Applied Mathematics Seminar, Spring 2021

## Mondays 4:00 PM - 5:00 PM MT, online (zoom information will be provided before the seminars)

• This seminar can be taken for credit: Students can get 1-3 credits by registering to the Applied Math Seminar class Math 7875 Section 010 for Spring 2021. Students should talk to the seminar organizer before taking it for a credit. Grading is based on attendance and giving a talk by presenting an applied-mathematics paper (not necessarily your own). Student talks will be appropriately labeled to distinguish them from visitor talks. The seminar organizer is available to review your slides, for dry-runs etc.
• Please direct questions or comments about the seminar (or its class) to Yekaterina Epshteyn (epshteyn (at) math.utah.edu) or to Akil Narayan (akil (at) sci.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizers for information about how to subscribe to this list.

January 25. Claremont & Utah Joint Applied Math Seminar.
Speaker: Chiu-Yen Kao, Department of Mathematical Sciences, Claremont McKenna College
Title: Minimization of the First Nonzero Eigenvalue Problem for Two-Phase Conductors with Neumann Boundary Conditions
Abstract: We consider the problem of minimizing the first nonzero eigenvalue of an elliptic operator with Neumann boundary conditions with respect to the distribution of two conducting materials with a prescribed area ratio in a given domain. In one dimension, we show monotone properties of the first nonzero eigenvalue with respect to various parameters and find the optimal distribution of two conducting materials on an interval under the assumption that the region that has lower conductivity is simply connected. On a rectangular domain in two dimensions, we show that the strip configuration of two conducting materials can be a local minimizer. For general domains, we propose a rearrangement algorithm to find the optimal distribution numerically. Many results on various domains are shown to demonstrate the efficiency and robustness of the algorithms. Topological changes of the optimal configurations are discussed on circles, ellipses, annuli, and L-shaped domains.

February 1. Claremont & Utah Joint Applied Math Seminar.
Speaker: Di Kang, Department of Mathematics and Statistics, McMaster University
Title: Searching for singularities in Navier-Stokes flows using variational optimization methods
Abstract: In the presentation we will discuss our research program concerning the search for the most singular behaviors possible in viscous incompressible flows. These events are characterized by extremal growth of various quantities, such as the enstrophy, which control the regularity of the solution. They are therefore intimately related to the question of possible singularity formation in the 3D Navier-Stokes system, known as the hydrodynamic blow-up problem. We demonstrate how new insights concerning such questions can be obtained by formulating them as variational PDE optimization problems which can be solved computationally using suitable discrete gradient flows. More specifically, such an optimization formulation allows one to identify "extreme" initial data which, subject to certain constraints, leads to the most singular flow evolution. In offering a systematic approach to finding flow solutions which may saturate known estimates, the proposed paradigm provides a bridge between mathematical analysis and scientific computation. In particular, it makes it possible to determine whether or not certain mathematical estimates are "sharp", in the sense that they can be realized by actual vector fields, or if these estimates may still be improved. In the presentation we will review a number of results concerning 1D and 2D flows characterized by the maximum possible growth of different Sobolev norms of the solutions. As regards 3D flows, we focus on the enstrophy which is a well-known indicator of the regularity of the solution. We find a family of initial data with fixed enstrophy which leads to the largest possible growth of this quantity at some prescribed final time. Since even with such worst-case initial data the enstrophy remains finite, this indicates that the 3D Navier-Stokes system reveals no tendency for singularity formation in finite time.

February 8. Claremont & Utah Joint Applied Math Seminar.
Speaker: Aaron Barrett, Department of Mathematics, University of Utah
Title: Complex Fluids in the Immersed Boundary Method: From Viscoelasticity to Blood Clots
Abstract: The immersed boundary method was first developed in the 1970s to model the motion of heart valves and has since been utilized to study many different biological systems. While the IB method has seen countless modifications and advancements from the perspective of fluid-structure interaction, the use of a Newtonian fluid model remains a fundamental component of many implementations. However, many biological fluids exhibit non-Newtonian responses to stresses, and as such, a Newtonian fluid model falls short to fully describe the system. In this talk, we will discuss models of two different systems: polymeric fluids and blood clotting, and we will address the numerical challenges associated with each system.

February 15. Claremont & Utah Joint Applied Math Seminar.
Speaker: Peter Hinow, Mathematical Sciences, University of Wisconsin, Milwaukee
Title: Modeling and Simulation of Ultrasound-mediated Drug Delivery to the Brain
Abstract: We use a mathematical model to describe the delivery of a drug to a specific region of the brain. The drug is carried by liposomes that can release their cargo by application of focused ultrasound. Thereupon, the drug is absorbed through the endothelial cells that line the brain capillaries and form the physiologically important blood-brain barrier. We present a compartmental model of a capillary that is able to capture the complex binding and transport processes the drug undergoes in the blood plasma and at the blood-brain barrier. We apply this model to the delivery of L-dopa, (used to treat Parkinson's disease) and doxorubicin (an anticancer agent). The goal is to optimize the delivery of drug while at the same time minimizing possible side effects of the ultrasound. In a second project, we present a mathematical model for drug delivery through capillary networks with increasingly complex topologies with the goal to understand the scaling behavior of model predictions on a coarse-to-fine sequence of grids.

February 22. Claremont & Utah Joint Applied Math Seminar.
Speaker: Christina Durón, Mathematics Department, The University of Arizona
Title: Heatmap Centrality: A New Measure to Identify Super-spreader Nodes in Scale-Free Networks
Abstract: The identification of potential super-spreader nodes within a network is a critical part of the study and analysis of real-world networks. Motivated by a new interpretation of the "shortest path" between two nodes, this talk will explore the properties of the recently proposed measure, the heatmap centrality, by comparing the farness of a node with the average sum of farness of its adjacent nodes in order to identify influential nodes within the network. As many real-world networks are often claimed to be scale-free, numerical experiments based upon both simulated and real-world undirected and unweighted scale-free networks are used to illustrate the effectiveness of the new "shortest path" based measure with regards to its CPU run time and ranking of influential nodes.

March 1. Claremont & Utah Joint Applied Math Seminar.
Speaker: Xiang Xu, Department of Mathematics & Statistics, Old Dominion University
Title: Blowup rate estimates of a singular potential in the Landau-de Gennes theory for liquid crystals
Abstract: The Landau-de Gennes theory is a type of continuum theory that describes nematic liquid crystal configurations in the framework of the Q-tensor order parameter. In the free energy, there is a singular bulk potential which is considered as a natural enforcement of a physical constraint on the eigenvalues of symmetric, traceless Q-tensors. In this talk we shall discuss some analytic properties related to this singular potential. More specifically, we provide precise estimates of both this singular potential and its gradient as the Q-tensor approaches its physical boundary.

March 8. Claremont & Utah Joint Applied Math Seminar.
Speaker: Henry Schellhorn, Department of Mathematics, Claremont Graduate School
Title: Optimal Control of the SIR Model in the Presence of Transmission and Treatment Uncertainty
Abstract: The COVID-19 pandemic illustrates the importance of treatment-related decision making in populations. This article considers the case where the transmission rate of the disease as well as the efficiency of treatments is subject to uncertainty. We consider two different regimes, or submodels, of the stochastic SIR model, where the population consists of three groups: susceptible, infected and recovered. In the first regime the proportion of infected is very low, and the proportion of susceptible is very close to 100%. This corresponds to a disease with few deaths and where recovered individuals do not acquire immunity. In a second regime, the proportion of infected is moderate, but not negligible. We show that the first regime corresponds almost exactly to a well-known problem in finance, the problem of portfolio and consumption decisions under mean-reverting returns (Wachter, JFQA 2002), for which the optimal control has an analytical solution. We develop a perturbative solution for the second problem. To our knowledge, this paper represents one of the first attempts to develop analytical/perturbative solutions, as opposed to numerical solutions to stochastic SIR models.

March 22. Claremont & Utah Joint Applied Math Seminar.
Speaker: Dmitry Pelinovsky, Department of Mathematics and Statistics, McMaster University
Title: Periodic travelling waves in nonlinear wave equations: modulation instability and rogue waves
Abstract: I will overview the following different wave phenomena in integrable nonlinear wave equations:
(1) universal patterns in the dynamics of fluxon condensates in the semi-classical limit;
(2) modulational instability of periodic travelling waves;
(3) rogue waves on the background of periodic and double-periodic waves.
Main examples include the sine-Gordon equation, the nonlinear Schroedinger equation, and the derivative nonlinear Schroedinger equation. For the latter equation, in collaboration with Jinbing Chen (South East University, China) and Jeremy Upsal (University of Washington, USA), we adapted the method of nonlinearization of the Lax system in order to characterize the existence and modulation stability of periodic travelling waves. We give precise information on the location of Lax and stability spectra, with assistance of numerical package based on the so-called Hill's method. Particularly interesting outcome is the explicit relation between the onset of modulation instability and the existence of a rogue wave (localized solution in space and time) on the background of periodic travelling waves.

March 29 (student's talk). Claremont & Utah Joint Applied Math Seminar.
Speaker: Dihan Dai, Department of Mathematics, University of Utah
Title: Hyperbolicity-Preserving Stochastic Galerkin Method for Shallow Water Equations
Abstract: The system of shallow water equations and related models are widely used in oceanography to model hazardous phenomena such as tsunamis and storm surges. Unfortunately, the inherent uncertainties in the system will inevitably damage the credibility of decision-making based on the deterministic model. The stochastic Galerkin (SG) method seeks a solution by applying the Galerkin method to the stochastic domain of the equations with uncertainty. However, the resulting system may fail to preserve the hyperbolicity of the original model. In this talk, we will discuss a strategy to preserve the hyperbolicity of the stochastic systems. We will also discuss a well-balanced hyperbolicity-preserving central-upwind scheme for the random shallow water equations and illustrate the effectiveness of our schemes on some challenging numerical tests.

April 12. Claremont & Utah Joint Applied Math Seminar.
Speaker: Traian Iliescu, Department of Mathematics, Virginia Tech
Title: Large Eddy Simulation Reduced Order Models
Abstract: In this talk, we present reduced order models (ROMs) for turbulent flows, which are constructed by using ideas from large eddy simulation (LES) and variational multiscale (VMS) methods. First, we give a general introduction to reduced order modeling and emphasize the connection to classical Galerkin methods (e.g., the finite element method) and the central role played by data. Then, we describe the closure problem, which represents one of the main obstacles in the development of ROMs for realistic, turbulent flows. To tackle the ROM closure problem, we use ROM spatial filters (e.g., the ROM projection and the ROM differential filter) and build new LES-ROMs that capture the large scale ROM features and model the interaction between these large scales and the small scale ROM features. Finally, we present results for these LES-ROMs in the numerical simulation of under-resolved engineering flows (e.g., flow past a cylinder and turbulent channel flow) and the quasi-geostrophic equations (which model the large scale ocean circulation).

April 19. Claremont & Utah Joint Applied Math Seminar.
Speaker: Ryan Murray, Department of Mathematics, North Carolina State University
Title: Adversarially robust classification via geometric flows
Abstract: Classification is a fundamental task in data science and machine learning, and in the past ten years there have been significant improvements on classification tasks (e.g. via deep learning). However, recently there have been a number of works demonstrating that these improved algorithms can be "fooled" using specially constructed adversarial examples. In turn, there has been increased attention given to creating machine learning algorithms which are more robust against adversarial attacks. In this talk I will describe a recently proposed framework for optimal adversarial robustness which is related to optimal transportation. I will then discuss some recent work, with Nicolas Garcia Trillos, which characterizes solutions of the optimal adversarial robust classification problem by using a geometric evolution equation. Surprisingly, this geometric evolution equation asymptotically takes the form of a weighted mean curvature flow, which suggests new analytical and computational approaches to the problem. I will also discuss a number of related open questions.

April 26. Claremont & Utah Joint Applied Math Seminar.
Speaker: Anna Little, Department of Mathematics, University of Utah
Title: Title Balancing Geometry and Density: Path Distances on High-Dimensional Data
Abstract: This talk discusses multiple methods for clustering high-dimensional data, and explores the delicate balance between utilizing data density and data geometry. I will first present path-based spectral clustering, a novel approach which combines a density-based metric with graph-based clustering. This density-based path metric allows for fast algorithms and strong theoretical guarantees when clusters concentrate around low-dimensional sets. However, the method suffers from a loss of geometric information, information which is preserved by simple linear dimension reduction methods such as classic multidimensional scaling (CMDS). The second part of the talk will explore when CMDS followed by a simple clustering algorithm can exactly recover all cluster labels with high probability. However, scaling conditions become increasingly restrictive as the ambient dimension increases, and the method will fail for irregularly shaped clusters. Finally, I will discuss how a more general family of path metrics, when combined with CMDS, give low-dimensional embeddings which respect both data density and data geometry. This new method exhibits promising performance on single cell RNA sequence data and can be computed efficiently by restriction to a sparse graph.

Seminar organizers: Yekaterina Epshteyn (epshteyn (at) math.utah.edu) and Akil Narayan (akil (at)sci.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148