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Past talks for Spring 2009

__ January 16 (Friday 3pm, LCB 219)__ Joint with the Stochastics Seminar.

Speaker: Nicolai Krylov, University of Minnesota

__ February 6 (Friday 3:05pm, LCB 215),__ Joint with Math Bio Seminar.

Speaker: Sarthok Sircar, University of South Carolina

**February 9**

Speaker: Yuliya Gorb, Texas A&M University, Mathematics Department

**Title: ** Multiscale Modeling and Simulation of Fluid Flows in Deformable
Porous Media

**Abstract: **
The main focus of this talk is on fluid flows in deformable elastic media and
associated multiscale problems. Many upscaling methods are developed for flows
in rigid porous media or deformable elastic media assuming infinitely small
fluid-solid interface displacements relative to the pore size. Much research is
needed for the most general and least studied problem of flow in deformable
porous media when the fluid-solid interface deforms considerably at the pore
level. We introduce a general framework for numerical upscaling of the
deformable porous media in the context of a multiscale finite element method.
This method allows for large interface displacements and significant changes in
pore geometry and volume. For linear elastic solids we present some analysis of
the proposed method.

__ February 11 (Wednesday 3:05pm, LCB 225)__ Joint with Math Bio Seminar.

Speaker: Margaret Beck, Brown University

Finally, remarks are made regarding the existence and stability of spatially periodic pulses, corresponding to successive heartbeats, and their relationship with alternans, irregular action potentials that have been linked with arrhythmia.

__ February 18 (Wed 3:05pm, LCB 215)__ Joint with Math Bio Seminar

Speaker: Pak-Wing Fok, California Institute of Technology

**March 2**

Speaker: Jichun Li, University of Nevada Las Vegas and IPAM, UCLA

**Title: ** Numerical study of Maxwell's equations in negative index metamaterials

**Abstract: ** Since 2000, there has been a growing interest in the study
of negative
index metamaterials across many disciplinaries. In this talk, I'll first
derive the Maxwell's equations resulting from such metamaterials. Then I'll
review some time-domain finite element methods recently developed for solving
these equations. After that, I'll discuss my most recent work on leap-frog
type finite element methods with succinct error estimates. Finally, some
numerical results and open issues will be discussed.

**March 6 (Friday 1pm, JWB 208, Thesis defense)**

Speaker: Lyubima Simeonova, University of Utah

**Title: ** Wave propagation through composite materials: Effective
properties and optimization

**Abstract: ** The effective properties of the complex permittivity for
waves in one-, two-, and three-dimensional random media are
investigated. When the wavelengths of the field are of the same order as
the size of the heterogeneities of the composite, scattering effects,
such as wave localization and cancellation, must be accounted for.
The effective dielectric coefficient is no longer a constant as in the
quasistatic case, but a function of the space variable. Since effective
dielectric coefficient cannot be calculated explicitly in general, to be
useful in applications it is important that we can bound both effective
dielectric coefficient itself, and some measure of the spatial
variations in effective dielectric coefficient. We have obtained such
bounds using novel methods that incorporate probability arguments and
the regularity properties of the solutions. The estimates hold in
bounded domains for any fixed frequency greater than 0 and show an
explicit dependence on the feature size and contrast of the random
medium. Pertinent numerical experiments are performed to illustrate the
results of the analytical proof. We also consider a related optimization
problem of finding the class of materials (described by a probability
density function) that minimizes the spatial average of the effective
dielectric coefficient. Existence and uniqueness of a minimizing
probability density function is proven, and numerical experiments are
performed to find the minimizing probability density function. Another
optimization problem where there is a restriction in the variability in
the medium is solved numerically. The dependence of the effective
dielectric coefficient on the contrast in the medium is explored, and
series expansion of the effective coefficient, that takes into account
the correlation functions of the medium, is derived. Every term in the
series is a constant provided the medium is stationary. For media with a
correlation function depending on position, the best approximation of
the effective dielectric coefficient is a function of the space variable.

The dissertation includes a problem in structural optimization, and in
particular optimization of periodic composite structures for
sub-wavelength focusing. A slab of material with negative refractive
index would act as a superlens, providing a perfect image of an object
in contrast to conventional lenses which are only able to focus a point
source to an image having a diameter of the order of the wavelength of
the incident field. We pose the question of what periodic dielectric
composite medium (described by dielectric coefficient with positive real
part) gives an optimal image of a point source. We show that a solution
exists provided the medium has small absorption. Solutions are
characterized by an adjoint-state gradient condition. We use techniques
of "topology optimization" in which material distribution is completely
arbitrary. We have demonstrated an optimized structure that gives a
focus with a spot size 0.284 of the wavelength, which is a significant
improvement to those previously obtained by using non-exotic materials.

**March 9**

Speaker: Vianey Villamizar, Brigham Young University, Department of Mathematics

**Title: ** Exact Nonreflecting Boundary Conditions for Multiple Scattering
on Generalized Curvilinear Coordinates

**Abstract: ** A multiple scattering problem modelled by the Helmholtz
equation is solved. Each arbitrarily shaped scatterer is enclosed by a
relatively close artificial boundary. Following [J. Comp. Phys. 201 (2004)
630-650], a DtN boundary condition is derived for several disjoint components
of the artificial exterior boundary. Then, a second order finite difference
method, combined with the novel Dirichlet-to-Neumann (DtN) non-reflecting
boundary condition on generalized curvilinear coordinates, is applied to the
inner regions. These inner regions are bounded by the physical scatterer
boundaries and the surrounding artificial scatterer boundaries. As a result,
the computational cost to obtain a numerical solution is greatly reduced. An
approximate solution for multiple scattering from two circular cylinders is
obtained using this method. Excellent convergence is obtained for this case
when compared to its exact solution. Approximate solutions for more general
scatterer configurations of two and three obstacles are also presented.
Finally, the radar cross section for various arbitrarily shaped scatterer
configurations are obtained.

**March 23**

Speaker: John Willis, DAMTP, University of Cambridge

**Title: ** Effective constitutive relations for waves in composites and metamaterials

**Abstract: ** The description of waves propagating through strongly-heterogeneous material requires some kind of averaging to be performed. Here, the material is taken to be random and ensemble averaging is considered; it is noted in this context that, although the ensemble average is not seen in any one realisation, it nevertheless provides a scaffold upon which the solution in any particular realisation can in principle be built. In practice, resort must be made to approximations. This work establishes exact variational principles which the ensemble averaged solutions must satisfy, and from which "effective constitutive relations" follow. It is demonstrated also that a similar variational structure follows if weighted ensemble averages are employed. Such weighted averages are relevant to so-called metamaterials which contain micro-resonators, whose displacements are best excluded from the averaging process.

**March 26 (Thursday 4:15pm, JWB 335, Joint with Math Department Colloquium)**

Speaker: Michael Vogelius, Rutgers University

**Title: **A survey of results concerning existence and blow up
for some nonlinear elliptic and parabolic problems related to corrosion
modelling

**Abstract: ** TBA

**March 30**

Speaker: Russell Richins, University of Utah (student talk)

**Title: ** Optimal Transportation

**Abstract: ** The problem that inspired the theory of optimal transportation was that of moving a pile of sand to fill a hole when the cost of moving any particle of sand from the pile to the hole is known. The problem is to minimize the cost of filling in the hole. I will discuss the mathematical formulation of this problem as well as some of the tools used to analyze it. I will also show how optimal transportation relates to composite materials in the problem of the optimal placement of conducting material inside an insulating background.

**April 6**

Speaker: Jingyi Zhu, University of Utah

**Title: ** Finite Difference PDE Approaches to Stochastic Volatility Models

**Abstract: ** Stochastic volatility models recognize that the volatility in
a stock price by itself is stochastic, and explore various features of the
process for the volatility, such as the mean-reverting property. Option prices
based on these models are much more realistic when compared to market data,
therefore they are widely used by sophisticated option traders. Traditional
pricing using stochastic volatility models typically relies on either
closed-form solutions or brute force Monte Carlo simulations, with their
obvious limitations. Finite difference approaches to solve the resulting
time-dependent PDE in two space dimensions provide a powerful alternative, with
the advantages such as easy accommodation of variable coefficients and fast
numerical convergence. However, one crucial factor that has been illusive is
the matter of boundary conditions for the volatility variable. We consider the
prototype model for stochastic volatility, the Heston model and its extended
forms, supply various boundary conditions accompanied by probabilistic
interpretations, and use finite difference techniques to solve the resulting
PDE problem in a bounded domain. We present different results, in terms of the
market observable "volatility smile" curve, to demonstrate the ramification of
the boundary conditions. Comparisons with other approaches such as Monte Carlo
simulations are also made to show the advantage of the finite difference
approaches.

**April 9 (Thursday 4:15pm, JWB 335, Joint with Math Department Colloquium)**

Speaker: Gang Bao, Michigan State University

**Title: **Distinguishability via Uncertainty Principle for inverse Scattering

**Abstract: ** The inverse scattering problem arises in diverse areas of industrial and military applications, such as nondestructive testing, seismic imaging, submarine detections, near-field or subsurface imaging, and medical imaging. A general model is concerned with a time-harmonic electromagnetic plane wave incident on a medium enclosed by a bounded domain. Given the incident field, the direct problem is to determine the scattered field for the known scatterer. The inverse medium scattering problem is to determine the scatterer from the boundary measurements of near field currents densities. Although this is a classical problem in mathematical physics, numerical solution of the inverse problem remains to be challenging since the problem is nonlinear, large-scale, and most of all ill-posed! The severe ill-posedness has thus far limited in many ways the scope of inverse problem methods in practical applications. In this talk, our recent results in mathematical analysis and computational studies of the inverse boundary value problems for the Maxwell equations will be reported. A novel continuation approach based on the uncertainty principle will be presented. By using multi-frequency or multi-spatial frequency boundary data, our approach is shown to overcome the ill-posedness for the inverse medium scattering problems. Convergence issues for the continuation algorithm will be examined. Our most recent progress on inverse source problems will also be discussed.

**April 13**

Speaker: Robert Palais, University of Utah

**Title: ** Rendering with Randomness, Rotating with Reflections

**Abstract: ** Mathematically generated point clouds enhance visualization of surfaces with multiple components along the line of view, provide well behaved grids for numerical methods, and optimize LIDAR analysis via synthesis and simulation. A result from integral geometry suggests an algorithm for representing implicitly defined surfaces. Implementing it involves obtaining uniform random directions on the unit sphere, and we will consider at least seven methods to do so. Converting these directions to uniformly distributed lines in space requires a rotation. We conclude with some surprising algorithms for performing rotations, and their consequences.

**April 16 (Thursday 4:15pm, JWB 335, Joint with Math Department Colloquium)**

Speaker: Robert Kohn, Courant Institute, New York University

**Title: **Price Bubbles from Heterogeneous Beliefs

**Abstract: ** Harrison and Kreps showed in 1978 how the heterogeneity of investor beliefs can drive speculation, leading the price of an asset to exceed its intrinsic value. By focusing on an extremely simple market model -- a finite-state Markov chain -- the analysis of Harrison and Kreps achieved great clarity but limited realism. My talk discusses joint work with Xi Chen, which achieves similar clarity with greater realism by considering an asset whose dividend rate is a mean-reverting stochastic process. Our investors agree on the volatility, but have different beliefs about the mean reversion rate. We determine the minimum equilibrium price explicitly; in addition, we characterize it as the unique classical solution of a certain linear differential equation. Our example shows, in a simple and transparent manner, how heterogeneous beliefs about the mean reversion rate can lead to everlasting speculation and a permanent "price bubble".

**April 20**

Speaker: Andrei Kouznetsov, Washington State University, Math Dept.

**Title: ** A Discrete Model of Phase Transitions in Solids

**Abstract:
** We study a discrete model of phase transitions in solids. Our model is a
network built of a finite number of nodes connected with non-linear links.

There are many works dedicated to this problem in 1D space. We work in
2D space and this makes the problem significantly more complicated,
since we need to satisfy compatibility conditions on the elongations of
the links of the network. This compatibility conditions are
automatically satisfied in 1D case.

In this presentation the focus is made on the set of deformations of
the model with no internal forces. This set is neither a linear space
nor a convex space and is very hard to work with. However, this set is
the key to understand the properties of materials built on our model.

The current presentation gives a description of the set of deformations
with no internal forces, compatibility conditions for elongations, and
explains main ideas and proofs of our theoretical results.

**April 27**

Speaker: Peg Howland, Utah State University, Department of Mathematics and Statistics

**Title: ** Using Generalized Discriminant Analysis and Factor Analysis
Approximations in Dimension Reduction

**Abstract: ** In applications ranging from text mining to face recognition,
dimension reduction is imperative for efficiency. Toward that end, we
extend classical linear discriminant analysis (LDA) so that class
separability is optimally preserved. The generalized singular value
decomposition (GSVD) provides the mathematical framework for this
method, and also for a two-stage approach that uses either principal
component analysis or QR decomposition before LDA. Further
algorithmic simplification can be achieved by applying a rank
reduction formula from factor analysis, and restricting the domain to
sign or binary vectors. We demonstrate the relationships between these
methods, as well as their relative accuracy and complexity, with
classification results on document and facial data.

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