epshteyn (at) math.utah.edu)September 19, Joint Stochastics/Applied Math/Math Biology Seminar, Note Time 3:00 - 4:00 pm, Room LCB 219
Speaker: Peter Bossaerts, Finance Department, University of Utah
Title: Human reaction to extreme events and its biological foundations
Abstract:The talk will discuss recent work on how humans react to extreme events. A deeper
understanding of human reactions requires proper definition of an extreme event
(linking it to the concept of an outlier in statistical analysis) and its nature
(distinguishing between leptokurtic and platykurtic settings, and whether the
outlier is transient or reflects more persistent changes). Evidently, the
noradrenergic system in the human brain provides crucial support for tracking
outliers, while signals in the anterior insula allow humans to distinguish between
transient and fundamental outliers. Overall, human reaction to outliers that signal
fundamental changes is better adapted than reaction to transient outliers. The
latter type of outlier is ubiquitous in modern large-scale social institutions,
however, such as financial markets, internet and air traffic.
September 22
Speaker: Tatyana Sorokina, Department of Mathematics, Towson University
Title: Dimension of splines and intrinsic supersmoothness
Abstract:The phenomenon, known as``supersmoothness" was first
observed for bivariate splines and attributed to the polynomial nature of
splines. Using only standard tools from multivatiate calculus it is easy to show
that if we continuously glue two smooth functions along a curve with a
``corner", the resulting continuous function must be
differentiable at the corner, as if to compensate for the singularity of the
curve. Moreover, locally, this property
characterizes non-smooth curves. We generalize this phenomenon to
higher order derivatives. Additionally, we show how supersmoothness helps to
find dimension of multivariate splines.
This talk is a part of the special session of CMDS13.
September 29
Speaker: Graeme Milton, Department of Mathematics, University of Utah
Title: The searchlight effect in hyperbolic media
Abstract:Hyperbolic media in which the dielectric tensor has both positive and negative
eigenvalues have been shown to defeat the
diff raction limit, and allow features at very small wavelengths to be resolved as
demonstrated through hyperlenses. Even
in quasistatics the underlying equation resembles a wave equation. Whereas a
circular hole in a dielectric media has
a simple dipolar field around it, we will see that a circular hole in an almost
lossless hyperbolic media, has surrounding
quasistatic fields which diverge along characteristic lines tangent to the hole,
and which have finite total energy absorption
along these lines, even as the loss in the media tends to zero. In a hyperbolic
medium a dipole with small polarizability
can dramatically influence the dipole moment of a distant polarizable dipole, if it
is appropriately placed. We call this the
searchlight e ffect, as the enhancement depends on the orientation of the line
joining the polarizable dipoles and can be
varied by changing the frequency. For some particular polarizabilities the
enhancement can actually increase the further
the polarizable dipoles are apart, like the way quarks interact more strongly the
further they are apart.
Graeme Milton, R.C.McPhedran, A. Sihvola
October 6
Speaker: Maxence Cassier, Department of Mathematics, University of Utah
Title: Analysis of two time-dependent wave propagation phenomena:
1) Space-time focusing on unknown scatterers;
2) Limiting amplitude principle in a medium composed of a dielectric and a
metamaterial.
Abstract:This talk consists of two independent parts related to my Ph.D research
thema.
In the first one, we are motivated by this challenging question: in a
propagative medium which contains several unknown scatterers, how can one
generate a wave that focuses selectively on one scatterer not only in
space, but also in time? In other words, we look for a wave that "hits
hard at the right spot". The technique proposed here is based on DORT
method (French acronym for Decomposition of the Time Reversal Operator)
which leads to space focusing properties in the frequency domain.
The second part is devoted to a transmission problem between a dielectric
and a metamaterial. The question we consider here is the following : does
the limiting amplitude principle hold in such a medium? This principle
defines the stationary regime as the large time asymptotic behavior of a
system subject to a periodic excitation. An answer is proposed here in the
case of an infinite two-layered medium composed of a dielectric and a
particular metamaterial (Drude model).
October 20
Speaker: Andrejs Treibergs, Department of Mathematics, University of Utah
Title: Which Electric Fields are Realizable in Conducting Materials?
Abstract:The usual problem is that we are given the conductivity $\sigma$ and seek the
electric field in the material, $\nabla u$,
by solving the conductivity equation for the potential
$$
\operatorname{div}(\sigma\nabla u) = 0.
$$
We ask instead, given a field $\nabla u$, is it possible to find a conductivity
scalar or matrix $\sigma$ to satisfy the conductivity equation? In other words, is
the field is realizable?
We show periodic fields are isotropically realizable by solving a first order PDE,
although the conductivity may
not be periodic. We can characterize fields that are isotropically realizable by a
periodic conductivity. We also
give conditions for periodic realizability by matrix conductivities.
This is joint work with Marc Briane and Graeme Milton.
October 27
Speaker: Elena Cherkaev, Department of Mathematics, University of Utah
Title: Spectral representation for composite materials in forward and inverse
homogenization problems
Abstract:The spectral representation of the effective properties of composites is a
Stieltjes analytic representation which relates the n-point
correlation functions of the microstructure to the moments of the spectral
measure
of an operator depending on the geometry of composite. The talk discusses an
inverse homogenization problem of deriving information about the
microgeometry from known effective properties, the approach is based on
reconstruction
of the spectral measure. The spectral measure which contains all
information about
the microstructure, can be uniquely recovered from effective measurements
known in
an interval of frequency. In particular, the volume fractions of materials
in the composite and an inclusion separation parameter, as well as the
spectral gaps
at the ends of the spectral interval, can be uniquely reconstructed. I
will discuss
identification of microstructural parameters from electromagnetic and
viscoelastic
effective measurements and show an extension to nonlinear composites.
November 3
Speaker: Braxton Osting, Department of Mathematics, University of Utah
Title: Geometric methods for graph partitioning
Abstract: Several geometric methods for graph partitioning have been
introduced in the past few years, with wide applications in clustering,
community detection, and image analysis. These methods, which I'll review,
are built on graph-based analogues of total variation, motion by mean
curvature, the Ginzburg-Landau functional, and the Merriman-Bence-Osher
threshold dynamics. In this talk, I'll discuss a new graph partitioning
method where the optimality criterion is given by the sum of the Dirichlet
eigenvalues of the partition components. The resulting eigenvalue
optimization problem can be solved by a rearrangement algorithm, which we
show to converge in a finite number of iterations to a local minimum of a
relaxed objective function. The method compares well to state-of-the-art
approaches when applied to clustering problems on graphs constructed from
synthetic data, MNIST handwritten digits, and manifold discretizations. The
model has a semi-supervised extension and provides natural representatives
for the clusters as well.
November 10
Speaker: Varun Shankar, Department of Mathematics, University of Utah
Title: A Radial Basis Function (RBF)-based Leray Projection
Method for the Incompressible Stokes and Navier-Stokes equations
Abstract: Traditionally, the unsteady incompressible Stokes and Navier-Stokes
equations have been solved by fractional-step projection methods. These
methods require a specification of numerical boundary conditions for the
pressure, in addition to boundary conditions for the velocity. Selecting
the wrong pressure boundary conditions leads to a large time-splitting
error. I will present a new method based on a discrete Leray projection
that avoids any time-splitting and poisson solves. The discrete Leray
projector is built with a divergence-free RBF interpolant. I will
demonstrate that the method shows high orders of convergence in both space
and time (6th and 4th orders) on the unsteady incompressible Stokes
equations, and show preliminary results for the Navier-Stokes equations as
well. This is a research-in-progress talk.
November 17
Speaker: Benjamin Webb, Department of Mathematics, Brigham Young University
Title: A New Type of Stability for Dynamical Networks
Abstract: In this talk the stability of a general class of dynamical networks is
considered. We begin by giving a brief introduction into the study of networks
including their graph structure and the variety of dynamics such systems exhibit.
Following this, we present a criteria for the global stability of a general
dynamical network and show that this type of stability is invariant with respect
to the addition and removal of time delays. Since time delays can have a
destabilizing effect on a system's dynamics this is a stronger form of stability,
which we call intrinsic stability. By using this notion of intrinsic stability we
show that it is possible to reduce a network by removing its "implicit delays".
The resulting smaller network can be used to obtain improved stability estimates
of the original unreduced network.
November 24
Speaker: Dongbin Xiu, Department of Mathematics and Scientific Computing and Imaging (SCI) Institute, University of Utah
Title: Multi-dimensional polynomial interpolation on arbitrary nodes
Abstract: Polynomial interpolation is well understood on the real line. In
multi-dimensional spaces, one often adopts a well established one-dimensional method
and fills up the space using certain tensor product rule. Examples
like this include full tensor construction and sparse grids construction.
This approach typically results in
fast growth of the total number of interpolation nodes and certain fixed geometrical structure of the
nodal sets. This imposes difficulties for practical applications,
where obtaining function values at a large number of nodes is
infeasible. Also, one often has function data from nodal locations that are not
by "mathematical design" and are ``unstructured''.
In this talk, we present a mathematical framework for conducting polynomial interpolation
in multiple dimensions using arbitrary set of unstructured nodes. The resulting method,
least orthogonal interpolation, is rigorous and has a straightforward numerical implementation.
It can faithfully interpolate any function data on any nodal sets, even on those that are considered
singular by the traditional methods. We also present a strategy to choose
``optimal'' nodes that result in robust
interpolation. The strategy is based on optimization of Lebesgue
function and has certain highly desirable mathematical properties.
December 1
Speaker: Andrej Cherkaev, Department of Mathematics, University of Utah
Title: Constraints in multiwell variational problems and recovering the functional from the optimal solution
Abstract: The talk deals with two problems:
1. The solution of quasiconvex envelopes for multiwell Lagrangians is an infinitely fast oscillating sequence that takes the values (Young's measures) in several wells. This values - supporting points of the envelope - are subject to certain constraints. There constraints are discussed and examples are provided.
2. In a number of applications, we could postulate that a design is optimal
(for instance, it is perfected by evolution), but the goal functional is
unknown. I will discuss the technique of recovering these functionals and
prove as an example that "All Beams are Created Optimal"
December 8
Speaker: Ben Adcock, Department of Mathematics, Simon Fraser University
Title: Function approximation via infinite-dimensional convex optimization
Abstract: In a number of applications, including uncertainty quantification and,
more generally, scattered data approximation, one is required to approximate a
smooth multivariate function from a small number of pointwise samples. Classically,
this task is carried out by methods such as interpolation or least-squares fitting.
Yet with the advent of compressed sensing there has been an increasing focus on the
use alternative techniques based on convex optimization. In this talk I will
describe an infinite-dimensional framework for function approximation from pointwise
samples via weighted l^1 minimization. The framework I will introduce is general,
and applies to arbitrary point sets and expansion systems. I will explain why
working in infinite dimensions is both theoretically and practically important, and
describe the critical role that weights play in the minimization. In the second
half of the talk I will address the following question: does weighted l^1
minimization always perform at least as well as classical least squares regression?
An affirmative answer to this question is of practical relevance, since it means
that weighted l^1 techniques should always be used in over classical techniques in
applications where the primary limitation is the amount of data available. I will
present a mathematical framework for examining this question, and answer it in the
affirmative for the case of polynomials approximations from structured and
unstructured data. Finally, I will discuss the role that sparsity plays this
framework, and present some open problems and challenges.
epshteyn (at) math.utah.edu).
Past lectures: 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.