Seminar in Applied Mathematics

Department of Mathematics and Statistics

Texas Tech University

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

image Thursday, September 8, 2pm, MATH 014 (Colloquium)
Passivity-Based Control and Optimization in Networked Robotics: From Human-Swarm Collaborations to Distributed Optimization
Takeshi Hatanaka, Tokyo Institute of Technology
In this talk, we highlight the role of passivity in human-swarm collaborations and distributed optimization. The former half addresses human operator enabled motion synchronization of an ensemble of kinematic robots to desired reference motion under distributed information exchanges among the robots and between the operator and robots. A feedback loop connecting the operator and the robots is designed based on passivity, and asymptotic synchronization is then demonstrated by assuming passivity of a human operator decision process. The aforementioned passivity assumption is further studied through experiments on a human-in-the-loop simulator and closed-loop system identification. In the later half, it is revealed that the control architecture presented above also provides a solution to a class of distributed optimization problems by just replacing the human operator block by a collection of the gradients of local cost functions. The passivity paradigm allows us not only to prove convergence to the optimal solution but also to integrate the other passive blocks while ensuring the convergence property. Finally, we show an application of the presented solution to 3-D human localization problem using pedestrian detection techniques.
image Thursday, September 15, 3:30pm, CHEM 113 (Physics-Math Joint Colloquium)
Huygens triviality of the time-independent Schrodinger’s equation
Arkady Kholodenko, Clemson University
It is well documented that Schrodinger's equation in its known form had emerged only in the 4th of Schrodinger's papers written in 1925. Development of quantum mechanics in the 1st of3 papers was guided by the Huygens ' principle. The same principle was also used by Jacques Hadamard in his studies of partial differential equations of hyperbolic type. Developments in physics and in mathematics of the same topics proceeded without overlap till late sixties and still are not complete. In our talk we address this deficiency being guided by the experimental observations. We apply the concept of Huygens triviality (put forward by Hadamard) to the time-independent Schrodinger's equation thus bringing Schrodinger's 4th installment on quantum mechanics in accord with 3 previous ones. As result, many new aspects of Schrodiner's formulation of quantum mechanics had emerged, e.g. its connection with twistor formalism, with Lie sphere geometry, with quaternionic harmonic analysis, etc. These connections allow us to reinterpret some current experiments and they paves the way towards entirely different formulation of quantum mechanics making it part of quantum field and string theory in the most logical way.
image Friday, September 16, 3pm, CHEM 113 (Colloquium)
Dual Huygens principle and its relation to works by Einstein-Rosen-Podolsky
Arkady Kholodenko, Clemson University
In this talk we extend the results described in the 1st talk to discuss more difficult time-dependent Schrodinger's equation. As by product of this study, a completely new vision of the famous EPR paper by Einstein Rosen and Podolsky written in 1935 is found. The logical developments of this topic are completely independent of those suggested by George Stewart Bell. Since they are complementary to his findings, it is hoped that upon further development they may further help to develop theories of teleportation and quantum entanglement.
image Wednesday, September 21
Convergence estimates for a multigrid algorithm with domain decomposition smoother
Giacomo Capodaglio, Texas Tech University
We study convergence estimates for a multigrid algorithm with domain decomposition smoother applied to symmetric elliptic PDEs. First, we reconsider a general convergence analysis on a class of multigrid algorithms, in a fairly general setting where no regularity assumptions are made on the solution. In this framework we explicitly highlight the dependence of the multigrid error bound on the number of smoothing steps. We observe that an increase in the number of smoothing steps can improve the convergence rate, a result that was already known in the literature although not yet shown under no regularity assumptions. A second purpose of the paper is to fit the domain decomposition theory in the multigrid context, choosing the smoother of the multigrid to be a multiplicative Schwarz type algorithm. We conclude with uniform and local refinement applications of the general theory where we explicitly derive bounds for the multigrid error.
image Wednesday, September 28
Potential theory for quasilinear elliptic equations
Dat Cao, Texas Tech University
In this talk, we study quasilinear elliptic equations of the type \begin{equation} - \Delta_p u = \sigma u^q \text{ in } \mathbb{R}^n, \end{equation} where $ \Delta_p u = \text{div} ( |\nabla u |^{p−2} \nabla u) $ is the $p$-Laplacian, $ 1 < p < \infty$, and $ \sigma \geq 0 $ is an arbitrary locally integrable function, or measure, in the case $ 0 < q < p−1 $. Necessary and sufficient conditions on $ \sigma $ for the existence of finite energy and weak solutions to (1) are given. Sharp global pointwise estimates of solutions are obtained as well. We also discuss the uniqueness and regularity properties of solutions. Our main tools are Wolff potential estimates, dyadic models, and related integral inequalities. Special nonlinear potentials of Wolff type associated with "sublinear" problems are constructed to obtain sharp bounds of solutions. We also treat equations with the fractional Laplacians $ (-\Delta)^{\alpha} $. This is a joint work with Igor Verbitsky.
image Wednesday, October 5
Optimal Obstacle Problems: The case of Harmonic Measure. (Joint Analysis - Applied Math Seminar)
Alexander Solynin, Texas Tech University
image Wednesday, October 12 (Joint Colloquium and Seminar)
Discovery of the Higgs Boson and Searches for Dark Matter at the Large Hadron Collider
Nural Akchurin, Texas Tech University
The experimental discovery of the Higgs boson was announced on the 4th of July 2012 and was based on data collected at the center-of-mass energy of 7-8 TeV. This discovery further solidified our confidence in the Standard Model of elementary particles. The Large Hadron Collider (LHC) has been colliding proton beams at 13 TeV in search of beyond Standard Model physics since the Higgs discovery. Naturally, the searches for dark matter are in full swing now, and I will provide a glimpse of what we are doing at the Compact Muon Solenoid (CMS) experiment, which is one of two general-purpose large detectors at the LHC.
image Tuesday, October 18, 3:30pm, CHEM 113 (Colloquium)
Time-domain astrophysics in the Advanced LIGO era
Alessandra Corsi, Texas Tech University
With Advanced LIGO announcing its first direct detection of gravitational waves (GWs) in February 2016, and with Virgo, KAGRA, and LIGO India detectors expected to join forces in the coming years, a new observational era in time-domain astrophysics has begun. Gravitational waves are expected to open a totally new view of the transient sky, and shed new light on the engines powering some of the most energetic stellar explosions and cosmic collisions. In this talk, I will discuss recent results, open challenges, and prospects for multi-messenger astrophysics with ground-based GW detectors, emphasizing the efforts that are being undertaken at TTU/Physics & Astronomy to shape this new era.
image Tuesday, October 25 - Thursday, October 27
Dayawansa Memorial Lecture Series
Magnus Egerstedt, Georgia Institute of Technology

image Wednesday, October 26, 11-11:50am, MATH 016 (National Mathematical Biology Colloquium)
Fiber Architecture (Differential Geometry) of the Heart and its Valves
Charles S. Peskin, Courant Institute of Mathematical Sciences
Cardiac tissue is highly anisotropic. The fibers that are responsible for this anisotropy are primarily the muscle fibers in the heart walls, and collagen fibers in the heart valve leaflets. The fiber architecture of the heart is remarkable. In the left ventricle, there are nested toroidal surfaces along which the muscle fibers run, following approximately geodesic spiral paths. In the aortic and pulmonic valves, the collagen fibers form a branching braided hammock-like structure that looks as if it might have a fractal character. The goal of the work described in this talk is to derive the fiber architecture of the heart from first principles. Our approach is to formulate partial differential equations for a system of fibers under tension supporting a pressure load, and then to see to what extent these equations predict the observed fiber architecture of the heart and its valves.
image Wednesday, November 2 (Joint Colloquium and Seminar)
Analysis of block preconditioners for the Rayleigh-Benard convection problems
Guoyi Ke, Texas Tech University
We use the notion of Field-Of-Values(FOV)-equivalence of matrices to analyze block preconditioners for different linearizations of the Rayleigh-B\'enard convection problem. We consider finite element discretizations based on the stability conditions (LBB conditions). First, sufficient conditions are given to establish norm- and FOV-equivalence of diagonal and triangular block preconditioners to the system matrices linearized with both Picard and Newton methods. Then, four upper triangular block preconditioners are investigated, and showed to be norm- or FOV-equivalent to the system matrices. Numerical results show that the GMRES convergence is robust with respect to the mesh size for these four preconditioned systems. We also offer a characterization of the performance of the Picard system and Newton system in terms of computational time.
image Thursday, November 10, 3:30pm, CHEM 113 (SIAM Colloquium)
Modeling Hurricane Storm Surge and Proposed Mitigation Systems for Floods in the Texas Coast
Clint Dawson, UT Austin
Since the 2005-2008 hurricane seasons, there has been extensive effort to understand and predict the impacts of hurricane-induced flooding on low-lying regions in the Gulf of Mexico. After Hurricane Ike in 2008, which made landfall at Galveston, TX, the Severe Storms Prediction, Education and Evacuation from Disasters (SSPEED) Center was formed, including a consortium of major universities in Texas. The focus of the center was to study the socio-economic/environmental/structural impact and possible mitigation of hurricane storm surge in the Houston-Galveston region. As part of this effort, our group at the University of Texas performed extensive studies using the Advanced Circulation (ADCIRC) modeling framework. In this talk, we will discuss this framework, the validation of this model for historical hurricanes, and the application of the model to the study of potential storm mitigation systems, including structural and non-structural options. We will also discuss a number of other issues that arise in these types of studies, including political, environmental and socio-economic considerations.
image Thursday, November 17, 3:30pm, CHEM 113 (Colloquium)
Challenges Arising in the Study of the Development and Spread of Antibiotic Resistant Bacteria
Mary Ann Horn, Vanderbilt University and National Science Foundation
Drug resistance has been an emerging problem since the discovery of penicillin. Resistance is now seen not only in clinical settings, but also increasingly in the community. Bacteria such as methicillin-resistant Staphylococcus aureus (MRSA) impacts healthy adults as well as patients in settings such as hospitals and nursing homes. MRSA is an example of a Gram-positive bacteria. More recently, antibiotic resistance in Gram-negative bacteria such as Escherichia coli and Salmonella aureus is an increasing problem. A primary difference between Gram-positive and Gram-negative bacteria is the composition of the cell walls, which results in the two types of bacteria having different mechanisms to acquire resistance. This talk will give an overview of some of our recent work on modeling of the development and spread of antibiotic resistance, ranging from common hospital acquired infections to those that have arisen in the community. (Joint work with Erika D'Agata, Shigui Ruan, Joanna Wares, and Glenn Webb.)
image Wednesday, November 30
Simulation of time-dependent PDE's with finite elements and high-order A-stable IRK timesteppers
Josh Engwer, Texas Tech University
Explicit Runge-Kutta (ERK) timesteppers such as Forward Euler and Classic RK4 are desired due to their relative ease of implementation and low computational cost. However, they tend to be unstable for most real-world time-dependent ODE's & PDE's unless the timestep is reduced to an impractically small value as dictated by possible constraints (such as the Courant-Friedrichs-Lewy condition for hyperbolic PDE's for example.) At best, this results in unbearably long running times in order to achieve a stable simulation with a meaningful time interval. At worst, the required timestep size is smaller than machine precision, effectively rounding down the timestep to zero! To help mitigate these potential issues, one is encouraged by Dalquist's 2nd Barrier to resort to A-stable implicit Runge-Kutta (IRK) timesteppers, the simplest of which is Backward Euler. Unfortunately, Backward Euler is only a 1st-order timestepper, meaning halving the timestep only halves the solution error. One can do far better than that as there are several classes of higher-order A-stable IRK timesteppers to choose from. For the purposes of this talk, the Gauss-Legendre family of IRK methods will be considered. Higher-order IRK timesteppers are typically introduced to solve certain stiff ODE's, and occasionally they are utilized together with finite difference methods (FDM's) in solving certain time-dependent PDE's. This talk will utilize these timesteppers with Lagrange finite elements in order to solve one of the most elementary time-dependent PDE's, the Heat Equation (both linear and non-linear) equipped with inhomogeneous Dirichlet boundary conditions. The simulation codes use the open-source FEniCS finite element framework to provide the necessary finite element method, and additional Python code is used to produce the timestepping, solution visualization, L^2 solution error, validation tests, and convergence rate plots of Heat Equation problems constructed using the 'Method of Manufactured Solutions'. Some remarks regarding implementation in FEniCS will be provided.