Seminar in Applied Mathematics

Department of Mathematics and Statistics

Texas Tech University

Spring 2018 - Wednesday, 4-5 pm - Room MATH 112

image Tuesday, January 23, 3:30pm, room CHEM 107
Backward SDE Methods for Nonlinear Filtering Problems
Feng Bao
We consider a dynamical system modeled by a stochastic differential equation with observational data available for the functional of the system state. The goal of the nonlinear filtering problem is to find the best estimate of the state of the dynamical system based on the observation. Some well-known approaches include extended Kalman filter, particle filter and Zakai filter. In this presentation, we shall present a new nonlinear filtering method, named the backward SDE filter. The backward SDE filter has the accuracy advantage of continuous filters such as the Zakai filter. In the meantime, it has the same sampling flexibility of discrete filters such as the particle filter. Both theoretical results and numerical experiments will be presented.
image Wednesday, January 31
Survival under Uncertainty
Dimitri Volchenkov
Stochastic modeling is used to estimate survival chances under uncertainty when factors arising due to volatile environments and those arising from subjective imperfections may evolve on different time scales. The most favorable survival statistics then obeys the Zipf power law, however survival is always fleeting in precarious environments, in line with the observations of Leigh Van Valen on that all groups of species go extinct (in million years) at a rate that is constant for a given group. Similar probability models may be used in order to understand the temporal patterns of interaction in daily human communications, as well as in searching and hunting behavior. We also demonstrate that the risk averse behavior (concavity of the utility function) naturally leads to the Pareto-like distributions of income (inequality) over the society. We report on the world-wide economic growth- inequality relation (U-curve) observed by us in the historical trends (1870-2014) for the first time. The observed trend in state secession suggests that half of presently extant states might break up by the end of the century. Perhaps, we are on the edge of global uncertainty.
image Wednesday, February 7
Numerical Index Computation for Free Boundary Minimal Hypersurfaces
Hung Tran
Free boundary minimal hypersurfaces locally minimize the area when the boundary is free to move on a given domain. The Morse index gives the number of distinctive deformations that would decrease the area to the second order. It is of most important to understand hypersurfaces with small indices and the Fraser-Schoen conjecture and its cousins are fundamental in this direction. As a consequence, we will discuss how numerical arguments can be used to address those conjectures.
image Tuesday, February 13, 3:30pm, room AGED 102
The embedding problem in differential geometry
Liviu Ornea
I shall make an overview of problems concerning the embedding of various types of real and complex manifolds, structured or not, in model spaces.
image Tuesday, February 20, 3:30pm, room AGED 102
Junior Scholar Minisymposium
image Tuesday, February 27, 3:30pm, room AGED 102
Junior Scholar Minisymposium
Amin Rahman; Josh Padgett
Amin Rahman, "Simple Models in a Complex World": When thinking of simple models, one may recall the logistic map by Robert May (Nature 1976); a difference equation with a quadratic right hand side. It is so simple that a middle school student can iterate the map to observe the dynamics, and yet it may be used for something as complex as modeling populations. In this talk we discuss models arising from wave dynamics, electronic circuits, fluid wave-particle interactions, and drug diffusion in tumors. In addition, some of the mathematical techniques used to analyze the models are also presented. --- Josh Padgett, "Operator Splitting Methods for Approximating Differential Equations": Splitting methods have been used to approximate solutions to problems arising in analysis, differential geometry, algebra, and differential equations. Classically, these methods have primarily been employed in linear problems, however. This talk will introduce the concept of operator splitting and then demonstrate its effective use in solving certain nonlinear differential equations. In particular, we will consider a singular solid-fuel ignition model and a generalized SKT predator-prey model. The latter portion of the talk will briefly discuss the potential to apply these splitting ideas to various nonlocal and stochastic problems.
image Thursday, March 1, 3:30pm, room AGED 102
Pseudo-integrable billiards and their topological, dynamical, and arithmetic properties
Vladimir Dragovic
We introduce a new class of billiard systems in the plane, with boundaries formed by finitely many arcs of confocal conics such that they contain some reflex angles. Fundamental dynamical, topological, geometric, and arithmetic properties of such billiards are studied. The novelty, caused by reflex angles on boundary, induces invariant leaves of higher genera and dynamical behavior different from Liouville--Arnold's Theorem. The billiard flow generates a measurable foliation defined by a closed 1-form w. Using the closed form, a transformation of the given billiard table to a rectangular cylinder is constructed and a trajectory equivalence between corresponding billiards has been established. A local version of Poncelet Theorem is formulated and necessary algebro-geometric conditions for periodicity are presented. It is proved that the dynamics depends on arithmetic of rotation numbers, but not on geometry of a given confocal pencil of conics. Examples of billiard trajectories having a fixed circle concentric with the boundary semicircles as the caustic, such that the rotation numbers with respect to the half-circles are different pairs of numbers r1 and r2 respectively, are presented. Are such billiard trajectories periodic, and what are all possible periods for given r1 and r2? This presentation is based on joint works with Milena Radnovic.
image Thursday, March 8, 3:30pm, room AGED 102
Kalyan Perumalla
image Wednesday, March 21
First-order traffic flow modeling in theory and practice
Jia Li
This talk will present a discussion of the first-order traffic flow modeling paradigm, i.e. the LWR theory and its extensions, with a particular focus on the gaps between the theory and practice and how they motivate my recent works. I will first present an application of the first-order theory to solve the faulty traffic sensor diagnosis problem. Then I will present a class of ‘ill-posed’ continuous-time dynamic point-queue models and provide a constructive procedure to derive their closed-form solutions. Last, I will discuss the challenges to model heterogeneous traffic flow and a pragmatic data-informed modeling framework to fill the gap.
image Thursday, March 29, 3:30pm, room AGED 102
Extremal eigenvalue problems and minimal surfaces
Richard Schoen
When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. In this talk we will survey two cases in which progress has been made focusing primarily on the case of surfaces with boundary. We will describe the geometric structure of the critical metrics which turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces.
image Thursday, April 5, 3:30pm, room AGED 102
Applications of Dynamics on Homogeneous Spaces to Number Theory
Dmitry Kleinbock
Dynamics on homogeneous spaces of Lie groups has been a useful tool in solving many previously intractable Diophantine problems. In this talk I will illustrate some well-known connections between homogeneous dynamics and number theory, and then talk about some more recent developments -- studying distribution of rational points on spheres and other quadric hypersurfaces. The new work is joint with Lior Fishman, Keith Merrill and Davis Simmons.
image Saturday, April 14
West Texas Applied Math Graduate Minisymposium II

image Thursday, April 19, 3:30pm, room AGED 102
Andrew Christlieb
image Wednesday, April 25
High-order Partitioned Implicit Runge-Kutta Timesteppers for Micromagnetics with Eddy Currents
Josh Engwer
Our research concerns the numerical solution of the Eddy Currents Equation coupled with the Landau-Lifschitz-Gilbert Equation. We construct a partitioned implicit Runge-Kutta (PIRK) timestepper with one component being L-stable and the other being quadratic invariant-preserving (QIP). Mixed (Nedelec,Lagrange)-elements are employed for spatial discretization. We discuss using the resulting scheme for simulations and validation tests. Remarks on software implementation will be provided.