All Seminars
| 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: |
| Title: Structured Low-Rank Approximation and the Proxy Point Method |
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| Seminar: Computational Math |
| Speaker: Mikhail Lepilov of Purdue University |
| Contact: Lars Ruthotto, lruthotto@emory.edu |
| Date: 2022-02-17 at 1:00PM |
| Venue: Online |
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| Abstract: Structured algorithms for large, dense matrices require efficient low-rank approximation methods to obtain their computational cost savings. There are many ways of obtaining such approximations depending on the type of matrix involved. For kernel matrices, analytic approximation methods such as truncated Taylor expansions or the proxy point method have been used in the Fast Multipole Method and other structured matrix algorithms. In this talk, we focus on the proxy point method, in which pairwise interactions between two separated clusters are approximated using the interactions of each cluster with a smaller chosen set of "proxy" points that separate the clusters. We perform a new accuracy analysis of this method when applied to 1D analytic kernels, and we then use it to devise a sublinear-time algorithm for constructing the HSS approximation of certain Cauchy and Toeplitz matrices. Finally, we extend this method its analysis to analytic kernels in several complex variables. |
| Title: Euler’s Polyhedron Formula and The Euler Characteristic |
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| Seminar: Mathematics |
| Speaker: Daniel Hess of University of Chicago |
| Contact: Bree Ettinger, betting@emory.edu |
| Date: 2022-02-04 at 10:00AM |
| Venue: MSC W201 and Zoom |
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| Abstract: A standard soccer ball is constructed using 12 regular pentagons and 20 regular hexagons. Is it possible to build one using only pentagons? How about only hexagons? It turns out that one of these is possible and one is not! The key to answering these questions is Euler’s Polyhedron Formula, which expresses a certain relationship between the number of vertices, edges, and faces in any convex polyhedron. In this talk, we will discuss this formula, the more general Euler characteristic, and applications such as the classification of the Platonic solids and triangulations of surfaces. |
| Title: Hamiltonian system and spectral inverse problems |
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| Seminar: Analysis Reading Seminar |
| Speaker: Guangqiu Liang of Emory University |
| Contact: Yiran Wang, yiran.wang@emory.edu |
| Date: 2022-02-04 at 3:00PM |
| Venue: MSC E406 |
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| Abstract: |
| Title: Geometric Group Theory and Untangling Earphones |
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| Seminar: Mathematics |
| Speaker: Neha Gupta of Georgia Institute of Technology |
| Contact: Bree Ettinger, betting@emory.edu |
| Date: 2022-02-02 at 10:00AM |
| Venue: MSC W201 and Zoom |
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| Abstract: Suppose you get your earphones entangled around a doughnut with two holes... an entirely probable scenario, right?! Then how "big" does your doughnut need to be, for you to successfully untangle your earphones? This is going to be a gentle introduction to groups, and how they connect to geometry. We will slowly build up to answering our original question. No prior experience with groups, geometry, or topology is required or assumed. This is joint work with Ilya Kapovich. |
| Title: Riemannian Geometry and Biomedical Data |
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| Seminar: Mathematics |
| Speaker: Sima Ahsani of Emory University |
| Contact: Bree Ettinger, betting@emory.edu |
| Date: 2022-01-28 at 10:00AM |
| Venue: MSC W201 and Zoom |
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| Abstract: Statistics is a science for everyone who wishes to collect, analyze, interpret, and understand data. Over the last few years, due to rapid technological developments, we handle increasingly large and complex data. For example, understanding biomedical data, that lie on matrix manifolds, in order to early detection of disease to prevent, control, or provide improved health care with low costs. Therefore, one of the most important steps in analyzing this type of data is understanding the structure of the surfaces where data live on them. To this end, differential geometry allows us to develop local methods to understand the global properties of surfaces that data lie on them. In this talk, after giving some examples of datasets that lie on curved spaces, I will provide an intuitive definition of Riemannian manifolds and their basic properties. Then, I will describe matrix manifolds and explain how to measure distances between two points and challenges that may arise when we want to calculate the mean of datasets and mention techniques that can be used to tackle these challenges. In the end, some resources and content will be provided to engage undergraduate students to know more about this research area to find the right way to develop their ideas and interest and take steps to build their future research areas. |