All Seminars
| Title: Direct Solvers for Elliptic PDEs |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Per-Gunnar Martinsson of UT Austin |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2021-04-23 at 1:30PM |
| Venue: https://emory.zoom.us/j/95900585494 |
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| Abstract: That the linear systems arising upon the discretization of elliptic PDEs can be solved efficiently is well-known, and iterative solvers that often attain linear complexity (multigrid, Krylov methods, etc) have proven very successful. Interestingly, it has recently been demonstrated that it is often possible to directly compute an approximate inverse to the coefficient matrix in linear (or close to linear) time. The talk will describe some recent work in the field and will argue that direct solvers have several advantages, including improved stability and robustness, the ability to solve certain problems that have remained intractable to iterative methods, and dramatic improvements in speed in certain environments. The talk will in particular focus on methods for solving elliptic PDEs with oscillatory solutions. These are particularly compelling targets for direct solvers, as it is notoriously difficult to attain fast convergence for iterative solvers in this environment. But they also pose additional challenges, as the inherent ill-conditioning of the physics of the problem require very high precision in both discretizing the PDE, and in solving the resulting linear system. |
| Title: The Hasse norm theorem and a local-global principle for multinorms |
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| Defense: Master's Thesis |
| Speaker: Yazan Alamoudi of Emory University |
| Contact: Yazan Alamoudi, yazan.mohammed.alamoudi@emory.edu |
| Date: 2021-04-20 at 4:00PM |
| Venue: https://tinyurl.com/YAlamoudi |
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| Abstract: In this defense, I will present the main result of the thesis, namely a local-global principle for multinorms from étale algebras associated to dihedral extensions of number fields of degree 2n. More precisely, the étale algebra obtained as the product of n field extensions which are the fixed fields under reflections satisfies the local-global principle for multinorms. A basic ingredient in the proof is the classical Hasse norm theorem and we shall present an outline of the proof of this theorem.\\ \\ Zoom Meeting: https://tinyurl.com/YAlamoudi. Passcode: qbdF0v |
| Title: Elliptic Curves and Moonshine |
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| Defense: Dissertation |
| Speaker: Maryam Khaqan of Emory University |
| Contact: Maryam Khaqan, maryam.khaqan@emory.edu |
| Date: 2021-04-16 at 12:00PM |
| Venue: https://tinyurl.com/MKhaqan |
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| Abstract: In this talk, I will describe the main results of my doctoral dissertation. The first main result of my thesis is a characterization of all infinite-dimensional graded modules for the Thompson group whose graded traces are certain weight 3/2 weakly holomorphic modular forms satisfying special properties. This characterization serves as an example of moonshine for the Thompson group.\\ \\ I will begin the talk by giving a brief history of moonshine, describing some of the existing examples of the phenomenon in the literature, and discussing how my work fits into the story. I will then demonstrate how we can use the aforementioned Thompson-modules to study geometric invariants (e.g., rank, p-Selmer groups, and Tate—Shafarevich groups) of certain families of elliptic curves. In particular, this serves as an example of using moonshine answer questions in number theory.\\ \\ Meeting ID: 984 6807 4730 Passcode: 196884 |
| Title: Randomization in Numerical Linear Algebra (RandNLA) |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Petros Drineas of Purdue University |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2021-04-09 at 1:30PM |
| Venue: https://emory.zoom.us/j/95900585494 |
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| Abstract: The introduction of randomization in the design and analysis of algorithms for matrix computations (such as matrix multiplication, regression, the Singular Value Decomposition (SVD), etc.) over the past 20 years provided a new paradigm and a complementary perspective to traditional numerical linear algebra approaches. These novel approaches were motivated by technological developments in many areas of scientific research that permit the automatic generation of Big Data, which are often modeled as matrices. In this talk, we will primarily focus on how such approaches can be used to design fast solvers for least-squares problems, ridge-regression problems, and even linear programs. |
| Title: Top-k Extreme Contextual Bandits with Arm Hierarchy |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Lexing Ying of Stanford University |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2021-03-26 at 1:30PM |
| Venue: https://emory.zoom.us/j/95900585494 |
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| Abstract: Contextual bandit is an online decision making framework that has found many applications in recommendation systems and search tasks. In this talk, we consider the extreme contextual bandit problem where the enormous number of arms poses the main theoretical and algorithmic challenges. This setting is particularly relevant to the mission of Amazon Search but yet rather under-explored in the literature. To address the large arm space, we introduce two new techniques. The first is an extension of the inverse gap weighting to the multiple arm feedback case, where the optimal regret bound is shown under realizability condition. The second is an arm space hierarchy that exploits the potential similarity between the arms. By combining these two techniques, we develop an extreme top-k contextual bandit algorithm that scales logarithmically in terms of the number of arms. Joint work with Rajat Sen, Alexander Rakhlin, Rahul Kidambi, Dean Foster, Daniel Hill, and Inderjit Dhillon. |
| Title: Isogenies of Elliptic Curves and Arithmetical Structures on Graphs |
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| Defense: Dissertation |
| Speaker: Tomer Reiter of Emory University |
| Contact: Tomer Reiter, tomer.reiter@emory.edu |
| Date: 2021-03-19 at 11:00AM |
| Venue: https://emory.zoom.us/j/92804829998?pwd=OFpZcWdlS2lrUFRLbDZQNklxZ3IwQT09 |
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| Abstract: In this defense, we prove two results that come from studying curves. The first is a classification result for elliptic curves. Let $\mathbf{Q}(2^{\infty})$ be the compositum of all quadratic extensions of $\mathbf{Q}$. Torsion subgroups of rational elliptic curves base changed to $ \mathbf{Q}(2^{\infty}) $ were classified by Laska, Lorenz, and Fujita. Recently, Daniels, Lozano-Robledo, Najman, and Sutherland classified torsion subgroups of rational elliptic curves base changed to $ \mathbf{Q} (3^{\infty})$, the compositum of all cubic extensions of $ \mathbf{Q} $. We classify cyclic isogenies of rational elliptic curves base changed to $\mathbf{Q}(2^{\infty}) $, for all but finitely many elliptic curves over $ \mathbf{Q}(2^{\infty}) $.\\ \\ Next, we turn to arithmetical structures, which Lorenzini introduced to model degenerations of curves. Let $G$ be a connected undirected graph on $n$ vertices with no loops but possibly multiedges. Given an arithmetical structure $(\textbf{r}, \textbf{d})$ on $G$, we describe a construction which associates to it a graph $G'$ on $n-1$ vertices and an arithmetical structure $(\textbf{r}', \textbf{d}')$ on $G'$. By iterating this construction, we derive an upper bound for the number of arithmetical structures on $G$ depending only on the number of vertices and edges of $G$. In the specific case of complete graphs, possibly with multiedges, we refine and compare our upper bounds to those arising from counting unit fraction representations. |
| Title: Applications of Fractional Operators from Optimal Control to Machine Learning |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Harbir Antil of George Mason University |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2021-03-19 at 1:30PM |
| Venue: https://emory.zoom.us/j/95900585494 |
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| Abstract: Fractional calculus and its application to anomalous diffusion has recently received a tremendous amount of attention. In complex/heterogeneous material mediums, the long-range correlations or hereditary material properties are presumed to be the cause of such anomalous behavior. Owing to the revival of fractional calculus, these effects are now conveniently modeled by fractional-order differential operators and the governing equations are reformulated accordingly. In the first part of the talk, we plan to introduce both linear and nonlinear, fractional-order differential equations. As applications, we will develop new physical models for geophysical electromagnetism and a new notion of optimal control will be discussed. In the second part of the talk, we will focus on novel Deep Neural Networks (DNNs) based on fractional operators. We plan to discuss the approximation properties and apply them to image denoising and tomographic reconstruction problems. We will establish that these DNNs are also excellent surrogates to PDEs and inverse problems with multiple advantages over the traditional methods. If time permits, we will conclude the talk by showing some of our initial results on chemically reacting flows using DNNs which clearly shows the effectives of the proposed approach. |
| Title: Randomized Fast Subspace Descent Methods |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Long Chen of University of California at Irvine |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2021-03-12 at 1:30PM |
| Venue: https://emory.zoom.us/j/95900585494 |
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| Abstract: In this talk, we propose randomized fast subspace descent (rFASD) methods and derive its convergence analysis. An outline of rFASD is as follows. Randomly choose a subspace according to some sampling distribution and find a search direction in that subspace. The update is giving by the subspace correction with such search direction and appropriated step-size. Convergence analysis for convex function and strongly convex function will be given. SGD, Coordinate Descent (CD), Block CD, and Block CD with Newton solver on each block can be viewed as examples in our framework. This is a joint work with Xiaozhe Hu and Huiwen Wu. |
| Title: ADAHESSIAN: An Adaptive Second Order Optimizer for Machine Learning |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Michael W. Mahoney of ICSI and Department of Statistics, UC Berkeley |
| Contact: Yuanzhe Xi, yxi26@emory.edu |
| Date: 2021-03-05 at 1:30PM |
| Venue: https://emory.zoom.us/j/95900585494 |
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| Abstract: Second order optimization algorithms have a long history in scientific computing, but they tend not to be used much in machine learning. This is in spite of the fact that they gracefully handle step size issues, poor conditioning problems, communication-computation tradeoffs, etc., all problems which are increasingly important in large-scale and high performance machine learning. A large part of the reason for this is that their implementation requires some care, e.g., a good implementation isn't possible in a few lines of python after taking a data science boot camp, and that a naive implementation typically performs worse than heavily parameterized/hyperparameterized stochastic first order methods. We describe ADAHESSIAN, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the Hessian. ADAHESSIAN includes several novel performance-improving features, including: (i) a fast Hutchinson based method to approximate the curvature matrix with low computational overhead; (ii) a spatial averaging to reduce the variance of the second derivative; and (iii) a root-mean-square exponential moving average to smooth out variations of the second-derivative across different iterations. Extensive tests on natural language processing, computer vision, and recommendation system tasks demonstrate that ADAHESSIAN achieves state-of-the-art results. The cost per iteration of ADAHESSIAN is comparable to first-order methods, and ADAHESSIAN exhibits improved robustness towards variations in hyperparameter values. |
| Title: Moduli of Wildly Ramified Covers of Curves |
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| Seminar: Algebra and Number Theory |
| Speaker: Andrew Kobin of University of California-Santa Cruz |
| Contact: David Zureick-Brown, dzureic@emory.edu |
| Date: 2021-02-22 at 1:00PM |
| Venue: https://emory.zoom.us/j/94062245183 |
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| Abstract: One of many hazards in the jungle of characteristic p algebraic geometry is the presence of wild ramification, which is in a loose sense what happens when objects possess automorphisms of order divisible by p. In this talk, I will tell an incomplete story of wild ramification for algebraic curves in characteristic p, starting with the fundamental example of Artin-Schreier curves. I will also describe some work in progress towards a description of the moduli stack of Artin-Schreier covers of curves, part of which relies on results in a recent preprint (arXiv:1910.03146). |