Seminar in Applied Mathematics

Department of Mathematics and Statistics

Texas Tech University

Spring 2016 - Wednesday, 4-5 pm - Room MATH 109

image Thursday, January 28, 11am, MATH 014
Colloquium Talk for Chair position
Magdalena Toda

image Thursday, February 4, 11am, MATH 014
Colloquium Talk for Chair position
T.Y. Tam

image Tuesday, February 9, 11am, MATH 014
Colloquium Talk for Chair position
Javier Rojo
image Wednesday, February 17
On the continuity of families of global, pullback and uniform attractors.
Luan T. Hoang, Texas Tech University
Let $\Lambda$ be a complete metric space, and let $\{S_\lambda(\cdot):\ \lambda\in\Lambda\}$ be a parametrized family of semigroups with global attractors $A_\lambda$. We assume that there exists a fixed bounded set $D$ such that $A_\lambda\subset D$ for every $\lambda\in\Lambda$. We show that the attractors $A_\lambda$ are continuous with respect to the Hausdorff distance at a residual set of parameters $\lambda$ in the sense of Baire Category. This result is then extended to the pullback and uniform attractors of a family of processes for non-autonomous systems. In applications, we consider the Lorenz system and two-dimensional Navier-Stokes equations. This is joint work with Eric Olson (University of Nevada, Reno) and James Robinson (University of Warwick).
image Wednesday, February 24
Slightly compressible fluids in heterogeneous porous media
Emine Celik, Texas Tech University
We study the generalized Forchheimer flows of slightly compressible fluids in heterogeneous porous media. The media's porosity and coefficients of the Forchheimer equation are functions of the spatial variables. The partial differential equation for the pressure is degenerate in its gradient and can be both singular and degenerate in the spatial variables. Suitable weighted Lebesgue norms for the pressure, its gradient and time derivative are estimated. The continuous dependence on the initial and boundary data is established for the pressure and its gradient with respect to those corresponding norms. Asymptotic estimates are derived even for unbounded boundary data as time tends to infinity. Moreover, we obtain the estimates for the $L^\infty$-norms of the pressure and its time derivative in terms of the initial and the time-dependent boundary data. This is a joint work with Luan Hoang.
image Wednesday, March 2
Green’s function asymptotics of periodic elliptic operators on Abelian coverings of compact manifolds
Minh Kha, Texas A&M University
Green’s function behavior near and at a spectral edge of a periodic operator is one of what was called by M. Birman and T. Suslina “threshold properties.” I.e., it depends upon the infinitesimal structure of the dispersion relation at the spectral edge. For a "generic" periodic second-order elliptic operators on a co-compact abelian cover, we will discuss the asymptotics at infinity of the Green's functions near and at the spectral gap edge as long as the dispersion relation of the operator has a non-degenerate extremum there. Previously, analogous results have been known for the Euclidean case only. One of the interesting features discovered is that the rank of the deck group plays more important role than the dimension of the manifold.
image Monday, March 7, 4pm, MATH 109
Propagation and Dynamics of Wave Envelope under Extreme Light-Matter Interaction
Alexey Sukhinin, Southern Methodist University, Dallas
Light filamentation in air is a field with two decades of research history. It is the phenomenon that describes the propagation of high intensity laser pulses over the long distances without diffraction due to the guiding mechanism of air ionization. Filament propagation is possible when self-focusing of high intensity beams is balanced with the de-focusing effect of ionized plasma channel. In this talk, I will describe mathematical models of filamentation in air and new trends in this field.
image March 14-18

Spring Break. No talks.

image Tuesday, March 22, 3:30pm, CHEM 101
SIAM Colloquium Talk
Dorothy Wallace, Dartmouth College

image Thursday, March 31
A discrete data assimilation scheme for the solutions of the 2D Navier-Stokes equations and their statistics
Cecilia Mondaini, Texas A&M University
The idea of data assimilation is to obtain a good approximation of the state of a certain physical system by combining observational data with dynamical principles pertaining to the underlying mathematical model. It is widely used in many fields of geosciences, mainly for oceanic and atmospheric forecasting. Recently, A. Azouani, E. Olson and E. Titi introduced a new continuous in time data assimilation algorithm inspired by ideas from control theory that is applicable to a wide range of dissipative evolution equations. In this talk, I will show how to adapt this previous data assimilation algorithm, in the case of the 2D Navier-Stokes equations, to the more realistic setting of observational measurements that are discrete in space and time and that may also be contaminated by systematic errors. We will see that, under suitable conditions on the relaxation parameter, the spatial resolution of the mesh and the time step between successive measurements, an asymptotic in time estimate of the difference between the approximating solution and the reference solution can be obtained, in an appropriate norm, which shows exponential convergence up to a term which depends on the size of the errors. A stationary statistical analysis of this spatio-temporal discrete data assimilation algorithm will also be provided. This is a joint work with C. Foias and E. Titi.
image Wednesday, April 6
Making Gravitational Wave CoCoA: A cross-correlation approach for detecting intermediate-duration gravitational waves
Rob Coyne, Department of Physics, TTU
The recent ground-breaking detection of gravitational waves (GWs) by Advanced LIGO marks the beginning of a new era in astronomy, but it is only the first step towards a greater understanding of the universe. Historically, searches for GW signals have fallen into two coarsely defined categories: those looking for short but powerful bursts of GWs, or those looking for long lived, continuous GW emission from weaker sources. However, there is reason to believe that many GW signals exist that fall into neither category, and for which traditional GW searches are poorly suited. Here we describe a novel application of a generalized cross-correlation technique (Coyne et al. 2016, Phys. Rev. D, submitted to PRD) optimized for the detection of such "intermediate duration” signals that will contribute to an entirely new class of next-generation GW searches.
image Wednesday, April 13 - Thursday, April 14
Distinguished Lecture Series
Reinhard Laubenbacher, University of Connecticut Health Center

image Wednesday, April 20
Iterative Learning Control for MIMO Systems
Rangana Jayawardhana, Texas Tech University
Iterative Learning Control (ILC) is based on the notion that a system that executes the same task repeatedly can learn from the previous executions to improve its performance. ILC algorithm is particularly useful for systems with model uncertainties and repeated disturbances as only a minimal knowledge about the system parameters is necessary for convergence. In practice, although it is straight forward to implement the ILC algorithm for SISO systems, it is not so for MIMO systems. In here we introduce an extension of the SISO ILC algorithm that can be applied to MIMO systems.
image Wednesday, April 27
Block triangular preconditioners for linearization schemes of the Rayleigh-Benard convection problem
Guoyi Ke, Texas Tech University
In this presentation, we compare two block triangular preconditioners for different linearizations of the Rayleigh- Benard convection problem discretized with finite element methods. The two preconditioners differ in the nested or non-nested use of a certain approximation of the Schur complement associated to the Navier-Stokes block. First, bounds on the generalized eigenvalues are obtained for the preconditioned systems linearized with both Picard and Newton methods. Then, the performance of the proposed preconditioners is studied in terms of computational time. This investigation reveals some inconsistencies in the literature that are hereby discussed. We observe that the non-nested preconditioner works best both for the Picard and for the Newton cases. Therefore, we further investigate its performance by extending its application to a mixed Picard-Newton scheme. Numerical results of two- and three-dimensional cases show that the convergence is robust with respect to the mesh size. We also give a characterization of the performance of the various preconditioned linearization schemes in terms of the Rayleigh number.
image Wednesday, May 4
Willmore energy and Generalized Willmore energy
Thanuja Paragoda, Texas Tech University
We study a Generalized Willmore flow for graphs and its numerical applications. First, we derive the time dependent equation which describes the geometric evolution of a Generalized Willmore flow in the graph case. This equation is recast in divergence form as a coupled system of second order nonlinear PDEs. Furthermore, we study finite element numerical solutions for steady-state cases obtained with the help of the FEMuS library (Finite Element Multiphysics Solver). We use automatic differentiation (AD) tools to compute the exact Jacobian of the coupled PDE system subject to Dirichlet boundary conditions. To validate our numerical algorithm, we compare steady-state Willmore flow solutions with known analytical solutions.