Analysis and Applications Seminar
Analysis and Applications
Analysis and Applications (AaA) Seminar
This seminar is intended to have a very informal format, and welcomes faculty and grad students from the Pikes Peak region who are interested in contemporary research in analysis (of all sorts) and applications. Areas covered include but are not limited to real and complex analysis, functional analysis, harmonic analysis, ODEs and PDEs, mathematical physics and applications to nonlinear phenomena, numerical analysis, scientific computation and other fields (too many to enumerate).
UC 309, UCCS campus
*** see abstracts below ***
|Feb 20, 2019||
|What’s new with the wave equation?|
|Mar 6, 2019||
Scalable semi-supervised learning with operators in Hilbert space
|Mar 20, 2019||
|The mathematics of (spatial) mobility|
|Apr 17, 2019||
|Exact dispersion relation for strongly nonlinear elastic wave propagation|
|May 8, 2019||
|The Curious Dynamics of Translating Bubbles: An Application of Perturbation Methods and Potential Flow Theory|
|Sep 12, 2018||
|Beach waves and KP solitons|
|Oct 3, 2018||
|An elementary trip from the Gauss hypergeometric function to the Poschl-Teller potential in quantum mechanics|
|Oct 17, 2018||
Geraldo de Souza,
|Fourier series, Wavelets, Inequalities, Geometry and Optimization|
|Nov 14, 2018||
|Semiclassical soliton ensembles|
|Dec 5, 2018||
|Discrete solitons for the focusing Ablowitz-Ladik equation with non-zero boundary conditions via inverse scattering transform|
|Apr 11, 2018||Greg Fasshauer
Colorado School of Mines
|An Introduction to Kernel-Based Approximation Methods|
|Mar 14, 2018||Ethan Berkove
|Short Paths and Long Titles: Travels through the Sierpinski carpet, Menger sponge, and beyond.|
|Feb 28, 2018||Radu Cascaval, UCCS||Traffic Flow Models. A Tutorial|
|Dec 8, 2017||Barbara Prinari, UCCS||Solitons and rogue waves for a square matrix nonlinear Schrodinger equation with nonzero boundary conditions|
|Nov 17, 2017||Oksana Bihun, UCCS||New properties of the zeros of Krall polynomials|
|Oct 27, 2017||Radu Cascaval, UCCS||What do Analysis and Scientific Computation have in common ...|
|Sep 29, 2017||Fritz Gesztesy, Baylor U||The eigenvalue counting function for Krein-von Neumann extensions of elliptic operators|
Please contact Dr. Radu Cascaval (firstname.lastname@example.org) if you are interested to join this seminar or need more info. Limited number of parking passes will be made available to non-UCCS individuals attending this seminar.
TITLES & ABSTRACTS:
Spring 2019 seminars:
May 8, 2019 Seminar
Speaker: Dr. Michael Calvisi, Mechanical and Aerospace Eng, UCCS
Title:The Curious Dynamics of Translating Bubbles: An Application of Perturbation Methods and Potential Flow Theory
When subject to an acoustic field, bubbles will translate and oscillate in interesting ways. This motion is highly nonlinear and its understanding is essential to the application of bubbles in diagnostic ultrasound imaging, microbubble drug delivery, and acoustic cell sorting, among others. This talk will review some of the interesting physics that occur when bubbles translate in an acoustic field, including Bjerknes forces, the added mass effect, and nonspherical shape oscillation. Such nonspherical shape modes strongly affect the stability and acoustic signature of encapsulated microbubbles (EMBs) used for biomedical applications, and thus are an important factor to consider in the design and utilization of EMBs. The shape stability of an EMB subject to translation is investigated through development of an axisymmetric model for the case of small deformations using perturbation analysis. The potential flow in the bulk volume of the external flow is modeled using an asymptotic analysis. Viscous effects within the thin boundary layer at the interface are included, owing to the no-slip boundary condition. The results of numerical simulations of the evolutions equations for the shape and translation of the EMB demonstrate the counterintuitive result that, compared to a free gas bubble, the encapsulation actually promotes instability when a microbubble translates due to an acoustic wave.
Apr 17, 2019 Seminar
Speaker: Dr. Mahmoud I. Hussein, Aerospace Eng, CU Boulder
Title:Exact dispersion relation for strongly nonlinear elastic wave propagation
Wave motion lies at the heart of many disciplines in the physical sciences and engineering. For example, problems and applications involving light, sound, heat or fluid flow are all likely to involve wave dynamics at some level. In this seminar, I will present our recent work on a class of problems involving intriguing nonlinear wave phenomena‒large-deformation elastic waves in solids; that is, the “large-on-small” problem.
Specifically, we examine the propagation of a large-amplitude wave in an elastic one-dimensional medium that is undeformed at its nominal state. In this context, our focus is on the effects of inherent nonlinearities on the dispersion relation. Considering a thin rod, where the thickness is small compared to the wavelength, I will present an exact formulation for the treatment of a nonlinearity in the strain-displacement gradient relation. As examples, we consider Green Lagrange strain and Hencky strain. The ideas presented, however, apply generally to other types of nonlinearities The derivation starts with an implementation of Hamilton’s principle and terminates with an expression for the finite-strain dispersion relation in closed form. The derived relation is then verified by direct time-domain simulations, examining both instantaneous dispersion (by direct observation) and short-term, pre-breaking dispersion (by Fourier transformations), as well as by perturbation theory. The results establish a perfect match between theory and simulation and reveal that an otherwise linearly nondispersive elastic solid may exhibit dispersion solely due to the presence of a nonlinearity. The same approach is also applied to flexural waves in an Euler Bernoulli beam, demonstrating qualitatively different nonlinear dispersive effects compared to longitudinal waves. Finally, I will present a method for extending this analysis to a continuous thin rod with a periodic arrangement of material properties. The method, which is based on a standard transfer matrix augmented with a nonlinear enrichment at the constitutive material level, yields an approximate band structure that accounts for the finite wave amplitude. Using this method, I will present an analysis on the condition required for the existence of spatial invariance in the wave profile.
This work provides insights into the fundamentals of nonlinear wave propagation in solids, both natural and engineereda problem relevant to a range of disciplines including dislocation and crack dynamics, geophysical and seismic waves, material nondestructive evaluation, biomedical imaging, elastic metamaterial engineering, among others.
Mar 20, 2019 Seminar
Speaker: Dr. Radu Cascaval, UCCS
Title: The mathematics of (spatial) mobility
Today’s society is faced with increasing challenges when it comes to mobility. Technological advances in transportation have encouraged urban sprawl and increased demands for moving people and goods around. At the same time, the pressure on the current infrastructure (roads, public and freight transit) is also increasing and thus calling for smarter approaches to critical issues. Short-term solutions are always being proposed to long-term problems. Recently, the concept of autonomy (from adaptive to full self-driving capabilities) has gained traction and is promising to address several of the issues, while raising new ones, such as safety. In this talk I will survey some of the mathematical challenges and opportunities that are relevant to this new paradigm, focusing on optimal control applied to self-driving maneuvers, trajectory optimization and problems arising when designing efficient transportation solutions, such as adaptive cruise control and platooning. Numerical simulations will be presented to support the models discussed.
Mar 6, 2019 Seminar
Speaker: Dr. Richard Wellman, Colorado College
Title: Scalable semi-supervised learning with operators in Hilbert space
There is preponderance of semi-supervised learning problems in science and industry, but there is a dearth of applicable semi-supervised algorithms. The LaplaceSVM Semi-Supervised Support Vector Machine is a learning algorithm that demonstrates state of the art performance on benchmark semi-supervised data sets. However this algorithm does not scale. In this talk we’ll discuss the mathematical foundations of the LaplaceSVM and show the kernel is a solution of a non-homogenous self-adjoint operator equation. It can be shown certain Galerkin spectral approximations are themselves valid reproducing kernels that encode the underlying Riemannian geometry. The spectral kernels have excellent scalability metrics and interesting mathematical properties. We discuss both the mathematics and experimental results of the resultant semi-supervised algorithm.
Feb 20, 2019 Seminar
Speaker: Dr. Daniel Appelo, Applied Math, CU Boulder
Title: What’s new with the wave equation?
The defining feature of waves is their ability to propagate over vast distances in space and time without changing shape. This unique property enables the transfer of information and constitutes the foundation of today’s communication based society. To see that accurate propagation of waves requires high order accurate numerical methods, consider the problem of propagating a wave in three dimensions for 100 wavelengths with 1% error. Using a second order method this requires 0.2 trillion space-time samples while a high order method requires many orders of magnitude fewer samples.
In the first part of this talk we present new arbitrary order dissipative and conservative Hermite methods for the scalar wave equation. The degrees-of-freedom of Hermite methods are tensor-product Taylor polynomials of degree m in each coordinate centered at the nodes of Cartesian grids, staggered in time. The methods achieve space-time accuracy of order O(2m). Besides their high order of accuracy in both space and time combined, they have the special feature that they are stable for CFL = 1, for all orders of accuracy. This is significantly better than standard high-order element methods. Moreover, the large time steps are purely local to each cell, minimizing communication and storage requirements.
In the second part of the talk we present a spatial discontinuous Galerkin discretization of wave equations in second order form that relies on a new energy based strategy featuring a direct, mesh-independent approach to defining interelement fluxes. Both energy-conserving and upwind discretizations can be devised. The method comes with optimal a priori error estimates in the energy norm for certain fluxes and we present numerical experiments showing that optimal convergence for certain fluxes.
Daniel Appelö holds a Ph. D. degree in Numerical Analysis from the Royal Institute of Technology in Sweden and is currently an Associate Professor in Applied Mathematics at University of Colorado, Boulder. Prior to joining CU he was an Associate Professor at University of New Mexico and held postdoctoral positions at California Institute of Technology and Lawrence Livermore National Laboratory.
Fall 2018 seminars:
Dec 5, 2018 Seminar
Speaker: Dr. Barbara Prinari, UCCS
Title: Discrete solitons for the focusing Ablowitz-Ladik equation with non-zero boundary conditions via inverse scattering transform
Abstract: Soliton solutions of the focusing Ablowitz-Ladik (AL) equation with nonzero boundary conditions at infinity are derived within the framework of the inverse scattering transform (IST). After reviewing the relevant aspects of the direct and inverse problems, explicit soliton solutions will be discussed which are the discrete analog of the Tajiri-Watanabe and Kuznetsov-Ma solutions to the focusing NLS equation on a finite background. Then, by performing suitable limits of the above solutions, discrete analog of the celebrated Akhmediev and Peregrine solutions will also be presented. These solutions, which had been recently derived by direct methods, are obtained for the first time within the framework of the IST, thus providing a spectral characterization of the solutions and a description of the singular limit process.
Nov 14, 2018 Seminar
Speaker: Dr. Robert Jenkins, Colorado State University - Fort Collins
Title: Semiclassical soliton ensembles
Abstract: Equations like the Korteweg-de Vries (KdV) and the nonlinear Schroedinger equation exhibit interesting and complicated dynamics when the dispersive length scales in the problem are small compared to those of the initial wave profile; this is the relevant scaling regime for many problem is optical fibers. In this talk I'll discuss one way to analyze such problems for integrable PDEs using the inverse scattering transform (IST) that approximates initial data by an increasingly large sum of solitons. I'll talk both about NLS and some more recent work of mine on the resonant three wave interaction equations. There will be lots of pictures to help clear up the technical details!
Oct 17, 2018 Seminar
Speaker: Dr. Geraldo de Souza, Auburn University
Title: Fourier series, Wavelets, Inequalities, Geometry and Optimization
Abstract: This talk will have two parts. In the first part, I will start with motivation and comments to some important problems in Analysis. Each problem has led to important discovery, such as Wavelets, technique of convergence of Fourier, among others. The second part I will talk about Inequalities. In general, I view the second part of this presentation as simple or perhaps an elementary approach to the subject (even though it is a new idea). On the other hand, this talk will show some interesting observations that are part of the folklore of mathematics. I will go over some very common and important inequalities in analysis that we see in the course of Analysis and even in Calculus. I will give some different views of different proofs, using Geometry, Graphing and some of them “a new analytic proof” by using optimization of functions of two variables (this is very interesting).
Oct 3, 2018 Seminar
Speaker: Dr. Robert Carlson, UCCS
Title: An elementary trip from the Gauss hypergeometric function to the Poschl-Teller potential in quantum mechanics
Abstract: A simple transformation takes the (G) equation for the Gauss hypergeometric function to the (J) equation for Jacobi polynomials. J has an (unusual) adjoint equation (H) (of Heun type) with an extra singular point. H has eigenfunctions that can be expressed in terms of the Gauss hypergeometric function. Another change of variables lets us rediscover a ‘solvable’ (Poschl-Teller) Schrodinger equation. The methods use the kinds of techniques we often teach in Math 3400.
Sep 12, 2018 Seminar
Speaker: Dr. Sarbarish Chakravarty, UCCS
Title: Beach Waves and KP Solitons
Abstract: In this talk, I will give a brief overview of the soliton solutions of the KP equation, and discuss how these solutions can describe shallow water wave patterns on long flat beaches.
Apr 11, 2018 Seminar
Speaker: Dr. Greg Fasshauer, Colorado School of Mines
Title: An Introduction to Kernel-Based Approximation Methods
Abstract: I will start with a few historical remarks, and then motivate the use of kernel-based approximation as a numerical approach that generalizes standard polynomial-based methods. Examples of kernels and their use in data fitting problems will be provided along with an overview of some of the concerns and issues associated with the use of kernel methods.
Mar 14, 2018 Seminar
Speaker: Dr. Ethan Berkove, Laffayette College (Joint with Rings and Wings Seminar)
Title: Short Paths and Long Titles: Travels through the Sierpinski carpet, Menger sponge, and beyond.
Abstract: Sierpinski carpet and Menger sponge are fractals which can be thought of as two and three dimensional versions of the Cantor set. Like the Cantor set, each is formed by starting with a shape (a square for the carpet, a cube for the sponge) and then recursively removing certain subsets of it. Unlike the Cantor set, what remains is connected in the following sense: given any two points s and f in the carpet or sponge, there is a path from s to f that stays in the carpet or sponge. In this talk, we’ll discuss what we know about the shortest path from s to f in the carpet, sponge, and even higher dimensional versions of these fractals. The proofs required a surprising (at least to us) breadth of techniques, from combinatorics, geometry, and even linear programming. (Joint work with Derek Smith)
Feb 28, 2018 Seminar
Speaker: Dr. Radu Cascaval, UCCS
Title: Traffic Flow Models. A Tutorial
Abstract: We present several traffic flow models, both at the micro- and macro-scale, including for multi-lane traffic. Problems of controlling the traffic will be described and numerical simulations will illustrate possible solutions.
Dec 8, 2017 Seminar
Speaker: Dr. Barbara Prinari, UCCS
Title: Solitons and rogue waves for a square matrix nonlinear Schrodinger equation with nonzero boundary conditions
Abstract: In this talk we discuss the Inverse Scattering Transform (IST) under nonzero boundary conditions for a square matrix nonlinear Schrodinger equation which has been proposed as a model to describe hyperfine spin F = 1 spinor Bose-Einstein condensates with either repulsive interatomic interactions and anti-ferromagnetic spin-exchange interactions, or attractive interatomic interactions and ferromagnetic spin-exchange interactions. Emphasis will be given to a discussion of the soliton and rogue wave solutions one can obtain as a byproduct of the IST.
Nov 17, 2017 Seminar
Speaker: Dr. Oksana Bihun, UCCS
Title: New properties of the zeros of Krall polynomials
Abstract: We identify a class of remarkable algebraic relations satisfied by the zeros of the Krall orthogonal polynomials that are eigenfunctions of linear differential operators of order higher than two. Given an orthogonal polynomial family p_n(x), we relate the zeros of the polynomial p_N with the zeros of p_m for each m <=N (the case m = N corresponding to the relations that involve the zeros of pN only). These identities are obtained by exacting the similarity transformation that relates the spectral and the (interpolatory) pseudospectral matrix representations of linear differential operators, while using the zeros of the polynomial p_N as the interpolation nodes. The proposed framework generalizes known properties of classical orthogonal polynomials to the case of non-classical polynomial families of Krall type. We illustrate the general result by proving new remarkable identities satisfied by the Krall-Legendre, the Krall-Laguerre and the Krall-Jacobi orthogonal polynomials.
Oct 27, 2017 Seminar
Speaker: Dr. Radu C. Cascaval, UCCS
Title: What do Analysis and Scientific Computation have in common ...
Abstract: Analysis, the world of the infinitesimally small, is thought to be one of the last standing outposts where humans can fight the computational invasion. In spite of this fact, computational sciences continue to benefit greatly from advances in analysis. This talk will illustrate this relationship, in particular functional analysis connections to numerical spectral methods, meshless methods, and their applications to numerical solutions to PDEs.
Sept 29, 2017 Seminar
Speaker: Dr. Fritz Gesztesy, Baylor University
Title: The eigenvalue counting function for Krein-von Neumann extensions of elliptic operators (Slides)
Abstract: We start by providing a historical introduction into the subject of Weyl-asymptotics for Laplacians on bounded domains in n-dimensional Euclidean space, and a brief introduction into the basic principles of self-adjoint extensions. Subsequently, we turn to bounds on eigenvalue counting functions and derive such a bound for Krein-von Neumann extensions corresponding to a class of uniformly elliptic second order PDE operators (and their positive integer powers) on arbitrary open, bounded, n-dimensional subsets \Omega in R^n. (No assumptions on the boundary of \Omega are made; the coefficients are supposed to satisfy certain regularity conditions.) Our technique relies on variational considerations exploiting the fundamental link between the Krein-von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform of the corresponding differential operator suitably extended to all of R^n. We also consider the analogous bound for the eigenvalue counting function for the corresponding Friedrichs extension. This is based on joint work with M. Ashbaugh, A. Laptev, M. Mitrea, and S. Sukhtaiev.