PDE & Applied math seminar

Wednesday 10:00 -10:50 am, https://ucr.zoom.us/j/97606227247

Organizers : Weitao Chen / Heyrim Cho / Yat Tin Chow / Qixuan Wang / Jia Guo / Mykhailo Potomkin

Past Organizers : Mark Alber / James Kelliher / Amir Moradifam

In Spring 2022, the PDE & Applied math seminar will be held both through zoom and in person. Specific information about format of each talk will be provided in the email announcement and posted below. If you're interested in attending the seminar, please contact Dr. Jia Gou (jgou@ucr.edu) and Mykhailo Potomkin (mykhailp@ucr.edu).

Spring 2022 Schedule

Mar 30 10:00 (Wed) Organizational meeting

Apr 6 10:00 (Wed) Hanliang Guo, University of Michigan, Ann Arbor

Apr 13 10:00 (Wed) Mykhailo Potomkin, UC Riverside

Apr 20 10:00 (Wed) Peng Chen, The University of Texas at Austin

Apr 27 10:00 (Wed) Stephan Wojtowytsch, Texas A&M

May 4 10:00 (Wed) Scott McCalla, Montana State University

May 11 10:00 (Wed) Adrian Moure, California Institute of Technology

May 18 10:00 (Wed) Daniel Gomez, University of Pennsylvania

May 25 10:00 (Wed) Iain Moyles, University of York

Jun 1 10:00 (Wed) James K. Alcala, UC Riverside

March 28, 2022, 10:00-10:50 AM PT

Dr. Jia Gou (UC Riverside) and Dr. Mykhailo Potomkin (UC Riverside)

Title: Organizational meeting

April 6, 2022, 10:00-10:50 AM PT

Dr. Hanliang Guo, University of Michigan, Ann Arbor

Title: Optimal swimming of ciliated microswimmer

Abstract: Cilia are microscopic hair-like organelles that protrude from epithelial cell-surfaces. Being one of the most conserved micro-structures in nature, cilia are important building-blocks of life. The best known functions of ciliary flows are locomotion for microorganisms and toxic particles removal for larger animals such as human being. Recent studies show that cilia are also critical in transporting cerebrospinal fluid in mammalian brains and in creating active flow environment to recruit symbiotic bacteria in a squid-vibrio system. In this talk, I will focus on the optimizations of microswimmers with densely packed cilia. In the first part of the talk, I will demonstrate a simple yet efficient optimization strategy for axisymmetric microswimmers with time-independent slip profile. A shape-based scalar metric is proposed to predict whether the optimal slip on a given shape makes it a pusher, puller or a neutral swimmer. I will then introduce an adjoint-based optimization procedure for displacement-based time-dependent slip profile. Our results show that adding a constraint to the cilia length can on average improve the efficiency for microswimmers with concave bodies.

April 13, 2022, 10:00-10:50 AM PT

Dr. Mykhailo Potomkin, UC Riverside

Title: How a microswimmer swims in nematic liquid crystal: a PDE approach

Abstract: Bacteria often swim in biofluids with properties different from isotropic Newtonian fluid but rather those of liquid crystal. Understanding how a bacterium navigates itself in such an environment is important for treatment strategies of many infectious diseases. I will present a nonlinear PDE system coupling liquid crystal hydrodynamics with the model of active microswimmer. The goal is to elucidate how the orientation order of liquid crystal affects the motion of an individual bacterium. In this talk, I will show that the model reveals how surface properties affect swimming direction. I will also discuss the emergence of topological defects around the microswimmer for large propulsion speeds. If time permits, I will present recent results on analysis of the model, its well-posedness, existence of steady states, as well as collective swimming in liquid crystal and how it can be described with the help of homogenization theory. This work is done jointly with I. Aronson (Penn State U.), L. Berlyand (Penn State U.), H. Chi (Penn State U.), A. Yip (Purdue U.), and L. Zhang (Shanghai Jiao Tong U.).

April 20, 2022, 10:00-10:50 AM PT

Dr. Peng Chen, The University of Texas at Austin

Title: Digital twin: Fast and scalable computational methods for learning

and optimization of complex physical systems under uncertainty

Abstract: In this talk, I will present some recent work on high-dimensional learning of complex models and model-constrained optimization (of control, design, and experiment) under uncertainty to create digital twin of physical systems. Tremendous computational challenges are faced for such problems when (1) the models (e.g., described by partial differential equations) are expensive to solve and have to be solved many times or in real time; and (2) the data, optimization, and uncertain variables are high-dimensional, bringing the curse of dimensionality for most conventional methods. We tackle these challenges by exploiting both data and model informed properties, such as smoothness, sparsity, correlation, intrinsic low-dimensionality or low-rankness, etc. I will present several new computational methods that achieve significant computational reduction (fast) and break the curse of dimensionality (scalable), including structure-exploiting model reduction, randomized high-order tensor decomposition, derivative informed deep learning, projected transport map, and functional Taylor approximations. I will also briefly talk about some applications of these methods in learning and optimal mitigation of infectious disease (COVID-19), optimal control of turbulent combustion, optimal design of stellarator for plasma fusion, and optimal experimental design for sensor placement.

April 27, 2022, 10:00-10:50 AM PT

Dr. Stephan Wojtowytsch, Texas A&M

Title: Neural network approximation in shallow and deep learning

Abstract: We will discuss some fundamentals of neural network approximation: Universal approximation theorems, the superiority of shallow neural networks over linear methods of approximation, and some aspects of depth separation phenomena, i.e. functions which can be approximated efficiently using deep, but not shallow neural networks. Time permitting, we will discuss some aspects of optimization for overparametrized function classes.

May 4, 2022, 10:00-10:50 AM PT

Dr. Scott McCalla, Montana State University

Title: Tracking bacterial growth using a nonlocal interfacial model

Abstract: Biological pattern formation has been extensively studied using reaction-diffusion and agent-based models. In this talk we will discuss nonlocal pattern forming mechanisms in the context of bacterial colony formation with an emphasis on arrested fronts. This will lead to a novel nonlocal framework to understand the interfacial motion in biological systems. We will then use this approach to model experiments for an interesting bacterial phenomenon.

May 11, 2022, 10:00-10:50 AM PT

Dr. Adrian Moure, California Institute of Technology

Title: Phase-field modeling and simulation in biomechanics: cell migration and other applications

Abstract: The phase-field method is a modeling technique that permits a simple and direct mathematical formulation of moving boundary problems. The phase-field method reformulates the moving boundary problem as a set of partial differential equations posed on a known and fixed computational domain. To solve the higher-order equations derived from the phase-field theory, we develop a numerical methodology based on isogeometric analysis – a generalization of the finite element method. In this talk, we will discuss phase-field modeling and numerical simulation of individual and collective cell migration. Cell motility, which is crucial in human health and development, represents an outstanding example of a problem with moving interfaces. The framework we developed captures the complex intra- and extra-cellular mechanochemical interactions that drive cell motion. I will show simulations of cell migration in different environments. In addition, I will briefly introduce applications of our modeling framework to other problems such as cancer growth and hydraulic fracturing.

May 18, 2022, 10:00-10:50 AM PT

Dr. Daniel Gomez, University of Pennsylvania

Title: Spike Solutions to the Singularly Perturbed Fractional Gierer-Meinhardt System

Abstract: The singularly perturbed Gierer-Meinhardt system is a model reaction-diffusion system commonly used to study the effects of short-range activation and long-range inhibition. In this singularly perturbed limit the activator has an asymptotically small diffusivity and this conspires with the longer range of the inhibitor to form localized spike solutions. In this talk I will discuss recent work on the formal asymptotic analysis of spike solutions in one-dimensional domains when both the activator and inhibitor exhibit Lévy flights. Mathematically this leads to a fractional reaction-diffusion system in which the classical Laplacian is replaced with the fractional Laplacian. The singular behavior of the corresponding fractional Green's function plays a crucial role in the asymptotic analysis of spike solutions and, depending on the fractional order, this leads to direct analogies with spike solutions to the classical Gierer-Meinhardt system in one-, two-, and three-dimensional domains.

May 25, 2022, 10:00-10:50 AM PT

Dr. Iain Moyles, University of York

Title: Timescale dynamics in an mRNA vaccine model

Abstract: We present an in-host mathematical model for vaccine delivery and antibody response from a liquid nanoparticle formulated mRNA vaccine. Through asymptotic model reduction, we show that five distinct timescales emerge representing priming through to antibody decay. We identify these timescales in a variety of datasets from two mRNA vaccines for COVID-19 and compare measured antibody response to model prediction showing good agreement. We exploit the different timescales to investigate the optimal time for second dosage.

June 1, 2022, 10:00-10:50 AM PT

James K. Alcala, UC Riverside

Title: Moving anchor acceleration methods in minimax problems

Abstract: Minimax problems, a classical problem setting within optimization, have recently gained more attention in the machine learning community due to their use in Generative Adversarial Networks (GANs) and adversarial training. A variety of algorithms exist to approach the optimal point of such problems, as do acceleration methods that improve their convergence rates. In this talk, we explore a novel acceleration technique, the so-called moving anchor, applied to a variant of the extra-gradient method. On the theoretical side, we demonstrate a comparable rate of convergence to state-of-the-art algorithms that achieve an O(1/k^2) rate of convergence on the squared gradient norm, with minimal assumptions. This is known to be the best rate possible in the family of first order algorithms. With optimal parameter choices, we expect our moving anchor algorithm to be faster than state-of-the-art algorithms by a constant factor. Numerical experiments illustrate the effectiveness of our moving anchor algorithm. We briefly explore a proximal version of the algorithm with some promise, and some other potential avenues of exploration.