounded by the Hausdorff distance between the Travel Time Data sets.
A5 Zhaiming Shen Georgia Institute of Technology Matrix Cross Approximation for Image Compression and Least Squares Approximation Matrix Cross Approximation is a method for low rank matrix approximation. Although it is not the best approximation, not as good as SVD, it has the interpretability of the physical meaning of the rows and columns of a matrix. In this talk, we present an estimate how a matrix cross approximation is close to the given matrix in Chebyshev norm which improves the classic result. In addition, I will explain a computational scheme to find a good matrix cross approximation. Finally, I shall present two examples to show how to use the matrix cross approximation for image compression and for solution of least squares problem.
B2 FNU Shumaila Miami University Computation of K-Functional for Sobolev Spaces on Riemannian Manifolds This mainly presents the computation of K-Functional and Real Interpolation of Sobolev Spaces on Riemannian manifolds. We have explored how Sobolev Spaces are defined on Riemannian Manifold then their Real Interpolation. So, mainly we have categorized it into the following described portions.
Firstly, we will define Sobolev space on Riemannian Manifolds. After the basic construction, we will introduce the K-functional and Real Interpolation of Sobolev Spaces on Riemannian Manifolds. In the end, we will modify the definition even on a bigger space which is Lorentz Space, and define the Sobolev Space over there. The Sobolev space over Rn is a vector space of functions that have weak
derivatives sitting inside Lp Spaces. The motivation for studying these spaces is that solutions of partial differential equations, when they exist, belong naturally to Sobolev spaces. The functions of Sobolev space is not easy to handle, we shall approximate these functions by smooth functions. We have calculated some inequalities in Sobolev space. With the help of these inequalities, we will embed the Sobolev spaces in some Lp spaces then extend it even to the bigger class which is the Lorentz spaces. Similarly, on the manifold using the covariant derivative, we define Sobolev Spaces over Riemannian Manifolds. Riemannian manifolds are natural extensions of Euclidean space with Riemannian Geometry.
A3 Byungjae Son Ohio Northern University On positive solutions to double phase problems with strong singular weights and nonlinearities We consider the singular double phase problem:
\begin{equation*}
\left\{
\begin{array}{c}
-\left(\alpha(t)\varphi_p(u')+\beta(t)\varphi_q(u')\right)'= \lambda h(t) f(u),~~t\in(0,1),\\
u(0)=0=u(1),
\end{array}
\right.
\end{equation*}
where $\lambda>0$, $1<p<q<\infty$, $\varphi_m(s):=|s|^{m-2}s$, and $\alpha$ and $\beta$ are nonnegative and continuous such that $\alpha(t)+\beta(t)>0$ almost everywhere. This model can have strong singular weights and nonlinearities such as $h(t)=t^{-\delta}$ with $\delta\geq 1$ and $f(u)=1+u^{-\gamma}$ with $\gamma\geq 1.$ This talk discusses the existence and multiplicity of positive solutions. We use approximation techniques to obtain the results.
B3 Jonathan Stanfill The Ohio State University Factorizations and Power Weighted Rellich and Hardy-Rellich-Type Inequalities We revisit and extend a variety of inequalities related to power weighted Rellich and Hardy-Rellich inequalities, including an inequality due to Schmincke. Our main tool is a factorization method that allows one to establish a multi-parameter family of inequalities whose parameters can be optimized appropriately. We will also discuss related factorizations being developed as part of a graduate student research project.

This is based on multiple joint works with Fritz Gesztesy, Michael M. H. Pang, and Bart Rosenzweig.
B7 Wasiu Sule Marshall University Gompertz distribution on time scales We shall investigate Gompertz dynamic equations within the context of time scales calculus. By exploring the mathematical foundations and applications of the Gompertz model, which is commonly used to describe growth phenomena in various fields such as biology and economics. This research seeks to analyze the Gompertz cumulative distribution functions (CDF) and probability density functions (PDF) across different time scales, including the real numbers R and integer multiples hN. Probability techniques will be used to derive the CDF and PDF associated with the Gompertz dynamic equations, and we will examine how varying the time scale impacts the characteristics of these distributions.

Through graphical representations, we intend to facilitate a comparative analysis that elucidates the relationship between time dynamics and growth patterns. This work aims to contribute to the field of applied statistics by providing insights into the integration of timescale calculus with Gompertz modeling, thereby establishing a foundation for further research into dynamic systems and their statistical properties.
A4 Mohyeedden Sweidan Concord University Analysis of the Shortley-Weller Scheme for Variable Coefficient Boundary Problems: Applications to Tumor Growth Modeling in Heterogeneous Environments In this talk, we explore the analysis of the Shortley-Weller scheme for solving two-point boundary value problems with variable coefficients, focusing on its application to problems where boundary points do not align with a uniform mesh. We establish that the scheme exhibits first-order local truncation error near the boundary, while demonstrating third-order accuracy in the boundary region and second-order accuracy elsewhere. The second part of the talk extends this numerical approach to the field of computational biology, where we apply the Shortley-Weller scheme to model tumor growth within heterogeneous microenvironments. Our study reveals that incorporating variable nutrient diffusion rates and extracellular matrix (ECM) permeability into the classic Darcy's law model allows for effective capture of chemotaxis and haptotaxis phenomena. Notably, the tumor growth patterns indicate a preference for areas with higher diffusion rates or lower ECM permeability. This dual application underscores the utility of the Shortley-Weller scheme in both theoretical and applied contexts, offering enhanced insights into complex boundary value problems and biological modeling.
B4 David Swigon University of Pittsburgh Qualitative inverse problems: mapping data to trajectory features of an ODE model Parameter fitting of mathematical models is a difficult problem, especially in biological and medical contexts. High dimensionality and nonlinearity of realistic models combined with general paucity of data makes it difficult to judge whether the model and data are compatible, find reasonable parametrizations, evaluate their accuracy and uniqueness, and quantify the predictive power of the model. In our recent work we focused on providing geometric conditions on trajectories that lead to parameter identifiability and descriptions of regions in data space that correspond to dynamical systems with specific behavior. These results will be described on an example two-variable Lotka-Volterra dynamical system.
C1 Changhui Tan University of South Carolina Sticky particle dynamics with alignment interactions In this talk, I will introduce the Euler-alignment system in collective dynamics, which models flocking behavior. The discussion will center on weak solutions, with the goal of isolating a unique solution through the use of an entropic selection principle. Notably, this selection principle aligns with the sticky particle rules applied in the agent-based Cucker-Smale dynamics. I will present an analytical convergence result and discuss the formation of both finite- and infinite-time clusters. This is joint work with Trevor Leslie.
A1 Maja Taskovic Emory University On the inhomogeneous wave kinetic equation and the associated hierarchy The wave kinetic equation is one of the fundamental models in the theory of wave turbulence, and provides a statistical description of weakly nonlinear interacting waves.

This talk will address the global in time well-posedness of the spatially inhomogeneous wave kinetic equation by applying techniques inspired by the analysis of the Boltzmann equation – another model of statistical physics that describes evolution of rarefied gases in which particles undergo predominantly binary interactions.

Time permitting, we will also discuss the well-posedness of the wave kinetic hierarchy – an infinite system of coupled equations closely related to the wave kinetic equation. Two essential tools for obtaining these results are the Hewitt-Savage theorem, which allows us to lift the existence result for the equation to the hierarchy, and the Klainerman-Machedon board game argument, which allows us to control the factorial growth of the Dyson series and consequently prove uniqueness of solutions.
A1 Ian Tice Carnegie Mellon University Traveling wave solutions to the free boundary Navier-Stokes equations While the existence of traveling wave solutions to the free boundary Euler equations is a classical subject, progress on the corresponding problem for the Navier-Stokes system has only recently begun. In this talk I will summarize joint work with Noah Stevenson on both the compressible and incompressible versions of this problem.
A4 Divine Wanduku Georgia Southern University Mean-field differential equation ecological models with general survival lifetime distributions in a renewal process Most traditional ecological models formulated with autonomous differential equations (DE’s) assume the exponential distribution for survival in lifetime stages. The non-flexible properties of exponential distribution including a constant hazard function (HRF) is unsuitable for modeling dynamic processes in ecological systems. In this study, a stochastic process for the growth of a species in a general ecological system and the corresponding mean-field dynamic model for the growth rate of the system are investigated. The general SVIS (Susceptible-Vaccinated-Infectious-Susceptible) family of epidemic models are explored as a case study, wherein the mean-field differential equations (MFDE’s) for the disease dynamics are obtained by applying a probability modeling approach, and survival lifetime distributions with more flexible properties are incorporated to represent complex disease scenarios e.g. complex vaccination strategies, where survival is a competing risk; to represent the nonlinear risk behaviors of death and recovery. The new MFDE models are non-autonomous, and the system coefficients are the HRFs of the lifetime distributions in the renewal process. Four asymptotic behaviors of the HRFs: a monotonic, a bathtub, a reverse bathtub, and a constant shape are explored in the models for disease control. Numerical simulation results are presented for different lifetime distributions.
B6 Zhuoran Wang University of Kansas Convergence analysis of GMRES with inexact block triangular preconditioning for saddle point systems with application to WG FE approximation of Stokes flow This talk presents a study on the convergence of the generalized minimal residual method (GMRES) for nondiagonalizable saddle point systems resulting from inexact block triangular Schur complement preconditioning.

It is demonstrated that GMRES convergence for the preconditioned system is primarily determined by the Schur complement and its approximation. As an example of application of this theoretical finding, the weak Galerkin finite element approximation of Stokes flow problems is examined. In this approximation, the resulting saddle point system is singular and inconsistent. A widely used regularization strategy that specifies the value of the pressure at a specific location is employed.

The nonsingularity of the regularized system is rigorously proven and bounds are derived for the eigenvalues of the preconditioned system as well as the GMRES residual. These bounds indicate that the convergence factor of GMRES is almost independent of the viscosity parameter and mesh size while the number of GMRES iterations needed to reach a prescribed level of residual depends on the parameters only logarithmically. Numerical results in two and three dimensions are presented to verify the theoretical findings.This work is a collaboration with Dr. Weizhang Huang.
A7 Richard Williams Marshall University The Heat Equation on Discrete Time Scales In 1988, Stefan Hilger introduced the theory of time scales, a unification between discrete in continuous analysis. Given a nonempty, closed subset of the real numbers (a "time scale"), functions inherit a so called delta derivative. This gives us an opportunity to study generalizations of differential equations and difference equations, called dynamic equations. Here, we will look at generalizations of the heat equation and investigate modern techniques in the literature to find solutions for given time scales, particularly discrete time scales.
C5 Stephan Wojtowytsch University of Pittsburgh The 'accelerated Allen-Cahn equation' on Euclidean spaces and in machine learning Momentum-based (‘accelerated’) methods of optimization play an important role in modern machine learning. We study the use of accelerated methods in the setting of semi-supervised learning for the Allen-Cahn equation on graphs. The ‘accelerated Allen-Cahn equation' is a hyperbolic PDE with different geometric properties compared to the regular (parabolic) Allen-Cahn equation. With a suitable convex-concave splitting, we are able to implement a time-stepping algorithm which improves on the performance of the Allen-Cahn equation with momentum in semi-supervised learning problems. This is joint work with Oluwatosin Akande, Patrick Dondl, Kanan Gupta and Akwum Onwunta.
B1 Lili Yan University of Minnesota, Twin Cities Inverse boundary problems for elliptic operators on Riemannian manifolds In an inverse boundary problem, one seeks to determine the coefficients of a PDE inside a domain, describing internal properties, from the knowledge of boundary values of solutions of the PDE, encoding boundary measurements. Applications of such problems range from medical imaging to non-destructive testing. In this talk, starting with the fundamental Calderon inverse conductivity problem, we shall first discuss a partial data inverse boundary problem for the Magnetic Sch\"odinger operator in the setting of compact Riemannian manifolds with boundary. Next, we discuss first-order perturbations of biharmonic operators in the setting of compact Riemannian manifolds with boundary. Specifically, we shall present a global uniqueness result as well as a reconstruction procedure for the latter inverse boundary problem on conformally transversally anisotropic Riemannian manifolds of dimensions three and higher.
A5 Nasir Yasin Old Dominion University Flow Patterns Behind Stationary and Moving Bluff Bodies Utilizing the SRC-Lattice Boltzmann Method Flow past bluff bodies is essential in engineering, yet the dynamics of staggered arrangements are not well understood. This study employs the single relaxation time lattice Boltzmann method (SRC-LBM) to analyze flow and force characteristics behind stationary and moving bluff bodies. We investigate the impact of varying Reynolds numbers (Re = 1-150) and gap spacing ratios (g* = 0.5-5). Our findings reveal distinct flow regimes, with larger gaps yielding regular vortex dynamics and smaller gaps leading to complex interactions. Additionally, we demonstrate that placing a rod in front of the bluff bodies effectively controls chaotic flow behavior. These results provide new insights for managing fluid dynamics by adjusting Reynolds number and gap spacing.