## ACM Seminar

The Applied and Computational Mathematics (ACM) seminar series since 2020.

Saturday, 18 March 2023 09:52

### Fiber coating dynamics: theory, algorithm, and applications

#### Speaker: Hangjie Ji (North Carolina State University)

Thin liquid films flowing down vertical fibers spontaneously exhibit complex interfacial dynamics, creating irregular wavy patterns and traveling liquid droplets. Such fiber coating dynamics is a fundamental component in many engineering applications, including mass and heat exchangers for thermal desalination and water vapor and particle capture. Through experiments and mathematical modelling, we demonstrate that flow regime transitions can be triggered by varying inlet geometries. Theoretical predictions, based on a full lubrication model and a weighted residual integral boundary-layer model, explain the experimentally observed velocity and stability of traveling droplets and their transition to isolated droplets. By coupling with the Marangoni effects, a similar regime transition can also be triggered by imposing a temperature field along the fiber. Using regularization techniques and a priori estimates for energy-entropy functionals, we prove the existence of non-negative weak solutions for a fiber coating PDE model and analytically study the traveling wave solutions. We will conclude by presenting our recent results on developing positivity-preserving numerical methods and optimal control for fiber coating dynamics.

Time: April 7, 2023 2:30pm-3:30pm
Location: LeConte 440
Host: Siming He and Changhui Tan

Friday, 03 March 2023 09:04

### Infinitely many solutions to the isentropic system of gas dynamics

#### Speaker: Cheng Yu (University of Florida)

In this talk, I will discuss the non-uniqueness of global weak solutions to the isentropic system of gas dynamics. In particular, I will show that for any initial data belonging to a dense subset of the energy space, there exists infinitely many global weak solutions to the isentropic Euler equations for any $$1 < \gamma \leq 1 + 2/n$$. The proof is based on a generalization of convex integration techniques and weak vanishing viscosity limit of the Navier-Stokes equations. This talk is based on the joint work with M. Chen and A. Vasseur.

Time: March 31, 2023 2:30pm-3:30pm
Location: LeConte 440
Host: Siming He and Changhui Tan

Friday, 03 March 2023 08:54

### Pressure robust scheme for incompressible flow

#### Speaker: Lin Mu (University of Georgia)

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.

Time: March 17, 2023 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Lili Ju

Friday, 17 February 2023 09:27

### The power of least squares in the worst-case and learning setting

#### Speaker: Felix Bartel (Technische Universität Chemnitz)

Least squares approximation is a time-tested method for function approximation based on samples. It is a natural to compare it to approximations based on arbitrary linear functionals as a benchmark. In this talk we will present recent results from the information-based complexity community showing the optimality of the least squares algorithm. We will consider the worst-case setting: drawing points which are good for a class of functions and the learning setting: we approximate an individual function based on possibly noisy samples. We support our findings with numerical experiments.

Time: February 24, 2023 2:30pm-3:30pm
Location: LeConte 440

Monday, 13 February 2023 09:41

### Leveraging computational modeling to understand biomedical diseases

#### Speaker: Wenrui Hao (Pennsylvania State University)

In this talk, I will explore the use of computational and mathematical modeling as critical tools in understanding and predicting the progression of biomedical diseases. I will present two recently developed modeling approaches, pathophysiology-driven modeling and data-driven modeling, and provide examples of each. For pathophysiology-driven modeling, I will introduce a mathematical model of atherosclerosis and discuss how it provides a personalized cardiovascular risk by solving a free boundary problem. This model also presents some interesting mathematical challenges that can deepen our understanding of cardiovascular risk. For data-driven modeling, I will use Alzheimer's disease as an example to illustrate the idea of learning a mathematical model from clinical data when the pathophysiology of a disease is not well understood. I will also discuss how this approach has been applied to personalized treatment studies of the recently FDA-approved Alzheimer's medication, aducanumab.

Time: March 3, 2023 2:30pm-3:30pm
Location: LeConte 440
Host: Yi Sun

Monday, 13 February 2023 09:17

### Modeling plug formation in films inside tubes: impact of surfactant, viscosity stratification, and slip

#### Speaker: Reed Ogrosky (Virginia Commonwealth University)

Viscous liquid films coating the interior of a tube occur in a variety of applications. If the film is thick enough, it may pinch off and form a plug, occluding the tube. In this talk I will discuss recent work examining the impact of surfactant, slip, and viscosity stratification on plug formation in a model for film flow. Implications for understanding occlusion in human airways will be discussed.

Time: February 17, 2023 2:30pm-3:30pm
Location: LeConte 440
Host: Paula Vasquez

Friday, 27 January 2023 09:48

### Machine learning of self organization from observation

#### Speaker: Ming Zhong (Illinois Institute of Technology)

Self organization (aka collective behaviors) occurs naturally in crystal formation, cell aggregation, social behaviors, etc. It is challenging and intriguing to understand self organization from the mathematical point of view. We offer a statistical/machine learning approach to explain self organization from observation data; moreover, our learning approach can aid in validating and improving the modeling of self organization. We develop a learning framework to derive physically meaningful dynamical systems to understand self organization from observation. We then investigate the steady state properties of our learned estimators. We also extend the learning approach for dynamical models constrained on Riemannian manifolds. We further improve our learning capability to infer interaction variables as well as interaction kernels. We study the effectiveness of our learning method on the NASA Jet Propulsion Laboratory's modern Ephemerides. Upon careful inspection of our model, we discover that it even captures potion of the general relativity effects. A complete learning theory on second-order systems is presented, as well as two new models on emergence of social hierarchy and concurrent emergence of flocking and synchronization.

Time: February 10, 2023 2:30pm-3:30pm
Location: LeConte 440
Host: Siming He and Changhui Tan

Thursday, 12 January 2023 21:18

### On the problem of emergence arising in hydrodynamic systems of collective behavior

#### Speaker: Roman Shvydkoy (University of Illinois Chicago)

Emergence is a phenomenon of formation of collective outcomes in systems where communications between agents has local range. In dynamics of swarms such outcomes often represent a globally aligned flock or congregation of aligned clusters. The classical result of Cucker and Smale states that alignment is unconditional in flocks that have global communication with non-integrable radial tails. Proving a similar statement for purely local interactions presents a major mathematical challenge. In this talk we will overview three programs of research directed on understanding the emergent phenomena: hydrodynamic topological interactions, kinetic approach based on hypocoercivity, and spectral energy method. We present a novel framework based on the concept of environmental averaging which allows us to obtain coercivity estimates leading to new flocking results.

Time: January 27, 2023 2:30pm-3:30pm
Location: LeConte 440
Host: Changhui Tan

Monday, 14 November 2022 11:49

### Quantitative steepness, semi-FKPP reactions, and pushmi-pullyu fronts

#### Speaker: Jing An (Duke University)

We will discuss the algebraic structure of a large class of reaction-diffusion equations and use it to study the long-time behavior of the solutions and their convergence to traveling waves in the pulled and pushed regimes, as well as at the pushmi-pullyu boundary. A new quantity named as the shape defect function is introduced to measure the difference between the profiles of the solution and the traveling waves. In particular, the positivity of the shape defect function, combined with a new weighted Hopf-Cole transform and a relative entropy approach, plays a key role in the stability arguments. The shape defect function also gives a new connection between reaction-diffusion equations and reaction conservation laws at the pulled-pushed transition. This is joint work with Chris Henderson and Lenya Ryzhik.

Time: November 18, 2022 2:30pm-3:30pm
Location: LeConte 205
Host: Siming He

Monday, 24 October 2022 10:12

### On the construction of 3D incompressible Euler equilibria by magnetic relaxation

#### Speaker: Federico Pasqualotto (Duke University)

Magnetic relaxation is a conjectured general procedure to obtain steady solutions to the incompressible Euler equations by means of a long-time limit of an MHD system. In some regimes, the magnetic field is conjectured to “relax” to a steady state of the 3D Euler equations as time goes to infinity.

In this talk, I will first review the classical problem of magnetic relaxation, connecting it to questions arising in topological hydrodynamics. I will then present a general construction of steady states of the incompressible 3D Euler equations by a long-time limit of a regularized MHD system. We consider the so-called Voigt regularization, and our procedure yields non-trivial equilibria on the flat 3D torus and on general bounded domains.

This is joint work with Peter Constantin.

Time: October 28, 2022 2:30pm-3:30pm
Location: LeConte 205
Host: Siming He

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