Applied Mathematics Seminar, Spring 2017
Boyd Building, 221, Mondays, 2:30 PM
1/30: Frank Gao. Department of Mathematics, University of Idaho
Abstract: Metric entropy, or logarithm of covering number, was first introduced by Kolmogorov in 1961 as a way of gauging geometric complexity of subsets in metric spaces. Over half a century, metric entropy has come to play an increasingly important role in a wide range of problems in mathematics including approximation theory, probability theory, information theory and statistics. In this talk, I will introduce some recent results on metric entropy estimates, and how these results can be used to determine optimal rates of convergence in estimation problems in statistics.
2/6: Daniel McKenzie, Department of Mathematics, University of Georgia: Title: A Compressive Sensing Approach to Graph Clustering
Abstract: Given a large graph G, it is interesting to determine the `cluster' to which a certain vertex belongs. One of the most accurate ways to determine this is by examining the eigenvectors of L(G), the graph Laplacian. Unfortunately determining the eigenvectors of a matrix requires O(n^3) operations, which is prohibitively slow for truly large graphs. In this talk we present a novel, compressed sensing inspired approach which is able to determine a cluster of interest in O(n_1 x n^2) operations, where n_1 is the size of said cluster. Some interesting examples will be shown.
2/13: Alexander Petukhov, Department of Mathematics, University of Georgia: Title: Deep Learning: Design and Applications.
Abstract: The neural networks trained by the deep learning algorithms suddenly became a part of our life. Self-driving cars, fast computer search and understanding of images and video are now reality.
Probably,so far the software products creating new objects of art (like paintings in style of the famous masters) and musical pieces are part of the entertainment industry rather than the actual art. However, this direction is so young and is being developed so actively that currently any fantasies become true very fast. While the neural networks cannot write papers for lazy scientists, they can read papers and report their summaries. The new technologies based on deep learning appear daily.
We will discuss the principles of the neural networks design and training. Some examples and demos showing how neural networks operate with data will be presented.
2/20: Clay Mersmann, Due to Departmental Cantrell Lectures in the afternoon, this seminar will be rescheduled in a later date.
2/27: Mu Lin, Ork Ridge National Laborary, Title: Weak Gelerkin Finite Element Methods and Numerical Applications
Weak Galerkin Finite Element Methods and Numerical Applications
Abstract: Weak Galerkin FEMs are new numerical methods that were first introduced by Wang and Ye for solving general second order elliptic PDEs. The differential operators are replaced by their weak discrete derivatives, which endows high flexibility such as almost arbitrary polygonal mesh and high order polynomial basis. This new method is a discontinuous finite element algorithm, which is parameter free, symmetric, symmetric, and absolutely stable. Furthermore, through the Schur-complement technique, an efficient implementation of the WG can be developed. Several applications of weak Galerkin methods will be discussed in this talk.
3/13: Zerotti Woods, Department of Mathematics, University of Georgia: Title: A Brief Introduction To Information Theory
"The fundamental problem of communication is that of reproducing at one point either exactly or approximately a message selected at another point. (Claude Shannon, 1948)
In this talk we will discuss ways to achieve nearly perfect communication over an imperfect, noisy communication channel. We will also discuss the limitations and the possibilities of communication over noisy channels. I will attempt to make this talk very friendly by giving a few simple examples.
3/20: Jay Lanterman, Department of Mathematics, University of Georgia: Title is "A construction of Hermite interpolatory Wachspress functions on some quadrilaterals."
Abstract: Generalized barycentric coordinates, especially Wachspress coordinates, have seen recent application in the development of polygonal splines, which are used as an analogue of splines to solve PDEs in a 2015 paper of Floater & Lai. While polygonal splines show some advantages over traditional splines, especially in the reduction of edges (and therefore degrees of freedom), one of their biggest relative drawbacks is their inability to enforce global smoothness. In this talk we'll detail a construction of local basis functions for a C^1 subspace of degree-5 Wachspress functions as a first step toward a more general C^1 polygonal spline space. The local basis functions are particularly well-suited for Hermite interpolation.
3/27: Clayton Mersmann, Department of Mathematics, University of Georgia: Title is: Spline Solutions of the Maxwell Equations
Abstract:The Maxwell equations are a system of first-order PDEs that describe all classical electromag- netic phenomena. We review common formulations of these equations for numerical study, and present a simple potential formulation for our spline analysis. In contrast with common finite element schemes, our spline solutions have an arbitrary degree of global smoothness; thus, we can obtain an accurate approximation of the electric and magnetic field quantities in question.
We conclude with some preliminary numerical results, including a computation of the electric field arising from a shielded microstrip.
4/3: Dr. Hari Shankar Mahato, College of Engineering, University of Georgia: Title: Upscaling of a System of Semilinear Diffusion‐Reaction Equations in a Heterogeneous Medium: Multi‐Scale Modeling and Periodic Homogenization
Abstract: A porous medium (concrete, soil, rocks, water reservoir, e.g.) is a multi‐scale medium where the heterogeneities present in the medium are characterized by the micro scale and the global behaviors of the medium are observed at the macro scale. The upscaling from the micro scale to the macro scale can be done via averaging methods. In this talk, diffusion and reaction of several mobile chemical species are considered in the pore space of a heterogeneous porous medium. The reactions amongst the species are modelled via mass action kinetics and the modelling leads to a system of multi‐species diffusion-reaction equations (coupled semi‐linear partial differential equations) at the micro scale where the highly nonlinear reaction rate terms are present at the right hand sides of the system of PDEs, cf. . The existence of a unique positive global weak solution is shown with the help of a Lyapunov functional, Schaefer’s fixed-point theorem and maximal Lp‐regularity, cf. [2, 3]. Finally, with the help of periodic homogenization and two‐scale convergence we upscale the model from the micro scale to the macro scale, e.g. [1, 3]. Some numerical simulations will also be shown in this talk, however for the purpose of illustration; we restrict ourselves to some relatively simple 2‐ dimensional situations.
4/10: Lihong Qiao, Department of Mathematics, University of Georgia: Title: AM-FM Modulation characters Extraction and its application
Abstract: We will first explain AM-FM modulation of one dimensional signals. Several approaches will be reviewed and difficulities will be pointed out. Then I will explain AM-FM modulation for two dimensional signals/images. Numerical results will be shown.
4/17: Kelly Black, Department of Mathematics, University of Georgia: Title: Study of Finite Time Blowup in a 3 Species Predator-Prey System Using Spectral Method by Kelly Black (UGA), Rana Parshad (Clarkson U.), Emmanuel Quansah (Clarkson U.)
Abstract: We analyze three variations on the Leslie-Gower predator prey model with an emphasis on the role of biological controls to attenuate explosive increase of invasive species. The system is modified to promote blow-up in the top most level predator species. The models are modified to allow for finite time blow-up to mimic rapid growth in populations. The variations are then used to explore possible remediation approaches to mitigate the impact of calamitous growth in the top level predator. The controls are based on the promotion of spatial dampening and do not make use of either chemical control or introduction of other non-native species.
The results shared here come from a numerical exploration of the system by Emmanuel Quansah. These results were part of his doctoral thesis. His focus was to explore the impact on blow-up times for the parameters in the resulting systems.
4/24: Abraham Varghese, Department of Mathematics, University of Georgia: Title: An Accurate Matrix Completion Algorithm,
ABSTRACT: We first review some existing algorithms to complete a matrix from its partial entries given. Usually these algorithms are either based on constraint minimization or unconstrained minimization with parameters. We propose a new unconstrained minimization without any parameters. The minimizing functional is based on a summation of gramian of some submatrices. We then use Newton iterative technique to obtain the minimizer with initial guessed matrix obtained from using the well-known Orthogonal Rank-One Matrix Pursuit (OR1MP). Numerical results show we are able to complete a matrix with much better accuracy than existing algorithms.