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Past talks for Fall 2012

**August 24 (Friday 4-5pm, LCB 222, student talk)**

Speaker: Andy Thaler, University of Utah, Mathematics Dept.,

**Title: ** Bounds on the Average Fields and Volume Fraction in Two-Phase Composite with Complex Permittivities

**Abstract: **
Raĭtum (1983) and Tartar (1995) derived bounds on the average
displacement current field in a composite with real permittivity.
Geometrically, these bounds are represented as a disk (ball) in two
(three) dimensions. The present work extends these bounds to the case of
complex-valued dielectric constants in composites with two isotropic
phases; the bounds we derive correlate the average electric and
displacement field values and the volume fractions of the phases. We
utilize an extension of the splitting method introduced by Milton and
Nguyen (2012) rather than variational principles. These bounds may have
important design applications related to directing fields.
The above bounds can also be used in an inverse fashion to bound the
volume fraction of one of the phases from measurements of the average
electric field and average displacement field. Our bounds generalize to
the case of a single inclusion in a body - in particular, boundary
measurements of the complex potential and flux can be used to estimate the
volume fraction of the inclusion. These bounds could have applications in
non-destructive testing and medicine, such as in the screening of organs
prior to transplantation.

**August 27**

Speaker: Yongtao Zhang, University of Notre Dame, Dept. of Applied and Computational Mathematics and Statistics

**Title: **Computational methods in pattern formation solutions

**Abstract: ** In this talk, I will present two kinds of numerical methods
for mathematical models in biological pattern formation problems. The first
method is the weighted essentially non-oscillatory (WENO) method for solving
the nonlinear chemotaxis models. Chemotaxis is the phenomenon in which cells or
organisms direct their movements according to certain gradients of chemicals in
their environment. Chemotaxis plays an important role in many biological
processes, such as bacterial aggregation, early vascular network formation,
among others. While WENO schemes on structured meshes are quite mature, the
development of finite volume WENO schemes on unstructured meshes is more
difficult. A major difficulty is how to design a robust WENO reconstruction
procedure to deal with distorted local mesh geometries or degenerate cases when
the mesh quality varies for complex domain geometry. In this work, we combined
two different WENO reconstruction approaches to achieve a robust unstructured
finite volume WENO reconstruction on complex mesh geometries. The second method
is the Krylov implicit integration factor (IIF) method for nonlinear
reaction-diffusion and advection-reaction-diffusion equations in pattern
formations. Integration factor methods are a class of "exactly linear
part" time discretization methods. Efficient implicit integration
factor (IIF) methods were developed for solving systems with both stiff linear
and nonlinear terms, arising from spatial discretization of time-dependent
partial differential equations (PDEs) with linear high order terms and stiff
lower order nonlinear terms. The tremendous challenge in applying IIF temporal
discretization for PDEs on high spatial dimensions is how to evaluate the
matrix exponential operator efficiently. For spatial discretization on
unstructured meshes to solve PDEs on complex geometrical domains, how to
efficiently apply the IIF temporal discretization was open. Here, I will
present our results in solving this problem by applying the Krylov subspace
approximations to the matrix exponential operator. We applied this novel time
discretization technique to discontinuous Galerkin (DG) methods on unstructured
meshes for solving reaction-diffusion equations. Then we extended the
Krylov IIF method to solve advection-reaction-diffusion PDEs and achieved high
order accuracy. Numerical examples are shown to demonstrate the accuracy,
efficiency and robustness of the methods.

**October 15**

Speaker: Andrej Cherkaev, University of Utah, Mathematics dept.

**Title: ** Lattice composites and damage

**Abstract: ** The talk discusses mesoscale behavior of inhomogeneous and
damageable lattice structures, compatibility conditions, damage spread, "lattice
composites", and a range of their effective properties.

**October 22**

Speaker: Gregory J. Rodin, The University of Texas at Austin, Institute for Computational Engineering and Sciences.

**Title: ** Effective properties of discrete and continuum systems

**Abstract: ** In multiscale modeling, the transition from fine to coarse
scales invariably gives rise to effective (or macroscopic) properties and
governing equations. This talk will address two topics. In the first part of
the talk, we consider coarse-graining of *n*-dimensional first order linear
systems, and demonstrate how the dynamics of eliminated degrees of freedom
(slaves) affects that of the masters. In particular, we establish that the
dynamics of coarse-grained system is governed by qualitatively different
properties and equations. In the second part of the talk, we discuss a
thermodynamic limit associated with the determination of the effective
properties of conducting composites formed by particles embedded in a matrix.
We establish that, under the assumption that the particles do not interact, the
effective properties are consistent with Maxwell-Clausius-Mossotti
formula, which is closely related to the variational bounds.

**October 29**

Speaker: Daniel Onofrei, University of Houston

**Title: ** Active control of acoustic and electromagnetic fields

**Abstract: ** The problem of controlling acoustic or electromagnetic fields is
at the core of many important applications such as, energy focusing,
shielding and cloaking or the design of super-directive antennas.
The current state of the art in this field suggests the existence of two
main approaches for such problems: passive controls, where one uses suitable
material designs to control the fields (e.g., material coatings of certain
regions of interest),
and active control techniques, where one employs active sources (antennas)
to manipulate the fields in regions of interest. In this talk I will first
briefly describe the main mathematical question and
its applications and then focus on the active control technique for the
scalar Helmholtz equation in a homogeneous environment. The problem can be
understood from two points of view, as a control question or as an inverse
source problem (ISP).
This type of ISP questions are severely ill posed and I will describe our
results about the existence of a unique minimal energy solution. Stability
of the solution and extensions of the results to the case of nonhomogeneous
environment and to the Maxwell system
are part of current work and will be described accordingly.

**October 31 (Oral exam, Wed 4-5pm in JTB 110)**

Speaker: Michał Kordy, University of Utah

**Title: ** Compatible finite element formulation of the equation for electric field in the frequency domain: crucial role of the divergence correction.

**Abstract: ** A novel method for the 3-D diffusive electromagnetic (EM)
forward problem is developed and tested. The method considers the standard curl
curl equation for the electric field E, a finite element formulation, and the
divergence correction. For E field, edge elements are considered which are
compatible with the curl operator. Elements are tetrahedra with straight edges,
which allow for a natural definition of scalar fields space, used in the
divergence correction -- the space of piecewise linear functions. Together with
edge elements, these spaces are part of a finite de Rham diagram. With the aid
of properties of this diagram, namely Hodge decomposition on a finite grid
level, the equation for E is decoupled into two equations: one on the range of
the gradient, and the other on a space orthogonal to it. The eigenvalues
associated with the equation of the range of the gradient are much smaller than
eigenvalues associated with the space orthogonal to it. As a result the system
matrix is ill-conditioned and Krylov solvers will focus on imposing an equation
on the space orthogonal to the gradient, yet will struggle with imposing the
equation on the range of the gradient. This highlights the importance of the
divergence correction which readdresses this tendency. Numerical study shows
that divergence correction on one hand acts as a preconditioner, speeding up
the convergence, and on the other hand corrects the obtained solution. This
correction is very important for low frequencies especially in the air, where
if the correction is not applied properly, the field differs greatly from the
true solution even if the residual of the equation appears small.

The talk is a part of an oral qualifying exam for the PhD student.

**November 5**

Speaker: Bei Wang, University of Utah, SCI Institute

**Title: ** Stratification Learning through Local Homology Transfer

**Abstract: ** Advances in scientific and computational experiments have
increased our ability to gather large collections of data points in
high-dimensional spaces, far outpacing our capacity to analyze and understand
them. For instance, in a large-scale simulation, one might want to understand
the relationships between a large number of input parameters and their effects
on a set of particular outcomes.

We approach the problem as follows, given a point cloud of data sampled
from some underlying space, can we infer the topological structure of the
space? Often we assume the support of the domain is either from a low-
dimensional space with manifold structure, or more interestingly, contains
mixed dimensionality and complexity. The former is a classic setting in
manifold learning. The latter can often be described by a stratifi ed set
of manifolds and becomes a problem of particular interest in the fi eld of
strati fication learning.

Stratified spaces, while not manifolds, can be decomposed into manifold
pieces that are glued together in some uniform way. An important tool
in strati cation learning is the study of local spaces, that is, the
neighborhoods surrounding singularities, where manifolds of
different dimensionality and complexity intersect.

We show that point cloud data can under certain circumstances be
clustered by strata in a plausible way. For our purposes, we consider
a stratified space to be a collection of manifolds of different
dimensions which are glued together in a locally trivial manner inside
some Euclidean space. To adapt this abstract definition to the world
of noise, we first define a multi-scale notion of stratified spaces,
providing a stratification at different scales which are indexed by a
radius parameter. We then use methods derived from kernel and cokernel
persistent homology to cluster the data points into different strata
based on local homology transfer maps. We prove a correctness
guarantee for this clustering method under certain topological
conditions. We then provide a probabilistic guarantee for the
clustering for the point sample setting - we provide bounds on the
minimum number of sample points required to state with high
probability which points belong to the same strata. Finally, we give
an explicit algorithm for the clustering.

If time permits, I would mention some other work in high-dimensional
data analysis and visualization.

**November 12 (exceptionally in LCB 219)**

Speaker: Michael Mascagni, Florida State University, Department of Computer Science

**Title: ** Novel Stochastic Methods in Biochemical Electrostatics

**Abstract: ** Electrostatic forces and the electrostatic properties of
molecules in solution are among the most important issues in understanding the
structure and function of large biomolecules. The use of implicit-solvent
models, such as the Poisson-Boltzmann equation (PBE), have been used with great
success as a way of computationally deriving electrostatics properties such
molecules. We discuss how to solve an elliptic system of partial differential
equations (PDEs) involving the Poisson and the PBEs using path-integral based
probabilistic, Feynman-Kac, representations. This leads to a Monte Carlo
method for the solution of this system which is specified with a stochastic
process, and a score function. We use several techniques to simplify the Monte
Carlo method and the stochastic process used in the simulation, such as the
walk-on-spheres (WOS) algorithm, and an auxiliary sphere technique to handle
internal boundary conditions. We then specify some optimizations using the
error (bias) and variance to balance the CPU time. We show that our approach
is as accurate as widely used deterministic codes, but has many desirable
properties that these methods do not. In addition, the currently optimized
codes consume comparable CPU times to the widely used deterministic codes.
Thus, we have an very clear example where a Monte Carlo calculation of a
low-dimensional PDE is as fast or faster than deterministic techniques at
similar accuracy levels.

**November 26**

Speaker: Qinghai Zhang, University of Utah, Mathematics Dept.

**Title: ** A Cocktail of ODE, Differential Geometry, Jordan Curve Theorem, Fluid Dynamics, Finite Volume Methods, Semi-Lagrangian Methods and More

**Abstract: ** Given a time interval (t_{n}, t_{n+k})
and a fixed simple curve LN
in the time-dependent velocity field,
which set of particles will pass through LN
and contribute to its flux within the time interval?
The locus of these fluxing particles at time t_{n}
is equivalent to the donating region (DR),
the dependence domain of the fixed curve
for the advection equation.
In this talk,
I propose an explicit, constructive, and analytical definition of DR
based on classical ODE theory, differential geometry, Jordan curve theorem,
and several well-known characteristic curves in fluid dynamics.
I will show that the DR contain,
and only contain, all the particles
that has a net effect of passing through LN once.

The second half of this talk will focus on
one computational aspect of DR,
namely the algorithms of Lagrangian flux calculation (LFC)
via algebraic quadratures
over the spline-approximated DRs.
As DR connects to both Eulerian and Lagrangian viewpoints,
LFC naturally leads to a conservative semi-Lagrangian method
whose time step size is free of the Eulerian stability constraint
of Courant number being less than one.
Various numerical tests on 2D advection tests
demonstrate high-order accuracies
from the 2nd order to the 8th order (both in time and in space)
and large Courant numbers from 1 to 10000.

**December 3**

Speaker: Semyon Tsynkov, North Carolina State University, Department of Mathematics

**Title: ** Dual carrier probing for spaceborne SAR imaging

**Abstract: ** Imaging of the Earth's surface by spaceborne synthetic
aperture radars (SAR) may be adversely affected by the ionosphere, as the
temporal dispersion of radio waves gives rise to distortions of signals emitted
and received by the radar antenna. Those distortions lead to a mismatch between
the actual received signal and its assumed form used in the signal processing
algorithm (known as the matched filter). In turn, the discrepancy between the
filter and the signal causes deterioration of the image.

To mitigate the ionospheric distortions, we propose to probe the terrain, and
hence the ionosphere, on two distinct carrier frequencies. The resulting two
images appear shifted with respect to one another, and the magnitude of the
shift allows one to evaluate the total electron content (TEC) in the
ionosphere. Knowing the TEC, one can correct the matched filter, and hence
improve the quality of the image. Robustness of the proposed approach can
subsequently be improved by applying an area-based image registration technique
to the two images obtained on two frequencies. The latter enables a very
accurate evaluation of the shift, which, in turn, translates into a very
accurate estimate of the TEC.

Time permitting, we will also discuss how the Ohm conductivity of the
ionosphere may affect the SAR resolution, what may be the effect of the
ionospheric turbulence, how to take into account the horizontal variation of
the ionospheric parameters, and what may be the role of the anisotropy due to
the magnetic field of the Earth.

Joint work with E. Smith and M. Gilman.

155 South 1400 East, Room 233, Salt Lake City, UT
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