In Memoriam Session 3
Title: Population Dynamics in Applied Ecology: Models & Experiments
Abstract: Understanding how and why populations are regulated is a central concern in applied ecology. Mathematical models, especially those parameterized with field- and laboratory- derived data, are a powerful tool for understanding population dynamics and the forces governing population regulation. I describe several examples taken from nearly two decades of collaborative research with the H.T. Banks lab in which field/lab data from ecological experiments were incorporated into population models in order to estimate parameters and derive insights into drivers of population dynamics in applied ecology. Ecological research case studies include topics ranging from tropical forest bird conservation to the support of ecosystem services such as pest control and pollination in agricultural systems.

Session K Chair
Session A
Title: Observability for Evolution Equations on Simplices
Abstract: In this talk, I will present some observability results for wave, Schrodinger, and heat equations on Euclidean simplices. A simplex has Lipschitz boundary, so standard results for smooth domains do not apply. However, the affine nature of a simplex allows the use of simple integration by parts arguments to prove asymptotic boundary observability for the wave equation from any face of the simplex. Similar results hold for the Schrodinger equation. Using a transmutation method, the observability for the wave equation provides an estimate on the cost of observability for solutions to the heat equation. This is joint work with subsets of Evan Stafford, Ziqing Lu, Sarah Carpenter, and Ameer Qaqish.

Session M
Title: Parameter identifiability for PDE models of fluorescence microscopy experiments
Abstract: The dynamics of intracellular proteins is key to many cellular functions, and is often investigated using microscopy experiments such as FRAP (fluorescence recovery after photobleaching). These experiments generate time-series data that average out spatial information about protein localization. Partial differential equations (PDE) models are often used to model the fluorescence dynamics and to infer parameters such as diffusion coefficients or reaction rates. However, it is not known whether these parameters can be identified based on such averaged time-series data. Here, we limitations of known methods in assessing parameter identifiability for PDE models, and propose custom methods for analyzing the dynamics of RNA binding proteins in the development of frog eggs.

Session J Chair
Session Q
Title: Multiscale models of SARS-CoV-2 infection
Abstract: Designing control strategies for the COVID-19 epidemic requires multi-scale understanding of individual infections, the probability of transmission through aerosol exposure, and the roles of vaccination and testing in limiting an outbreak at the population level. In the first part of this talk, I will present several studies investigating within-host and aeosol dynamics in animals and humans infected with SARS-CoV-2. They will focus on the tradeoff between viral infectiousness and viral positivity, as well as the biases induced by the scarcity of data early in the individual’s infection. In the second part, I will connect the virus profile of infected individuals with transmission, testing strategies, and vaccination at the population level through multi-scale immuno-epidemiological models. Using the multi-scale models, we will predict the best testing-vaccination combinations for limiting an outbreak with variants of increased transmissibility. Our findings can improve interventions.

In Memoriam Session 3
Title: Data-Driven Mathematical Modeling of Addiction Among Individuals with Mild to Moderate Alcohol Use Disorder
Abstract: Alcohol use disorder (AUD) comprises a continuum of symptoms and associated problems that has led AUD to be a leading cause of morbidity and mortality across the globe. Consideration is being given to helping individuals with AUD to moderate or control their drinking at low-risk levels. Because so much remains unknown about the factors that contribute to successful moderated drinking, we use dynamical systems modeling to identify mechanisms of behavior change. We apply a discrete dynamical system model to clinical data using mixed effects parameterization in order to identify both population and individual level parameters. We demonstrate how conducting a parameter sensitivity analysis can assist in identifying optimal targets of intervention at the patient-specific level.

Session O Chair
Session R
Title: Entrainment dominated effects in the long residence times of a sphere settling in stratified fluids
Abstract: We study the delayed settling dynamics of spheres going through stably stratified sharp density variations in viscous-dominated regimes. In particular, we focus on cases with long residence times at the interface rivaling the ones observed under similar configurations for marine snow aggregates in the ocean. To capture the most significant aspects of the system, we solve a first principle model that reduces to a highly coupled system. Taking the far field asymptotic approach speeds up the numerics in the appropriate region of validity. The entrainment dominated regime, however, requires the exact solution for the sphere exterior problem of the Stokes equations. In this regime, we discuss the numerical challenges in computing a convolution integral and the effects of diffusion.

In Memoriam Session 2 Chair
In Memoriam Session 1
Title: Control of a consumer-resource agent-based model using partial differential equation approximation
Abstract: Agent-based models (ABMs) are increasing in popularity as tools to simulate and explore many biological systems. Successes in simulation lead to deeper investigations, from designing systems to optimizing performance. The typically stochastic, rule-based structure of ABMs, however, does not lend itself to analytic and numerical techniques of optimization the way traditional dynamical systems models do. The goal of this work is to illustrate a technique for approximating ABMs with a partial differential equation (PDE) system to design some management strategies on the ABM. We propose a surrogate modeling approach, using differential equations that admit direct means of determining optimal controls, with a particular focus on environmental heterogeneity in the ABM. We implement this program with both PDE and ordinary differential equation (ODE) approximations on the well-known rabbits and grass ABM, in which a pest population consumes a resource. The control problem addressed is the reduction of this pest population through an optimal control formulation. After fitting the ODE and PDE models to ABM simulation data in the absence of control, we compute optimal controls using the ODE and PDE models, which we them apply to the ABM. The results show promise for approximating ABMs with differential equations in this context.

Session I Chair
Session S
Title: Noise-Induced Tipping in Stochastic Piecewise-Smooth Systems
Abstract: We present a theory for computing most probable transition paths for systems of stochastic differential equations with piecewise-smooth drift and additive noise. In particular, we consider systems with a switching manifold in the drift that forms a co-dimension one hyperplane and investigate noise-induced transitions between metastable states on either side of the switching manifold. For systems with a smooth drift, such paths correspond to minimizers of the Freidlin-Wentzell functional and thus are solutions to a corresponding system of Euler-Lagrange equations with appropriate boundary conditions. However, for piecewise-smooth systems the Euler-Lagrange equations are only weakly defined and can contain a continuum of solutions. To obtain a rigorous theory in the piecewise smooth limit, we mollify the drift and use gamma-converge to derive an appropriate rate functional for the system in the piecewise-smooth limit. The resulting functional consists of the standard Freidlin–Wentzell rate functional, with an additional contribution due to times when the most probable path slides in a crossing region of the switching manifold. We explore implications of our results through case studies, which exhibit notable phenomena such as non-unique most probable paths and noise-induced sliding in a crossing region.

Session Q Chair
Session J
Title: Volume Scavenging: A Model of Fluid-Dynamical Competition
Abstract: The Florida tortoise beetle (Hemisphaerota cyanea) is a purplish-blue beetle which feeds on palm trees and is common to the Southeastern United States. It would be rather unremarkable if it were not for its ingenious defense mechanism against predatory ants: When attacked, the beetle clings to the ground with more persistence than the ants muster in their assault. The beetle achieves this strong adhesion by excreting an oily liquid through thousands of openings at its leg endings, thus forming liquid bridges with the substrate. In this way the beetle can withstand pulling forces up to 60 times and more of its body mass. Motivated by this defense mechanism, I present a dynamical system modeling surface tension-induced flows of liquids in networks of interconnected channels. These channel flows are driven by surface-tension induced pressure imbalances between fluid droplets of varying sizes (“volume scavenging”). These pressure differences push liquid from one droplet to another along the network of channels. The analysis of the nonlinear dynamics will be accompanied by animations, highlighting the similarities of microfluidic behavior and the emergence of inequality in socioeconomic competition.

Session P Chair
Session T
Title: A mathematical model for wetting and drying in filter membranes
Abstract: A filter membrane may be frequently used during its lifetime, with wetting and drying processes occurring in the porous medium for several cycles. During these cycles, the concentration distribution of molecules or contaminants and the medium morphology evolve. As a consequence, the filter performance ultimately deteriorates after several cycles. In this work, we formulate a coupled mathematical model for the wetting and drying dynamics in a porous medium occurring consecutively. Our model accounts for the porous medium internal morphology (internal structure, porosity, etc.), the contaminant deposition, and the evolution of dry/wet interfaces due to evaporation. The model provides insights to the overall porous medium evolution over cycles of wetting and drying processes and predicts the timeline to discard the filter based on its optimum performance.

In Memoriam Session 2
Title: Physiologically Based Pharmacokinetic Models and Applications in Chemical Risk Assessments
Abstract: Because many observable health effects associated with chemical exposures are believed to be more directly related to internal measures of dose (e.g., tissue concentration) than to external ones (e.g., amount ingested), the chemical risk assessment process often involves translating applied doses or environmental exposures into internal dose metrics. Physiologically based pharmacokinetic (PBPK) models, which use systems of ordinary differential equations to describe the biological processes of absorption, distribution, metabolism, and excretion, provide a means for calculating internal dose metrics. PBPK models can also be used to compare and combine data from toxicological studies that involve different animal species, different dosing regimens, and different routes of exposure, and to estimate human exposure levels that result in internal doses associated with adverse health outcomes. Thus, PBPK models can be used to support chemical risk assessments in a variety of ways. This presentation will include a description of the fundamental concepts underlying PBPK modeling and a discussion of applications of PBPK models in human health chemical risk assessments such as those conducted by the Integrated Risk Information System (IRIS) Program of the U.S. Environmental Protection Agency.

Session A Chair
Session U
Title: A mixed boundary value problem for \(u_{xy}=f(x,y,u,u_x,u_y)\)
Abstract: Consider a single hyperbolic PDE \(u_{xy}=f(x,y,u,u_x,u_y)\), with locally prescribed data: \(u\) on a non-characteristic curve \(M\) and \(u_x\) along a non-characteristic curve \(N\). We assume that \(M\) and \(N\) are graphs of one-to-one functions, intersecting only at the origin, and located in the first quadrant of the \((x,y)\)-plane. It is known that if \(M\) is located above \(N\), then there is a unique local solution, obtainable by successive approximation. We show that in the opposite case, when \(M\) lies below \(N\), the uniqueness can fail in the following strong sense: for the same boundary data, there are two solutions that differ at points arbitrarily close to the origin. In the latter case, we also establish existence of a local solution (under a Lipschitz condition on the function \(f\)). The construction, via Picard iteration, makes use of a careful choice of additional \(u\)-data which are updated in each iteration step. This is a joint work with Kris Jenssen from Penn State University,

Session U Chair
Session K
Title: Recover All Coefficients in Second-order Hyperbolic Equations from Finite Sets of Boundary Measurements
Abstract: In this talk we consider the inverse hyperbolic problem of recovering all spatial dependent coefficients, which are the wave speed, the damping coefficient, potential coefficient and gradient coefficient, in a second-order hyperbolic equation defined on an open bounded domain with smooth enough boundary. We show that by appropriately selecting finite pairs of initial conditions as well as a boundary condition we can uniquely and stably recover all those coefficients from the corresponding boundary measurements of their solutions. The proofs are based on sharp Carleman estimate, continuous observability inequality and regularity theory for general second-order hyperbolic equations.

Session C Chair
Session I
Title: Degenerate Dispersive Equations
Abstract: We discuss recent work on some quasilinear toy models for the phenomenon of degenerate dispersion, where the dispersion relation may degenerate at a point in physical space. In particular, we discuss the stationary states, as well as existence and uniqueness of solutions for degenerate KdV and NLS-type equations using a novel change of variables. This is joint work with Pierre Germain and Ben Harrop-Griffiths. Given time, we will discuss some work on Gibbs measures for a discrete version of this problem that is joint with with Jonathan Mattingly and Dana Mendelson.

In Memoriam Session 2
Title: Population model for the decline of the invasive pest Homalodisca vitripennis while in the presence of the parasitoid Cosmocomoidea ashmeadi
Abstract: The glassy-winged sharpshooter, Homalodisca vitripennis, is an invasive pest which presents a major economic threat to the grape industries in California by spreading a disease-causing bacteria, Xylella fastidiosa. Recently a common enemy of H. vitripennis, certain mymarid parasitoid species including Cosmocomoidea ashmeadi and Cosmocomoidea morrilli, have been studied to use in place of insecticides as a control method. We create a time and temperature dependent mathematical model to analyze data and answer the question: Does the implementation of C. ashmeadi as a biological control method cause a significant decrease in the population of H. vitripennis?

Session X Chair
Session B
Title: Pressure Robust Scheme for Incompressible Flow
Abstract: In this talk, we shall introduce the recent development regarding the pressure robust finite element method (FEM) for solving incompressible flow. We shall take weak Galerkin (WG) scheme as the example to demonstrate the proposed enhancement technique in designing the robust numerical schemes and then illustrate the extension to other finite element methods. Weak Galerkin (WG) Method is a natural extension of the classical Galerkin finite element method with advantages in many aspects. For example, due to its high structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations on the general meshing by providing the needed stability and accuracy. Due to the viscosity and pressure independence in the velocity approximation, our scheme is robust with small viscosity and/or large permeability, which tackles the crucial computational challenges in fluid simulation. We shall discuss the details in the implementation and theoretical analysis. Several numerical experiments will be tested to validate the theoretical conclusion.

In Memoriam Session 1
Title: TBA
Abstract: TBA

In Memoriam Session 3
Title: Modeling Immunosuppression in Renal Transplant Recipients
Abstract: In 2021, a new record was set in the United States for the number of kidney transplant operations performed in a year with over 24,000 procedures taking place, more than half of all solid organ transplants that year. As of 2018, more than 200,000 Americans are renal allograft hosts and this number grows while continuing to set records annually. The recipients of donor kidneys are met with two difficult biological challenges: 1) in almost all cases, allograft tissues are targeted and attacked by the host’s immune system thereby requiring suppression of the host’s immune system and 2) immunosuppression treatments weaken the host’s defense allowing latent pathogens to proliferate, such as the BK virus which infects cells within the nephrons of the kidney causing permanent damage and irreversible loss of kidney function. These patients require a balanced immunosuppression regimen which still allows the immune system to protect the body while preventing acute rejection of the allograft. The treatment protocols for kidney recipients vary among the over 250 renal transplant programs while an optimal treatment regimen has yet to be determined. In a collaboration between the Center for Research in Scientific Computation (NCSU) and the Duke Transplant Center, we are developing a mathematical model to assist in determining personalized optimal immunosuppression treatments for individual renal transplant recipients. A system of ordinary differential equations describe interactions between the allograft, BK virus and the immune system through six state variables including creatinine, a biomarker for kidney health and functioning. Determining optimal drug efficacy levels is performed by a receding horizon controller, and to personalize the treatment, Kalman filtering is utilized as a feedback mechanism to introduce current individual patient data into the model. Our initial efforts demonstrate the utility of taking this modeling approach.

Session G Chair
Session L
Title: A Carleman-based numerical method for the 3D inverse scattering problem
Abstract: We study the inverse scattering problem for the three-dimensional Helmholtz equation with multi-frequency back scattering data. Our approach relies on a new derivation of a boundary value problem for a system of coupled quasi-linear elliptic partial differential equations. We solve this coupled system by developing the Carleman convexification method. Using the Carleman weight function, we construct a globally strictly convex cost functional and prove the global convergence to the exact solution of the gradient projection method. The Lipschitz stability estimate of the Carleman convexification method is proved also via a Carleman estimate. Finally, our theoretical finding is verified via several numerical tests with computationally simulated data and experimental data. These tests demonstrate that we can accurately recover all three important components of targets of interest: locations, shapes, and dielectric constants.

In Memoriam Session 1
Title: Accurate angular integration with only a handful of neurons
Abstract: To flexibly navigate, many animals rely on neural activity that tracks the animal’s movements relative to its surroundings. In mammals, this signal is carried by large populations of cells (N ~ O(10^4) – O(10^6)) that maintain a persistent internal representation of the animal’s head direction when standing still and accurately integrate the animal’s angular velocity when turning. Such neural dynamics can be realized by ring attractor networks: a class of networks that generate a continuum of marginally stable states along a ring-like manifold. Although there are many variants of ring attractor networks, they all rely on large numbers of neurons as the ring attractor solution emerges in the limit as network size becomes infinite. Surprisingly, in the fruit fly, Drosophila Melanogaster, a head direction representation is maintained by a much smaller number of neurons whose dynamics and connectivity resemble those of a ring attractor network. These findings challenge our understanding of ring attractors and their putative implementation in neural circuits. Here, we analyze the ability of small systems to integrate an angular velocity input using threshold linear networks (though simulations suggest our results hold for more general nonlinear systems as well). We construct an energy function to determine the system’s encoding ability in the absence of input and find that certain parameterizations lead to perfect encodings (i.e., ring attractors) and hence perfect integration as well. Such optimal parameterizations correspond to singularities in the above threshold subsystems, suggesting that ring attractors emerge in small systems as a discrete set of line attractors “stitched together”. We then analyze the network dynamics away from these optimal parameterizations and calculate the rate at which the system’s ability to integrate accurately degrades. This work shows how even small networks can accurately track an animal’s movements to guide navigation, and it informs our understanding of the functional capabilities of discrete systems more broadly.

Session M Chair
Session C
Title: Image guided 1D fluid mechanics model for blood flow
Abstract: This study discusses the use of one-dimensional fluid dynamics models for prediction of flow, pressure, shear stress, and wave propagation in pulmonary vascular networks obtained from images. The model presented here includes predictions at multiple scales, including the large arteries and veins, arterioles and venules, and capillaries. The large arteries and veins are represented by a directed graph extracted from computed tomography images, whereas the network of arterioles and venules are represented by structured trees with parameters informed by data. The capillary network modeled using a sheet approximation, is coupled to the network of arterioles and veins in a ladderlike figuration. In the large vessels, we solve the 1D Navier Stokes equations, while in the network of small vessels and capillaries we solve linearized equations, which are coupled to large vessels via outflow boundary conditions. The model is calibrated to a healthy control, and we progressively increase disease severity via vessel stiffening and narrowing To differentiate healthy and diseased networks, we account for remodeling by altering vessel stiffness, vessel geometry (length and radius), and resistance provided by vessels in the microcirculation. This study addresses the importance of these features comparing hemodynamic predictions in healthy controls and control patients with disease.

Session D Chair
Session W
Title: On a class of linear parabolic equations with degenerate coefficients
Abstract: We investigate a class of linear parabolic equations in divergence form with degenerate diffusion coefficients. Our study is motivated on the questions about the finer regularity of solutions to a class of degenerate viscous Hamilton-Jacobi equations. Suitable weighted spaces are found in which the existence and regularity estimates of solutions are proved under some partially VMO condition on the leading coefficients. The talk is based on the joint work with Hongjie Dong (Brown University) and Hung Tran (University of Wisconsin – Madison).

Session N Chair
Session D
Title: Up-to-the-Neumann-Boundary Regularity for a Free Boundary Problem in Two Dimensions
Abstract: In this talk, we will discuss the regularity properties of a free boundary problem up to a Neumann fixed boundary in two dimensions. This will extend prior work of the authors by demonstrating that the free boundary itself must be smooth and intersect with the fixed boundary at a right angle.

Session B Chair
Session X
Title: Solutions of Generalized KdV type Equations with Various Limits
Abstract: In this talk, we will consider the generalized Korteweg-de Vries (KdV) equation \(u_t+au_x+2buu_x+cu_{xxx}-du_{xx}=0\) involving four parameters a, b, c, and d in a periodic domain. This equation has many physical applications including modeling the propagation of water waves. We will present some of the analytical and numerical results in various limiting cases of the parameters.

Session W Chair
Session N
Title: A uniqueness result for a p-Laplacian Dirichlet problem
Abstract: We present a uniqueness for positive solutions to a p-Laplacian Dirichlet problem when the forcing term in the equation is non-monotone and allowed to be singular at the origin.

Session T Chair
Session E
Title: Self-organized dynamics: aggregation and flocking
Abstract: Self-organized behaviors are commonly observed in nature and human societies, such as bird flocks, fish swarms and human crowds. In this talk, I will present some celebrated mathematical models, with simple small-scale interactions that lead to the emergence of global behaviors: aggregation and flocking. In particular, I will focus on the Euler-alignment system, and present some recent progress on the global wellposeness theory, large time behaviors, as well as interesting connections to classical equations in fluid mechanics.

Session F Chair
Session O
Title: On the binary-ternary Boltzmann equation
Abstract: The Boltzmann equation provides a statistical description of a dilute gas with predominantly binary particle interactions. When gas is dense enough, higher order interactions are more likely to happen. An equation that captures ternary particle interactions, as well as the binary ones, is called the binary-ternary Boltzmann equation. In this talk we present moment estimates and global well-posedness results for this equation. Moments estimates, in particular, show that the binary-ternary Boltzmann equation behaves better than the classical Boltzmann equation in the sense that it can generate exponential moments of higher order. The talk is based on a joint work with Ampatzoglou, Gamba and Pavlovic.

Session E Chair
Session P
Title:Global existence and blowup of solutions of stochastic Keller–Segel-type equation
Abstract: In this talk I will consider a stochastic Keller–Segel-type equation, perturbed with random noise. For random pertubations in divergence form, the equation has a global weak solution for small initial data. This result is consistent with the deterministic case. However, in stark contrast with the deterministic case, if the noise is not in a divergence form, the solution has a finite time blowup with nonzero probability for any nonzero initial data. This is a joint project with Alex Misiats (VCU).

Session R Chair
Session F
Title:Amplitude Blowup in Radial Euler Flows
Abstract: We show that the full compressible Euler system admits unbounded solutions. The examples are radial flows of similarity type and describe a spherically symmetric and continuous wave moving toward the origin. At time of focusing, the primary flow variables suffer amplitude blowup at the origin. The flow is continued beyond collapse and gives rise to an expanding shock wave. We verify that the resulting flow provides a genuine weak solution to the full, multi-dimensional compressible Euler system. While unbounded radial Euler flows have been known since the work of Guderley (1942), those are at the borderline of the regime covered by the Euler model: the upstream pressure field vanishes identically (either because of vanishing temperature or vanishing density there). In contrast, the solutions we build exhibit an everywhere strictly positive pressure field, demonstrating that the geometric effect of wave focusing is strong enough on its own to generate unbounded values of primary flow variables. This is joint work with Helge Kristian Jenssen (PSU).

Session L Chair
Session G
Title:A variational quasi-reversibility method for a time-reversed nonlinear parabolic problem
Abstract: This talk is about a modified quasi-reversibility method for computing the exponentially unstable solution of a terminal-boundary value parabolic problem with noisy data. As a PDE-based approach, this variant relies on adding a suitable perturbing operator to the original PDE and, consequently, on gaining the corresponding fine stabilized operator. The designated approximate problem is a forward-like one. This new approximation is analyzed in a variational framework, where the finite element method can be applied. For each noise level, the Faedo-Galerkin method is exploited to study the weak solvability of the approximate problem. Relying on the energy-like analysis coupled with a suitable Carleman weight, convergence rates in \(L^2\)–\(H^1\) of the proposed method are obtained when the true solution is sufficiently smooth.

Session J
Title: The diffusive Lotka-Volterra competition model in fragmented patches I: Coexistence
Abstract: We consider a lotka-volterra competition model where two species compete in a domain with the strengths of competitions b1 and b2. A parameter \lambda in the model represents the square of the patch size and two parameters \gamma1 and \gamma2 represent the matrix hostility. We analyze the positive solutions of the model as the parameters b1, b2 and \gamma1, \gamma2 vary.

Session W
Title: Maximal and minimal weak solutions for elliptic problems with nonlinearity on the boundary
Abstract: We consider the existence of weak solutions for semilinear elliptic equation with nonlinearity on the boundary. We establish the existence of a maximal and a minimal weak solution between an ordered pair of sub- and supersolution. To prove the result, we utilize the surjectivity of a pseudomonotone and coercive operator, Zorn’s lemma and a version of Kato’s inequality.

Session P
Title: Asymptotic Behaviors For The Compressible Euler System With Nonlinear Velocity Alignment
Abstract: We consider the pressureless compressible Euler system with a family of nonlinear velocity alignment. The system is a nonlinear extension of the Euler – alignment system in collective dynamics. I will show the asymptotic emergent phenomena of the system: alignment and flocking. Different types of the nonlinearity and the nonlocal communication protocols are investigated, resulting a variety of different asymptotic behaviors.

Session N
Title: Dual-Wind Discontinuous Galerkin Method for a Parabolic Obstacle Problem and an Optimal Control Problem
Abstract: In this talk, a dual-wind discontinuous Galerkin method (DWDG) and its application to a parabolic obstacle problem and an optimal control problem (OCP) is discussed. \\ A fully discrete scheme to solve the parabolic obstacle problem with a general obstacle function in $\mathbb{R}^2$ that uses a symmetric dual-wind discontinuous Galerkin discretization in space and a backward Euler discretization in time is proposed and analyzed. The convergence of numerical solutions in $L^\infty(L^2)$ and $L^2(H^1)$ like energy norms is established and the rates are computed. Next, the DWDG method is used to discretize an infinite dimensional linear elliptic PDE constrained optimal control problem with control constraints to a finite dimensional optimization problem. The finite dimensional optimization problem is then solved with a primal-dual active set strategy to find the numerical solution $(\overline{y_h},\overline{u_h})$ which is an optimal state and control pair. Several numerical tests are provided to demonstrate the robustness and effectiveness of the proposed methods to both problems. This is a joint work with Tom Lewis, Aaron Rapp and Yi Zhang.

Session F
Title: Superdiffusive Fractional In time NLS: Nonlinear Theory
Abstract: We consider a new class of partial differential equations dubbed Superdiffusive Fractional-In-Time Nonlinear Schrödinger equations on bounded domains. This equation will serve as a proper interpolation between the nonlinear Schrödinger equation and the Klein-Gordon Wave equation by making use of the (nonlocal in time) Caputo fractional time derivative of order $1<\beta< 2$. In space, we consider nonnegative $L^2(X)$ self-adjoint operators $A$ which include Laplacians and the (nonlocal in space) fractional Laplace operators with appropriate boundary conditions. In this talk, we present recent advances in the nonlinear theory, behavior of solutions, and potential applications as a control problem. Such models will serve appropriately in the important study of nonlinear optics through dispersive media and in the study of Bose-Einstein condensates.

Session R
Title: Global well-posedness and scattering in nonlinear wave equations
Abstract: In this talk we will consider the wave equation in odd space dimensions with energy-supercritical nonlinearity, in the radial setting. We will review the concentration compactness and rigidity arguments starting from the earlier work by Kenig-Merle and Duyckaerts-Kenig-Merle in the energy-critical and energy-supercritical cases, and outline the key ideas behind the proof of global well-posedness and scattering results in 3d and higher odd dimensions

Session G
Title: Application of Adomian Decomposition Method to certain Partial Differential Equations
Abstract: he Adomian decomposition method was introduced and developed by G. Adomian. A unique feature of this method is, that it deals directly 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 result in the reduction of any differential equation into a set of recursive relation for the Adomian solution series. We first present the methodology and show its application to a couple of PDEs that model fluid flow.

Session D
Title: Some uniqueness results for strongly singular problems
Abstract: We consider a semilinear problem with strongly singular nonlinearity and discuss Brezis-Oswald type uniqueness results for positive solutions. Specifically, we will present uniqueness results using comparison arguments, behavior of positive solution near the boundary and appropriate modification of test functions. This is a joint work with Francesca Faraci.

Session Q
Title: On the impact of super spreaders on COVID-19 dynamics
Abstract: Superspreading phenomenon has been observed in many infectious diseases and contributes significantly to public health burden in many countries. Superspreading events have recently been reported in the transmission of the COVID-19 pandemic. The present study uses a set of nine ordinary differential equations to investigate the impact of superspreading on COVID-19 dynamics. The model developed in this study addresses the heterogeineity in infectiousness by taking into account two forms of transmission rate functions for superspreaders based on clinical (infectivity level) and social or environmental (contact level). The basic reproduction number has been derived and the contribution of each infectious compartment towards the generation of new COVID-19 cases is ascertained. Data fitting was performed and parameter values were estimated within plausible ranges. Numerical simulations performed suggest that control measures that decrease the effective contact radius and increase the transmission rate exponent will be greatly beneficial in the control of COVID-19 in the presence of superspreading phenomena.

Session P
Title: Doubly Non-Local Cahn Hilliard Equation with a Fractional Time Derivative
Abstract: We consider a doubly nonlocal Cahn-Hilliard equation (dnCHE) which describes phase separation of a binary system but replace the classical time derivative with a Caputo fractional time derivative. In doing so, this modification can be used to model dynamic processes in which particles are thought to have some ‘memory’. We establish both the existence and uniqueness of a solution to this modified equation. Then, using a combination of a forward Euler scheme and a convolution quadrature rule we numerically approximate our mild solution. We establish convergence of mild solutions to that of the mild solution of the dnCHE when the order of the fractional derivative approaches 1.

Session F
Title: Application of Quasilinearization method to unsteady nanofluid flow problem
Abstract: The unsteady boundary layer flow of nanofluid in a free stream is investigated . A new self-similar solutions is obtained and the resulting system of nonlinear ordinary differential equations is solved numerically using an implicit finite difference scheme in combination with the quasilinearization method. Numerical results are presented for the skin friction coefficient, the local Nusselt number and the local Sherwood number as well as for the velocity, temperature and the nanoparticle volume fraction profiles.

Session A
Title: Metric entropy for scalar conservation laws
Abstract: Inspired by a question posed by Lax in 2002, the study of metric entropy for nonlinear partial differential equations has received increasing attention in recent years. This talk demonstrates methods to obtain sharp upper and lower bounds on the metric entropy for a class of bounded total generalized variation functions taking values in a general totally bounded metric space. Thereafter we use this result to establish metric entropy estimates for the set of entropy admissible weak solutions to a scalar conservation law with weakly genuinely nonlinear flux. Estimates of this type could provide a measure of the order of resolution of a numerical method required to solve the equation.

In Memoriam Session 2
Title: Stability conditions for linear neutral delay differential equations with distributed delays
Abstract: We discuss stability conditions, of both delay-independent and delay-dependent type, for systems of linear neutral delay equations with distributed delays. Some examples and numerical results will be given.

Session J
Title: Modeling effects of matrix heterogeneity on population persistence at the patch-level
Abstract: We study the structure of positive solutions to steady state reaction diffusion equation of the form: \begin{equation*} \left\lbrace \begin{matrix} -u” =\lambda u(1-u);~~ (0,1) \\ -u'(0) + \sqrt{\lambda} \gamma_1 u(0) = 0 \\ ~~ u'(1) + \sqrt{\lambda} \gamma_2 u(1) = 0 \end{matrix} \right. \hspace{.05in} , \end{equation*} where $\lambda>0$ is a parameter that encompasses patch size and $\gamma_1$, $\gamma_2$ are positive parameters related to the hostility at the boundaries $0$ and $1$, respectively. Note here that the parameter $\lambda$ influences both the equation and the boundary conditions. In this paper, we establish the existence, nonexistence, and uniqueness results for this model. In particular, we establish exact bifurcation diagrams for this model, first when $\gamma_2$ is fixed and $\gamma_1$ is evolving, and then when the Dirichlet boundary condition is satisfied at $x = 0$ ($u(0)=0$) and $\gamma_2$ is evolving. In each case, our results are established by combining a quadrature method and the method of sub-super solutions. Finally, we present some numerical results that we obtained for this model. Here, we numerically simulate how the parameters $\gamma_1$ and $\gamma_2$ affect the minimum size for $\lambda$ beyond which a positive solution exists. This model arises in the study of steady states for a population satisfying a logistic growth reaction and diffusing in a region surrounded by two exterior hostile matrices, and $\lambda$ is related to the minimum patch size for the existence of a positive steady state.

Session B
Title: Localized method of particular solutions using polynomial basis functions for solving two-dimensional nonlinear partial differential equations.
Abstract: The localized method is one of the popular approaches in solving large-scale problems in science and engineering. In this paper, we implement the localized method of particular solutions using polynomial basis functions for solving various nonlinear problems. To validate our proposed numerical method, we present four numerical examples in regular and irregular domains which are solved by using localized method of particular solution with polynomial basis functions. We compared our numerical method with localized method of particular solutions using multiquadric radial basis function and numerical results clearly show that our numerical method is highly accurate, efficient, and outperformed the method using multiquadric radial basis function.

Session A
Title: Sharp critical thresholds for a class of nonlocal traffic flow models
Abstract: I will introduce a class of macroscopic traffic flow models with nonlocal look-ahead inter- actions. The global regularity of solutions depend on the initial data. I will then present sharp critical threshold conditions that distinguish the initial data into a trichotomy: Subcritical initial conditions lead to global smooth solutions, while two types of supercritical initial conditions lead to two kinds of finite time shock formations. The existence of non-trivial subcritical initial data indicates that the nonlocal look-ahead interactions can help avoid shock formations, and hence prevent the creation of traffic jams. This is joint work with my advisor, Dr. Changhui Tan Key words: Macroscopic traffic flow, partial differential equations, nonlocal conservation law, continuity equations, phase plane analysis, fluid dynamics, critical threshold

Session C
Title: Modeling the dynamics of Usutu virus infection in birds
Abstract: Nora Heitzman-Breen, Jacob Golden, Sarah C. Kuchinsky, Francesca Frere, Christa F. Honaker, Paul B. Siegel, Tanya LeRoith, Nisha K. Duggal, Stanca Ciupe Usutu virus is a mosquito-borne flavivirus maintained in wild bird populations, causing high avian mortality rates and occasional severe neurological disorders in humans. It has been hypothesized that increased Usutu virus replication in birds and/or decreased bird immune competence leads to increased mosquito infection and increased spillover in humans. To provide insight into the intrinsic complexity of host-virus processes in birds, we use within–host mathematical models to characterize the mechanisms responsible for virus expansion and clearance in juvenile chickens challenged with four Usutu virus strains. Several virus strains are co-circulating in the wild, and we find heterogeneity between the virus strains, with the time between cell infection and viral production varying between 16 h and 23 h, the infected cell lifespan varying between 48 min and 9.5 h, and the basic reproductive number varying between 12.05 and 19.49. The strains with high basic reproductive number have short infected cell lifespan, indicative of immune responses. The virus strains with low basic reproductive number have lower viral peaks and longer lasting viremia, due to lower infection rates and high infected cell lifespan. These results can be used to better determine which virus strain is the most likely to spillover in the human population. We also investigate the effect of antibody on virus dynamics by fitting the models to chickens that were genetically engineered to have low and high antibody count; and show that the viral clearance rate is a stronger mitigating factor for USUV viremia than neutralizing antibody response in this avian model.

Session O
Title: Σ-shaped bifurcation curves for classes of reaction diffusion equations with non-linear boundary conditions
Abstract: \noindent We analyse positive solutions of the steady state reaction diffusion equation of the form: \begin{equation*} \left\lbrace \begin{matrix}-\Delta u=\lambda f(u) ;~\Omega \\ \frac{\partial u}{\partial \eta}+\sqrt{\lambda}g(u) u=0; ~ \partial\Omega \end{matrix} \right. \end{equation*} \noindent where $\lambda>0$ is a positive parameter, $\Omega$ is a bounded domain in $\mathbb{R}^N$; $N > 1$ with smooth boundary $\partial \Omega$ or $\Omega=(0,1)$, $\frac{\partial u}{\partial \eta}$ is the outward normal derivative of $u$, $f \in C^1([0,\infty),[0, \infty)) $ is an increasing function such that $f(0)=0$, $f'(0) > 0,$ and $ \lim\limits_{s \rightarrow \infty}\frac{f(s)}{s}=0$ (sublinear at infinity). Further, we assume that $g \in C([0,\infty), (0, \infty))$ is a non-increasing function such that $\lim\limits_{s \rightarrow \infty} g(s) = g_\infty > 0$. We discuss the existence of multiple positive solutions for certain ranges of $\lambda$ leading to the occurrence of $\Sigma$-shaped bifurcation diagrams. We establish our results via the method of sub-supersolutions.

Session E
Title: Positive weak solutions of nonlocal parabolic problems with logistic reaction term
Abstract: We study a parabolic reaction-diffusion equation with logistic reaction term and the fractional Laplacian as the diffusion operator. We discuss existence of a positive weak solution by constructing appropriate ordered sub- and supersolutions.

Session U
Title: A GBC Approache for Smooth Bijections between two domains.
Abstract: We present a construction of the bijection of the harmonic GBC map transforming from one arbitrary polygonal domain V to another arbitrary polygonal domain W. In addition, we shall point out that the harmonic GBC map is also a diffeomorphism over the interior of V to the interior of W. Finally, we remark on how to construct a harmonic GBC map from V to W when the number of vertices of V is different from the number of vertices of W and how to construct harmonic GBC functions over a polygonal domain with a hole or holes. We also point out that it is possible to use the harmonic GBC map to deform a nonconvex polygon V to another nonconvex polygon W by a good arrangement of the boundary map between ∂V and ∂W. Several numerical deformations based on images are presented to show the effectiveness of the map based on bivariate spline approximation of the harmonic GBC functions.

Session S
Title: Nonlinear Schrödinger Equation with Repulsive Potentials and Rotations
Abstract: We will introduce the nonlinear Schrödinger equation (NLS) with repulsive harmonic potentials and rotations, and we will give a transform that connects this to the classical NLS in the mass-critical case. Then we will give some results on global well-posedness and blowup rates.

Session R
Title: Strong Ill-Posedness in L∞ for the Riesz Transform Problem
Abstract: We prove strong ill-posedness in L∞ for linear perturbations of the 2d Euler equations of the form: ∂tω+u⋅∇ω=R(ω), where R is any non-trivial second order Riesz transform. Namely, we prove that there exist smooth solutions that are initially small in L∞ but become arbitrarily large in short time. Previous works in this direction relied on the strong ill-posedness of the linear problem, viewing the transport term perturbatively, which only led to mild growth. In this work we derive a nonlinear model taking all of the leading order effects into account to determine the precise pointwise growth of solutions for short time. Interestingly, the Euler transport term does counteract the linear growth so that the full nonlinear equation grows an order of magnitude less than the linear one.

Session P
Title: On the Riccati dynamics of 2D Euler-Poisson equations with attractive forcing
Abstract: The multi-dimensional Euler-Poisson(EP) system describes the dynamic behavior of many important physical flows. In this talk, a Riccati system that governs pressureless two-dimensional EP equations is discussed. The evolution of divergence is governed by the Riccati type equation with several nonlinear/nonlocal terms. Among these, the vorticity accelerates divergence while others further amplify the blow-up behavior of a flow. The growth of these blow-up amplifying terms are related to the Riesz transform of density, which lacks a uniform bound makes it difficult to study global solutions of the multi-dimensional EP system. We show that the Riccati system can afford to have global solutions, as long as the growth rate of blow-up amplifying terms is not higher than exponential, and admits global smooth solutions for a large set of initial configurations. Several recent works in a similar vein will be reviewed.

Session X
Title: A Multivariate Spline based Collocation Method for Numerical Solution of Partial Differential Equations
Abstract: We propose a collocation method based on multivariate polynomial splines over triangulation or tetrahedralization for numerical solution of partial differential equations. We start with a detailed explanation of the method for the Poisson equation and then extend the study to the second order elliptic PDE in non-divergence form. We shall show that the numerical solution can approximate the exact PDE solution very well. Then we present a large amount of numerical experimental results to demonstrate the performance of the method over the 2D and 3D settings.

Session X
Title: A parameter-free enriched Galerkin method for the Stokes equations
Abstract: We propose a parameter-free enriched Galerkin (EG) method for the Stokes equations using modified weak Galerkin (mWG) bilinear forms. The discrete inf-sup stability condition is a fundamental property of numerical methods for the Stokes equations, and it has been successfully achieved by recent work on a new EG method whose numerical velocity is a discontinuous piecewise linear function. In the EG method, interior penalty discontinuous Galerkin (IPDG) approaches have been adopted to treat the discontinuity of numerical velocity, and the approaches require a sufficiently large penalty parameter. In this work, for the same numerical velocity, we use mWG bilinear forms compatible with discontinuous piecewise polynomials without the penalty parameter. Therefore, our EG method is a parameter-free EG method for the Stokes equations. We prove that our parameter-free EG method guarantees the optimal convergence rates, and pressure-robustness is also achieved by employing a velocity reconstruction operator on the load vector of the right-hand side. Furthermore, the theoretical results are verified by various numerical examples.

Session T
Title: Training neural networks to learn the dynamics of partial differential equations
Abstract: Neural networks have recently become a very prominent tool for the study of partial differential equations (PDEs). This is partially due to the success of physics-informed neural networks (PINN), where a neural network is trained to model the solution of a PDE by satisfying all the physical constraints put on the equation. However, what do we do if we are uncertain of the equations of motion? In this talk, we present a data-driven method to learn the dynamics of PDEs with little knowledge of the underlying physics. Given solution data from the PDE, we train a neural network to learn how the PDE evolves from one time step to the next. Since our method learns from local data instead of global data, it can simulate the solution from a large set of initial conditions. We show how our method can outperform classical time-stepping methods, such as the finite volume method, in both accuracy and run time on problems in fluid dynamics.

Session L
Title: Narrow-Stencil Approximation Methods for Fully Nonlinear Elliptic Boundary Value Problems
Abstract: This talk will introduce a new convergent narrow-stencil finite difference method for approximating viscosity solutions of second order fully nonlinear elliptic partial differential equations including Hamilton-Jacobi-Bellman equations. The proposed finite difference method naturally extends the Lax-Friedrichs method for first order problems to second order problems by introducing a key stabilization term called a numerical moment. By abandoning the standard monotonicity assumption, the new methods do not require the use of wide-stencils. The new narrow-stencil methods are easy to formulate and implement, and they formally have higher-order truncation errors than monotone methods when first-order terms are present in the PDE. We will also discuss a new discontinuous Galerkin method that formally extends the narrow-stencil finite difference methods to achieve higher order accuracy.

Session K
Title: Tracing and Forecasting Metabolic Indices of Cancer Patients Using Patient-Specific Deep Learning Models
Abstract: With the advancements in medical research ad practice, more and more cancer patients can live a long time after their cancer treatments, making the quality of life and toxicity management during and post-treatments the primary focus of healthcare providers and cancer patients. How to detect early health anomalies and predict potential adverse effects from some measurable biomarkers or signals for an individual cancer patient is of great importance in cancer patient care. Developing the ability to track a patient’s health status and to monitor the evolution of the specific disease intelligently would provide an enormous benefit to both the patient and the healthcare provider, enabling faster responses to deal with adverse effects and more precise and effective treatments and interventions. With the longitudinally collected time-series data of the patient at multiple time points before, during and after cancer treatments, it is becoming increasingly plausible to have an intelligent tools or device for continuous monitoring, tracking, and forecasting of cancer patients’ health status based on statistical, causal, and mechanistic modeling of patient phenotypes and various biomarkers in the time series data. We develop a patient-specific dynamical system model from the time series data of the cancer patient’s metabolic panel taken during the period of cancer treatment and recovery. The model consists of a pair of stacked long short-term memory (LSTM) recurrent neural networks and a fully connected neural network in each unit. It is intended to be used by physicians to trace back and look forward at the patient’s metabolic indices, to identify potential adverse events, and to make short-term predictions. Once a master model is built, the patient-specific models can be calibrated through transfer learning.

Session U
Title: Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies
Abstract: We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator \begin{equation} \label{eq_int_1} (\Delta^2-k^4 +q)u =0\quad \text{in}\quad \Omega \end{equation} from the knowledge of the partial Cauchy data set. Here $\Omega\subset \{x = (x_1, x_2, \dots, x_n) \in \mathbb{R}^n: x_n<0\}$, $n\ge 3$, be a bounded open set with $C^\infty$ boundary. Assume that $\Gamma_0:=\partial \Omega\cap \{x_n=0\}$ is non-empty, and let us set $\Gamma=\partial \Omega\setminus \Gamma_0$. That is, our geometric setting is that of a domain whose inaccessible portion of the boundary is contained in a hyperplane, and we are given the Cauchy data set on the complement. The uniqueness and logarithmic stability for this problem were established earlier, respectively. We establish H\”older type stability estimates in the high frequency regime, with an explicit dependence on the frequency parameter, under mild regularity assumptions on the potentials, sharpening those of earlier results. In particular, no continuity of the potentials in the main result is assumed, and the required Sobolev regularity assumptions are fairly mild and are independent of the dimension.

Session Q
Title: Optimal multiclass classification and prevalence estimation with applications to SARS-CoV-2 antibody assays
Abstract: Antibody tests are routinely used to identify past infection, with examples including Lyme disease and, of course, COVID-19. An accurate classification strategy is crucial to interpreting diagnostic test results and includes problems with more than two classes. Classification is further complicated when the relative fraction of the population in each class, or generalized prevalence, is unknown. We present a prevalence estimation method that is independent of classification and an associated classification scheme that minimizes false classifications. This work hinges on constructing probability models for data that are inputs to an optimal-decision theory framework. As an illustration, the method is applied to antibody data with SARS-CoV-2 naïve, previously infected, and vaccinated classes.

Session A
Title: A quantitative analysis on total number of shocks for scalar conservation laws
Abstract: In this talk, I will present a quantitative version of the transversality theorem. More precisely, given a continuous function \(g \in C([0,1]^d,\mathbb{R}^m)\) and a smooth manifold \(W \subset \mathbb{R}^m\) of dimension p, one establishes a quantitative estimate on the (d + p − m)-dimensional Hausdorff measure of the set \(Z^g_W ={x \in [0,1]^d : g(x) \in W}\) over \(\varepsilon\)-perturbations in \(C^0\). The obtained result is applied to bound the total number of shock curves in weak entropy solutions to scalar conservation laws with uniformly convex fluxes in one space dimension.

Session M
Title: Mathematical Model of Immune-inflammatory Response in COVID-19 Patients
Abstract: Investigating the immune-inflammatory response characteristics in COVID-19 patients can predict disease severity and progression. There is evidence that severity of SARS-CoV-2 infection is linked to the dysregulation of the inflammatory immune response. Previous studies have shown the neutrophil-to-CD$8^+$T cell ratio (N8R) can be used in predicting disease severity in COVID-19 patients. Specifically, N8R increases as COVID-19 disease severity worsens. We developed a mathematical model (system of ordinary differential equations) to describe the inflammatory response to SARS-CoV-2 infection. Specifically, the model incorporates cellular and cytokine populations of neutrophils, macrophages, pro-inflammatory mediators, anti-inflammatory mediators, and CD$8^+$T cells. The model is fit to longitudinal data from COVID-19 patients confirmed to have either mild or severe disease. We identify and classify steady states, and conduct bifurcation analysis to determine parameters required for (1) the resolution of inflammation, (2) the persistence of inflammation (chronic inflammation), and (3) alterations in the neutrophil-to-CD$8^+$T cell ratio (used as a predictor of disease severity). This mathematical model can be used to identify markers for severe COVID-19 and give insight into effective treatment protocols and interventions directed at reducing inflammation and the neutrophil-to-CD$8^+$T cell ratio.

Session W
Title: A Constructive Definition of the Fourier Transform over a Separable Banach Space
Abstract: Gill and Myers proved that every separable Banach space, denoted $\mathcal{B}$, has an isomorphic, isometric embedding in $\mathbb{R}^{\infty}=\mathbb{R}\times\mathbb{R}\times\cdots$\ . They used this result and a method due to Yamasaki 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. \\ \hspace{2in}\\ The objective of this talk is to apply this theory to developing a constructive definition of the Fourier transform on $L^1[\mathcal{B}]$. Our approach is constructive in the sense that this Fourier transform is defined as an integral on $\mathcal{B}$, 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 Fourier transform on $\mathcal{B}$, by means of finite-dimensional approximation. As an application, we will apply the familiar properties of the transform to solving the heat equation on $\mathcal{B}$.

Session Q
Title: Towards understanding the effect of fibrinogen interactions on fibrin gel structure
Abstract: Fibrin polymerization involves the conversion of fibrinogen molecules to fibrin monomers which polymerize to form a gel that is a major structural component of a blood clot. Fibrinogen plays a dual role in fibrin polymerization; it can occupy binding sites by binding to fibrin, inhibiting gelation, and fibrinogen can be converted to fibrin. A kinetic polymerization model is proposed, involving two types of monomers, each of which has two reaction sites that participate in different binding reactions. With a moment generating function approach, we track the temporal dynamics of a closed system of moment equations up until finite time blow-up. We examine the impact of fibrinogen-fibrin binding and fibrinogen conversion to fibrin on whether a gel forms and the resulting branch point density, if it does.

Session C
Title: Identifiability and optimal control analysis of HIV infection and opioid addiction model.
Abstract: Based on the growing association between opioid addiction and HIV infection, a compartmental model is developed to study dynamics and optimal control of two epidemics: opioid addiction and HIV infection. We show that the disease-free-equilibrium is locally asymptotically stable when the basic reproduction number R_0= max 〖(R〗_(0 , )^u R_(0 )^v) 1 and it is locally asymptotically stable when the invasion number of the opioid addiction is R_(inv )^u 1 and it is locally asymptotically stable when the invasion number of the HIV infection is R_(inv )^v< 1. We study structural identifiability of the parameters, estimate parameters employing yearly reported data from Central for Disease Control and Prevention (CDC), and study practical identifiability of estimated parameters. We observe the basic reproduction number ¬¬¬¬R_0 using the parameters. Next, we introduce four distinct controls in the model for the sake of control approach, including treatment for addictions, health care education about not sharing syringes, highly active anti-retroviral therapy (HAART), and rehab treatment for opiate addicts who are HIV infected. US population using CDC data, first applying a single control in the model, and observing the results, we better understand the influence of individual control. After completing each of the four applications, we apply them together at the same time in the model and compare the outcomes using different control bounds and state variable weights. We conclude the results by presenting several graphs.

Session E
Title: Unique Maximum of Expected Score in Adaptive Algorithm- A New Approach for Secretary Problem
Abstract: Secretary problem appeared in the late 1950s and early 1960s. Starting from the idea of choosing the best secretary from a given number of candidates, the variants of this famous math-based decision-making have related to certain practical problems such as investment procedures and atomic bomb inspection programs. The Adaptive Algorithm suggested in Zhou et al. (2021) has drawn the main objective of Secretary Problem to maximize a numerical score for each candidate evaluation. The main goal of this project is to establish the existence and optimality of Adaptive Algorithm strategy by proving the existence of a unique maximum of expected score in this algorithm as well as limit of sequence of unique maximizer {xn} of each expected score function when n approaches infinity.

Session O
Title: An exact bifurcation diagram for a p-q Laplacian boundary value problem
Abstract: We study positive solutions to the $p$-$q$ Laplacian two-point boundary value problem: \begin{align*} \begin{cases} -\mu[(u’)^{p-1}]’ – [(u’)^{q-1}]’ = \lambda u(1-u) ; \hspace{.10in} (0,1) \\ u(0) = 0 = u(1) \end{cases} \end{align*} when $p = 4$ and $q=2$. Here $\lambda>0$ is a parameter and $\mu \geq 0$ is a weight parameter influencing the higher-order diffusion term. When $\mu = 0$ (the Laplacian case) the exact bifurcation diagram for a positive solution is well-known, namely, when $\lambda \leq \pi^2$ there are no positive solutions, and for $\lambda > \pi^2$ there exists a unique positive solution $u_{\lambda,\mu}$ such that $||u_{\lambda,\mu}||_{\infty} \rightarrow 0$ as $\lambda \rightarrow \pi^2$ and $||u_{\lambda,\mu}||_{\infty} \rightarrow 1$ as $\lambda \rightarrow \infty$. Here, we will prove that for all $\mu > 0$ similar bifurcation diagrams preserve, and they all bifurcate from $(\lambda,u) = (\pi^2,0)$. Our results are established via the method of sub-super solutions and a quadrature method. We also present computational evaluations of these bifurcation diagrams for various values of $\mu$ and illustrate how they evolve when $\mu$ varies.

Session G
Title: Simulations of Photonic Crystal Ring Resonators using Domain Decomposition and Difference Potentials
Abstract: A photonic crystal ring resonator (PCRR) is a micro-scale optical device that combines a closed-loop waveguide with a light input and output. PCRRs are constructed with periodically placed scattering rods where the exclusion of rods is used to form the path of a waveguide. We simulate PCRRs numerically using a non-iterative domain decomposition approach that is insensitive to jumps in material properties, in particular, those between the scattering rods and surrounding medium. To approximate the governing Helmholtz equation, we use a compact fourth order accurate finite difference scheme combined with the method of difference potentials (MDP). The MDP renders exact coupling between the decomposition subdomains and maintains high order accuracy for non-conforming boundaries/interfaces on regular grids.

Session J
Title: On the controllability of some systems modeling cell dynamics related to leukemia
Abstract: The subject I choose to approach in this talk refers to my latest research topic: two control problems for a model of cell dynamics related to leukemia. The first control problem is in connection with classical chemotherapy, which indicates that the evolution of the disease under the treatment should follow a prescribed trajectory assuming that the drug works by increasing the cell death rates of both malignant and normal cells. In the second control problem, as for targeted therapies, the drug is assumed to work by decreasing the multiplication rate of malignant (leukemic) cells only. The control objective is that the disease state reaches the desired endpoint. The solvability of these two problems is proved by using a general method of analysis, and the solution of these problems is obtained within an abstract scheme, by using operator-based techniques. Numerical simulations are included to illustrate the theoretical results and prove their applicability.

Session I
Title: The Coupling Conduction Effects on Natural Convection Flow along a Vertical Flat Plate with Joule Heating and Heat Generation
Abstract: The out-turn of heat generation and Joule heating on natural convection flow through a vertical flat plate have been investigated in the given article. Joule heating and heat conduction due to wall thickness ‘b’ are esteemed as well in this analysis. With the intent to obtain similarity solutions to the problem being constituted, the evolved equations are made dimensionless employing appropriate transformations. The non-dimensional equations are then modified into non-linear equations by bringing into being a non-similarity transformation. The out-turn non-linear alike equations confine with their commensurate boundary conditions formed on conduction and convection are solved numerically applying the finite difference method accompanied by Newton’s linearization approximation. The numerical outcomes in terms of the skin friction coefficient, velocity, temperature, and surface temperature profiles are shown both graphically and in tabular forms for the different values of the parameters correlated with the problem.

Session E
Title: A variational approach to multi-dimensional scaling
Abstract: Dimension reduction is a crucial pre-processing step in many learning algorithms, and can be seen as a particular form of metric or graph embedding problem. Despite the widespread use of these algorithms, many of the non-linear versions of these algorithms lack rigorous theory. This talk will discuss recent work on multi-dimensional scaling, one of the most basic forms of non-linear dimension reduction. In particular, we pose this as a variant of an optimal transportation problem with a free marginal in the target space; in particular this can be seen as a Gromov-Wasserstein projection problem. We utilize a variational approach to derive regularity properties of the optimal embeddings selected by a simple version of multi-dimensional scaling, and provide new asymptotic consistency results for the algorithm.

Session X
Title: A Symmetric Banded Matrix Isospectral Flow on Real Space
Abstract: In this talk, we discuss an isospectral flow of the form $\displaystyl \dot{P} = [[P’, P]_{du}, P]$ in the space of real matrices, where, $\dot{P}$ is the time derivative of matrix $P$, $P’$ is the transpose of matrix $P$, and $[P’, P]_{du}$ is the same as matrix $[P’, P]$ on the upper triangular elements and all elements below the diagonal are zero. We prove that if the initial condition $P_0$ is a banded matrix having lower bandwidth $p$ with a simple and real spectrum, then $P(t)$ converges as $t \to \infty$ to a banded symmetric matrix having bandwidth $p$, isospectral to $P_0$. Also, the limit point has the same sign pattern in the $p^{th}$ subdiagonal elements as in $P_0$. We provide simulation results to highlight the transient behavior of $p^{th}$ subdiagonal elements under the flow.

Session B
Title: A New Unstaggered Locally Divergence-Free Finite Volume Scheme for Ideal and Shallow Water Magnetohydrodynamics
Abstract: Many mathematical models of fundamental interest are constrained by the infamous divergence-free condition of magnetic fields or velocity profiles. Seen within a wide range of astrophysical, geophysical, and engineering applications, such divergence-free constraints are physically and analytically exact. On a discrete level, however, an improper treatment of an identically-zero divergence may lead to large instabilities in solutions – even when applying methods that already successfully simulate fluids without this constraint. Thus, a careful algorithmic construction is required to ensure these constraints are exactly preserved within numerical approximations. In this talk, we restrict our attention to an important sub-class of divergence-free systems – the magnetohydrodynamic equations, keeping in mind that the developed method can be extended to other models constrained by zero-divergence. We present a new method that exactly preserves the divergence-free condition of the magnetohydrodynamic system on an unstaggered mesh. The designed method has been successfully tested on several examples.

Session N
Title: A Fredholm Alternative for elliptic equations with interior and boundary nonlinear reactions
Abstract: This talk treats two-parameter problems for a triple (a, b,m) of continuous, symmetric bilinear forms on a real separable Hilbert space V that are used for large classes of elliptic PDEs with nontrivial boundary conditions. First, a Fredholm alternative for the associated linear two-parameter eigenvalue problem is developed, and then this is used to construct a nonlinear version of the Fredholm alternative. The Steklov-Robin Fredholm equation is used to exemplify the abstract results.

Session S
Title: Global attractors for a wave equation subject to nonlinear boundary dissipation and nonlinear interior/boundary sources with critical exponents
Abstract: In this talk we shall consider a wave model in 3D on a bounded domain which contains nonlinear sources with critical exponent in the interior/boundary and nonlinear feedback dissipation on the bound- ary. Similar models with simpler nonlinear boundary terms have been already studied broadly whereas the generality of our model is not only the presence of nonlinear interior and boundary damping but also nonlinear boundary source. Boundary actuators are easily accessible to external manipulations- hence feasible from the engineering point of view and practically implementable. On the other hand, the underlying mathematics is challenging. Boundary actions are represented by unbounded, unclosable operators, hence not treatable by perturbation theory(even from the point of view of well-posedness the- ory.) Our main result shows that a suitably calibrated boundary damping prevents the blow up of the waves, and allows to contain wave asymptotically (in time) in a suitable attracting set which is compact.

Session M
Title: Cholera Transmission Dynamic Model with Environmental Impacts of Plankton Reservoirs
Abstract: Cholera is an acute disease that is a global threat to the world and can kill people within a few hours if left untreated. In the last 200 years, seven pandemics occurred, and, in some countries, it remains endemic. The World Health Organization (WHO) declared a global initiative to prevent cholera by 2030. Cholera dynamics are contributed by several environmental factors such as salinity level of water, water temperature, presence of plankton especially zooplankton such as cladocerans, rotifers, copepods, etc. Vibrio cholerae (V. cholerae) bacterium is the main reason behind the cholera disease and the growth of V. cholerae depends on its host in the water reservoir which is the zooplankton because they share a symbiotic relationship. Investigating plankton bloom could be one of the key indicators for predicting cholera outbreaks. Though there are lots of models for cholera transmission dynamics, there are few existing models focused on the environmental impacts of plankton reservoirs. In this work, we have formulated a model of cholera transmission dynamics with the environmental impacts of plankton reservoirs. We have derived the basic reproduction number and discussed various alternative threshold parameters using the next generation matrix approach. Next, we have considered the existence and stability of the disease-free and positive equilibria. Our model analysis could be helpful for scientists to better understand the impact of environmental factors on cholera outbreaks and eventually for a possible prediction of the timing and location of the next cholera outbreak.

Session K
Title: Evaluating impacts of physiological variability on human equivalent doses using PBPK models
Abstract: Evaluating impacts of physiological variability on human equivalent doses using PBPK models CM Schacht1, AE Meade2,3, AS Bernstein3,4, B Prasad4, PM Schlosser4, HT Tran1, DF Kapraun4 1 Center for Research in Scientific Computation, North Carolina State University, Raleigh, N.C. 2 Center for Computational Toxicology and Exposure, Office of Research and Development, U.S. Environmental Protection Agency, Research Triangle Park, N.C. 3 Oak Ridge Institute for Science and Education, Oak Ridge, T.N. 4 Center for Public Health and Environmental Assessment, Office of Research and Development, U.S. Environmental Protection Agency, Research Triangle Park, N.C. Physiologically based pharmacokinetic (PBPK) models, which are typically expressed as systems of ordinary differential equations, are regularly used to inform human health risk assessments of chemicals. By performing simulations with a PBPK model, one can estimate human exposure levels that result in internal doses equal to those predicted for laboratory animals exposed to substances according to specific experimental dosing regimens. For chemical risk assessment, doses associated with adverse health outcomes are typically considered. Using scalar parameter values representing an “average” adult human, one can apply a PBPK model to estimate a scalar “human equivalent dose” (HED), which refers to the human concentration (for inhalation exposure) or dose (for oral exposure) of a substance that is expected to induce the same magnitude of toxic effect for a human as that observed for laboratory animals exposed to a known concentration or dose. However, such scalar values do not address variability among humans or uncertainty in parameter values. The World Health Organization International Programme on Chemical Safety (IPCS) has proposed a chemical hazard characterization approach, APROBA, that seeks to incorporate these and other elements of uncertainty to generate probabilistic reference values for chemicals. A key assumption in the APROBA approach is that various underlying distributions, including distributions of HEDs, are lognormal. We sought to evaluate this assumption by performing simulations using published PBPK models for dichloromethane and chloroform. We investigated how the shapes of HED distributions were impacted when we made different assumptions about the distributions of PBPK model parameters. To account for pharmacokinetic (PK) variability in humans, we used Monte Carlo methods to randomly draw sets of values for the PBPK model parameters based on distributions that describe uncertainty and human variability. We then used reverse dosimetry to obtain samples of HEDs. Using the Royston normality test, we found that while some HED distributions were lognormal, this depended on the distributions chosen to represent parameter variability as well as the applied doses. For higher doses (which generally coincide with higher internal dose metrics), HED distributions were less likely to be lognormal. Also, while lognormal parameter distributions produced mainly lognormal HED distributions, uniform parameter distributions produced dramatically less lognormal results. In the future, our conclusions about HED distributions and the impact of parameter distributions may be generalized by investigating other PBPK models to better characterize uncertainty in reverse dosimetry calculations.

Session H
Title: Infinitely many entire solutions to the curl-curl problem with critical exponent
Abstract: We prove the existence of an unbounded sequence of solutions to the curl-curl problem with critical exponent via a variational approach and suitable group actions to recover compactness.

Session T
Title: A Primal-Dual Method For Topological Changes in Adversarial Classification
Abstract: While the robustness of machine learning algorithms to data perturbing adversaries has been an important topic in the literature, our understanding of how to achieve robustness remains limited. We consider an adversary with the power to perturb the input data within an $\varepsilon$ neighborhood. In this work, we describe an algorithm to construct an optimal classifier for every adversarial power $\varepsilon >0$. Prior work has shown that a set of uncoupled ODEs govern the evolution of the optimal adversarial classifier in one dimension for small enough $\varepsilon$. We find that the optimal classifier is governed by the same ODEs except for a finite number of instantaneous changes in topology or discontinuous movements in endpoints of classification intervals. We rely on a novel primal dual method to prove the optimality of our algorithm.

Session L
Title: A second-order symplectic method for an advection-diffusion-reaction problem in Bioseparation
Abstract: An advection-diffusion-reaction problem with non-homogeneous boundary conditions is considered that modes the chromatography process, a vital stage in bioseparation. We prove stability and error estimates for both constant and affine adsorption, using the midpoint method for time discretization and finite elements for spatial discretization. In addition, we did the stability analysis for nonlinear, explicit adsorption in the continuous case. The numerical tests are performed that validate our theoretical results.

Session D
Title: Optimal mass described by a Sturm-Liouville problem with eigenparameter in the boundary condition.
Abstract: We find an optimal mass of a structure described by a Sturm-Liouville (S-L) problem with a spectral parameter in the boundary conditions. While previous work on the subject focused on a somewhat simplified model, we consider a more general S-L problem. We use the calculus of variations approach to determine a set of critical points of the corresponding functional – yet these “predesigns” themselves do not represent meaningful solutions. We additionally introduce a set of solvability conditions on the data of the S-L problem which confirm that these critical points do represent meaningful solutions we refer to as designs.

Session N
Title: Optimal Control in Fluid Flows through Deformable Porous Media
Abstract: We consider an optimal control problem subject to an elliptic-parabolic coupled system of partial differential equations that describes fluid flow through biological tissues. Our goal is to optimize the fluid pressure and solid displacement using distributed or boundary control. We first show existence and uniqueness of an optimal control and then present necessary optimality conditions. The optimal controls can be approximated numerically.

Session W
Title: Solitons and collapses in the models based on focusing nonlinear Schrodinger Equation.
Abstract: Nonlinear Schrodinger Equations with focusing nonlinearity plays an important role in nonlinear optics. It describes the self-focusing of light due to the Kerr effect. In this talk, I will discuss dynamics and stability properties of soliton solutions of models based on coupled NLS equations. These solutions were obtained numerically for various frequency ratios in both resonant and non-resonant regimes. These findings could lead to a better understanding of critical power for self-similar multi-color collapse and could lead to novel types of laser filamentation.

Session R
Title: An Isoperimetric Sloshing Problem in a Shallow Container with Surface Tension
Abstract: In 1965, B. A. Troesch solved the isoperimetric sloshing problem of determining the container shape that maximizes the fundamental sloshing frequency among two classes of shallow containers: symmetric canals with a given free surface width and cross-sectional area, and radially symmetric containers with a given rim radius and volume. We extend these results in two ways: (i) we consider surface tension effects on the fluid free surface, assuming a flat equilibrium free surface together with a pinned contact line, and (ii) we consider sinusoidal waves traveling along the canal with wavenumber $\alpha\ge 0$. Generalizing our recent variational characterization of fluid sloshing with surface tension to the case of a pinned contact line, we derive the pinned-edge linear shallow sloshing problem, which is an eigenvalue problem for a generalized Sturm-Liouville system. In the case without surface tension, we show that the optimal shallow canal is a rectangular canal for any $\alpha > 0$. In the presence of surface tension, we solve for the maximizing cross-section explicitly for shallow canals with any given $\alpha\ge 0$ and shallow radially symmetric containers with m azimuthal nodal lines, $m=0,1$. Our results reveal that the squared maximal sloshing frequency increases considerably as surface tension increases. Interestingly, both the optimal shallow canal for $\alpha = 0$ and the optimal shallow radially symmetric container are not convex. As a consequence of our explicit solutions, we establish convergence of the maximizing cross-sections, as surface tension vanishes, to the maximizing cross-sections without surface tension. This is joint work with Christel Hohenegger and Braxton Osting.

Session S
Title: On the almost sure exponential convergence of a stochastic process in a family of stochastic differential equation multi-population HIV/AIDS epidemic models with random delays for ART treatment
Abstract: A class of stochastic nonlinear multi-population HIV/AIDS models is studied. The multi-population structure represents behavioral changes in the population, in response to the information and education campaigns (IECs) against HIV/AIDS. The epidemic dynamics is subject to Brownian motion perturbations in the random supply of official developmental assistance (ODAs), and random poverty rates overtime. The impacts of the IECs, the supply of ODAs, early treatment and poverty rates are investigated by conducting stochastic analysis of the almost sure exponential convergence of the stochastic process in the HIV/AIDS models. The behavioral change and the noise induced basic reproduction numbers are obtained.

Session F
Title: Microlocal control estimates for 0th order operators
Abstract: Internal waves are gravity waves in density stratified fluids. In 2D aquaria, Maas et al predicted and then observed in experiments the internal wave attractors. Colin de Verd\`iere–Saint-Raymond used 0th order pseudodifferential operators with Morse–Smale dynamical assumptions as microlocal models for internal waves. In this talk, I will show how the dynamical assumptions on the classical flow of 0th order operators allow us to obtain microlocal control estimates for 0th order operators.

Session G
Title: Sticky particle Cucker-Smale dynamics and numerical simulations
Abstract: The 1D Euler-alignment system can be effectively described by a nonlocal scalar balance law. The entropy conditions for the scalar balance law serves as the selection principle that determines the unique weak solution of the 1D Euler-Alignment system, which corresponds to the sticky particle collision rule in the Cucker-Smale dynamics. In this talk, I will introduce a numerical algorithm to approximate the sticky particle Cucker-Smale dynamics based on established well-posedness and stability result. Under a reasonable assumption about the time step and collision states, we analyze the approximation error and demonstrate our method with plenty of numerical examples.

Session B
Title: Convergence Analysis of Non-Monotone Finite Difference Methods for Approximating Viscosity Solutions of Stationary Hamilton-Jacobi Equations
Abstract: A new non-monotone finite difference (FD) approximation method is proposed for stationary Hamilton-Jacobi problems with Dirichlet boundary condition. The FD method has local truncation errors that are above the first order Godunov barrier for monotone methods, and it is proved to converge to the unique viscosity solution of the underlying first order fully nonlinear partial differential equation. A stabilization term called a numerical moment is used to ensure the proposed scheme is admissible, stable, and convergent. Numerical tests are provided that compare the accuracy of the proposed scheme to that of the Lax-Friedrich’s method.

Session U
Title: A la Dhont method for solving Stokes equation with prescribed boundary conditions
Abstract: The motion of a sphere in stratified fluids is of importance to environmental applications involving the ocean and the atmosphere. Modeling this problem involves solving the Navier-Stokes equation with variable density field. In the viscous limit, the Oseen tensor provides the fundamental function solution to the Stokes equations for the flow external to a sphere located at the origin. Motivated by the difficulty that arises from the three dimensional convolution of the Oseen tensor and the forcing term, we attempt to derive the solution through a different method that would be more suitable for numerical implementation. We computed a new solution for the fluid flow applying La Dhont Method to the Stokes equation with prescribed boundary conditions at the sphere and at infinity. Our new tensor is a result of solving the PDE and matching the restrictions of the boundary conditions, and the rate of convergence of the solution at infinity. Comparisons between the Oseen tensor and the new solution using La Dhont Method approach will be discussed.

Session I
Title: Stability by Fixed Point Theory for Nonlinear Delay and Fractional Differential Equations
Abstract: In this paper we study the stability properties of nonlinear differential equations with variable delays and give conditions to ensure that the zero solution is asymptotically stable by applying Krasnoselskii’s Fixed Point Theorem. These conditions do not require the boundedness of delays, nor do they ask for a fixed sign on the coefficient functions. An asymptotic stability theorem with a necessary and sufficient condition is proved. The same technique is also applied to some nonlinear fractional differential equations of Caputo type.

Session L
Title: Carleman Linearization of Nonlinear Systems and Its Finite-section Approximations
Abstract: The Carleman linearization is one of the mainstream approaches to lift a finite-dimensional nonlinear dynamical system into an infinite-dimensional linear system. Finite-section approximation of the lifted system has been widely used to study dynamical and control properties of the original nonlinear system. In this context, some of the outstanding problems are to determine under what conditions, as the finite-section order increases, the trajectory of the resulting approximate linear system from the finite-section scheme converges to that of the original nonlinear system, and whether the time interval over which the convergence happens can be quantified explicitly. In this talk, I will discuss explicit error bounds for the finite-section approximation and prove that the convergence is indeed exponential with respect to the finite-section order. For a class of nonlinear systems, it is shown that one can achieve exponential convergence over the entire time horizon up to infinity. Moreover, the proposed error bound estimates can be used to compute proper truncation lengths for a given application, e.g., determining the proper sampling period for model predictive control and reachability analysis for safety verifications. This talk is based on the joint paper, Carlemen linearization of nonlinear systems and its finite-section approximations, with Amini, Sun and Motee, arXiv 2207.07755.