All Seminars
| Title: Improving Multigrid Methods with Deep Neural Networks |
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| Defense: Dissertation |
| Speaker: Ru Huang of Emory University |
| Contact: Ru Huang, ru.huang@emory.edu |
| Date: 2022-03-28 at 10:30AM |
| Venue: MSC N215 |
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| Abstract: Multigrid methods are one of the most efficient techniques for solving large sparse linear systems arising from Partial Differential Equations (PDEs) and graph Laplacians from machine learning applications. There are two key components of multigrid, smoothing which aims at reducing high-frequency errors on each grid level, and coarse grid correction which interpolates the solution at the coarse grid. However, finding optimal smoothing algorithms is problem-dependent and can impose challenges for many problems. Meanwhile, as the multigrid hierarchy is formed, coarse-grid operators have significantly more nonzeros per row than the original fine-grid operator, which generates high parallel communication costs on coarse-levels. In this talk, I will first talk about my research on developing an efficient adaptive framework for learning optimal smoothers from operator stencils in the form of convolutional neural networks (CNNs). I will also talk about our deep learning framework for sparsifying coarse grid operators. I will demonstrate how these techniques can be used for challenging anisotropic rotated Laplacian problems, variable coefficient diffusion problems, and linear elasticity problems. |
| Title: Generating Graphs with Deep Learning and Graph Theory |
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| Defense: Dissertation |
| Speaker: Yuliang Ji of Emory University |
| Contact: Yuliang Ji, yuliang.ji@emory.edu |
| Date: 2022-03-25 at 9:00AM |
| Venue: MSC E406 |
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| Abstract: Deep generative models attract lots of attention in recent years. With deep neural networks and specific designs, deep generative models can generate high-quality realistic data. In this dissertation, I focus on combining the deep generative models with the traditional graph theory algorithms to reduce the dependence on the volume of the training data and improve the quality of the generated graphs. I first propose a deep learning method to improve the Havel-Hakimi graph realization algorithm to generate doppelganger graphs from a single graph. Second, I present a few new architectures of normalizing flow models with improved performance and theoretical guarantees. Finally, I develop a permutation invariant method via leveraging graph theory and denoising diffusion models for generating molecular graphs. |
| Title: The Laplace and Heat Operators on Quantum Graphs |
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| Defense: Dissertation |
| Speaker: Kenny Jones of Emory University |
| Contact: Wesley Jones, wesley.kenderdine.jones@emory.edu |
| Date: 2022-03-24 at 3:00PM |
| Venue: Emerson E363 |
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| Abstract: This presentation will introduce general strategies, techniques, and results for differential operators on quantum graphs. The focus of the talk will be on new results presented in my doctoral dissertation. The first result is a sharp diameter bound on the spectral gap for quantum graphs. Followed by a new technique for bounding the heat kernel on quantum graphs and several bounds for the heat kernel. Finally, I will present an original equation and derivation of the mean value theorem for the heat equation on quantum graphs and give a bound for the mean value theorem. |
| Title: Codes from Fiber Products of Curves |
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| Seminar: Algebra |
| Speaker: Mckenzie West of University of Wisconsin - Eau Claire |
| Contact: David Zureick-Brown, dzureic@emory.edu |
| Date: 2022-03-22 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: Error correcting codes are used to store information efficiently while still allowing for recovery in the case of partial loss. Recently work has begun to construct codes using the algebraic geometric properties of curves over finite fields. In this talk, we'll introduce the general construction of codes from fiber products curves and provide a few explicit examples of locally recoverable codes created using these methods. This collaborative work started at the inaugural Rethinking Number Theory workshop in October 2020. |
| Title: Weak degeneracy of graphs |
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| Seminar: Discrete Math |
| Speaker: Anton Bernshteyn of Georgia Tech |
| Contact: Liana Yepremyan, liana.yepremyan@EMORY.EDU |
| Date: 2022-03-16 at 4:00PM |
| Venue: MSC E406 |
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| Abstract: Motivated by the study of greedy algorithms for graph coloring, we introduce a new graph parameter, which we call weak degeneracy. This notion formalizes a particularly simple way of "saving" colors while coloring a graph greedily. It turns out that many upper bounds on chromatic numbers follow from corresponding bounds on weak degeneracy. In this talk I will survey some of these bounds as well as state a number of open problems. This is joint work with Eugene Lee (Carnegie Mellon University). |
| Title: Scale Inverse Problems: Low-Rank Approximations and Optimization |
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| Defense: Dissertation |
| Speaker: Chang Meng of Emory University |
| Contact: Chang Meng, chang.meng@emory.edu |
| Date: 2022-03-03 at 12:00PM |
| Venue: MSC W501 |
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| Abstract: Inverse problems can be found in a variety of scientific applications, and the development of efficient and reliable methods remain an essential and challenging task. In this thesis, we introduce novel low-rank solvers for linear systems that arise from large scale inverse problems, which are usually ill-posed and require the use of regularization to obtain meaningful solutions. The new methods are developed around the concept of regularization: i) the low-rank, Kronecker product based forward model approximation method involves the approximation of a truncated singular value decomposition; and ii) the low-rank Krylov subspace methods are based on nuclear norm regularization. We explore the performance of these novel low-rank methods in various imaging applications such as image deblurring, inpainting and computer tomography. Besides applications where the forward model is known and fixed, we also consider an extended application, where the forward model is not exactly known and requires calibration. In this context, we are able to not only apply our new low-rank methods, but also propose a new hybrid machine learning and block coordinate descent algorithm that effectively improves solution accuracy. |
| Title: Introduction to Quantum Graphs |
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| Seminar: Analysis Reading |
| Speaker: Haozhe Yu of Emory University |
| Contact: Yiran Wang, yiran.wang@EMORY.EDU |
| Date: 2022-02-25 at 3:00PM |
| Venue: MSC E406 |
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| Abstract: We will introduce basic concepts of quantum graphs. The major focus will be the spectral gap of quantum graphs because it is one of the most important properties of the graph. We will present examples to illustrate this idea and introduce techniques to estimate the spectral gap. |
| Title: RISING: A stable and reliable approach to the solution of Inverse Problems with Neural Networks |
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| Seminar: Numerical Analysis and Scientific Computing |
| Speaker: Davide Evangelista of University of Bologna |
| Contact: Yuanzhe Xi, yuanzhe.xi@emory.edu |
| Date: 2022-02-24 at 10:00AM |
| Venue: MSC W201 |
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| Abstract: Solving Inverse Problems usually requires inverting underdetermined and ill-conditioned linear operators. Classically, this is obtained by solving a regularized variational problem. Unfortunately, computing the solution usually requires a huge amount of time and computational resources. On the other hand, the use of data-driven approach such as Neural Networks permit to compute the solution of the Inverse Problems in a relatively small amount of time, since it does not require the explicit computation of the forward linear operator. Moreover, the results computed by Neural Networks show an extraordinary visual quality, usually order of magnitude greater than the state-of-the-art variational models. Unfortunately, it is known that classical Neural Network approaches are extremely unstable, and the quality of the obtained results can be easily reduced by adversarial examples. We propose RISING, an hybrid approach that permits stable and reliable solutions of Inverse Problems via Neural Networks. |
| Title: Probabilistic Bezout Over Finite Fields, and Some Applications |
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| Seminar: Math Colloquia |
| Speaker: Bhargav Narayanan of Rutgers University |
| Contact: Liana Yepremyan, liana.yepremyan@emory.edu |
| Date: 2022-02-23 at 4:00PM |
| Venue: MSC W303 |
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| Abstract: What is the distribution of the number of distinct roots of k random polynomials (of some fixed degree) in k variables? I will talk about a recently proved Bezout-like theorem that gives us a satisfactory answer over (large) finite fields. This result can be used to construct several interesting families of “extremal graphs”. I shall illustrate this method by 1) discussing the easiest applications in detail, reproving some well-known lower bounds in extremal graph theory, and 2) outlining how this method has recently found applications in establishing hardness results for a few basic computational problems. |
| Title: Harmonic Analysis on Cosphere Bundle |
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| Seminar: Analysis Reading |
| Speaker: Guangqiu Liang of Emory University |
| Contact: Yiran Wang, yiran.wang@EMORY.EDU |
| Date: 2022-02-18 at 3:00PM |
| Venue: MSC E406 |
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| Abstract: |