All Seminars

Title: Geometric and Statistical Approaches to Shallow and Deep Clustering
Seminar: Numerical Analysis and Scientific Computing
Speaker: James M. Murphy of Tufts University
Contact: Elizabeth Newman, elizabeth.newman@emory.edu
Date: 2021-11-05 at 12:30PM
Venue: MSC W201
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Abstract:
We propose approaches to unsupervised clustering based on data-dependent distances and dictionary learning. By considering metrics derived from data-driven graphs, robustness to noise and ambient dimensionality is achieved. Connections to geometric analysis, stochastic processes, and deep learning are emphasized. The proposed algorithms enjoy theoretical performance guarantees on flexible data models and in some cases guarantees ensuring quasilinear scaling in the number of data points. Applications to image processing and computational chemistry will be shown, demonstrating state-of-the-art empirical performance.
Title: A Multilevel Subgraph Preconditioner for Linear Equations in Graph Laplacians
Seminar: Numerical Analysis and Scientific Computing
Speaker: Junyuan Lin of Loyola Marymount University
Contact: Elizabeth Newman, elizabeth.newman@emory.edu
Date: 2021-10-29 at 12:30PM
Venue: https://emory.zoom.us/j/94914933211
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Abstract:
We propose a Multilevel Subgraph Preconditioner (MSP) to efficiently solve linear equations in graph Laplacians corresponding to general weighted graphs. The MSP preconditioner combines the ideas of expanded multilevel structure from multigrid (MG) methods and spanning subgraph preconditioners (SSP) from Computational Graph Theory. To start, we expand the original graph based on a multilevel structure to obtain an equivalent expanded graph. Although the expanded graph has a low diameter, which is a favorable property for the SSP, it has negatively weighted edges, which is an unfavorable property for the SSP. We design an algorithm to properly eliminate the negatively weighted edges and theoretically show that the resulting subgraph with positively weighted edges is spectrally equivalent to the expanded graph. Then, we adopt algorithms to find SSP, such as augmented low stretch spanning trees, for the positively weighted expanded graph and, therefore, provide an MSP for solving the original graph Laplacian. MSP is practical to find thanks to the multilevel property and has provable theoretical convergence bounds based on the support theory for preconditioning graphs.
Title: Variations on a theme of Shinzel and Wójcik
Seminar: Algebra and Number Theory
Speaker: Matthew Just of Emory University
Contact: David Zureick-Brown, dzureic@emory.edu
Date: 2021-10-19 at 4:00PM
Venue: MSC W301
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Abstract:
Let $\alpha$ and $\beta$ be rational numbers not equal to 0 or $\pm 1$. How does the order of $\alpha$ (mod $p$) compare to the order of $\beta$ (mod $p$) as $p$ varies? A result of Shinzel and W\'ojcik states that there are infinitely many primes $p$ for which the order of $\alpha$ (mod $p$) is equal to the order of $\beta$ (mod $p$). In this talk, we discuss the problem of determining whether there are infinitely many primes $p$ for which the order of $\alpha$ (mod $p$) is strictly greater than the order of $\beta$ (mod $p$). This is joint work with Paul Pollack.
Title: Galerkin Transformer
Seminar: Numerical Analysis and Scientific Computing
Speaker: Shuhao Cao of Washington University in St. Louis
Contact: Yuanzhe Xi, yuanzhe.xi@emory.edu
Date: 2021-10-15 at 12:30PM
Venue: https://emory.zoom.us/j/94914933211
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Abstract:
Transformer in "Attention Is All You Need" is now THE ubiquitous architecture in every state-of-the-art model in Natural Language Processing (NLP), such as BERT. At its heart and soul is the "attention mechanism". We apply the attention mechanism the first time to a data-driven operator learning problem related to partial differential equations. Inspired by Fourier Neural Operator which showed a state-of-the-art performance in parametric PDE evaluation, an effort is put together to explain the heuristics of, and to improve the efficacy of the attention mechanism. It is demonstrated that the widely-accepted "indispensable" softmax normalization in the scaled dot-product attention is sufficient but not necessary. Without the softmax normalization, the approximation capacity of a linearized Transformer variant can be proved rigorously for the first time to be on par with a Petrov-Galerkin projection layer-wise. Some simple changes mimicking projections in Hilbert spaces are applied to the attention mechanism, and it helps the final model achieve remarkable accuracy in operator learning tasks with unnormalized data. The newly proposed simple attention-based operator learner, Galerkin Transformer, surpasses the evaluation accuracy of the classical Transformer applied directly by 100 times, and betters all other models in concurrent research. In all experiments including the viscid Burgers' equation, an interface Darcy flow, an inverse interface coefficient identification problem, and Navier-Stokes flow in the turbulent regime, Galerkin Transformer shows significant improvements in both speed and evaluation accuracy over its softmax-normalized counterparts and other linearizing variants such as Random Feature Attention (Deepmind) or FAVOR+ in Performer (Google Brain). In traditional NLP benchmark problems such as IWSLT 14 De-En, the Galerkin projection-inspired tweaks in the attention-based encoder layers help the classic Transformer reach the baseline BLEU score much faster.
Title: PDE Models of Infectious Disease: Validation Against Data, Time-Delay Formulations, Data-Driven Methods, and Future Directions
Seminar: Numerical Analysis and Scientific Computing
Speaker: Alex Viguerie of Gran Sasso Science Institute
Contact: Alessandro Veneziani, avenez2@emory.edu
Date: 2021-10-08 at 12:30PM
Venue: MSC W201
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Abstract:
In the wake of the COVID-19 epidemic, there has been surge in the interest of mathematical modeling of infectious disease. Most of these models are based on the classical SIR framework and follow a compartmental-type structure. While the majority of such models are based on systems of ordinary differential equations (ODEs), there have been several recent works using partial differential equation (PDE) formulations, in order to describe epidemic spread across both space and time. This talk will focus on the application of such PDE models, and discuss different PDE formulations, the advantages and disadvantages, and assess their performance against measured data. Emphasis is placed on the incorporation of time-delay formulations and the application of modern data-driven techniques to further inform and enhance the performance of such models.
Title: Turán density of cliques of order five in 3-uniform hypergraphs with quasirandom links, Part 2
Seminar: Combinatorics
Speaker: Mathias Schacht of The University of Hamburg and Yale University
Contact: Dwight Duffus, dwightduffus@emory.edu
Date: 2021-10-08 at 3:00PM
Venue: MSC E408
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Abstract:
We continue with the proof that 3-uniform hypergraphs with the property that all vertices have a quasirandom link graph with density bigger than 1/3 contain a clique on five vertices. This time we focus on the structure of holes in reduced hypergraphs, which leads to a restricted problem that is easier to solve.
Title: Geometric equations for matroid varieties
Seminar: Algebra and Number Theory
Speaker: Ashley Wheeler of Georgia Institute of Technology
Contact: David Zureick-Brown, dzureic@emory.edu
Date: 2021-10-05 at 4:00PM
Venue: MSC W301
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Abstract:
Each point $x$ in $Gr(r, n)$ corresponds to an $r \times n$ matrix $A_x$ which gives rise to a matroid $M_x$ on its columns. Gel'fand, Goresky, MacPherson, and Serganova showed that the sets $\{y \in Gr(r, n)\,|\,M_y = M_x\}$ form a stratification of $Gr(r, n)$ with many beautiful properties. However, results of Mn\"ev and Sturmfels show that these strata can be quite complicated, and in particular may have arbitrary singularities. We study the ideals $I_x$ of matroid varieties, the Zariski closures of these strata. We construct several classes of examples based on theorems from projective geometry and describe how the Grassmann--Cayley algebra may be used to derive non-trivial elements of $I_x$ geometrically when the combinatorics of the matroid is sufficiently rich.
Title: Inference, Computation, and Games
Seminar: Numerical Analysis and Scientific Computing
Speaker: Florian Schaefer of Georgia Institute of Technology
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2021-09-24 at 12:30PM
Venue: MSC W201
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Abstract:

In this talk, we develop algorithms for numerical computation, based on ideas from competitive games and statistical inference.

In the first part, we propose competitive gradient descent (CGD) as a natural generalization of gradient descent to saddle point problems and general sum games. Whereas gradient descent minimizes a local linear approximation at each step, CGD uses the Nash equilibrium of a local bilinear approximation. Explicitly accounting for agent-interaction significantly improves the convergence properties, as demonstrated in applications to GANs, reinforcement learning, and computer graphics.

In the second part, we show that the conditional near-independence properties of smooth Gaussian processes imply the near-sparsity of Cholesky factors of their dense covariance matrices. We use this insight to derive simple, fast solvers with state-of-the-art complexity vs. accuracy guarantees for general elliptic differential- and integral equations. Our methods come with rigorous error estimates, are easy to parallelize, and show good performance in practice.

Title: Clusters and semistable models of hyperelliptic curves
Seminar: Algebra and Number Theory
Speaker: Jeffrey Yelton of Emory University
Contact: David Zureick-Brown, dzureic@emory.edu
Date: 2021-09-21 at 4:00PM
Venue: MSC W301
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Abstract:
For every hyperelliptic curve $C$ given by an equation of the form $y^2 = f(x)$ over a discretely-valued field of mixed characteristic $(0, p)$, there exists (after possibly extending the ground field) a model of $C$ which is \emph{semistable} -- that is, a model whose special fiber (i.e. the reduction over the residue field) consists of reduced components and has at worst very mild singularities. When $p$ is not $2$, the structure of such a special fiber is determined entirely by the distances (under the discrete valuation) between the roots of $f$, which we call the \emph{cluster data} associated to $f$. When $p = 2$, however, the cluster data no longer tell the whole story about the components of the special fiber of a semistable model of $C$, and constructing a semistable model becomes much more complicated. I will give an overview of how to construct``nice" semistable models for hyperelliptic curves over residue characteristic not $2$ and then describe recent results (from joint work with Leonardo Fiore) on semistable models in the residue characteristic $2$ situation.
Title: Steklov-Poincaré analysis of the basic three-domain stent problem
Seminar: Numerical Analysis and Scientific Computing
Speaker: Irving Martinez of Emory University
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2021-09-17 at 12:30PM
Venue: MSC W201
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Abstract:
The Steklov-Poincaré problem was previously considered in the artery lumen and wall setting with a single interface. Here the analysis is expanded to incorporate solute behavior in the presence of a fixed-volume, solid, simple stent. In this geometry, a third domain is added to the two-domain structure of artery wall and lumen. Through this intersecting domain volume setting there are three interfaces: lumen-wall, stent-lumen, and wall-stent. Steady-state incompressible Navier-Stokes equations are used to explain the behavior of blood through the lumen, while advection-diffusion dynamics are considered for the solute mechanics across the lumen, wall, and stent. Having a fixed blood velocity value, Steklov-Poincaré decomposition of the advection-diffusion equations is applied locally to each of the interfaces. To unify these instances on a global scale, their overall intersection is explored in a smaller manifold, reducing the problem to one previously solved by Quarteroni, Veneziani, and Zunino. Through finite element analysis (FEM), the solution is discretized and found to be convergent. Finally, computational simulations with one, three, and five stent rings, placed between the volumes of inner and outer cylindrical meshes, were performed using NGSolve, confirming the convergence of the solution and its relation to the coarseness of the mesh.