Applied Math Seminar

On Nonlocal Dispersal Evolution Equations

04/20/2015 - 12:00pm
04/20/2015 - 1:00pm
Wenxian Shen (Auburn University)

The current talk is concerned with some dynamical issues in nonlocal dispersal evolution equations. First, I will present some spectral theory for nonlocal dispersal operators with time periodic dependence. I will then consider the asymptotic dynamics of nonlocal dispersal evolution equations/systems in bounded media. Finally, I will give some discussion on the front propagation dynamics of nonlocal dispersal evolution equations in unbounded media.

Kravis 100

Recovery from Linear Measurements via Denoising and Approximate Message Passing

02/04/2015 - 1:15pm
02/04/2015 - 2:15pm
Dror Baron (NC State)


Approximate message passing (AMP) decouples linear inverse problems into iterative additive white 
Gaussian noise (AWGN) scalar channel denoising problems. The first part of the talk provides a 
tutorial-style overview of AMP, including advantages, a convenient design methodology, and 
possible pitfalls. The second part describes how we have designed denoising algorithms that 
recover signals from AWGN, and applied these denoisers within AMP to reconstruct signals from 
linear mixing channels. Examples include parametric denoising for parametric signals, universal 
denoising for stationary ergodic signals, and image denoising for natural images. Our favorable 
numerical results indicate that AMP is a promising tool for solving linear inverse problems.
Dror Baron received the B.Sc. (summa cum laude) and M.Sc. degrees from
the Technion - Israel Institute of Technology, Haifa, Israel, in 1997
and 1999, and the Ph.D. degree from the University of Illinois at
Urbana-Champaign in 2003, all in electrical engineering.
From 1997 to 1999, he worked at Witcom Ltd. in modem design. From 1999
to 2003, he was a research assistant at the University of Illinois at
Urbana-Champaign, where he was also a Visiting Assistant Professor in
2003. From 2003 to 2006, he was a Postdoctoral Research Associate in
the Department of Electrical and Computer Engineering at Rice
University, Houston, TX. From 2007 to 2008, he was a quantitative
financial analyst with Menta Capital, San Francisco, CA. From 2008 to
2010 he was a visiting scientist in the Department of Electrical
Engineering at the Technion. Dr. Baron joined the Department of
Electrical and Computer Engineering at North Carolina State University
in 2010 as an assistant professor. His research interests include
information theory and statistical signal processing.



Kravis Center 166

Mathematical models of incompressible viscoelastic media

01/26/2015 - 3:30pm
01/26/2015 - 4:30pm
V. V. Pukhnachev (Lavrentyev Institute of Hydrodynamics, Siberian Division of RAS, and Novosibirsk State University)

This research was stimulated by experiments carried out in Yekaterinburg (Apakashev and Pavlov, 1997) and in Chelyabinsk (Korenchenko and Beskachko, 2008). Authors found that at small strain rates ordinary water and similar liquids demonstrate not only viscous properties but also elastic ones. At the same time, we can neglect liquid compressibility. As a result, we lose the important property of hyperbolicity for the governing PDE system, which is typical for many mathematical models in continuum mechanics. There is no general theory of initial boundary value problems for these systems. However we can obtain a lot of valuable information analyzing exact solutions of corresponding systems in one and two dimensions. In particular, we study problem of filling in a spherical cavity in Maxwell and Kelvin-Voigt viscoelastic media and the analog of the classical Couette problem in plane and cylindrical geometries for both of them. We found a deep analogy between Kelvin-Voigt model and model of acoustics of viscous gas. In contrast, equations of incompressible Maxwell model are similar to equations of inviscid gas with non-convex constitutive law. It means that existence of weak or strong discontinuities is possible in the motion of incompressible Maxwell medium. We give examples of this phenomenon for motion with plane waves.

CGU Math South Conference Room, 710 N. College Ave

Actin polymerization and subcellular mechanics drive nonlocal excitable waves in living cells

02/02/2015 - 12:00pm
02/02/2015 - 1:00pm
Jun Allard (University of California, Irvine)

Traveling waves of the actin have recently been reported in many cell types. Actin is a protein that forms polymers that power cell crawling in immune cells during immune surveillance, skin cells during wound healing, and cancer cells during invasion. Fish keratocyte cells, which usually exhibit rapid and steady crawling, exhibit traveling waves of protrusion when plated on highly adhesive surfaces. We hypothesize that waving arises from a competition between actin polymerization and mature adhesions for VASP, a protein that associates with growing actin barbed ends. We developed a mathematical model of actin protrusion coupled with membrane tension, adhesions and VASP. The model is formulated as a system of partial differential equations with a nonlocal integral term and noise. These equations reveal a mathematical structure in which the system dynamically undergoes a bifurcation between oscillatory and excitable states. Simulations of this model lead to a number of predictions, for example, that overexpression of VASP prevents waving, but depletion of VASP does not increase the fraction of cells that wave. Further experiments confirmed these predictions and provided quantitative data to estimate the model parameters. We thus conclude that the waves are the result of competition between actin and adhesions for VASP, rather than a regulatory biochemical oscillator or mechanical tag-of-war. We hypothesize that this waving behavior contributes to adaptation of cell motility mechanisms in perturbed environments. This is work in collaboration with Erin Barnhart, Julie Theriot, and Alex Mogilner.

Kravis Center 100


12/15/2014 - 1:15pm
12/15/2014 - 2:15pm
Ellis Cumberbatch (CGU)

The ODE studied (first-order Abel of second kind) is part of a solution approach to the PDEs governing the flow of electrons in a double gate transistor. The ODE has exact solutions that are quite complicated. The application requires simple, accurate formulae for the current flow, and to this end the matched asymptotic expansion technique is used. This technique involves approximations in various regions. The full ODE cannot be avoided for the solution in one layer, however. So the exact solutions were explored and they show surprising behaviour. Profs M. Chugunova and S. L. Smith (UCSD) are co-authors.

CGU Burkle 24

Fast integral-equation based solutions of the Laplace eigenvalue problems with mixed boundary conditions

12/01/2014 - 12:00pm
12/01/2014 - 1:00pm
Eldar Akhmetgaliyev (Caltech)

In this talk we present a range of numerical methods which, based on use of Green functions and integral equations, can be applied to produce solution of Laplace eignvalue problems with arbitrary boundary conditions (including, e.g., Dirichlet/Neumann mixed boundary conditions) and in arbitrary domains (including e.g. domains with corners). As part of our presentation we present newly obtained characterizations of the singularities of solutions and eigenfunctions which arise at transition points where Dirichlet and Neumann boundary conditions meet; the numerical methods mentioned above rely on use of these characterizations in conjunction with the novel Fourier Continuation technique to produce solutions with a high order of accuracy. In particular, the resulting method exhibits spectral convergence for smooth domains (in spite of the solution singularities at Dirichlet/Neumann junctions) and prescribed high-order convergence for non-smooth domains.

A point of interest concerns the search algorithm in our eigensolver, which proceeds by searching for frequencies for which the integral equations of the problem admit non-trivial kernels. As it happens, the “minimum-singular- value” objective function gives rise to a challenging nonlinear optimization problem. To tackle this difficulty we put forth an improved objective functional which can be optimized by means of standard root-finding methods.

The methods above were also applied to modal analysis problems in electromagnetics: our calculation of TE and TM modes (eigenfunctions) of the cross sections of specifically designed quadruple-ridged flare horn microwave (astrophysical) antenna have been applied to the problem of optimization of the antenna parameters. The resulting eigensolutions are produced with such high accuracy that it becomes possible to track the eigenvalue/eigenfunction evolution with shape changes even as eigenvalues cross—a capability that is necessary for the antenna-design application, and which existing commercial software packages were not able to deliver.

Other applications will also be mentioned, including methods for evaluation of transmission eigenvalues that arise in the field of inverse problems and computation of Laplace eigenfunctions as a basis for spectral decomposition of functions in the space of square integrable functions—with application to, e.g., highly accurate separation-of-variables solution of time-dependent problems (including diffusion and wave-propagation) in arbitrary, possibly singular spatial domains and with possibly mixed boundary conditions.


KRV 164

Conductivity imaging from minimal interior measurements and weighted least gradient problems

11/17/2014 - 12:00pm
11/17/2014 - 1:00pm
Amir Moradifam (UC Riverside)

I will discuss the problem of recovering an isotropic conductivity outside of some perfectly conducting or insulating inclusions from knowledge of the magnitude of one current density vector field. This problem is closely related to uniqueness of minimizers of certain weighted least gradient problems and theory of minimal surfaces. We prove that the conductivity outside of the inclusions as well as the shape and position of the inclusions are uniquely determined by the magnitude of the current generated by imposing a given boundary voltage.

CMC, Kravis Center, KRV 164

Active water vapor absorption in the mealworm Tenebrio molitor: modeling water and ion fluxes

11/10/2014 - 12:00pm
11/10/2014 - 1:00pm
Jonathan Wright (Biology, Pomona College)

Several arthropod families have evolved specialized water abstraction mechanisms that are amenable to mathematical modeling. One example is the larvae of tenebrionid beetles, including mealworms, which are able to absorb water vapor from sub-saturated humidity. Such water vapor absorption (WVA) provides an effective means of rehydration for insects living in arid microclimates or exploiting foods of low water content. The mealworm uptake mechanism employs an elaborate cryptonephric system ensheathing the rectum. Active transport of potassium ions into this structure generates high osmotic concentrations and colligative lowering of water vapor pressure. In sufficiently high relative humidity, the vapor pressure gradient is inward, causing water vapor to diffuse from the rectal lumen into the cryptonephric system. Key unresolved questions with this system include: (i) how potassium is cycled from the cryotonephric system into the blood and back again, and (ii) how water is ultimately moved into the blood against a large osmotic gradient. This talk will discuss our current understanding of the ion gradients and WVA kinetics, and how we might model ion and water fluxes during steady-state uptake.

CMC, Kravis Center, KRV 164

The neural ring: using algebraic geometry to analyze neural codes

11/03/2014 - 12:00pm
11/03/2014 - 12:50pm
Nora Youngs (Mathematics, Harvey Mudd College)

Navigation is one of the most important functions of the brain. This year, the Nobel Prize in Medicine and Physiology was awarded for the discovery of place cells and grid cells, the neurons responsible for this ability. Though the external observed correspondence of these neurons to 2D receptive fields has been carefully recorded and proven, the animal itself navigates the world without access to these mapsAn important problem confronted by the brain is to infer what properties of a stimulus space can - in principle - be extracted from the stimulus space. This motivates us to define the neural ring and a related neural ideal, algebraic objects that encode the combinatorial data of a neural code. We find that these objects can be expressed in a "canonical form'' that directly translates to a minimal description of the receptive field structure intrinsic to the neural code, and present an algorithm to compute this canonical form.

CMC, Kravis Center, KRV 164

Modeling Equine Infectious Anemia Virus Infection: Virus Dynamics, Immune Control and Escape

10/13/2014 - 1:15pm
10/13/2014 - 2:15pm
Elissa J. Schwartz (Washington State University)

Equine Infectious Anemia Virus (EIAV) is a retrovirus that establishes a persistent infection in horses and ponies. The virus is in the same lentivirus subgroup that includes human immunodeficiency virus (HIV). The similarities between these two viruses make the study of the immune response to EIAV relevant to research on HIV. I will present mathematical models of within-host EIAV infection dynamics that include immune responses. Models are fit to data from horses with severe combined immunodeficiency to estimate viral dynamic parameters. Analysis of the models yields results on thresholds that would be necessary for immune responses to successfully control infection. Furthermore, model results predict the conditions under which multiple competing strains coexist or a subdominant viral strain escapes antibody neutralization and dominates the infection. Numerical simulations are presented to illustrate the results. These findings have the potential to lead to immunological control measures for lentiviral infection.

HMC Shanahan 3461 (across from the SkyCube)
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