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# Applied and Computational Mathematics

DMS Applied Mathematics Seminar
Sep 24, 2021 03:00 PM
ZOOM

Speaker: Lars Ruthotto, Emory University

DMS Applied Mathematics Seminar
Sep 17, 2021 03:00 PM
228 Parker Hall

Speaker: Stephen Shipman,  Louisiana State University (LSU)

Title: Inverse problem for a spectral asymmetry function

Abstract: For the Schrödinger equation $$−u'' + q(x)u = λu$$ on a finite x-interval, there is defined an “asymmetry function” $$a(λ;q)$$, which is entire of order 1/2 and type 1 in $$λ$$.  The main result identifies the classes of square-integrable potentials $$q(x)$$ that possess a common asymmetry function.  For any given $$a(λ)$$, there is one potential for each Dirichlet spectral sequence.  This has applications to the spectral theory of multi-layer quantum graphs.  (Collaboration with Malcolm Brown, Karl Michael Schmidt, and Ian Wood)

DMS Applied Mathematics Seminar
Sep 10, 2021 03:00 PM
ZOOM

Speaker: Sigal Gottlieb, University of Massachusetts, Dartmouth

Title: Developing high order, efficient, and stable time-evolution methods using a time-filtering approach.

Abstract: Time stepping methods are critical to the stability, accuracy, and efficiency of the numerical solution of partial differential equations. In many legacy codes, well-tested low-order time-stepping modules are difficult to change; however, their accuracy and efficiency properties may form a bottleneck. Time filtering has been used to enhance the order of accuracy (as well as other properties) of time-stepping methods in legacy codes. In this talk I will describe our recent work on time filtering methods for the Navier Stokes equations as well as other applications. A rigorous development of such methods requires an understanding of the effect of the modification of inputs and outputs on the accuracy, efficiency, and stability of the time-evolution method. In this talk, we show that time-filtering a given method can be seen as equivalent to generating a new general linear method (GLM). We use this GLM approach to develop an optimization routine that enabled us to find new time-filtering methods with high order and efficient linear stability properties. In addition, understanding the dynamics of the errors allows us to combine the time-filtering GLM methods with the error inhibiting approach to produce a third order A-stable method based on alternating time-filtering of implicit Euler method. I will present our new methods and show their performance when tested on sample problems.

DMS Applied Mathematics Seminar
Sep 03, 2021 03:00 PM
ZOOM

Speaker: Maria Ntekoume, Rice University

Title: Symplectic non-squeezing for the KdV flow on the line

Abstract: In this talk we prove that the KdV flow on the line cannot squeeze a ball in $$\dot H^{-\frac 1 2}(\mathbb R)$$ into a cylinder of lesser radius. This is a PDE analogue of Gromov’s famous symplectic non-squeezing theorem for an infinite dimensional PDE in infinite volume.

DMS Applied Mathematics Seminar
Aug 27, 2021 03:00 PM
228 Parker Hall

Speaker: Le Chen (Auburn)

Title: Exact asymptotics of the stochastic wave equation with time-independent noise

Abstract: In this talk, I will report a recent joint work with Raluca Balan and Xia Chen. In this work, we study the stochastic wave equation in dimensions $$d\leq 3$$, driven by a Gaussian noise $$\dot{W}$$ which does not depend on time. We assume that either the noise is white, or the covariance function of the noise satisfies a scaling property similar to the Riesz kernel. The solution is interpreted in the Skorohod sense using Malliavin calculus. We obtain the exact asymptotic behaviour of the p-th moment of the solution either when the time is large or when p is large. For the critical case, that is the case when d=3 and the noise is white, we obtain the exact transition time for the second moment to be finite.

DMS Applied Mathematics Seminar
Apr 23, 2021 02:00 PM
ZOOM Join from PC, Mac, Linux, iOS or Android: https://auburn.zoom.us/j/88918338743

CANCELED               CANCELED                  CANCELED

Speaker: Weiwei Hu (University of Georgia, https://www.math.uga.edu/directory/people/weiwei-hu)

Title: Optimal Control Design for Fluid Mixing: Analysis and Computation

Abstract: The question of what velocity fields effectively enhance or prevent transport and mixing, or steer a scalar field to a desired distribution, is of great interest and fundamental importance to the fluid mechanics community. In this talk, we mainly discuss the problem of optimal mixing of an inhomogeneous distribution of a scalar field via active control of the flow velocity, governed by the Stokes or the Navier-Stokes equations. Specifically, we consider that the velocity field is steered by a control input which acts tangentially on the boundary of the domain through the Navier slip boundary conditions. This is motivated by mixing within a cavity or vessel by rotating or moving walls. Our main objective is to design a Navier slip boundary control that optimizes mixing at a given final time. Non-dissipative scalars governed by the transport equation will be of our main focus in this talk. In the absence of molecular diffusion, mixing is purely determined by the flow advection. This essentially leads to a nonlinear control and optimization problem. A rigorous proof of the existence of an optimal controller and the first-order necessary conditions for optimality will be derived. Numerical experiments will be presented to demonstrate our ideas and control designs.

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DMS Applied Mathematics Seminar
Apr 16, 2021 02:00 PM
ZOOM https://auburn.zoom.us/j/88918338743

Speaker: Ms. Shuwen Xue (Auburn University)

Title: Dynamics of chemotaxis models with logistic source

Abstract: Chemotaxis models are used to describe the evolution of mobile species subject to chemical substances. In this talk, I will introduce the chemotaxis models with logistic source first. Then, I will talk about parabolic-elliptic chemotaxis model with logistic source. The effects of chemotaxis on spreading speeds and traveling waves will be investigated. Biologically, spreading speeds can be understood as the asymptotic rate at which a species, initially introduced in a bounded range, expands its spatial range as time evolves, while traveling waves describe the propagation of  species as a wave with a fixed shape and a fixed speed. Next, parabolic-elliptic chemotaxis model under climate change will be discussed. The persistence criterion for species and the existence of forced waves will be presented. Forced waves illustrate how the species under consideration would die out at every point. Some numerical simulations will be presented to demonstrate the existence of forced waves.

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DMS Applied Mathematics Seminar
Apr 09, 2021 02:00 PM
ZOOM https://auburn.zoom.us/j/88918338743

Speaker: Prof. John Schotland (Yale University)

Title: Quantum Optics in Random Media

Abstract: Quantum optics, the quantum theory of light-matter interactions, is primarily concerned with systems consisting of a small number of atoms. In this talk I will describe recent work on many-body problem in quantum optics, focusing on the case of random media. In this setting, there is a close relation to kinetic equations for PDEs with random coefficients. Applications to the propagation of entangled two-photon states will be described.

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DMS Applied Mathematics Seminar
Apr 02, 2021 02:00 PM
ZOOM https://auburn.zoom.us/j/88918338743

Speaker: Mr. Jiaqi Cheng (Auburn University)

Title:  The lottery model for ecological competition in non-stationary environments

Abstract: In this work, we study the two-species lottery competition model with non-stationary reproduction and mortality rates of  both species is studied.   First a diffusion approximation for the fraction of sites occupied by each adult species is derived as the continuum limit of a classical discrete-time lottery model.   Then a non-autonomous stochastic differential equation on sites occupied by the species, as well as a Fokker-Planck equation on its transitional probability are developed.   Existence, uniqueness, and dynamics of solutions for the resulting stochastic differential equation are investigated, from which sufficient conditions for the existence of a time-dependent limiting process and coexistence of species in the sense of stochastic persistence are established.   A unique classical solution to the Fokker-Planck equation is also proved to exist and shown to be a probability density.  Numerical simulations are presented to illustrate the theoretical results.

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DMS Applied Mathematics Seminar
Mar 26, 2021 02:00 PM
ZOOM

This week’s talk will be given by three speakers on an interdisciplinary collaboration.

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Speakers:

Hoa Nguyen, Trinity University, San Antonio, TX (https://www.trinity.edu/directory/hnguyen5)

Orrin Shindell, Trinity University, San Antonio, TX (https://www.trinity.edu/directory/oshindel)

Bruce Rodenborn, Centre College, Danville, KY (https://www.centre.edu/directory/name/bruce-rodenborn/)

Title: Experiments and Simulations of Bacterial Motility Near Surfaces

Abstract: Motile bacteria commonly experience hydrodynamic effects of a nearby surface in their fluid environment. We use biological experiments, numerical simulations, and dynamically similar macroscopic experiments to quantify the forces and torques on microorganisms that swim near a boundary using a helical flagellum. Using the computational method of images for regularized Stokeslets with parameter values measured from biological experiments, we compute the forces and torques exerted on bacteria of different body sizes and helical geometries moving near a boundary. These results are compared with dynamically similar macroscopic experiments and theoretical predictions. Our numerical simulations are verified for rotating cylinders and rotating helices of different wavelengths. We calculate the propulsive efficiency of bacterial motion and find it increases for all helical wavelengths near a boundary and that the maximum increase in efficiency occurs for biologically relevant helical wavelengths. These results suggest that hydrodynamic boundary effects are an important factor in bacterial evolution.

DMS Applied Mathematics Seminar
Mar 12, 2021 02:00 PM
ZOOM

Speaker: Dr. Songting Luo (Iowa State University, https://orion.math.iastate.edu/luos/CV/index.html)

Title: Asymptotic Green's function methods for high frequency wave propagation

Abstract: We will present asymptotic methods for simulating high frequency wave propagation. The Huygens' principle or the Feymann's path integral is used as the time propagator for the wavefunction, where the Green's functions are approximated by asymptotic approximations. Upon obtaining analytic approximations for the phase and amplitude of the Green's functions, the resulting integral can be evaluated by fast Fourier transform after appropriate low rank approximations.  The perfectly matched layer method is incorporated to restrict the computation onto a bounded domain of interest. Numerical example will be presented to demonstrate the asymptotic Green's function methods.

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Last Updated: 09/25/2015