Math Colloquium - Spring 2022

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January 21

Patricia Ning (Department of Statistics, University of Michigan, Ann Arbor)

Location: CULM 611 and WebEx

High-dimensional Parameter Learning over General Graphical State Space Models: Beating the Curse of Dimensionality

Disease transmission systems are highly nonlinear and stochastic and are imperfectly observable. However, conducting high-dimensional parameter learning for partially observed, nonlinear, and stochastic spatiotemporal processes is a methodological challenge and is an open problem so far. We propose the iterated block particle filter (IBPF) algorithm for learning high-dimensional parameters over graphical state space models with general state spaces, measures, transition densities, and graph structure. Theoretical performance guarantees are obtained on beating the curse of dimensionality (COD), algorithm convergence, and likelihood maximization. Experiments on a highly nonlinear and non-Gaussian spatiotemporal model for measles transmission reveal that the iterated ensemble Kalman filter algorithm (Li et al. (2020), Science) is ineffective and the iterated filtering algorithm (Ionides et al. (2015), PNAS) suffers from the COD, while our IBPF algorithm beats COD consistently across various experiments with different metrics.

January 28

Thi-phong Nguyen (Department of Mathematics, Purdue University)

Location: WebEx

Fast and efficient numerical methods for inverse scattering problems in complex media

Inverse scattering problems arise in many real life applications such as non-destructive evaluation, medical imaging, geophysical prospecting, etc. For instance, illuminating a probed domain with some (acoustic, electromagnetic, elastic) waves and measuring the response (scattered waves) at some distance, the inverse problem is to identify the presence of defects, such as cracks, and if possible, finding their location and reconstructing their shape.  

In this talk, I will discuss the study of the so-called Sampling Methods for solving such inverse problems and their application for two particular problems. The first one is the reconstruction of closed fractures inside a finite solid body using seismic waves. The second one is related to the reconstruction of local defects in an infinite periodic layer without any prior knowledge on the periodic structure. I will also present some open problems and future research directions in connection with these problems.

February 4

Chrysoula Tsogka (Department of Applied Mathematics, University of California Merced)

Location: WebEx

Fast Signal Recovery from Quadratic Measurements

We present a novel approach for recovering a sparse signal from quadratic measurements corresponding to a rank-one tensorization of the data vector. Such quadratic measurements, referred to as interferometric or cross-correlated data, naturally arise in many fields such as remote sensing, spectroscopy, holography and seismology. Compared to the sparse signal recovery problem that uses linear measurements, the unknown in this case is a matrix, $X=\rho \rho^*$, formed by the cross correlations of $\rho \in C^K$. This creates a bottleneck for the inversion since the number of unknowns grows quadratically in $K$. The main idea of the proposed approach is to reduce the dimensionality of the problem by recovering only the diagonal elements of the unknown matrix, $| \rho_i|^2$, $i=1,\ldots,K$.  The contribution of the off-diagonal terms $\rho_i \rho_j^*$ for $i \neq j$ to the data is treated as noise and is absorbed using the Noise Collector approach introduced in [Moscoso et al, The noise collector for sparse recovery in high dimensions, PNAS 117 (2020)]. With this strategy, we recover the unknown  by solving a convex linear problem whose cost is similar to the one that uses linear measurements. The proposed approach provides exact support recovery when the data is not too noisy, and there are no false positives for any level of noise.

February 11

Darren Crowdy (Department of Mathematics, Imperial College London)

Location: WebEx

To Slip, or Not to Slip?

I will present theoretical results on two different problems in low-Reynolds-number fluid dynamics where the nature of the boundary condition on a surface is not necessarily clear.

The first problem is motivated by recent experimental observations of equilibrium tilt of graphene nanoplatelets in a simple shear flow. We will discuss how a delicate interplay between particle shape anisotropy and intrinsic surface slip can cause this steady tilt instead of periodic Jeffery-type orbits.

A second problem surveys the role of surfactants in affecting the effective boundary condition in a given flow situation. In particular, we will demonstrate some new analytical techniques for studying the dynamical formation of so-called “stagnant cap” equilibria, and how surface reaction kinetics can modify these. The latter results rely on a new theoretical connection made between Marangoni flows and the complex Burgers equation, a link that will be shown to lead to new classes of exact time-evolving solutions for the surfactant dynamics.

February 18

Krasimira Tseneva (College of Engineering, Mathematics and Physical Sciences, University of Exeter)

Location: WebEx

Mathematical-based Microbiome Analytics for Clinical Translation

Traditionally, human microbiology has been based on laboratory focused cultures of microbes isolated from human specimens in patients with acute or chronic infection. These approaches primarily view human disease through the lens of a single species and its relevant clinical setting however such approaches fail to account for the surrounding environment and wide microbial diversity that exists in vivo. Given the emergence of next generation sequencing technologies and advancing bioinformatic pipelines, researchers now have unprecedented capabilities to characterise the human microbiome in terms of its taxonomy, function, antibiotic resistance and even bacteriophages. Despite this, an analysis of microbial communities has largely been restricted to ordination, ecological measures, and discriminant taxa analysis. This is predominantly due to a lack of suitable computational tools to facilitate microbiome analytics. In this talk I will introduce the available and emerging analytical techniques including integrative analysis, microbial association networks, topological data analysis (TDA) and mathematical modelling. I will then present our recently developed approach to the multi-biome that integrates bacterial, viral and fungal communities in bronchiectasis through weighted similarity network fusion. Bronchiectasis is a progressive chronic airway disease characterised by microbial colonisation and infection. We demonstrate that Integrative microbiomics captures microbial interactions to determine exacerbation risk, which cannot be appreciated by the study of a single microbial group. 

February 25

Lexing Ying (Department of Mathematics and Institute for Computational and Mathematical Engineering, Stanford University)

Location: WebEx

Prony's method, analytic continuation, and quantum signal processing

Prony's method is a powerful algorithm for identifying frequencies and amplitudes from equally spaced signals. It is probably not as well-known as it should have been. In the first part of the talk, we will review the Prony's method. In the second part of the talk, we use the ideas of the Prony's method to solve two problems: (1) analytic continuation from noisy samples and (2) stable factorization of the phase factors for quantum signal processing.

March 4

Cancelled

March 11

Monika Nitsche (Department of Mathematics and Statistics, University of New Mexico)

Location: CULM LECT II

Accurate near-interface velocity evaluation in vortex sheet and multi-nested Stokes flow

Boundary integral formulations yield efficient numerical methods to solve elliptic boundary value problems.  They are the method of choice for interfacial fluid flow in either the inviscid vortex sheet limit, or the viscous Stokes limit. The fluid velocity at a target point is given by an integral over all interfaces. However, for target points near, but not on the interface, the integrals are near-singular and standard quadratures lose accuracy. While several accurate methods exist for planar geometries, they do not generally apply to the non-analytic case that arises in axisymmetric geometries. We present a method based on Taylor series expansions of the integrand about basepoints on the interface that accurately resolve a large class of integrals, and apply it to solve the near-interface problem in planar vortex sheet flow, axisymmetric Stokes flow, and Stokes flow in 3D. The application to multi-nested Stokes flow uses a novel representation of the fluid velocity.

March 18

Spring Recess - No Colloquium

March 25

Alejandro Aceves (Department of Mathematics, Southern Methodist University)

Location: CULM LECT II 

Modeling climate change: A dynamical systems approach

It is not surprising that research on climate is front and center; the latest Nobel Prize in Physics attests to this. In this talk I will give my own perspective of models based on a dynamical systems approach. As it will be noticed, while these models are relatively simple, they can prove to be useful in assessing possible climate events. Emphasis will be placed in the studies of the Atlantic Meridional Overturning Circulation (AMOC).

April 1

Dimitri Papageorgiou (Department of Mathematics, Imperial College London)

Location: WebEx

Evolution PDEs arising in multiphase-multiphysics flows

Viscous multilayer shear flows of immiscible liquids can become unstable due to the presence of interfaces. The nonlinear patterns that emerge are complex and extreme events such as interface pinching and/or wall touching are possible. It is of interest, therefore, to develop reduced dimension and reduced parameter nonlinear models that can be used to study the dynamics and quantify solutions and allied phenomena. I will describe how multiscale asymptotics can be used to derive classes of equations that couple thin and thick regions and produce nonlocal PDEs. The physical origins can be free-stream shear, electric or magnetic fields etc. The particular case of two-layer viscous flows in channels will be considered in detail and comparisons of the nonlocal equations with experiments will be shown that confirm their usefulness. Finally I will discuss models that describe multilayer extrusion processes that in many cases support slip at the interface between viscous fluids. Such slip is the viscous analogue of classical inviscid slip that leads to Kelvin-Helmholtz instability. It will be shown that the viscous case also supports short-wave instabilities, albeit not catastrophic. Nonlinear aspects will also be discussed.

April 8

Richard Braun (Department of Mathematical Sciences, University of Delaware)

Location: WebEx and CULM LECT II

Data Extraction and Math Modeling for Tear Breakup via in vivo Fluorescence (FL) Imaging

The tear film is a thin fluid multilayer left on the eye surface after a blink.  A good tear film is essential for health and proper function of the eye, yet millions have a condition called dry eye disease (DED) that inhibits vision and may lead to inflammation and ocular surface damage.  There is little quantitative data about tear film failure, often called tear breakup (TBU).  Currently, it is not possible to directly measure important variables such as tear osmolarity (roughly, saltiness) within areas of TBU.  We present automatic methods via deep convolutional neural network to extract data from video of FL imaging in healthy eyes, and subsequently use to estimate important variables like osmolarity within regions of TBU.    Not only is new data obtained, but far more data, enabling statistical methods to be applied.  Though partial differential equations models have been used for detailed initial exploration, using simplified ordinary differential equation models are a key component for generating significantly more data than previous approaches.  So far, the approach provides quantitative baseline data for TBU in healthy subjects; future work will produce data from DED subjects.

April 22

Zachary Kilpatrick (Department of Applied Mathematics, University of Colorado Boulder)

Location: WebEx 

How heterogeneity shapes the efficiency of collective decisions and foraging

Many organisms regularly make decisions regarding foraging, home-site selection, mating, and danger avoidance in groups ranging from hundreds up to millions of individuals. These decisions involve evidence-accumulation processes by individuals and information exchange within the group. Moreover, these decisions take place in complex, dynamic, and spatially structured environments, which shape the flow of information between group mates. We will present a statistical inference model for framing evidence accumulation and belief sharing in groups and some examples of how interactions shape decision efficiency in groups. Our canonical model is of Bayesian agents deciding between two equally likely options by accumulating evidence to a threshold. First passage times and error rates can be accurately estimated using asymptotics for order statistics in the limit of large group sizes. When neighbors only share their decisions with each other, groups comprised of individuals with a distribution of decision thresholds make more efficient decisions than homogeneous ones. To conclude, we will briefly discuss specific examples of the impacts of spatial and communication heterogeneity on collective decision making in foraging animal groups like honey bees and primates.

April 29

Antoine Mellet (University of Maryland)

Location: WebEx and CULM LECT II

Free boundary problems for cell motility

We discuss some free boundary problems modeling the crawling motion of cells on a substrate. In particular, we derive a new model as the singular limit of a diffuse interface approximation of Cahn-Hilliard type with a chemo-repulsive potential (which account for the formation of protrusions along the membrane of the cell). The resulting free boundary problem is a Hele-Shaw flow which combines the (regularizing) effects of surface tension with the (destabilizing) effects of the chemo-repulsive potential. It exhibits interesting properties (e.g. symmetry breaking and hysteresis phenomena) which are in good agreement with observations.

Updated: April 18, 2022