Prof. Joseph E. Pasciak
Texas A&M University
Dept. of Mathematics
"Numerical Modeling of Photonic Crystals"

Abstract:
Photonic crystals are structures constructed of
dielectric materials arranged in a periodic array.
Such structures have been found to exhibit interesting spectral
properties with respect to classical electromagnetic wave propagation,
including the appearance of band gaps. In this talk, a mixed method for
computing band structures of three-dimensional photonic crystals will be 
described.  The method combines a finite element discretization on a
uniform grid with a fast Fourier transform preconditioner and a
preconditioned subspace eigenvalue iteration algorithm.  Numerical
examples illustrating the behavior of the method will also be presented. 


------------------------------------------------------------------------


Speaker: Michael C. Fu, University of Maryland at College Park Robert H.
Smith School of Business and Institute for Systems Research

Title: Pricing of American-Style Options via Monte Carlo Simulation
(based on joint work with Jian-Qiang Hu, Scott B. Laprise, Dilip B. Madan,
Yi Su, Rongwen Wu)

Abstract:
We address the application of Monte Carlo simulation to the problem of
pricing financial derivatives that allow early exercise opportunities.  Up
until 1993, it was commonly accepted in the finance community that
simulation could not be applied to these American-style options. However,
this belief has been all but refuted by a flurry of activity since then that
has produced a number of simulation-based algorithms. We will briefly review
the literature, and then present our approaches, which are based on
converting the stochastic dynamic programming problem into a parametric
optimization problem by characterizing the early exercise boundary.
Stochastic approximation techniques are then applied to obtain a
simulation-based algorithm. The related gradient estimation problem is
addressed in this talk, and work in progress will also be described.


------------------------------------------------------------------------


Speaker: Elmira Popova, OR/IE Group, UT Austin 

Title: MARKOV CHAIN MONTE CARLO METHODOLOGIES APPLIED TO MORTGAGE
PREPAYMENT RATES 
Joint work with Ivilina Popova, Purdue University and Edward George, MSIS,
UT Austin 

Abstract.
This research proposes a novel approach for modeling prepayment rates of
pools of mortgages. Our goal is to establish a model that gives a good
prediction for prepayment rates for individual pools of mortgages and
incorporates the empirical evidence that prepayment is past dependent. We
construct a model that both describes and forecasts the prepayment rates
successfully. There are many factors that influence the prepayment behavior
and for many of them there is no available (or impossible to gather)
information. We implement this issue by creating a mixture model with a
Markov Chain Monte Carlo estimation algorithm. Numerical results using a
large data set from the Bloomberg Database will be presented. 


------------------------------------------------------------------------


       "Neural Networks in Structural Engineering - Some New Results and
                           Prospects of Applications"

                                               Prof. Zenon Waszczyszyn
                               Cracow University of Technology, Poland

Fundamentals of multilayer, Back-Propagation Neural Networks (BPNNs)
are shortly presented. Then BPNN applications in analysis of the
following problems are discussed: 1) simulation of Return Mapping
Algorithm, and BPNN supported, FE analysis of the plane stress
problem, 2) estimation of fundamental vibration periods of real
buildings, 3) detection of a damage in a steel beam, 4) identification
of loads applied to an elastoplastic beam. A regularization network
with RBFs (Radial Basis Functions) is briefly discussed and its
application to estimation of concrete fatigue durability is shown. As
an alternative to RBF neural networks, an application of ANFIS
(Adaptive Neuro-Fuzzy Inference System) is given with respect to a
problem of biomechanics, corresponding to estimation of proximal
femurs strength. A modified Hopfield network is used to the analysis
of elastic and elastoplasic plates with unilateral boundary
conditions. We conclude the presentation with a short discussion of
prospects of neurocomputing applications in structural engineering.


------------------------------------------------------------------------


Speaker: Patrick Henaff, Koch Industries 

Title: REAL OPTIONS & THE VALUATION OF NATURAL GAS STORAGE FACILITIES 
Abstract.
The availability of a storage facility grants the operator the option to
inject or withdraw gas. With appropriate trading strategies involving gas
for immediate delivery and futures contracts, the operator can synthesize
options on various spreads. This paper first identifies the options embedded
in a storage facility, and then develops a model to price these options. The
framework is a two-factor stochastic dynamic programming model. Illustrative
numerical results are next provided. In particular, we explore the
sensitivity of the economic value to various design parameters of the
storage cavern, and show that the model provides insights to evaluate
alternative investment plans. 


------------------------------------------------------------------------


          Special Seminar: Global Curvature, Ideal Knots and Models 
          of DNA Self-Contact 
          by Oscar Gonzalez in RLM 6.116


Experiments on knotted DNA molecules suggest that certain physical
properties of DNA knots can be predicted from a corresponding ideal
shape. Intuitively, when a given knot in a piece of string is pulled
tight, it always achieves roughly the same geometrical configuration,
with a minimum length of string within the knot. Such a configuration
is called an ideal shape for the knot, and approximations of ideal
shapes in this sense have been found via a series of computer
experiments. These shapes have intriguing physical features and have
been shown to capture average properties of knotted polymers.  But
when does a shape satisfy the intuitive geometrical definition for
ideality? In this talk I show that ideal shapes can be understood
using only elementary (but new!) mathematics. In particular, I show
that global curvature, a very natural and simple generalization of the
classic concept of local curvature, leads to a simple characterization
of an ideal shape and to a necessary condition for ideality. Another
application of global curvature can be found in characterizing the
equilibria of knotted curves or rods, which may exhibit self-contact
after sufficient twisting. Here global curvature provides a simple way
to formulate the constraint that prevents a rod from passing through
itself.


------------------------------------------------------------------------


The Random Projection Method for Hyperbolic Conservation Laws with
                                              Stiff Reaction Terms"

                                                       Dr. Shi Jin
             School of Mathematics, Georgia Institute of Technology

Hyperbolic systems with source terms arise in the modeling of
chemically reacting flows. In these problems, the chemical time scale
may be orders of magnitude faster than the fluid dynamical time
scales, making the problem numerically stiff. The numerical difficulty
of this problem is classical -- one always gets the wrong shock speed
unless one fully resolves the small chemical scale numerically. We
introduce a novel numerical method -- the random projection method-
-that is able to capture the correct shock speed without resolving the
small scale. The idea is to replace the ignition temperature by a
uniformly distributed random variable in a suitable domain. The
statistical average of this method corrects the spurious shock speed,
as will be proved with a scalar model problem and demonstrated by a
wide range of numerical examples in inviscid denotation waves in both
one and two space dimensions. This is a joint work with W. Bao.


------------------------------------------------------------------------


Speaker: John Chadam, University of Pittsburgh

Title: The Exercise Boundary for an American Put Option: Analytical and
Numerical Approximations

ABSTRACT: Recently there has been considerable activity centered on
obtaining accurate approximations near expiry for the early exercise
boundary for American puts (Kuske and Keller; Bunch and Johnson; and
Stamicar, Sevcovic and Chadam). In this talk we will summarize these
results in a common framework and develop three new approximations which
are not only more accurate but are also valid over much longer time
intervals. Two are implicit formulas and the last arises as the solution of
a naturally occurring ordinary differential equation which is extremely
easy to solve numerically. All of the new approximations are derived from
the Black and Scholes partial differential equation fromulation of the
problem using free boundary methods. (Joint work with Xinfu Chen,
University of Pittsburgh and Rob Stamicar, University of British Columbia
and FinancialCad)


------------------------------------------------------------------------


Do Y. Kwak (Department of Mathematics, KAIST, Taejon, Korea) will be giving
a seminar at 1:00 today (2/14) in SHC 329.  Everyone is invited.  The title
and abstract are listed below.

"V-cycle multigrid convergence of cell-centered difference"

In this talk, we present a new multigrid algorithm which is suitable for
cell-centered finite difference method for elliptic problems.  Convergence
of most multigrid methods depends on two properties:  Elliptic regularity
and Energy norm of the prolongation operator.  A brief survey of
conventional multigrid theory will be followed by an analysis for
nonconforming element, then cell-centered finite difference case will be
dealt with.  We design a new prolongation operator for this case and show
the energy norm is less than equal to 1 and conclude the V-cycle
convergence (with certain regularity).  Numerical experiment shows that
multigrid convergence with this modified prolongation operator both for
rectangular grid and triangular grid are significantly improved over the
natural injection studied earlier.


------------------------------------------------------------------------


Dr. Wing Lok (Justin) Wan
Scientific Computing/Computational Mathematics (SCCM) Program,
Stanford University

Title: Scalable Multilevel Algorithms in Image Restoration

Abstract: In image restoration problems, one wants to restore the
original image from the given blurred and noisy image obtained from,
for instance, experimental data. Restoration approaches based on
traditional methods such as FFT, wavelets, typically introduce
oscillations (Gibbs phenomenon) near the edges of the restored image,
sometimes known as the ringing effect. Moreover, the deblurring
process is highly ill-posed, and regularization techniques are needed
to obtain meaningful images. In PDE based approaches, images are
viewed as two-dimensional functions and edges as discontinuities. The
total variation regularization method is able to preserve sharp
edges. However, the resulting discretization matrix, composed of a
differential and an integral operator, leads to challenging large
scale numerical computations. These operators are ill-conditioned with
highly nonlinear and varying coefficients. In this talk, we propose a
multigrid method based on energy minimization which is able to handle
differential operators with highly varying coefficients. This
multigrid method is scalable and applicable to any dimensions and very
general geometry. We also describe a cosine transform preconditioner
for solving the integral operator efficiently. Finally, we combine the
two methods to obtain a fast solver for the image problem.


------------------------------------------------------------------------


Speaker: Hui Wang, Columbia University

Title: Utility Maximization in Incomplete Markets with Random Endowment 
Abstract 
This paper solves a long-standing open problem in mathematical finance: to
find a solution to the problem of maximizing utility from terminal wealth of
an agent with random endowment process, in the general, semimartingale model
for incomplete markets, and to characterize it via the associated dual
problem. We show that this, indeed, is possible, if the dual problem and its
domain are carefully defined. More precisely, we show that the optimal
terminal wealth is equal to the inverse of marginal utility evaluated at the
solution to the dual problem, which is in the form of the regular part of an
element of $(L^\infty)^*$ (the dual space of $L^\infty$). 


------------------------------------------------------------------------


Speaker: Jitka Dupacova, Department of Probability and Mathematical
Statistics, Charles University, Prague, Czech Republic

Title: Portfolio Optimization via Stochastic Programming: Methods of Output
Analysis

Abstract 

The decisions based on solution of portfolio optimization problems are often
influenced by errors or misspecifications due to approximation, estimation
and incomplete information. Selected methods for analysis of results
obtained via stochastic programming are presented and their scope
illustrated on generic examples -- the Markowitz model, a multiperiod bond
portfolio management problem and a general strategic investment problem. The
methods are based on asymptotic and robust statistics and on the moment
problem.


------------------------------------------------------------------------


"Distributed Order Differential Equations and Their Solution Techniques"

Dr. Ron Bagley
Division of Engineering, University of Texas at San Antonio

I will present a new class of differential equations that arise in the
mathematical description of the mechanical response of polymers and
ceramics. These differential equations contain an infinite number of
derivatives of finite order, where the order of differentiation varies
smoothly over a finite domain. Dr Bagley will also present a general
solution technique for these equations. These techniques include the
use of a new integral transform domain, the employment of a little
known generalized function, similar to the Dirac delta function, and
the use of a generalized series representation for the differential
equation's solution.


------------------------------------------------------------------------


Chi-Wang Shu
Division of Applied Mathematics
Brown University

"On the Gibbs Phenomenon and Its Resolution"

Abstract: We discuss the joint work with David Gottlieb (part of the
early work was also joint with Solomonoff and Vandeven) on the Gibbs
phenomenon and its resolution.  A prototype problem is: Given the
first N Fourier coefficients of a analytic but not periodic function
(i.e. the function has one discontinuity with known location and
unknown jump), can one recover the point values of the function
everywhere (in the maximum norm) with exponential accuracy?  The
answer is yes and we provide both a proof and a constructive procedure
which gives good accuracy even for relatively small N.  This work has
been generalized to the cases of unknown discontinuity locations and
several discontinuities, expansions in Legendre, Chebyshev and other
Gegenbauer basis, and collocation rather than Galerkin expansions.

Numerical results will be shown.


------------------------------------------------------------------------


PS. You may also be interested in the Physics colloquium given on Wednesday
April 5th, from 4-5 pm in RLM 4.102, by Nigel Goldenfeld, from U.
Illinois-Urbana, on My Manhattan Project: A Physicist's Adventure on Wall
Street. http://www.ph.utexas.edu/~walkerdl/cal.html


------------------------------------------------------------------------


Computational Finance Seminar, for Friday Arpil 7th 
Time: Friday, April 7th, 2000 11-12 am 
Speaker: Jeffrey D. Oldham, Trinity University 

Title: Accurate Approximations for Asian Options 

Abstract An option is the right to buy or sell an item during a
specified time period and at a specific price. Individuals and
businesses purchase options as insurance against asset price
changes. Sellers of options receive a payment in exchange for assuming
the risk of possible price movements. Both buyers and sellers of
options need to be able to accurately value options to ensure they are
not paying too much or are assuming too much risk. Asian options, also
called average-price options, protect against changes in the average
asset price over a specified period. For example, an airline company
regularly buying aviation fuel could purchase an Asian option
protecting against large changes in fuel costs.  In this talk, we
present an asymptotic fully-polynomial approximation scheme for
pricing Asian options. We model asset prices using an n-step
recombinant binary tree. For an option with strike price X, we give a
O(kn^2) time algorithm with one-sided error of nX/k, for any positive
integer k. Our technique works for both European Asian and American
Asian options and is powerful enough to extend to other path-dependent
options with similar properties. This is the first polynomial-time
algorithm for Asian options to give guaranteed error bounds.  Joint
work with Donald Aingworth and Rajeev Motwani.


------------------------------------------------------------------------


Dr. Valery I. Levitas
Texas Tech University, Department of Mechanical Engineering,
Lubbock, Texas 79409-1021, U.S.A.
ph: (806) 742 3777; fax:  (806) 742 3540
E-mail: valery.levitas@coe.ttu.edu

"General Approach for the Description of Structural Changes in Inelastic
Materials  with Application to Phase Transitions, Ductile Fracture and
Strain-Induced Chemical Reactions"

For the description of phase transformations (PT), chemical reaction
(CR) and fracture in elastic materials the well-known energetic
methods can be applied. For such structural changes (SC) in inelastic
materials corresponding methods were lacking.  A new unified approach
to various SC in inelastic materials has been developed for small and
large strains.  The theory developed includes a non-local
thermodynamic criterion of SC, a scale-dependent kinetic equation for
the SC time, principle of minimum of SC time for determination of
shape and volume of transformed region and global SC criterion based
on stability analysis. The following problems are solved analytically
or numerically.

1. PT and CR under compression and shear of materials by Bridgman
(diamond) anvils. A number of experimental results are explained (e.g.
a significant reduction in PT pressure due to plastic shear, pressure
self-multiplication effect and the obtaining of fundamentally new
materials which are impossible to produce without additional plastic
strains), and some of the interpretations are completely unexpected.

2. Martensite nucleation at the intersection of shear-band is modeled
and applied for interpretation of experimental results for TRIP
steels. It is shown that the fulfillment of the thermodynamic PT
criterion is not sufficient for the occurrence of PT, because solution
without PT for elastoplastic material is always possible. To choose
the best solution and consequently to describe the competition between
plasticity and PT, a global PT criterion based on stability analysis
is applied.

3. Initiation of CR in energetic materials inside the shear band. The
phenomenon of reaction-induced plasticity (RIP), similar to
transformation-induced plasticity (TRIP) at phase transitions is
predicted and modeled. The reasons of the effect of plastic strain on
thermodynamics and kinetics of CR are found. The solutions illustrate
the fundamental difference in SC conditions for the homogeneously
distributed pressure and shear stresses in this problem and strongly
non-homogeneous pressure distribution in the Bridgman anvils.
Consequently each experimental situation should be simulated carefully
before any conclusion is made.

4. Void nucleation and crack propagation in elastoplastic material is
studied analytically and numerically. Competition between fracture and
PT is investigated using global PT and fracture criteria.


------------------------------------------------------------------------


Title: "Smoothed Particle Hydrodynamics and the Numerical Gas Dynamical
        Simulation of Galaxy Formation in Cosmology"

Paul Shapiro, Dept. of Astronomy

The formation of structure in the universe --- galaxies, clusters, and
large-scale structure --- involves the complex interplay of the physics of
compressible, inviscid gas dynamics, microscopic radiative processes, and
gravitational N-body dynamics.  It is believed that the structure we see today
resulted from the gravitational amplification of an initial spectrum of
small-amplitude Gaussian random noise density fluctuations over a cosmological
time span of more than 10 billion years.  The nonlinear outcome of this
amplification process resulted in supersonic motions which led to shock waves
which dissipated the kinetic energy of gravitational collapse and caused
radiation to be released, allowing the ordinary baryon-electron plasma which
constitutes only a small fraction of the average matter density of the universe
to compress to the point of becoming self-gravitating and able to form stars.
This extreme multi-scale problem involves many orders of magnitude of density,
temperature, and pressure contrast which must be resolved simultaneously, a
severe challenge to numerical methods and existing hardware.  We will discuss
some recent attempts along these lines.


------------------------------------------------------------------------


Murray Carlson, Finance Department
(joint work with Zeigham Khokher and Sheridan Titman)

An Equilibrium Analysis of Exhaustible Resource Investments 

We develop a general equilibrium model of an extractable resource market where
both the prices and extraction choices are determined endogenously.  With
plausible parameters the model generates prices that are roughly consistent with
observed oil and gas forward and option prices as well as with the two-factor
price processes that were calibrated in Schwartz (1997). However, the subtle
differences between the endogenous price process determined within our general
equilibrium model and the exogenous processes considered in earlier papers can
generate significant differences in both financial and real option values.


------------------------------------------------------------------------


Per-Gunnar Martinsson
"Shape From Shading"

He will begin by giving a presentation of the "shape from shading"-problem,
which is one component of the more general problem of restoring the shape of a
3D object from a 2D image of said object.

The mathematical equations at the core of this problem take the form of first
order, fully non-linear PDEs. As the most typical example he will discuss the
"eikonal equation", (du/dx)^2 + (du/dy)^2 = f, where f = f(x,y) is a given data
function and we want to determine u = u(x,y).

He will present some new, surprising results for such equations obtained by
employing methods of differential geometry. In particular, the stable manifold
theorem and the Sternberg normal form were used.


------------------------------------------------------------------------


                Dr. Paul Saylor
"How the Scattering Amplitude Yields a Comparison between
       Conjugate Gradient Error Estimates"

The conjugate gradient (CG) method minimizes the A-norm of the error. (Matrix A
is assumed to be SPD.) This quantity is not available during execution of CG
whereas the Euclidean norm of the residual is available. A straightforward
estimate yields an upper bound on the A-norm of the error in terms of the
residual. A different approach to estimating the A-norm of the error has been
taken by Golub et al. Their approach is to observe that the A-norm of the error
is a Stieltjes integral and then obtain an upper bound from a Gauss quadrature
combined with a Gauss-Radau approximation. The two bounds can be compared side
by side through making some further approximations with Gauss quadrature.


------------------------------------------------------------------------


Timothy Walsh
The University of Texas at Austin, TICAM

"Boundary Element Modeling of the Acoustical Transfer Properties of the
Human Head/Ear - Goal Oriented Adaptivity with Multiple Load Vectors"

The head-related transfer function (HRTF), used by acousticians and audiologists
in understanding the localization process, as well as in the design of hearing
aids, virtual acoustic simulators, and in tests for hearing system damage, is
normally measured in an anechoic chamber using a rotating frame.  This procedure
is time consuming and expensive, and does not allow for parametric studies of
the effects of various geometrical features on the overall response.  Numerical
simulations of the HRTF provide the advantage that parametric studies can be
performed simply by changing the input parameters to the code.  However, the
standard adaptive procedure would require the generation of a separate adaptive
mesh for each frequency and each right hand side, which simply makes the problem
intractable.  Here we will present an adaptivity procedure based on
goal-oriented adaptivity for the case of multiple right hand sides, all in the
context of the hp boundary element method.


------------------------------------------------------------------------


We are privileged to have Dr. Franz Winkler with us tomorrow.  He has agreed to
give a talk on "What we can and (still) can not do in symbolic computation"


------------------------------------------------------------------------


Roberto Roveda

A Combined Discrete Velocity/Particle
Based Numerical Approach for
Continuum/Rarefied Flows

A 27-speed Adaptive Discrete Velocity (ADV) Euler solver has been coupled with
the direct simulation Monte Carlo (DSMC) method to create a single, adaptive
hybrid solver that models mixed rarefied-continuum flows efficiently. Novel
features such as ghost cells, a temperature correction factor, and the free-cell
scheme were implemented in DSMC while the ADV method has been upgraded to model
gases having internal rotational energy.  A second step consisted of devising a
property-based coupling for information exchange at the interface.  Finally, an
adaptive procedure that tracks non-equilibrium regions with disconnected patches
of DSMC was implemented to resolve complicated transient flow structures.  The
two dimensional algorithm was then applied to study a pressure-driven impulsive
jet that issues from a slit. Simulations were performed of a freely expanding
plume and a plume impinging normally on a flat plate.


------------------------------------------------------------------------


        The first TACG of the semester will be next Tuesday.  I will be
discussing 2 recent (July 20, 2000) Nature articles on packing of DNA: Best
packing in proteins and DNA (Stasiak and Maddocks) and Optimal shapes of compact
strings (Maritan et al).  This should serve both as a continuation of our
discussion on ideal knots and catenanes, and as a (gentle) introduction to the
work of Dr. Oscar Gonzalez, who's joined the UT Math faculty this fall.


------------------------------------------------------------------------