BYU-Brigham Young University

Department of Mathematics

Applied Mathematics Seminar

Primary Organizer: Jared P. Whitehead

Schedule of Talks:

Winter 2016

Unless otherwise noted, all talks are held Thursdays at noon in 301 TMCB

 

Thursday, April 7, 2016

Title of Talk: TBA

Abstract:

 

Previous Talks: Winter 2016

 

Thursday, January 7, 2016

Title of Talk: Studying and Controlling Material Properties Using
Ultrafast Laser Spectroscopy
Jeremy Johnson, Brigham Young University (Chemistry and Biochemistry)

Abstract: New and future technologies rely heavily on the development of new materials with properties optimized for specific applications. Light can be a wonderful tool to measure and even control all sorts of fascinating material properties, but there is one important truth all spectroscopists keep in mind: light only cares about the optical properties of a material!  In order to use light to study a host of material properties, the radiation must couple to the material property of interest.  But oftentimes the optical properties are coupled to many material properties and understanding what we see can be difficult.  Therefore, making measurements more “selective” to the property or dynamics of interest is crucial.  In ultrafast spectroscopy, short pulses of light are used to pump (induce a change in) a material and subsequently probe the response. Spectroscopists have employed many means to increase the selectivity in experiments, from simply changing laser spot sizes or shapes or pulse durations, to using specific wavelengths for either the pump or probe beam, and additionally having multiple pump and probe beams incident on the sample with a variety of incidence angles and time sequences. Perhaps the most straightforward is by simply changing the wavelength (color) of electromagnetic radiation we use, from x-rays to radio waves.  In the Johnson Spectroscopy Lab, we focus on experiments using ultraviolet, visible, and infrared radiation.  In addition we have a strong emphasis on using terahertz (THz) radiation, an exciting region of the electromagnetic spectrum that lies just beyond the infrared, with wavelengths from 3 mm to 30 μm corresponding to frequencies from 0.1 to 10 THz (1 THz = 1012 Hz).  These frequencies are associated with the time scales of atomic vibrations in solids, the lifetimes of excited electronic carriers in some materials, electronic spin which gives rise to magnetism, and other dynamic properties we can study in solids, liquids, and gases. In this talk, I will give a brief overview of ultrafast spectroscopy – what it is and what it is good for – and include examples of various spectroscopic studies that we have begun in the Johnson Spectroscopy Laboratory here at BYU. I will present some questions in regards to analyzing data that could prompt possible collaborative projects.

 

Thursday, January 14, 2016

Title of Talk: Viscous detonations in the reactive Navier-Stokes equations
Joshua Lytle, Brigham Young University (Mathematics)

Abstract: A detonation is a type of traveling wave arising in combustion problems, and consists of an exothermic reaction driving a shock with supersonic speed. The standard practice is to study detonations with the reactive Euler or ZND equations, and considers viscous effects to be inconsequential. However, recent work has revealed new behavior in viscous detonations of the reactive Navier-Stokes equations. Evans function computation shows that unstable eigenvalues restabilize as activation energy increases. This phenomenon is termed ‘viscous hyperstabilization’, and poses a challenge for experimenters. Our research extends this Evans function study into the multi-dimensional setting.

 

Thursday, January 21, 2016

Title of Talk: Integral equation modeling for nonlocal diffusion and mechanics
Max Gunzburger, Florida State University

Abstract: We use the canonical examples of fractional Laplacian and peridynamics equations to discuss their use as models for nonlocal diffusion and mechanics, respectively, via integral equations with singular kernels. We then proceed to discuss theories for the analysis and numerical analysis of the models considered, relying on a nonlocal vector calculus to define weak formulations in function space settings. In particular, we discuss the recently developed asymptotically compatible families of discretization schemes. Brief forays into examples and extensions are made, including obstacle problems and wave problems.

 

Friday, January 29, 2016

(NOTE: different place and time)

Title of Talk: A hybrid framework with analysis for stochastic wave propagation models
Mahadevan Ganesh, Colorado School of Mines

Abstract: We consider a class of wave propagation models with aleatoric and epistemic uncertainties. Using mathematical analysis-based, shape-independent, a priori parameter estimates, we develop offline/online strategies to compute statistical moments of a key quantity of interest in such models. We present an efficient reduced order model (ROM) and high performance computing (HPC) framework with analysis for quantifying aleatoric and epistemic uncertainties in the propagation of waves through a stochastic media comprising a large number of three dimensional particles. Simulation even for a single deterministic three dimensional configuration is inherently difficult because of the large number of particles. The aleatoric uncertainty in the model leads to a larger dimensional system involving three spatial variables and additional stochastic variables. Accounting for epistemic uncertainty in key parameters of the input probability distributions leads to prohibitive computational complexity. Our hybrid ROM and HPC framework can be used in conjunction with any computational method to simulate a single particle deterministic wave propagation model.

 

Thursday, February 18, 2016

Title of Talk: On the rotation-two-component Camassa-Holm system modeling the equatoriall water waves
Hongjun Gao, Nanjing Normal University

Abstract: In this talk, a modified two-component Camassa–Holm system with the effect of the Coriolis force in the rotating fluid is derived, which is a model in the equatorial water waves. The effects of the Coriolis force caused by the Earth's rotation and nonlocal nonlinearities on blow-up criteria and wave-breaking phenomena are then investigated. Our refined analysis relies on the method of characteristics and conserved quantities and is proceeded with the Riccati-type differential inequality. Conditions which guarantee the permanent waves are obtained by using a method of the Lyapunov function. Finally, solitary-wave solutions are discussed.

 

Thursday, February 4, 2016

Title of Talk: TBA
Canceled this week

Abstract:

 

Thursday, February 25, 2016

Title of Talk: An Algorithm for Melnikov Functions and Application to a Chaotic Rotor
Weinian Zhang, Sichuan University

Abstract: This talk is concerning a chaotic oscillation of a nonlinear model of turbine rotor. Melnikov’s method is one of the most important methods in determining chaos, but the practical model prevents the application of Melnikov’s method because its complicated nonlinearity makes difficulties to count the number of equilibria, not mentioning the determination of stability and the computation of heteroclinic orbits. In this work, a numerical algorithm is given to compute Melnikov functions with an idea of avoiding the computation of . heteroclinic orbits. The convergence of the algorithm and the estimates of errors help us to give a parameter region for chaotic oscillation of the rotor.

 

Thursday, March 3, 2016 (10AM, 292 TMCB)

Title of Talk: Analysis and computation of some singular perturbation problems in fluid mechanics
Gung-Min Gie, University of Liousville

Abstract: In most practical applications of fluid mechanics, it is the interaction of the fluid with the boundary that is most critical to understanding the behavior of the fluid. Physically important parameters, such as the lift and drag of a wing, are determined by the sharp transition the air makes from being at rest on the wing to flowing freely around the airplane near the wing. Mathematically, the behavior of such flows at small viscosity is modeled by the Navier-Stokes equations. In this talk, we discuss some recent results on the boundary layers of the Navier-Stokes equations as well as some related numerical computations.

 

Thursday, March 10, 2016

Title of Talk: Two new direct search algorithms for constrained multifidelity optimization problems.
Mark Abramson, Brigham Young University (Mathematics)

Abstract: Multifidelity optimizations occur in engineering design when multiple engineering simulation codes are available at different levels of fidelity. An example of this might be in aerodynamic optimization of the shape of a wing, in which the computation of aerodynamic quantities, such as lift and drag, can be computed using a full Navier-Stokes solver, or an Euler solver, or a linearized potential code. High fidelity simulations are more accurate, but also more computationally expensive – sometimes prohibitively so. The goal of this work is the design of algorithms that optimize with respect to the high-fidelity simulation, but exploit the use of lower fidelity codes as much as possible. Two algorithms will be described, which are based on the surrogate management framework (SMF) and the class of mesh adaptive direct search (MADS) algorithms. In both cases, interpolating surrogates are constructed and updated from previously evaluated iterates of the algorithm to speed convergence. The first algorithm is to solve the problem recursively, in which the Search step of each recursion level is the optimization of a surrogate function constructed from the next lower fidelity level simulation augmented with an interpolating surrogate that accounts for the difference between adjacent levels of fidelity. The second approach is an augmentation of the optimization problem, in which the fidelity level is incorporated as a variable in the problem, and a relaxable constraint is added to force the solution to be at the highest level of fidelity. Preliminary numerical results are presented.

 

Thursday, March 17, 2016

Practice Session for Student Research Conference
TBA

Abstract:

 

Thursday, March 24, 2016

Title of Talk: Navier-Stokes equations, stochastic cascades and the issue of symmetry breaking
Radu Dascaluic, Oregon State University

Abstract: We will discuss a probabilistic cascade structure that can be naturally associated with the 3D Navier-Stokes equations. In particular, our aim will be to see how the explosion properties of such cascades can help establish a connection between the uniqueness of symmetry-preserving (self-similar) solutions and the uniqueness of the general problem.

 

Thursday, March 31, 2016

Title of Talk: Immersed particle dynamics in fluids with memory
Christel Hohenegger, University of Utah

Abstract: Passive microrheology is an experimental which tracts the motion of passive tracer particles in various fluids and uses statistical quantities to extract mechanical fluid properties out of recorded paths data. In a viscous fluid, the theoretical background relies on the Stokes-Einstein relation for the drag of a sphere and on the description of Brownian motion. However, for a fluid with memory or a fluid which exhibits both viscous and elastic properties, questions remain as to which mechanical properties can be reconstructed. We present a first step towards this goal by describing a fluctuating Stokes model for an immersed particle passively advected by a Maxwellian fluid. We include the memory's correlations in a thermal stress and show that our formulation satisfies equipartition of energy. We then describe the resulting stochastic partial differential equations for the non-Markovian, stationary fluid velocity process. We also present a covariance based numerical method for generating particle paths and discuss the different sources of errors, as well as implications for inverse characterization.

Previous Talks: Fall 2015

Tuesday, September 1, 2015

Title of Talk: Energy Conservation in Numerical Models of the Atmosphere
Thurs, 1/09/2015-2:00pm, TMCB 292
James Kent, University of South Wales

Abstract: The numerical methods that solve the governing equations in an atmospheric dynamical core are designed to dissipate potential enstrophy and prevent the buildup of kinetic energy at the grid scale. A side effect of this is the dissipation of total energy which should be conserved. Different approaches have been adopted to conserve energy in dynamical cores, such as energy fixers (which replace the dissipated energy by modifying the temperature in the thermodynamic equation), and stochastic backscatter schemes. In this talk I will present the first steps towards designing a deterministic energy conserving restoration scheme that considers the conversion of kinetic energy to heat, replacing kinetic energy due to model error, and the backscatter of kinetic energy.

The energy conserving restoration scheme (ECRS) is presented in the context of the shallow water equations on the sphere. It is designed to be used with any existing shallow water equation scheme (called the preliminary scheme) which can adequately dissipate potential enstrophy. For each prognostic variable a pattern is chosen; this is added to the preliminary scheme solution, and the amount added is calculated to ensure energy conservation. Results from a series of idealized test cases will be presented

 

Tuesday, September 8, 2015

Canceled due to holiday on September 7, 2015.
Tues, 8/09/2015-2:00pm, TMCB 292
TBA

Abstract:

 

Thursday, September 17, 2015

Title of Talk: On some special properties of solutions to the generalized Korteweg-de Vries equation
Thurs, 17/09/2015-2:00pm, TMCB 292
Gustavo Ponce, University of California Santa Barbara

Abstract: (download pdf) We shall study the special properties of solution to the initial value problem (IVP) for the k-generalized Korteweg-de Vries equation. More precisely we shall discuss (very briefly) the well-posedness of this IVP, unique continuation and propagation of regularity results concerning solutiosn of the same.

 

Tuesday, September 22, 2015

Title of Talk: TBA
Tues, 22/09/2015-2:00pm, TMCB 292
Djoko Wirosoetisno, University of Durham

Abstract: We discuss several timestepping schemes used to integrate fluid equations over long timescales. Of particular interest is how stability conditions in suitable spaces may lead to convergence of objects of physical interest, such as the global attractor and invariant measures supported on it.

 

Tuesday, September 29, 2015

Title of Talk: The Role of Applied Mathematics in Engineering Design Optimization as Applied to Wind Energy Systems
Tues, 29/09/2015-2:00pm, TMCB 292
Andrew Ning, Brigham Young University (Mechanical Engineering)

TBA

Abstract: Wind turbines and aircraft are complex systems intersecting several disciplines: aerodynamics, structures, controls, economics, etc. Small configuration changes can have strong interdisciplinary interactions. We use optimization and uncertainty quantification methods to assess trade-offs and explore new conceptual designs. The problems we are interested are multidisciplinary, nonlinear, and often contain hundreds or thousands of design variables. In this talk we will explore various wind energy optimization-driven design problems at a high level, with a focus on the importance of applied math in solving large systems problems.

 

Tuesday, October 6, 2015

Title of Talk: Improving Automatic Speech Recognition with Non-Negative Tensor Factorization
Tues, 06/10/2015-2:00pm, TMCB 292
David Wingate, Brigham Young University (Computer Science)

Abstract: Non-negative matrix factorization (NMF) decomposes a non-negative matrix into the product of two (usually smaller) non-negative matrices. This technique is a standard tool for audio source separation, where non-negative matrices such as magnitude spectrograms can be decomposed into "dictionaries" and "activations"; the process implicitly constructs a model of a sound source, which can be used for a variety of purposes.

In this talk I will describe the multi-dimensional extension of NMF, called non-negative tensor factorization (NTF), and illustrate its application in the context of audio source separation. The move to higher-order tensors allows us to include additional sources of information, such as directionality derived from multi-microphone arrays. I will illustrate how this additional information, combined with our more powerful framework, can be used to improve real-world automatic speech recognition in noisy and far-field situations. Time permitting, I will also discuss real-world deployment challenges.

 

Tuesday, October 13, 2015

Title of Talk: Wave mean-flow interaction in geophysical fluid dynamics (GFD), the influence on the global attractor
Tues, 13/10/2015-2:00PM, JKB 1126
Jared Whitehead, Brigham Young University (Mathematics)

Abstract: The first accurate numerical weather prediction of the 1950s relied on an asymptotic reduction of the full equations of motion for atmospheric flow. This and other asymptotic reductions rely on the fact that fast waves have little to no effect on the slower evolving mean flow (often referred to as the slow manifold). Eliminating the fast waves, Jule Charney and colleagues developed a system of equations amenable to numerical integration.

In this talk we review the progress in the past 60+ years to put these concepts on a rigorous mathematical footing. Using direct numerical simulations we explore the validity of several distinct asymptotic limits in the 3D rotating, stratified Boussinesq equations, and the relative accuracy of the asymptotic approximation as the small parameter remains finite. These simulations validate and motivate the investigation of long-time dynamics of similar models from GFD wherein we show that the slow manifold is arbitrarily close to the global attractor. We close with some recent calculations and open questions regarding the precise geometry of the global attracting set for these types of systems.

 

Tuesday, October 20, 2015

Title of Talk: Mathematics in Materials Science: Symmetries and Properties of Grain Boundaries and Deformation in Amorphous Metals
Tues, 20/10/2015-2:00pm, 1126 JKB
Eric Homer, Brigham Young University (Mechanical Engineering)

Abstract: The crystallography of grain boundaries is an important determinant of structure and properties. As a result, grain boundaries influence a variety of properties including strength, ductility, corrosion resistance, conductivity, etc. However, the field is currently limited by its ability to connect properties with the structure of the grain boundaries. Recent work has demonstrated that a representation that naturally accounts for the symmetries of the crystallographic space simplifies representation and leads to new structure property relationships that have not previously been discovered.

Amorphous metals are a relatively recently discovered material with impressive mechanical properties. However, they have a major flaw, these metals shatter like a glass upon failure. We utilize a collection of mathematical approaches to model the deformation in the materials and enable statistical analysis of the results for meaningful insight into the nature of deformation.

Tuesday, October 27, 2015

Title of Talk: Time Reversal Acoustics: Communications, Energy Focusing, and Source Reconstruction
Tues, 27/10/2015-2:00pm, 1126 JKB
Brian Anderson, Brigham Young University (Physics)

Abstract: Time reversal is a method of focusing energy to a location in space. It has been used in acoustics applications for communications, medical applications, nondestructive evaluation, and source localization and characterization. Time reversal requires two steps, a forward step and a backward step. During the forward step, sound travels from a source to a receiver along potentially many reflected paths. The recorded sound at the receiver is then reversed in time and can this reversed signal can then be broadcast from the source to the receiver again or from the receiver to the source. In either case, a focusing of energy occurs along with a temporal reconstruction of the originally emitted sound signal. This talk will explain the physics of time reversal and show examples of how it works, followed by an introduction to how it is being used at BYU for various applications.

 

Tuesday, November 3, 2015

Title of Talk: Local High Order Absorbing Boundary Conditions in terms of Farfield Expansions
Tues, 2/11/2015-2:00pm, 1126 JKB
Vianey Villamizar, Brigham Young University (Mathematics)

Abstract: The appropriate definition of absorbing boundary conditions (ABC) for wave scattering problems in un- bounded domain plays a key role in wave scattering. The main reason is that the equations modeling the wave phenomena in fields such as geophysics, oceanography, and acoustics among others, are defined on an infinite domain and they require a numerical treatment. A popular approach consists of truncating the infinite domain high order of approximation to the exact solution and it is computationally less expensive than those currently used in the applications.

 

Title of Talk: Weighted T-spline with Applications in Isogeometric Analysis
Thurs, 12/11/2015-2:00pm, JFSB B142
Jessica Zhang, Carnegie Mellon University

Abstract: To facilitate isogeometric analysis, we present a new type of T-spline named the weighted T-spline, which introduces a new weighting idea to T-spline basis functions. Weighted T-spline basis functions satisfy partition of unity and are linearly independent. Less geometrical constraint is applied to the T-mesh, and the number of control points can be decreased. Weighted T-splines are applied to reparameterize trimmed NURBS patches and handling extraordinary nodes, demonstrating that it can generate complicate models for isogeometric analysis. Knot interval duplication is used to extract knot vectors for the vicinity around extraordinary nodes, based on which bicubic weighted T-spline basis functions are defined. After that we modify the extracted Bezier coefficients to obtain a gap-free T-spline surface. Then we use biquartic Bezier basis functions with optimized coefficients to elevate the surface continuity from C0 to G1. Comparison with other methods shows that our method generates T-spline surfaces with better surface continuity. Furthermore, we have recently extended weighted T-spline basis functions to support hybrid degree and local p-refinement. Volumetric weighted T-splines are constructed with parametric mapping and sweeping methods. Linear elasticity and heat transfer problems are solved, demonstrating that the generated surface and volumetric models are suitable for isogeometric analysis.