# Applied Mathematics Seminar

The Applied Mathematics Seminar (AMS) typically meets on Thursdays in Room 164 of the Adel Mathematics Building. Any faculty, students, or friends of the department are welcome to attend. Seminar talks are typically rotated between faculty (and on occasion students or visitors) with research interests related to applied mathematics, widely defined to include almost anything from ordinary and partial differential equations, dynamical systems, nonlinear and linear functional analysis, numerical analysis, optimization, operations research, scientific computing, modeling, and advanced physics.

### Fall 2021 Schedule

In Fall 2021 we are meeting from 3:00 to 4:00 in room 164 of the Adel Mathematics Building.

December 2: Ryan Kelly, “Finite Difference Methods from Birkhoff Polynomials”
Given a differential equation, one can approximate solutions with a collection of polynomials that satisfy the differential equation, and have internal continuity conditions. These conditions, when well posed, naturally give rise to finite difference matrices. We study a $C^3$ spline which enforces a second order ODE at midpoints and breakpoints. We attempt to analytically establish the numerically observed O(h^6) convergence to solutions of $y’’ = -y^3$ and $y’’ = \lambda y$ with zero Dirichlet boundary conditions. We first analyze the local behavior of each (Birkhoff) polynomial in the spline, then extend this to global behavior.

November 25: Thanksgiving

November 18: Ye Chen, “An introduction to adaptive MCMC for parameter estimation of nonlinear dynamical systems”
In our recent work of COVID-19 case number prediction, a comprehensive SEIRV (susceptible, exposed, infected, recovered, vaccinated) model is developed to explore the disease dynamic in US. Parameter estimation of this nonlinear SEIRV model (in general, any nonlinear dynamical system) remains a very challenging inverse problem due to its nonconvexity and ill-conditioning. In particular, Markov chain Monte Carlo (MCMC) methods have become increasingly popular as they allow for a rigorous analysis of parameter and prediction uncertainties without the need for assuming parameter identifiability. A broad spectrum of MCMC algorithms have been proposed, including various adaptive MCMC algorithms. I am going to review the theoretical criteria and the framework of stochastic approximation, and analyses the properties of an adaptive MCMC algorithm with vanishing adaptation.

November 11: Veterans Day

October 28, November 4: Jim Swift, “Finite Difference Approximations of the Eigenvalues of the Laplacian”
We focus on eigenvalues the Laplacian on the interval, and on regions in the plane, with either 0 Dirichlet or 0 Neumann boundary conditions. (This means that either u = 0, or the normal derivative of u = 0.) An outline of a way to get an order h^2 computation of the Neumann eigenvalues for a general region in the plane is presented. One of our former Master’s students tried and failed to do this.

October 14, 21: Rachel Neville, “Topological Data Analysis”

October 7: Mikhail Baltushkin, “A Numerical Investigation of the Double Pendulum”

September 30: Shafiu Jibrin, “The Weighted Analytic Center for Linear Matrix Inequalities”
We study the problem of computing the weighted analytic center for linear matrix inequality constraints. We apply and compare conjugate gradient (CG) methods to find the weighted analytic center. We use Newton’s method exact line search and Quadratic Interpolation inexact line search. The results indicate that both line searches work well with the methods, but exact line search handles weights better than the inexact line search when some weight is relatively much larger than the other weights.

September 23: John Neuberger, “Spline ODE Solvers”

September 16: Arthur Boggs, “The Mathematics behind Modern Digital Communications”
We apply radio theory, information theory, sampling theory, and coding theory to modern digital communications, which, understandably, relies heavily on digital signal processing. One might say Henry Morse sent the first digital signal in 1844 via his invention of the telegraph. Soon after, James Clerk Maxwell laid out the theory of electromagnetic radiation in 1865 with his famous set of partial differential equations combining electricity and magnetism. Heinrich Hertz applied Maxwell’s equations and built the first digital radio, the spark-gap transmitter and receiver in 1889. This leads us to the modern era where wireless communications are ubiquitous. We now use most of the electromagnetic spectrum from DC to light for all manner of communications. We will combine the relevant mathematics behind radio theory, information theory, sampling theory, and coding theory to gain a mathematical understanding of modern digital communications.

September 9: Jonathan Olson, “Tricomi’s Equation and the (so-called) Polar Wave Equation”

September 2: Ryan Kelly, “One-Dimensional Birkhoff Interpolation”

Contact the AMS organizers John Neuberger or Jim Swift if you have questions.