Talks in Spring 2003
January 27
Speaker: Hongmei Zhang (Department of Statistics,
Iowa State University)
Title: Sample Size Calculation for Finding
Unseen Species
Abstract: This talk concentrates on inference
and study design for applications
where objects of different types are observed. The example
I carry
through with concerns observations of animals of different species,
a
famous problem in the statistical literature. Approaches to
estimating
the number of unseen species are considered. We apply a Bayesian
model to
estimate the total number of species in a region. A new methodology
to
determine the needed sample size for future data collection is developed.
The method uses Monte Carlo simulation to determine how large an
extra
sample is needed to guarantee that a certain proportion of total
species
can be collected with a specified confidence. We conduct a
simulation
study and apply the method to an application in a DNA sequencing
approach.
January 31
Speaker: Anthony Almudevar (Department
of Mathematics & Statistics, Acadia University)
Title: Inference of Multiple Pedigree Relationships
based on Genotypic Data
Abstract: The estimation of pedigree relationships
between individuals is a problem
of some interest in the biological, medical and forensic sciences.
Such estimation is
commonly done using microsatellite genetic markers. The inference
has a statistical
basis in the rules of Mendelian inheritance, with controllable error
rates. While the
problem of determining the relationship between pairs of individuals
is well understood,
the problem of reconstructing a pedigree among numerous individuals
introduces
considerable computational challenges. Although the likelihood
of genetic data for a
specific pedigree can be calculated, the large number of putative
pedigrees rules out
an enumerative approach for all but the smallest samples.
I will discuss a general approach to this problem, in which a class
of constraints on
the admissible set of pedigrees is defined in such a way
that a constrained optimization of the pedigree likelihood is
computationally feasible. A simulated annealing algorithm is
then used to
determine the constraint yielding the global maximum. This
approach will
be demonstrated for two types of problem. In the first, it
is assumed
that parents of all nonfounders are represented in the sample, and
that
the founders themselves are unrelated. The number of generations
is
arbitrary. In the second, the parents of siblings groups need
not be
represented in the sample, as will typically be the case when sampling
from natural populations. In this case pedigrees are restricted
to two
generations.
February 24
Speaker: Karen Kafadar (Department
of Mathematics, University of Colorado-Denver)
Title: Statistical Analysis of Microarrays
Abstract: Microarray technology has made available
large
data sets that can provide information on gene activity
when cells are subjected to various treatments. Before
proceeding with a formal statistical analysis, many
biological and procedural aspects should be cosidered.
These aspects may guide the analysis and subsequent
statistical inference. Several of these issues are
discussed in connection with the analysis of oligonucleotide
and cDNA microarray experiments. The particular focus
in this article is on effects caused by the cDNA slide
manufacturing process, appropriate transformations of the
data, and on adjustments for background. A prescription
for the analysis of microarray data is proposed and
demonstrated using data from a cDNA experiment comparing
the genetic expressions in two mouse cell lines; a
candidate set of genes is identified for further study.
The prescription may be modified for oligonucleotide
microarray data.
March 12 (in Jones 131, 4:00pm) (note the unusual
time and place!)
Speaker: Alexander Pankov (Department
of Mathematics, Texas A&M University)
Title: Nonlinear Schroedinger equations with periodic
potential
Abstract: Such equations appear in many applications:
nonlinear optics, condensed
matter physics (Bose-Einstein condensation), etc. We present a survey of
recent results on existence of nontrivial solutions decaying exponentially
fast at infinity. Such solutions can be interpreted as localized light
beams in nonlinear optics, or condensed states of matter trapped in a
bounded region.
March 24
Speaker: Kellie J. Archer (Department of Biostatistics,
Virginia Commonwealth University)
Title: Expression Summaries for Oligonucleotide Microarrays
and Their Implications on Multiple Hypothesis Testing
Abstract: Gene expression can be quantified using the
Affymetrix Genechip technology,
which is characterized by the ability to simultaneously study the
expression of thousands of genes. Formally, DNA (deoxyribonucleic
acid) is
composed of two single stranded polymers, where each strand is composed
of
sequences of four distinct bases, adenine (A), thymine (T), guanine (G),
and cytosine (C). The two DNA strands are held together in a double
helix
by hydrogen bonding such that complementary bases (A binds with T and C
binds with G) interact. Hybridization is the process whereby two single
stranded nucleic acid chains bond to form a stable double helix. The
Affymetrix Genechip technology exploits the hybridization process by
affixing several small single strands of DNA (i.e., probes) to the surface
of a chip in precisely defined locations and then allowing a sample of DNA
or RNA isolated from a biological source to bind to their complementary
strands. Rather then an entire gene being placed in a spot, the Affymetrix
Genechip is an oligonucleotide array consisting of a several perfect match
(PM) and their corresponding mismatch (MM) probes that interrogate for a
single gene. There are roughly 11-20 PM/MM probe pairs that interrogate
for each gene/EST, depending upon the chip used, called a probe set.
Gene
expression is obtained when the intensity values for the probe set are
combined to provide an overall signal for the probe set. The Affymetrix
software, Microarray Suite 5.0, calculates a measure of gene expression
using the PM and MM data. Other methods of calculating gene expression
for
Affymetrix data are also available. Three methods of calculating gene
expression for Affymetrix microarrays will be presented and the impact of
the choice of expression summaries on multiple hypothesis testing will be
explored.
March 31
Speaker: Arthur Sherman (Acting Chief, Mathematical Research Branch,
National Institutes of Health, NIDDK)
Title: The Chay-Keizer Model for Pancreatic Beta-Cells:
A 20-Year Retrospective
Abstract: Insulin secretion in mammals and man occurs
in an oscillatory fashion, driven by oscillations of membrane potential and
calcium in pancreatic beta-cells. This year marks the twentieth anniversary
of the first effective model for these oscillations, developed by Chay and
Keizer. We will discuss the accomplishments and predictions of the model
and how it has been modified to account for more recent experimental data.
The central hypothesis was that slow, negative-feedback by calcium itself
shuts off calcium entry into the cell and leads to oscillations. Though
now considered too simple, this model was already able to account for whole-body
glucose homeostasis by a control mechanism analogous to a thermostat.
We will describe an improvement due to Chay involving an additional known
calcium compartment, the endoplasmic reticulum (ER). Calcium concentration
has not yet been measured dynamically in the ER in beta-cells, but oscillations
therein appear to be necessary to explain the variation of the experimentally
accessible cytosolic compartment. Comparisons with pituitary cells, which
lack this modulation by the ER, will be given. Finally, we describe
an additional compartment, the calcium subspace, whose existence itself is
yet only a theoretical prediction, but which equally appears to be necessary
to explain recent, more refined data.
April 7th
Speaker: Omar Ghattas (Carnegie Mellon University)
Title: Towards Forward and Inverse Earthquake Modeling
on Terascale Computers
Abstract: Toward our goal of modeling strong earthquakes
in seismic regions, we are interested in determining mechanical properties
of sedimentary basins (such as the greater Los Angeles Basin) and descriptions
of earthquake sources from seismograms of past earthquakes. This gives rise
to very large inverse problems of recovering the coefficients and source of
the elastic wave equation from boundary observations of the response. Our
current forward simulations involve 100 million finite elements; over the
next several years the desired increase in resolution and growth in basin
size will require an order of magnitude increase in number of unknowns. Inversion
of such forward models provides a major challenge for inverse methods. It
is imperative that these methods be able to scale algorithmically to $O(10^9)$
grid points, to highly-resolved (e.g. grid-based) elastic material models
of large seismic basins, and to parallel architectures with thousands of processors.
I will discuss prototype parallel algorithms for the earthquake material
and source inversion problem. Tikhonov and total variation regularization
treat ill-posedness associated with rough components of the model, while multiscale
grid/frequency continuation addresses multiple minima associated with smooth
components. Parallel inexact Gauss-Newton-Krylov methods are used to solve
the optimality conditions. CG matrix-vector products are computed via checkpointed
adjoints, which involve forward and adjoint wave equation solutions at each
iteration. Preconditioning is via limited memory BFGS updates, initialized
with approximate inverses of an approximation to the Gauss-Newton Hessian.
Experience on problems with up to several million grid points suggests near
mesh-independence of Newton and CG iterations, good parallel efficiency, and
distinct speedups over a quasi-Newton method. However, significant difficulties
remain, and I will conclude with a discussion of these, along with possible
avenues for addressing them.
This work is joint with V. Akcelik, J. Bielak, and I. Epanomeritakis at
Carnegie Mellon, G. Biros at Courant, and other members of the Quake Project
(www.cs.cmu.edu/~quake).
April 14
Speaker: Michael R. Taaffe (Department of Industrial
and Systems Engineering, Virginia Tech)
Title: General Time-Dependent Infinite-Server
Queueing Networks
Abstract: Infinite-server queueing models have been
the subject of much Operations
Research work in recent years, partially because of their central role in
approximating systems with many servers. Time-dependent arrival/service
processes have also received increasing attention from the OR community
because few real-world systems are truly time homogeneous. The
intersection of these two topics is our focus today.
Applications for models of time-dependent, infinite-server queueing
systems and network systems come from a surprisingly large variety of
fields, including population processes in biology, migration processes,
immigration processes, and epidemiology. Perhaps the most interesting
OR
application for such models are engineering and management problems found
in the wireless data/telecommunication industry. Time-dependent,
infinite-server queueing networks have become a standard model for analysis
of mobile cellular telecommunication system design and management problems.
We develop numerically exact methods for evaluating the time-dependent
mean, variance and higher-order moments of the distribution of the number
of entities in the [Ph_t/Ph_t/infinity] queueing system and for the
multiclass [Ph_t/Ph_t/infinity]^K queueing network system, as well
as at
the individual network nodes. The Ph-type distribution and the Ph_t-type
process are essentially re-parameterizations of *general* distributions and
processes. We also allow for time-dependent deterministic entity
arrivals/deletions (for instance time-based deterministic policies for
"releasing" jobs into the system). We allow for multiple, independent,
time-dependent entity classes and develop time-dependent performance
measures, by entity class, at the nodal and network levels. We also
demonstrate a numerically exact method for evaluating, by entity-class, the
time-dependent distribution function and moments of virtual sojourn time
through the network.
A further generalization of the stationary Ph-type renewal process is the
MAP (non-renewal) process. Again we generalize to the time-dependent
case and present algorithmically exact results for [MAP_t/Ph_t/infinity]
and [MAP_t/Ph_t/infinity]^K queueing models.
For all of the models mentioned we have developed MAPLE software and have
put it in downloadable form on the Internet.
(http://users.iems.nwu.edu/$\sim$nelsonb/PhPhInf/)
April 21
Speaker: Michael Fagan (Department of Computational
& Applied Mathematics, Rice University)
Title: Computer Model Augmentation: Automatic
Differentiation and Beyond
Abstract: A great deal of modern science and engineering
relies on
computer simulation models. The typical computer simulation
function, however, computes only the 'model' outputs. Many
other quantities of interest, such as sensitivities and error
bounds, must be computed some other way. One way to compute
(or at least approximately compute) these additional quantities
is through the techniques of *program augmentation*.
This talk comprises two parts. Part I addresses augmentation
techniques in general. Part II illustrates some specific
program augmentation techniques and applications. The specific
augmentation techniques addressed in this talk are:
a) Automatic differentiation,
b) Interval computation,
c) 'Stochastic' automatic differentiation.
Talks in Fall 2002
September 23
Speaker: M. Saleet Jafri
(School of Computational Sciences, George Mason University)
Title: Modeling the mechanism of Calcium sparks
in the heart
Abstract: Calcium sparks are the elementary calcium
(Ca) release events
that underlie excitation-contraction coupling in heart and
skeletal
muscle and play an important role in regulating tone in smooth
muscle.
The properties and significance of Ca sparks have been studied
in
detail, yet the mechanism for Ca spark termination remains
unclear. A
mathematical model based on recent experimental data was
developed to
study the mechanism of calcium spark generation and termination
in
heart.
Ryanodine
receptors (RyRs) are Ca channels in the sarcoplasmic
reticulum (SR) that control Ca release from the SR and are
thus
responsible for Ca sparks. RyRs are activated by Ca
itself and, in
heart muscle, are triggered to release calcium by the influx
of Ca
through voltage-gated L-type Ca channels (dihydropyridine
receptors,
DHPRs) that are located close to the RyRs. The DHPRs
are found in the
sarcolemma and transverse tubular (T-tubule) membrane.
The DHPRs are
very close to RyRs, sharing a common very narrow "subspace"
that exists
between the T-tubular membrane and the SR (~15 nm wide).
Three recent
results are important for the proposed model. First,
it was discovered
that RyRs in heart are packed into arrays of up to hundreds of RyRs.
Second, a physical coupling among adjacent RyRs has been
shown to
affect gating of RyRs. Third, the open probability
of the RyRs has
been shown to decline with lower SR Ca content.
The model
includes an array of RyRs (up to 100) that share a
subspace with one DHPR. Each RyR is modeled as
an independent two-
state channel that is activated by subspace calcium.
The RyR open
probability (Po) is influenced by the SR lumenal [Ca], with
the Po
decreasing as SR lumenal [Ca] declines. Additionally,
the state of one
RyR influences the state of other RyRs through a co-operativity
factor. This last feature is used to simulate "coupled
gating" of RyRs.
Our model
of Ca sparks nicely reproduces many features of Ca
sparks that have been observed experimentally in heart.
Ca sparks are
activated at a very low rate under resting conditions but
Ca sparks
are "triggered" when there is an influx of Ca into the subspace,
as
occurs when DHPRs are activated. Ca spark durations,
amplitudes, and
profiles are similar to experiment. When the RyR dependence
on SR
lumenal Ca is removed, Ca sparks fail to terminate.
Ca spark durations
also increase when the coupling is reduced; this mimics the effect
of
agents such as FK506 that disrupt coupled gating. The
remarkable
success of this "sticky cluster" model of the cardiac Ca
spark,
suggests that the two central novel features of our model
(coupled
gating of RyRs and the dependence of RyR Po on SR lumenal
[Ca]) may
play an important role in regulating SR Ca release in heart.
September 30
Speaker: Sebastian Schreiber (Department
of Mathematics William & Mary)
Title: Coevolution in Host-Parasitoid Systems
Abstract Parasitoids are insects (usually flies
or wasps) that lay their eggs in other insect hosts. These eggs develop
into larvae that consume the host and lead to its eventual death. Parasitoids
are important as they regulate pests in many agricultural systems, constitute
as much as 25% of all insect species (10% of all metazoans), and are valuable
tools in testing many ecological and evolutionary theories.
Parasitoids and hosts often live in patchy environments. From
the perspective of the host, patches may vary in plant nutritional quality,
plant defenses, and microclimate. From the perspective of the parasitoid,
patches may vary in host abundance, host size, and microclimate. In the
first part of this talk, I will discuss theoretical results about the coevolution
of host and parasitoid patch preferences and under what conditions coevolution
stabilizes host-parasitoid interactions.
An important component of parasitoid biology is their haplo-diploid
mechanism of sex determination: males are haploid and develop from unfertilized
eggs; females are diploid, developing from fertilized eggs. Female parasitoids
can control whether particular eggs are fertilized, and thus determine
the sex of each offspring. In the second part of this talk, I will discuss
theoretical results about the evolution of parasitoid sex allocation (i.e.
the probability they lay female vs. male eggs in encountered hosts) in
patchy environments. By integrating the preceding two theories, conclusions
can be drawn about the coevolution of sex allocation and patch preferences.
The talk will conclude by presenting field and experimental
studies and discussing the potential implications for agricultural systems.
This talk is based on joint work with Laurel R. Fox (University
of California, Santa Cruz) and Wayne M. Getz (University of California,
Berkeley).
October 21, at 4:00pm
Speaker: Constance M. Schober,
(Department of Mathematics, Old Dominion University)
Title: Multi-symplectic Methods for Hamiltonian
PDEs
Abstract Recent results on spectral and finite
difference multi-symplectic schemes for one and two dimensional PDEs
are discussed. The new schemes are shown to satisfy discrete multi-symplectic
conservation laws. These discrete conservation laws are used to obtain
error estimates for the proposed schemes. The conservation of local energy
and momentum is examined as well as preservation of several global invariants.
October 28, at 4:00pm
Speaker: Roland Duduchava, (A. Razmadze
Mathematical Institute, Tbilisi, Georgia)
Title: The Mathematical Theory of Cracks: The
Wiener-Hopf Method
Abstract In linear elasticity, an elastic medium
with a crack is modelled by the
Neumann boundary value problem (BVP) for the linear Lame equation
(or for a
homogeneous partial differential equation with constant coefficients
of
order 2, when the medium is anisotropic). We apply the potential
method in
combination with the Wiener-Hopf method (based on factorization
of matrix
symbols) to describe detailed asymptotics of the displacement and
stress
vector fields. The approach is implemented in the following
steps:
i. Applying the potential method, the BVP is reduced
to a boundary
pseudodifferential equation (BPsDE) on an open surface that coincides
with
the crack surface.
ii. The full asymptotic of a solution to the PsDE on
the open surface is
obtained, including an explicit description of the exponents and
the
logarithmic terms.
iii. A detailed spatial asymptotic of the solution to
the BVP is derived.
Explicit connections between coefficients of the surface and the
spatial
asymptotic are found. The obtained information is important,
e.g. for
developing fracture criteria for elastic materials with cracks.
Compared with the alternative method of V. Kondeatjev (to which
V. Maz'ja,
B. Plamenevsky, B.W. Schulze, S. Nazarov, P. Grisward, M. Dauge,
V. Kozlov,
M. Costabel and others have contributed), the Wiener-Hopf method
produces
more refined asymptotics, but (so far) is more limited in its applications.
Here are some results established by the Wiener-Hopf method:
I. In some problems, e.g. cracks in anisotropic elastic
materials with
free crack faces, logarithmic terms in the asymptotic are absent.
II. In the crack problem for anisotropic elastic body
with mixed
conditions, when on one side of the crack surface the displacement
vector and on the other side the stress vectors are prescribed,
solutions are oscillating and the exponents of the asymptotic decay
by
step 1/2, as opposed to step 1 in the non-mixed problem (M. Costabel,
M.
Dauge, R. Duduchava, and D. Natroshvili).
III. For cracks on the interface between two anisotropic
three-dimensional
bodies, the displacement and the stress vector field are oscillating
near
the crack front. An explicit criterion, derived from an investigation
of
eigenvalues of the 3-by-3 matrix symbol, prevents oscillatory solutions
(R. Duduchava, A.M. Sandig, and W.L. Wendland).
November 4, at 4:00pm
Speaker: David E. Keyes, (Richard F.
Barry Professor, Departments of Mathematics & Statistics and Computer
Science, Old Dominion University & Institute for Scientific Computing
Research, Lawrence Livermore National Laboratory, & ICASE NASA Langley
Research Center)
Title: Solvers for Real People
Abstract Like the theoretical peak performance
of a computer system, theoretical
efficiency for algorithms is rarely closely approached for real
applications. While the quest for the "textbook efficiency"
continues on
many fronts, real users need to have their solver capabilities upgraded
today to exploit the platform potential to run more highly resolved
computations. The Terascale Optimal PDE Simulations (TOPS) project
of the
U.S. DOE SciDAC initiative is working on both fronts --- attempting
to
make fundamental advances in numerical algorithms that will be integrated
into tomorrow's scalable solver software while attempting to be
of service
to SciDAC application developers and others at the outset of the
initiative.
In this talk, we dwell on some practical aspects of migrating
from a
legacy (usually operator-split) nonlinear solver for evolutionary
or
equilibrium systems of PDEs to a Jacobian-free Newton-Krylov framework
that provides strong controls on splitting error while still incorporating
physically-based operator-split methodology where possible. It is
emphasized that to support even a single application from development
through production use on various platforms, contemporary solver
libraries
must offer a menu of flexibly combinable and tunable components
to allow
application-specific and architecture-specific trade-offs (e.g.,
memory
versus flops, synchronization frequency versus stability, robustness
versus efficiency). We mention recent results on nonlinear Schwarz
preconditioning and pseudo-transient continuation methods for PDEs.
We
also briefly discuss recent successes with the M3D extended
magnetohydrodynamics code of our SciDAC partners in fusion energy
research, which is designed to underscore the desirability of being
able
to draw from a broad family of solvers within a single application.
This talk is partially based on a review article of Jacobian-Free
Newton-Krylov methods co-authored with Dana Knoll of Los Alamos.
November 11, at 4:00pm
Speaker: James R. Wilson, (Department
of Industrial Engineering, North Carolina State University)
Title: ASAP2: An Improved Batch Means Procedure
for Steady-State Simulation Output Analysis
Abstract: We introduce ASAP2, an improved variant
of the batch-means
algorithm ASAP for steady-state simulation output analysis.
ASAP2 operates as follows: the batch size is progressively
increased until the batch means pass the Shapiro-Wilk test for
multivariate normality; and then ASAP2 delivers a
correlation-adjusted confidence interval. The latter adjustment
is based on an inverted Cornish-Fisher expansion for the
classical batch means t-ratio, where the terms of the expansion
are estimated via a first-order autoregressive time series model
of the batch means. ASAP2 is a sequential procedure designed
to
deliver a confidence interval that satisfies a prespecified
absolute or relative precision requirement. An experimental
performance evaluation on a set of difficult test problems
suggests that ASAP2 compares favorably to ASAP and the well-known
procedures ABATCH and LBATCH with respect to half-length and
coverage probability of the delivered confidence intervals.
November 18, at 4:00pm
Speaker: Dirk Gillespie, (School
of Medicine, University of Miami)
Title: THE PHYSICS OF ION SELECTIVITY IN BIOLOGICAL
ION CHANNELS
Abstract: Ion channels are proteins that serve
as a conduit for ions (mainly Na+, K+,
Ca2+, and Cl-) to move across biological membranes. Charge
movement of ions
through channels accounts for the majority of electrical activity
in the human
body and is responsible for the conduction of action potentials
in nerves,
muscle contraction, and fluid balance in the lungs and kidneys,
as well as
many other phenomena. When studied as single molecules, an
ion channel can
discriminate between different kinds of ions. For example, the L-type
calcium
channel conducts equal amounts of Ca2+ and Na+ even when there is
10^4 times
more NaCl than CaCl2 in the baths surrounding the channel.
Other kinds of
channels have different ion preferences. To study the physics
of ion
selectivity, we apply modern theories of electrolyte solutions to
channels.
Specifically, by modeling the ions as charged, hard spheres in a
confined
geometry (the channel) we find that the complex selectivity phenomena
exhibited by channels stem from the interaction of two phenomena:
1) the ions
seek to neutralize the charge on the channel protein and 2) the
ions (as hard
spheres) must complete with each other and the protein atoms for
space inside
the channel. The implementation of these ideas in both
equilibrium and
nonequilibrium situations will be discussed.
November 25, at 4:00pm
Speaker: Jeffrey L. Solka, (Advanced Computation
Technology Group, Naval Surface Warfare Center, Dahlgren, VA)
Title: Classifier Optimization Via Graph Complexity
Measures
Abstract: This talk will examine some of our
recent work in classifier
optimization via metric space adaptation based on graph theoretic
classifier complexity measures. The talk will describe our newly
developed
methodologies and present some preliminary results on both synthesized
data sets, an artificial olfactory data set, and a gene expression
data set. (This is joint work with D. A. Johannsen.) |
