Applied Math Collective

Applied Math Collective was initiated by my advisors and Fernando Guevara Vasquez. The aim is to provide an informal platform where the speaker discusses general-interest "SIAM review"-style applied math papers, led by either faculty or graduate student. We meet Thursdays at 2pm in LCB 222. Please contact me if you would like to attend or give a talk so that I can add you to the mailing list.

➜ Spring 2019

January 10
Speaker: Chee Han Tan
Title: The Perron-Frobenius theorem
Abstract: In real word situations, matrices more often than not have nonnegative entries. For a real square positive matrix, Oskar Perron proved in 1907 that it has a unique largest real eigenvalue amd the corresponding eigenvector can be chosen to have positive components. Georg Frobenius generalised this result in 1912 to certain classes of nonnegative matrices. In this talk we will present a linear algebra proof of Perron's theorem and discuss the crucial observation of Frobenius in generalising Perron's theorem.

January 17
Speaker: Chee Han Tan
Title: A resolvent proof of the Perron-Frobenius theorem
Abstract: This is a continuation of the previous talk. We will prove the Perron-Frobenius theorem using the idea of resolvent operator, defined as $R_\lambda(A) = (\lambda I - A)^{-1}$ for a given square matrix $A$ and $\lambda$ in the resolvent set. We will recall several facts about $R_\lambda(A)$ and demonstrate how to compute the Laurent expansion of $R_\lambda(A)$ about the eigenvalues of $A$, which is the key ingredient in the proof. This talk is largely based on the SIAM-review article "The many proofs and applications of Perron’s theorem" by C. R. MacCluer and the American Mathematical Monthly article "Linear Algebra via Complex Analysis" by A. P. Campbell and D. Daners.

January 24
Speaker: Ryeongkyung (R.K.) Yoon
Title: Universal convergence theorem and application in artificial neural network
Abstract: The universal convergence theorem is used in artificial neural network. I will state and prove this theorem following the paper "Approximation by Superpositions of a Sigmoidal Function" by G. Cybenkot and briefly introduce application to artificial neural network.

February 7
Speaker: Nathan Willis
Title: Block Operators and Spectral Discretizations
Abstract: In this talk I will present the J. L. Aurentz and L. N. Trefethen 2017 SIAM Review paper "Block Operators and Spectral Discretizations". The authors impress upon the reader the utility of block operator diagrams before discretizing the problem and illustrate the strength of this idea through several examples. We will work through a portion of these examples where we see the solutions to boundary value problems, variable-coefficient problems, eigenvalue problems, optimization problems, and problems with nonlinear operators. After setting up the problems as block operators they will be solved in Matlab exploiting the Chebfun package.

February 14
Speaker: Huy Dinh
Title: Anomalous Diffusion in Sea Ice Dynamics
Abstract: Sea ice has rich behavior observed across multiple scales of time and space. Current models do not incorporate interactions between scales. Specifically, this talk will focus on the how advection and collisions between ice floes gives rise to anomalous diffusion of the ice pack. In order, this talk will present how anomalous diffusion is characterized, a smaller scale model which reproduces the behavior and how continuous time random walks fractional PDEs can be used to model the larger scale dynamics. Visualizations included!

February 21
Speaker: China Mauck
Title: Magnetic Resonance Electrical Impedance Tomography
Abstract: Electrical Impedance Tomography (EIT) is a noninvasive medical imaging technique that measures electrical conductivity and permittivity, making it possible to differentiate between types of biological tissue. The inverse problem of EIT image reconstruction is highly ill-posed. Magnetic Resonance Electrical Impedance Tomography (MREIT) was proposed in the early 1990s in response to this ill-posedness. We will examine the mathematical framework and the model for MREIT, associated image reconstruction algorithms, matters of uniqueness and convergence, and experimental results. This talk is based on the 2011 SIAM Review article "Magnetic Resonance Electrical Impedance Tomography (MREIT)" by Jin Keun Seo and Eung Je Woo.

February 28
Speaker: Rebecca Hardenbrook
Title: Optimization and Machine Learning
Abstract: In this talk I will be partially reviewing an extensive 2018 SIAM Review by L. Bottou, F.E. Curtis, and J. Nocedal titled "Optimization Methods for Large-Scale Machine Learning". I will introduce scenarios in which optimization problems arise from machine learning problems and discuss stochastic and batch methods for these optimization problems, two well-researched methods for solving such problems. Much of the discussion on current and future directions will be saved for a later talk.

March 7
Speaker: Dihan Dai
Title: Google's PageRank
Abstract: PageRank, the best known algorithms of Google, is used to rank the importance of web pages according to an eigenvector of a weighted link matrix. In this talk, I will present the linear algebra idea as well as the algorithm of the method, and perhaps some other applications of the PageRank methods. The talk is mainly based on 2006 SIAM review "The $25,000,000,000 Eigenvector: The Linear Algebra behind Google" by Kurt Bryan and Tanya Leise.

March 21
Speaker: Zexin Liu
Title: Compute the recurrence coefficients for a general non-negative measure
Abstract: Orthogonal polynomials have proven so mathematically foundational over the centuries that families associated to special measures $\mu$ are endowed with storied eponyms: Among them are the Legendre, Chebyshev, Hermite, Laguerre, Gegenbauer, Jacobi, and Bessel polynomials. Many theoretical results and computational algorithms owe their existence to the three-term recurrence relation. It is therefore of enormous importance that the recurrence coefficients an and bn be computed stably and accurately. For even modestly complicated measures $\mu$, this task is quite difficult. When $\mu$ falls within a certain collection of special forms, the recurrence coefficients can be computed using technical mathematics. However, when $\mu$ is not in this collection, then it can be quite difficult to compute these coefficients. So we aim to construct an algorithm so that we can compute the recurrence coefficients when $\mu$ is not in this collection.

March 28
Speaker: Ryleigh Moore
Title: An Introduction to Numerical Simulation of Stochastic Differential Equations
Abstract: In this talk, I will give a practical and (hopefully) accessible introduction to numerical methods for stochastic differential equations (SDEs). I will begin by introducing Brownian motion and providing an overview of integration with respect to such a motion using Itô and Stratonovich integrals. I will then discuss how the Euler-Maruyama and Milstein methods are used to simulate SDEs. This talk is based on the 2001 SIAM review article "An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations" by Desmond J. Higham.

April 4
Speaker: Fernando Guevara Vasquez
Title: The Cherkaev and Gibiansky variational principle
Abstract: The variational principles of Cherkaev and Gibiansky [1] allows to reformulate a system with losses (for example the conductivity equation with complex conductivity) as either a saddle point variational problem or a minimization variational problem. If time permits, we shall look at the generalization by Milton, Seppecher and Bouchitte [2].

[1] Cherkaev and Gibiansky, "Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli", Journal of Mathematical Physics 35, 127 (1994).
[2] Milton, Seppecher, Bouchitte, "Minimization variational principles for acoustics, elastodynamics, and electromagnetism in lossy inhomogeneous bodies at fixed frequency", Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465 (2009), no. 2102, 367–396.

April 11
Speaker: Huy Dinh
Title: Eigenfaces, a Perspective on a Machine's Perspective
Abstract: Eigenfaces was the first method used to allow computers to recognize faces without the excessive supervision from humans. Facial images carry an enormous quantity of information which humans condense into a few features to identify a face or person. We will give a friendly explanation on how principal component analysis can be used statistically to determine a compact representation of images. The Eigenfaces will be presented with commentary on the history and impact of the method.

April 18
Speaker: Christel Hohenegger
Title: Nineteen Dubious Ways to Compute the Exponential of a Matrix
Abstract: We will present various ways of computing a matrix exponential by means of examples. This talk is based on the 2003 SIAM review article "Nineteen Dubious Ways to Compute the Exponential of a Matrix, Twenty-Five Years Later" by Cleve Moler and Charles Van Loan.