Applied Math. Seminar, Fall 2006

**August 30: SPECIAL TIME, DATE, and LOCATION, 2:15 - 3:05 PM, LCB 225**

**Speaker: **Amy Shen, Washington University in St. Louis - Dept. of Mechanical and Aerospace Engineering

**Title: **Hydrodynamics of complex fluids at small length-scales

**Abstract: **Understanding fluid transport and interfacial phenomena of
complex fluids at small length-scales is crucial to understanding how to design
and exploit of micro- and nano-fluidic devices. I will present two
examples. The first studies evaporation driven self-assembly to
synthesize nanoporous thin films. A combination of experimental
measurement and modeling using lubrication theory shows how
self-assembly influences coating film thickness. The second example
studies how length-scale and fluid elasticity affect droplet pinch-off
of "simple" polymeric liquids in microfluidic
environments. Boger fluids (viscoelastic liquids with nearly constant
shear viscosity) are pumped into microchannels and pinched off to form
droplets in an immiscible oil
phase. We find a power law relation between the dimensionless
capillary pinch-off time and the so-called elasticity number, E, of the
fluid. Theoretical models that neglect the extensional viscosity of the
fluid become increasingly more inaccurate as the fluid elasticity
increases.

**September 5: SPECIAL TIME, DATE, and LOCATION, 3:05 - 3:55 PM,
LCB 115**

** Joint with the Approximation Theory Seminar **

** Speaker: **Peter Alfeld, University of Utah

**Title: **The Bernstein Bézier Form of a Multivariate Polynomial

**Abstract: **The Bernstein-Bézier form (or just B-form) of a polynomial
is a highly successful and widely used way of representing polynomials,
particularly polynomials in more than one variable. Its power stems
from the fact that algebraic issue, such as two polynomials joining
smoothly, can be studied and interpreted geometrically. There is also
a close geometric connection between the coefficients of a polynomial
and the shape of its graph. In this talk I will define the B-form of a
polynomial and discuss some of its properties. This will serve as the
foundation for several future talks this semester. The talk will
include computer demonstrations.

**September 11: - Joint with the Stochastics Seminar**

Speaker: Eulalia Nualart, University of Paris 13

**Title: **Potential theory for non-linear stochastic heat equations

**Abstract: **In this talk we develop potential theory for a system of
non-linear stochastic
heat equations in spatial dimension one and driven by a space-time white
noise.
In particular, we prove upper and lower bounds on hitting probabilities of
the process which is solution
of this system of equations, in terms of respectively Hausdorff measure and
Newtonian capacity.
These estimates make it possible to discuss polarity for points and to
compute the Hausdorff dimension
of the range and the level sets of this process. In order to prove the
hitting probabilities estimates,
we need to establish Gaussian type bounds for the bivariate density of the
process in order to quantify its
degenerance. For this, we use techniques of Malliavin calculus.

**September 18:**

Speaker: Joel Tropp, University of Michigan at Ann Arbor

**Title: **Sparse solutions to underdetermined linear systems

**Abstract: **A central problem in electrical engineering,
statistics, and applied mathematics is
to solve ill-conditioned systems of linear equations. Basic linear
algebra forbids
this possibility in the general case. But a recent strand of
research has
established that certain ill-conditioned systems can be solved
robustly with
efficient algorithms, provided that the solution is sparse (i.e.,
has many zero
entries). This talk describes a popular method, called l1
minimization or Basis
Pursuit, for finding sparse solutions to linear systems. It details
situations where
the algorithm is guaranteed to succeed. In particular, it describes
some new work on
the case where the matrix is deterministic and the sparsity pattern
is random. These
results are currently the strongest available for general linear
systems.

**September 25: - Joint with the Approximation Theory
Seminar**

Speaker: Yuliya Babenko, Sam Houston State University

**Title: **On asymptotically optimal methods of adaptive spline interpolation

**Abstract: **In this talk we shall present the exact asymptotics of the optimal error in the weighted $L_p$-norm, $1\leq p \leq \infty$, of linear spline interpolation of an arbitrary function $f \in C^2([0,1]^2)$. The connections with the problem of approximating the convex bodies by polytopes and the problem of adaptive mesh generation for finite element methods will also be discussed.
We shall present review of existing results as well as a series of new ones.
Proofs of these results lead to algorithms for construction of asymptotically optimal sequences of triangulations for linear interpolation. Similar results are obtained for some other classes of splines. We shall also discuss the analogous multivariate results as well.

**October 2:**

Speaker: Andrej Cherkaev, University of Utah

**Title: **New bounds for multiphase conducting composites

**Abstract: **New bounds for effective properties tensors of multimaterial composites are
suggested. These bounds complement the translation bounds or
Hashin-Shtikman bounds. We show that the bounds are exact for
three-material composites and determine optimal microstructurs of them.
The bounds are obtained using the theory of "localized polyconvexity" which
will be also discussed.

**October 9: SPECIAL TIME and LOCATION, 12:55 - 1:45 PM, LCB 225
Joint with the Stochastics Seminar**

Speaker: Pierre Seppecher, University of Toulon

**October 16:**

Speaker: Graeme Milton, University of Utah

**Title: **Cloaking: a New Phenomena in Electromagnetism and Elasticity

**Abstract: **Since my talk last semester, there have been quite a number of
developments with regards to the theory of cloaking (making an object
invisible). Not only developments with respect to cloaking associated
with superlenses, as I had discussed, but also with proposals
by Pendry, Schurig and Smith and Leonhardt, for designing a shield
which cloaks objects to electromagnetic waves. This work is related
to the earlier work of Greenleaf, Lassas and Uhlmann, on cloaking
for conductivity. Here we will review these developments and also
discuss how cloaking might be extended to elasticity using these
ideas. This requires new materials, in particular materials with
anisotropic density. We show how such materials can be made.

**October 23:**

Speaker: Jorge Balbas, University of
Michigan at Ann Arbor - Dept. of Mathematics

**Title: **Non-oscillatory Central Schemes for One-dimensional Shallow Water Flows along
Channels with Non-uniform Rectangular Cross-sections

**Abstract: **We present a new high-resolution, non-oscillatory semi-discrete central
scheme for one-dimensional shallow water flows along channels with non-uniform rectangular
cross sections. The scheme extends existing central semi-discrete schemes for hyperbolic
conservation laws by incorporating a discretization of the source terms appearing in shallow
water equations so that nonlinear fluxes are balanced
for steady-state solutions. We also incorporate exact information in the polynomial
reconstruction of the wet area, improving the control of spurious oscillations. Along with
the scheme, we present a systematic approach to calculate exact steady-state solutions for the
balance law. This allows us to validate the scheme by comparing the approximate numerical
solutions to the exact ones.

**October 30:**

Speaker: Adam Oberman, Simon Fraser University - Dept. of Mathematics

**Title: **Fully nonlinear elliptic PDEs: models, applications, and solution methods.

**Abstract: **This will be an accessible talk about modeling using fully nonlinear elliptic
PDEs.
Modern applications of these PDEs are to Image Processing and Math Finance.
As well as the Level Set Method for curve evolution, Optimal Control and
Stochastic Control.
I'll discuss some interesting models, overview the relevant theory, and then
show how to solve these equations.
Examples will include: level set motion by mean curvature, the convex hull,
the infinity Laplacian, as well as examples from math finance and control
theory.
We will present results which allow schemes to be built for a wide class of
equations.

**November 6:**

Speaker: Jonathan Kaplan, Stanford University - Dept. of Mathematics

**Title: **The Morphlet Transform: A Multiscale Transform for Diffeomorphisms

**Abstract: **Diffeomorphisms are a classical tool and object of study in
theoretical mathematics. Recently, there has been an increase in
the use and study of diffeomorphisms in applied mathematics. In
particular, diffeomorphisms have appeared as a new and potent tool
in image analysis. There is a growing interest in understanding
computationally efficient mechanisms for representing and
manipulating diffeomorphisms. Inspired by the success of wavelets
in signal processing, we describe a multiscale transform acting on
diffeomorphisms. This transform is defined on dyadic samples and
is nonlinear. Its design draws from the theory of interpolating
wavelet transforms and nonlinear subdivision schemes. We call
this transform the morphlet transform.

**November 13:**

Speaker: Coralia Cartis, Rutherford Appleton Laboratory - Computational Science & Engineering Dept.

**Title: **Some challenges in interior point methods for linear programming

**Abstract: **Through the depth of their theory and the span of their successful applications,
interior point methods have sparked a veritable "revolution" in convex
optimization. Now, fifteen years after their landmark discovery, interior point
methods have become highly successful at solving (very) large-scale linear
programming problems, with millions of variables and constraints not uncommon.
Nonetheless, some important questions at the interface of theory and practice
remain open and I will address three such topics in this talk. In particular, I
will present a new way of initializing these algorithms which overcomes the
fundamental assumptions underlying interior point methods theory that require the
set of admissible solutions to be full-dimensional and that are rarely satisfied by
real-life problems (this is joint work with Nick Gould, Oxford University).
Furthermore, addressing the lack of theoretical reliability of the interior point
algorithm implemented in most commercial and public software, I show on an example
what may go wrong and then describe a theoretically reliable alternative. As
interior point methods have made linear programming solvable in polynomial time,
complexity is a crucial aspect of this area. We set up a new general framework in
which we perform such a complexity analysis, that attempts to be more practical and
insightful than existing, highly-constructive, techniques by employing stiffness
analysis of vector fields, a concept traditionally associated with ordinary differential
equations (this is joint work with Raphael Hauser, Oxford University).

**November 14: SPECIAL TIME, DATE, and LOCATION, 3:00 - 4:00 PM LCB 215**

** Special Seminar - Joint with Bio-Math **

Speaker: Doron Levy, Stanford University - Mathematics Dept.

**Title: **Modeling the Dynamics of the Immune Response to Chronic Myelogenous Leukemia

**Abstract: **
Chronic Mylogenous Leukemia (CML) is a blood cancer with a common acquired
genetic defect resulting in the overproduction of malformed white blood
cells. The cause of CML is an acquired genetic abnormality in
hematopoietic stem cells in which a reciprocal translocation between
chromosomes 9 and 22 occurs. It is this abnormality that leads to
dysfunctional regulation of cell growth and survival, and consequently to
cancer. Treatment and control of CML underwent a dramatic change with the
introduction of the new drug, Gleevec, which was shown to be an effective
treatment available to nearly all CML patient. Nevertheless, by now it
is widely agreed that Gleevec does not represent a true cure for CML,
with many patients beginning to relapse despite of continued therapy.
The only known treatment that can potentially cure CML is a bone-marrow
(or stem-cell) transplant.

In this talk we will describe our recent works in modeling the interaction
between the immune system and cancer cells in CML patients. One model
follows this dynamics after a stem-cell transplant. A second model follows
the immune-cancer dynamics in patients treated with Gleevec. Related
mathematical questions and possible exciting applications of the models will
be discussed. This is a joint work with Peter Kim and Peter Lee
(Hematology, Stanford Medical School).

**November 17: SPECIAL TIME, DATE, and LOCATION, 3:30 - 4:20 PM LCB 121**

** Special Seminar - Joint with the Approximation Theory Seminar **

Speaker: Frank Stenger, U. of Utah - Computer Science Dept.

**Title: **SINC-PACK Enables Separation of Variables

**Abstract: **
This talk is mainly for mathematicians. It consists of a "proof-part"
of Stenger's SINC-PACK computer package (an approx. 400-page tutorial
+ about 250 Matlab programs) that one can always achieve separation of
variables when solving linear elliptic, parabolic, and hyperbolic PDE
(partial differential equations) via use of Sinc methods.

Some examples illustrating computer solutions via Sinc-Pack will
nevertheless be given in the talk. For example, in one dimension,
Sinc-Pack enables the following, over finite, semi--infinite, infinite
intervals or arcs: interpolation, differentiation, definite and
indefinite integration, definite and indefinite convolution, Hilbert
and Cauchy transforms, inversion of Laplace transforms, solution of
ordinary differential equation initial value problems, and solution of
convolution-type integral equations. The methods of the package are
especially effective for problems with (known or unknown - type)
singularities, for problems over infinite regions, and for PDE
problems.

In more than one dimension, the package enables solution of linear and
nonlinear elliptic, hyperbolic, and parabolic partial differential
equations, as well as integral equations and conformal map problems,
in relatively short programs that use the above one-dimensional
methods. The regions for these problems can be curvilinear, finite,
or infinite. Solutions are uniformly accurate, and the rates of
convergence of the programs of SINC-PACK are exponential.

In Vol 1. of their 1953 text, Morse and Feshbach prove for the case of
3-dimensional Poisson and Helmholtz PDE that separation of variables
is possible for essentially 13 different types of coordinate systems.
A few of these (rectangular, cylindrical, spherical) are taught in our
undergraduate engineering-math courses. We prove in the talk that one
can ALWAYS achieve separation of variables via use of Sinc-Pack, under
the assumption that calculus is used to model the PDE.

**November 20: - Joint with the Approximation Theory
Seminar**

Speaker: Tatyana
Sorokina, University of Georgia

**Title: **Quasi-Interpolation by Multivariate Piecewise Polynomials.

**Abstract: **Quasi-interpolants provide an alternative to finite elements
in multivariate approximation. While there are reliable tools for studying
classical finite elements, there is no theory of quasi-interpolation in
several variables. We will discuss some theoretical aspects of
quasi-interpolation, consider explicit bivariate and trivariate schemes,
and state open problems.

**November 27:**

Speaker: Paul Fife, University of Utah

**Title: **The structure of turbulent flow near boundaries

**Abstract: **
The problem of deriving key features of steady turbulent flow adjacent
to a wall has drawn the attention of some of the most noted fluid
dynamicists of all time. Standard examples of such features are found
in the mean velocity profiles of turbulent flow in channels, pipes or
boundary layers. Possibly the best known of the elementary
theoretical efforts along this line, and certainly the simplest, is
the argument (obtained independently) by Izakson (1937) and Millikan
(1939) regarding the mean velocity profile. They showed that if an
inner scaling and an outer scaling for the profile are valid near the
wall and near the center of the flow (or the edge of the boundary
layer), respectively, and if there is an overlap region where both
scalings are valid, then the profile must be logarithmic in that
common region. That piece of theory has been used and expanded upon by
innumerable authors for over 60 years, and at the present time is
still rightly enjoying popularity.

The main foci of the talk will be on (i) a careful examination of the
Izakson-Millikan argument, and (ii) an account of a newer approach to
gaining theoretical understanding of the mean velocity and Reynolds
stress profiles, due to Klewicki, McMurtry, Metzger, Wei and
myself. It has similar goals but entirely different methods. The
question will be how, and how well, do these arguments supply insight
into the structure of the mean flow profiles? Answering the question
WHY? is even more important than WHAT?

**December 4:**

Speaker: Patrick J. Wolfe, Harvard University - Department of Statistics

**Title: **Time-Frequency Representations and Statistical Models for Speech:

Exploring the Space of Acoustic Waveform Variation

**Abstract: **Time-frequency representations are ubiquitous in audio signal processing,
their use being motivated by both auditory physiology and the mathematics of
Fourier analysis. Indeed, information-carrying natural sound signals can
often be meaningfully represented as a superposition of translated, modulated
versions of a simple window function exhibiting good time-frequency
concentration. In combination with statistical models formulated in the
space of time-frequency coefficients, such an approach provides a principled
way of decomposing sounds into their constituent parts, as well as an
effective means of exploiting the local correlation present in the
time-frequency structure of natural sound signals such as speech. In
addition to applications involving the reconstruction of noisy and missing
audio data, this talk will describe the ways in which generative statistical
models provide a means of exploring the space of acoustic waveform variation,
and how in doing so they point toward a new way forward in a variety of
speech processing and discrimination tasks.

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