epshteyn (at) math.utah.edu)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.
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.
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.
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
Speaker:
Vladimir Druskin, Schlumberger
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.
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.
epshteyn (at) math.utah.edu).
Past lectures: Fall 2014, Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, Spring 2011, Fall 2010, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006, Spring 2006, Fall 2005, Spring 2005, Fall 2004, Spring 2004, Fall 2003, Spring 2003, Fall 2002, Spring 2002, Fall 2001, Spring 2001, Fall 2000, Spring 2000, Fall 1999, Spring 1999, Fall 1998, Spring 1998, Winter 1998, Fall 1997, Spring 1997, Winter 1997, Fall 1996, Spring 1996, Winter 1996, Fall 1995.