# All Seminars

Title: Initial Guesses for Sequences of Linear Systems in a GPU-accelerated Incompressible Flow Solver
Seminar: Numerical Analysis and Scientific Computing
Speaker: Anthony Austin of Naval Postgraduate School
Contact: Yuanzhe Xi, yxi26@emory.edu
Date: 2021-04-30 at 1:30PM
Venue: https://emory.zoom.us/j/95900585494
Abstract:
We revisit the projection method of Fischer for generating initial guesses when iteratively solving a sequence of linear systems, showing that it can be implemented efficiently in GPU-accelerated PDE solvers. We specifically consider such a solver for the incompressible Navier--Stokes equations and study the effectiveness of the method at reducing solver iteration counts. Additionally, we propose new methods for generating initial guesses based on stabilized polynomial extrapolation and show that they are generally competitive with projection methods while requiring only half the storage and performing considerably less data movement and communication. Our implementations of these algorithms are freely available as part of the libParanumal collection of GPU-accelerated flow solvers.
Title: Direct Solvers for Elliptic PDEs
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
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
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
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
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
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)
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
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
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
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
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
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
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
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
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
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.