The Euler Alignment system is a hydrodynamic PDE version of the celebrated Cucker-Smale ODE's of collective behavior. Together with Changhui Tan, we developed a theory of weak solutions in 1D, which provide a uniquely determined way to evolve the dynamics after a blowup. Inspired by Brenier and Grenier's work on the pressureless Euler equations, we show that the dynamics of our system are captured by a nonlocal scalar balance law. We generate the unique entropy solution of a discretization of this balance law by introducing the "sticky particle Cucker-Smale" system to track the shock locations. Our approximation scheme for the density converges in the Wasserstein metric; it does so with a quantifiable rate as long as the initial velocity is at least Holder continuous. In this talk, we will discuss the limiting configurations, or "flocking states," that arise in this system, and how to predict them from the initial data.

Time: April 21, 2023 2:30pm-3:30pm

Location: LeConte 440

Host: Changhui Tan

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]]>Live cell imaging and single particle tracking techniques have become increasingly popular amongst the mathematical biology community. Lysosomes, known for endocytosis, phagocytic destruction, and autophagy, move about the cell along microtubules. Intracellular transport of lysosomes is carried out in membrane-bound vesicles by motor proteins. Single particle tracking methods utilize stochastic models to simulate intracellular transport and give rise to rigorous analysis of the resulting properties, specifically related to transitioning between inactive to active states. We find confidence in our methodology and develop a framework to understand how these properties play a role in determining an optimal frame rate for capturing live cells.

Biological systems are traditionally studied as isolated processes (e.g. regulatory pathways, motor protein dynamics, transport of organelles, etc.). Although more recent approaches have been developed to study whole cell dynamics, integrating knowledge across biological levels remains largely unexplored. In experimental processes, we assume that the state of the system is unknown until we sample it. Many scales are necessary to quantify the dynamics of different processes. These may include a magnitude of measurements, multiple detection intensities, or variation in the magnitude of observations. The interconnection between scales, where events happening at one scale are directly influencing events occurring at other scales, can be accomplished using mathematical tools for integration to connect and predict complex biological outcomes. In this work we focus on building statistical inference methods to study the complexity of the cytoskeleton from one scale to another by relying on two main components facilitating intracellular transport; that is microtubule network organization and cargo transport.

Time: April 14, 2023 2:30pm-3:30pm

Location: LeConte 118

Host: Paula Vasquez

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]]> 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

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]]>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

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]]> 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

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]]> 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

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]]> 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

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]]> 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

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]]> 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

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]]> 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

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