Applied Mathematics Seminar, Spring 2015

Mondays 3:55 PM - 5:00 PM, LCB 219

• This seminar can be taken for credit: Students can get 1-3 credits by registering to the Applied Math Seminar class Math 7875 Section 010 for Spring 2015. Grading is based on attendance and giving at least one talk presenting an applied-mathematics paper (not necessarily your own). Student talks will be appropriately labeled to distinguish them from visitor talks. The seminar organizer is available to review your slides, for dry-runs etc.
• Please direct questions or comments about the seminar (or its class) to Yekaterina Epshteyn (epshteyn (at) math.utah.edu)
• Talks are announced through the applied-math mailing list. Please ask the seminar organizer for information about how to subscribe to this list.

January 23. Note: 3:55pm - 5:00pm, Room LCB 225
Speaker: Alexander Mamonov, Schlumberger
Title: Reduced order models for seismic full waveform inversion.
Abstract:We present a framework for the numerical solution of the seismic full waveform inversion (FWI) problem using the reduced order models (ROMs). In FWI one determines the spatial distribution of the acoustic or elastic properties of the subsurface from the surface or well-bore measurements of the seismic data induced by sources. In our approach the ROM is a projection of the acoustic or elastic PDE operator on the subspace spanned by the snapshots of the solutions to the forward problem. The ROM can be found directly from the measured time domain seismic data. The use of the ROM in inversion is twofold. First, after the transformation to the block tridiagonal (finite difference) form the ROM misfit can be used as an objective functional for optimization. Such functional is more convex than the conventional data misfit and thus is less prone to common issues like abundant local minima (cycle skipping), multiple reflection artifacts, etc. Second, if a background kinematic model is available the projected PDE operator can be backprojected to obtain a seismic image directly. This leads to a non-linear migration algorithm that recovers not only the locations of discontinuities (reflectors) but also their relative strength, the process known as the true amplitude migration.

Joint work with V. Druskin and M. Zaslavsky.

January 26 (Student Talk)
Speaker: Predrag Krtolica, Department of Mathematics, University of Utah
Title: Compatibility Conditions in Discrete Structures
Abstract:The talk will be about the analysis of compatibility conditions of 2-dimensional hexagonal frameworks, which, in the limit, lead to the continuum compatibility condition, suggesting the equivalence. This fact has many applications, as it is much easier to study brittleness of materials on a discrete model than on a continuum.

In addition, I will briefly talk about the statics of hexagonal grids, assuming infinitesimal deformation, and their possible applications.

February 2
Speaker: Ching-Shan Chou, Department of Mathematics, Ohio State University
Title: Computer Simulations of Yeast Mating Reveal Robustness Strategies for Cell-Cell Interactions
Abstract: Cell-to-cell communication is fundamental to biological processes which require cells to coordinate their functions. A simple strategy adopted by many biological systems to achieve this communication is through cell signaling, in which extracellular signaling molecules released by one cell are detected by other cells via specific mechanisms. These signal molecules activate intracellular pathways to induce cellular responses such as cell motility or cell morphological changes. Proper communication thus relies on precise control and coordination of all these actions.

The budding yeast Saccharomyces cerevisiae, a unicellular fungi, has been a model system for studying cell-to-cell communication during mating because of its genetic tractability. In this work, we performed for the first time computer simulations of the yeast mating process. Our computational framework encompassed a moving boundary method for modeling cell shape changes, the extracellular diffusion of mating pheromones, a generic reaction-diffusion model of yeast cell polarization, and both external and internal noise. Computer simulations revealed important robustness strategies for mating in the presence of noise. These strategies included the polarized secretion of pheromone, the presence of the alpha-factor protease Bar1, and the regulation of sensing sensitivity; all were consistent with data in the literature. In summary, we constructed a framework for simulating yeast mating and cell-cell interactions more generally, and we used this framework to reproduce yeast mating behaviors qualitatively and to identify strategies for robust mating.

February 9
Speaker: Hyeonbae Kang, Department of Mathematics, Inha University
Title: Spectral theory of Neumann-Poincare operator and applications
Abstract:The Neumann-Poincare (NP) operator is a boundary integral operator which arises naturally when solving boundary value problems using layer potentials. It is not self-adjoint with the usual inner product. But it can symmetrized by introducing a new inner product on $H^{-1/2}$ spaces using Plemelj's symmetrization principle. Recently many interesting properties of the NP operator have been discovered. I will discuss about this development and various applications including solvability of PDEs with complex coefficients and plasmonic resonance.

February 23
Speaker: Lajos Horvath, Department of Mathematics, University of Utah
Title: Statistical inference from panel data
Abstract:We consider N panels with T observations in each panel. The panels are time series, the dependence between the panels is modeled by unobservable common factors. We provide a CUSUM type tests as well as estimators for the location of the change in the means of the panels. We obtain several limit theorems for the CUSUM tests under the no change null and the exactly one change alternative. We establish the asymptotics for difference between the true value of the time of change and its estimator. We illustrate the testing method with data on the Gini index (measure of inequality) and on the corruption index. We apply the limit results to construct confidence intervals for the time of change in the exchange rates of the currencies of 23 countries with respect to the US dollar and for the change in the the GDP/capita in 113 countries.

February 27, Special Joint Time Series/Stochastics/Applied Math Seminar

Note: Time 3pm, Room LCB 219

Speaker: Alexander Aue, Department of Statistics, University of California, Davis
Title: Spectral analysis of high-dimensional time series
Abstract:This talk is concerned with extensions of the classical Marcenko-Pastur law to time series. Specifically p-dimensional linear processes are considered with are built from innovation vectors with independent, identically distributed (real- or complex-valued) entries possessing zero mean, unit variance and finite fourth moments. The coefficient matrices are assumed to be simultaneously diagonalizable. In this setting, the limiting behavior of the empirical spectral distribution of both sample covariance and symmetrized sample autocovariance matrices is determined in the high dimensional setting for which dimensionality and sample size diverge to infinity at the same rate. The results extend existing contributions in the literature for the covariance case and are (among) the first of their kind for the autocovariance case.
The talk is based on joint work with H. Liu and D. Paul.

March 9
Speaker: Jay Gopalakrishnan, Department of Mathematics, Portland State University
Title: The least-squares properties of Discontinuous Petrov-Galerkin methods
Abstract:This talk introduces a class of least squares methods that utilize discontinuous finite element spaces crucially, namely the DPG (Discontinuous Petrov Galerkin) methods. These methods minimize a residual in a computable dual norm. Simple conditions can be laid out under which a priori and a posteriori error analyses can be obtained. The features of the new method that make it an interesting alternative for certain problems will be highlighted.
This talk is a part of the special session of CMDS13.

March 13
Note: Room LCB 225
Title: Reduced order models for large scale wave problems
Abstract:Reduced order models approximate transfer functions of large-scale linear dynamical systems by small equivalent ones. Their matrices can be geometrically interpreted as finite-difference operators discretized on so-called optimal grids, a. k. a. spectrally matched grids or finite-difference Gaussian quadrature rules. In this talk we discuss some recent applications of this powerful approach to numerical solution of hyperbolic problems in the time and frequency domains. They include optimal discretization of perfectly matched layers and multi-scale elastic wave propagation. Time permitting, I will discuss another recent model reduction approach for wave propagation in unbounded domains, based on scattering resonance representation.

Contributors: Mikhail Zaslavsky; Alex Mamonov; Leonid Knizhnerman; Stefan Güttel and Rob Remis.

March 23
Speaker: Becca Thomases, Department of Mathematics, University of California, Davis
Title: Microorganism locomotion in viscoelastic fluids
Abstract: Low Reynolds number swimming of microorganisms in Newtonian fluids is an extensively studied classical problem. However, many biological fluids such as mucus are mixtures of water and polymers and are more appropriately described as viscoelastic fluids. Recently, there have been many studies on locomotion in complex fluids. Both experiments and theory have exhibited that viscoelasticity can lead to either an enhancement or retardation of swimming, but a complete understanding of this problem is lacking. A computational model of finite-length undulatory swimmers is used to examine the physical origin of the effect of elasticity on swimming speed. We show that both favorable stroke asymmetry and swimmer elasticity contribute to a speed-up, but a substantial boost results only when these two effects work together. Additionally, we examine a reduced model of an oscillatory bending beam in a viscoelastic fluid, and identify a threshold in amplitude related to the development of large elastic stresses. We relate this transition to previously studied bifurcations in steady extensional flows of complex fluids. This reduced model sheds some light on properties of swimmer gaits that are related to either elastic enhancement or hindrance.

March 27
Note: Room LCB 225
Speaker: Hee-Dae Kwon, Department of Mathematics, Inha University
Title: Applications of optimal control theory to a model of HIV infection
Abstract:In this talk, a model of HIV infection is considered with various compartments, including target cells, infected cells, viral loads and immune effector cells, to describe HIV type 1 infection. We show that the proposed model has one uninfected steady state and several infected steady states and investigate their local stability by using a Jacobian matrix method. We obtain equations for adjoint variables and characterize an optimal control by applying Pontryagin's Maximum Principle. In addition, we apply techniques and ideas from linear optimal control theory in conjunction with a direct search approach to derive on-off HIV therapy strategies. The results of numerical simulations indicate that hybrid on-off therapy protocols can move the model system to a "healthy" steady state in which the immune response is dominant in controlling HIV after the discontinuation of the therapy.

March 30
Speaker: Maxence Cassier, Department of Mathematics, University of Utah
Title: The limiting amplitude principle in a medium composed of a dielectric and a metamaterial
Abstract: For wave propagation phenomena, the limiting amplitude principle holds if the time-harmonic regime represents the large time asymptotic behavior of the solution of the evolution problem with a time-harmonic excitation. In this talk, we study a transmission problem between a dielectric and a metamaterial. The question we consider is the following: does the the limiting amplitude principle hold in such a medium? An answer is proposed in the case of a two-layered medium composed of a dielectric and a particular metamaterial (Drude model). In this context, we reformulate the time-dependent Maxwell's equations as a Schrödinger equation and perform its complete spectral analysis. This permits a quasi-explicit representation of its solution via the generalized diagonalization" of its associated unbounded self-adjoint operator. As an application of this study, we show finally that the limiting amplitude principle holds except for a particular frequency, called the plasmonic frequency, characterized by a ratio of permittivities and permeabilities equal to -1 across the interface. This frequency constitutes an unusual example of resonance in an unbounded medium and the response of the system to this excitation blows up linearly in time.

April 13 (Student Talk)
Speaker: Ornella Mattei, Visiting Department of Mathematics, University of Utah
Title: Variational formulations for the linear viscoelastic problem in the time domain
Abstract: The talk deals with the derivation of new variational formulations for the linear viscoelastic hereditary problem in the time domain. Such formulations are obtained by dividing the time domain into two subintervals of equal length, with the resulting doubling of the unknowns (displacement, strain and stress fields) and the consequent decomposition of the equations governing the problem (constitutive law, balance and compatibility equations). In particular, following some energetic arguments, we prove that a sub-operator of the split constitutive law operator is positive definite, since the related quadratic form physically represents a free energy (if the relaxation tensor is completely monotonic). We then derive a global minimum principle, which allows one to seek bounds of the mechanical properties of a composite material with viscoelastic phases.

April 27
Speaker: Brian Simanek Department of Mathematics, Vanderbilt University
Title: Paraorthogonal Polynomials and Electrostatics on the Unit Circle
Abstract: In 1885, Stieltjes showed that the zeros of certain Jacobi polynomials mark the equilibrium position of electrons confined to an interval. The proof relies on the fact that Jacobi polynomials solve a certain second order differential equation. In this talk, we will consider an analogous problem on the unit circle. The main tool we will use is paraorthogonal polynomials on the unit circle, and we will show that the zeros of such polynomials mark the equilibrium position of electrons confined to the unit circle under the influence of an external field. As in the case of an interval, the proof will depend on establishing these polynomials as solutions to second order differential equations.

May 16 (Snowbird Workshop as a part of CMDS-13)
Speaker: Kirill Cherednichenko, Department of Mathematical Sciences, University of Bath
Title: Homogenisation of the system of high-contrast Maxwell equations
Abstract:I shall discuss the system of Maxwell equations for a periodic composite dielectric medium with components whose dielectric permittivities $\epsilon$ have a high degree of contrast between each other. I assume that the ratio between the permittivities of the components with low and high values of $\epsilon$ are of the order $\eta^2,$ where $\eta>0$ is the period of the medium. I determine the asymptotic behaviour of the electromagnetic response of such a medium in the homogenisation limit", as $\eta\to0,$ and derive the limit system of Maxwell equations in ${\mathbb R}^3.$ The results that I shall present extend a number of conclusions of the paper [Zhikov ,V. V., 2004. On the band-gap structure of the spectrum of some divergent-form elliptic operators with periodic coefficients, St.Petersburg Math.J.] to the case of the full system of Maxwell equations. This is joint work with Shane Cooper (Bath).

May 18, Note Room LCB 215
Speaker: Scott McCalla, Department of Mathematical Sciences, Montana State University
Title: Existence and Stability of Radially Symmetric Solutions to the Swift--Hohenberg Equation
Abstract:The existence, stability, and bifurcation structure of localized radially symmetric solutions to the Swift--Hohenberg equation is explored both numerically through continuation and analytically through the use of geometric blow-up techniques. The bifurcation structure for these solutions is elucidated by formally treating the dimension as a continuous parameter in the equations. This reveals a family of solutions with an anomalous amplitude scaling that is far larger than expected from a formal scaling in the far field. One key advantage of the geometric blow-up techniques is that a priori knowledge of this scaling is unnecessary as it naturally emerges from the construction. The stability of these patterned states will also be discussed.

Seminar organizer: Yekaterina Epshteyn (epshteyn (at) math.utah.edu).

155 South 1400 East, Room 233, Salt Lake City, UT 84112-0090, T:+1 801 581 6851, F:+1 801 581 4148