epshteyn (at)
math.utah.edu) or to
Akil Narayan (akil (at) sci.utah.edu) 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.
epshteyn (at)
math.utah.edu) and Akil Narayan (akil (at)sci.utah.edu).
Past lectures: Fall 2020, Spring 2020, Fall 2019, Spring 2019, Fall 2018, Spring 2018, Fall 2017, Spring 2017, Fall 2016, Spring 2016, Fall 2015, Spring 2015, Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000, Spring 2000, Fall 1999, Spring 1999, Fall 1998, Spring 1998, Winter 1998, Fall 1997, Spring 1997, Winter 1997, Fall 1996, Spring 1996, Winter 1996, Fall 1995.