All Seminars

Title: Rethinking regularization in modern machine learning and computational imaging
Colloquium: Computational Mathematics
Speaker: Gregory Ongie of
Contact: James Nagy, jnagy@emory.edu
Date: 2020-01-13 at 4:00PM
Venue: MSC W303
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Abstract:
Optimization is central to both supervised machine learning and inverse problems in computational imaging. These problems are often ill-posed and some form of regularization is necessary to obtain a useful solution. However, new paradigms in machine learning and computational imaging necessitate rethinking the role of regularization, as I will illustrate with two examples. First, in the context of supervised learning with shallow neural networks, I will show how a commonly used form of regularization has a surprising reinterpretation as a convex regularizer in function space. This yields novel insights into the role of overparameterization and depth in learning with neural networks having ReLU activations. Second, I will discuss a novel network architecture for solving linear inverse problems in computational imaging called a Neumann network. Rather than using a pre-specified regularizer, Neumann networks effectively learn a regularizer from training data, outperforming classical techniques. Beyond these two examples, I will show how many open problems in the mathematical foundations of deep learning and computational imaging relate to understanding regularization in its many forms.
Title: Data Assimilation for State and Parameter Estimation in Hurricane Storm Surge Modeling
Colloquium: Computational Mathematics
Speaker: Talea L. Mayo of University of Central Florida
Contact: James Nagy, jnagy@emory.edu
Date: 2020-01-09 at 1:30PM
Venue: MSC W301
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Abstract:
Numerical hydrodynamic models are frequently used within the coastal science and engineering communities to simulate tides, waves, and hurricane storm surges. The applications of these simulations are vast, and include hindcasts of historical events, forecasts of impending hurricanes, and long-term flood risk assessment. However, like most numerical models, they are subject to epistemic and aleatoric uncertainties, due to factors including the approximation of relevant physical processes by mathematical models, the subsequent numerical discretization, uncertain boundary and initial conditions, and unknown model parameters. Quantifying and reducing these uncertainties is essential for developing reliable and robust hydrodynamic models. Data assimilation methods can be used to estimate uncertain model states (e.g. water levels) by informing model output with observations. I have developed these methods for hurricane storm surge modeling applications to reduce uncertainties resulting from coarse spatial resolution (i.e. limited computational resources) and uncertain meteorological conditions. While state estimation is beneficial for accurately simulating the storm surge resulting from a single, observed hurricane, broader contributions can be made by estimating uncertain model parameters. To this end, I have also developed these methods for parameter estimation. In this talk, I will discuss applications of data assimilation methods for both state and parameter estimation in hurricane storm surge modeling.
Title: Witt vectors and perfectoid rings
Seminar: Algebra
Speaker: Christopher Davis of UC Irvine
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-12-10 at 4:00PM
Venue: MSC W303
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Abstract:
This talk will introduce (p-typical) Witt vectors and the de Rham-Witt complex. Classically these are evaluated on rings of characteristic p, but our focus will be on rings which are p-torsion-free. In particular, we hope to discuss special features of Witt vectors and the de Rham-Witt complex when they are evaluated on p-torsion-free perfectoid rings. This is joint work with Irakli Patchkoria, and is based on earlier work of Hesselholt and Hesselholt-Madsen.
Title: Recent advances in the fast simulation of the Steady and Unsteady Incompressible Navier-Stokes Equations.
Seminar: Computational Math
Speaker: Alessandro Veneziani of Emory University
Contact: Yuanzhe Xi, yuanzhe.xi@EMORY.EDU
Date: 2019-12-06 at 2:00PM
Venue: MSC W303
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Abstract:
The efficient numerical solution of the Steady Incompressible Navier-Stokes equations is receiving more attention recently, driven by some applications where steadiness is solved as a surrogate of time average (see, e.g., [Tang, Chun Xiang, et al., JACC: Cardiov Imag (2019)]). The efficient numerical solution is challenged by the absence of the time derivative that makes the algebraic structure of the problem more problematic. In this talk, we cover some recent advances considering smart algebraic factorizations to mimic splitting strategies popular in the unsteady case [A. Viguerie, A. Veneziani, CMAME 330 (2018)], new stabilization techniques inspired by turbulence modeling [A. Viguerie, A. Veneziani, JCP 391 (2019)] and the treatment of nonstandard boundary conditions emerging in computational hemodynamics [A. Veneziani, A. Viguerie, in preparation (2019)], that inspired this research. In the latter case, the focus will be on the so-called backflow and inflow instabilities [H. Xu et al., to appear in JCP 2020] occurring in defective problems (i.e., problems where the data available are incomplete to make the mathematical formulation well-posed). Dedicated to the memory of Dr. G. Zanetti (1959-2019). The NSF Project DMS-1620406 supported this research.
Title: One trick with two applications
Seminar: Combinatorics
Speaker: Mathias Schacht of The University of Hamburg and Yale University
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2019-12-06 at 4:00PM
Venue: MSC W303
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Abstract:
We discuss a recent key lemma of Alweiss, Lovett, Wu and Zhang which led to big improvement for the Erdos-Rado sunflower problem. Essentially the same lemma was also crucial in the recent work of Frankston, Kahn, Narayanan, and Park showing that thresholds of increasing properties of binomial random discrete structures are at most a log-factor away from the so-called (fractional) expectation threshold. This fairly general result gives a new proof of the Johansson-Kahn-Vu theorem for perfect matchings in random hypergraphs.
Title: Relative local-global principles
Seminar: Algebra
Speaker: Danny Krashen of Rutgers University
Contact: David Zureick-Brown, dzb@mathcs.emory.edu
Date: 2019-12-03 at 4:00PM
Venue: MSC W303
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Abstract:
In various contexts, the Hasse principle can be used to transfer questions of rational points and triviality of Galois cohomology classes from global fields to local fields. Some such results have been extended, for example, in the work of Kato, Bayer-Flukiger-Parimala, Parimala-Preeti, Parimala-Sujatha, to apply to function fields over global fields. In this talk, I will discuss recent joint work with David Harbater and Alena Pirutka in which we examine to what extent local-global principles for one field extend to local-global principles for a function field over this field. We focus particularly on the case where one starts with a semiglobal field (a function fields over discretely valued fields).
Title: Swimming bacteria: Mathematical modelling and applications
Seminar: Computational Math
Speaker: Christian Esparza-Lopez of University of Cambridge
Contact: Irving Martinez, irving.martinez@emory.edu
Date: 2019-11-22 at 1:00PM
Venue: MSC W201
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Abstract:
Miniaturisation of actuators and power sources are two of the biggest technical challenges in the design and fabrication of microscopic robots. As it is often the case, Nature can offer insight into overcoming some of these challenges. Swimming bacteria, such as the well-studied flagellated E. coli, are known to be efficient swimmers with intricate sensing capabilities. They have thus inspired scientists to mimic them to improve the design of artificial micro-robots, often with biomedical purposes such as targeted drug delivery. This talk will consist of two parts. After a brief introduction to the study of swimming bacteria I will review the random walk model for bacterial diffusion and chemotaxis, and will show how to use it to describe the diffusive behaviour of artificial micro-swimmers propelled by swimming bacteria. In the second part of the talk I will address the problem of non-flagellated swimming bacteria. Specifically, we will study a minimal model to describe the dynamics of Smeliferum, a helical bacterium that swims by progressively changing the handedness of its body.
Title: Statistical Data Assimilation for Hurricane Storm Surge Modeling
Seminar: Numerical Analysis and Scientific Computing
Speaker: Talea L. Mayo of University of Central Florida
Contact: James Nagy, jnagy@emory.edu
Date: 2019-11-22 at 2:00PM
Venue: MSC W303
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Abstract:
Coastal ocean models are used for a variety of applications, including the simulation of tides and hurricane storm surges. As is true for many numerical models, coastal ocean models are plagued with uncertainty, due to factors including but not limited to the approximation of meteorological conditions and hydrodynamics, the numerical discretization of continuous processes, uncertainties in specified boundary and initial conditions, and unknown model parameters. Quantifying and reducing these uncertainties is essential for developing reliable and robust storm surge models. Statistical data assimilation methods are often used to estimate uncertain model states (e.g. storm surge heights) by combining model output with uncertain observations. We have used these methods in storm surge modeling applications to reduce uncertainties resulting from coarse spatial resolution. While state estimation is beneficial for accurately simulating the surge resulting from a single, observed storm, larger contributions can be made with the estimation of uncertain model parameters. In this talk, I will discuss applications of statistical data assimilation methods for both state and parameter estimation in coastal ocean modeling.
Title: Connected Fair Detachments of Hypergraphs
Seminar: Combinatorics
Speaker: Amin Bahmanian of Illinois State University
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2019-11-22 at 4:00PM
Venue: MSC W303
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Abstract:
Let $G$ be a hypergraph whose edges are colored. A $(u,n)$-detachment of $G$ is a hypergraph obtained by splitting a vertex $u$ into $n$ vertices, say $u_1,\dots, u_n$, and sharing the incident edges among the subvertices. A detachment is fair if the degree of vertices and multiplicity of edges are shared as evenly as possible among the subvertices within the whole hypergraph as well as within each color class. In this talk we solve an open problem from 1970s by finding necessary and sufficient conditions under which a $k$-edge-colored hypergraph $G$ has a fair detachment in which each color class is connected. Previously, this was not even know for the case when $G$ is an arbitrary graph. We exhibit the usefulness of our theorem by proving a variety of new results on hypergraph decompositions, and completing partial regular combinatorial structures.
Title: (-1)-homogeneous solutions of stationary incompressible Navier-Stokes equations with singular rays
Seminar: Analysis and PDEs
Speaker: Xukai Yan of Georgia Institute of Technology
Contact: Maja Taskovic, maja.taskovic@emory.edu
Date: 2019-11-21 at 3:00PM
Venue: MSC E308A
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Abstract:
In 1944, L.D. Landau first discovered explicit (-1)-homogeneous solutions of 3-d stationary incompressible Navier-Stokes equations (NSE) with precisely one singularity at the origin, which are axisymmetric with no swirl. These solutions are now called Landau solutions. In 1998 G. Tian and Z. Xin proved that all solutions which are (-1) homogeneous, axisymmetric with one singularity are Landau solutions. In 2006 V. Sverak proved that with just the (-1)-homogeneous assumption Landau solutions are the only solutions with one singularity. Our work focuses on the (-1)-homogeneous solutions of 3-d incompressible stationary NSE with finitely many singularities on the unit sphere. In this talk we will first classify all (-1)-homogeneous axisymmetric no-swirl solutions of 3-d stationary incompressible NSE with one singularity at the south pole on the unit sphere as a two dimensional solution surface. We will then present our results on the existence of a one parameter family of (-1)-homogeneous axisymmetric solutions with non-zero swirl and smooth on the unit sphere away from the south pole, emanating from the two dimensional surface of axisymmetric no-swirl solutions. We will also present asymptotic behavior of general (-1)-homogeneous axisymmetric solutions in a cone containing the south pole with a singularity at the south pole on the unit sphere . We also constructed families of solutions smooth on the unit sphere away from the north and south poles, and will have obtained some asymptotic stability result of these solutions. This is a joint work with Professor Yanyan Li and Li Li.