A
Sampling of Research Accomplishments During the IMA Annual Programs
198990 through 199596
A
SAMPLING OF RESEARCH ACCOMPLISHMENTS IN NONLINEAR WAVES 198990
Important research results were obtained by many of the longterm
participants in the IMA program, and much significant research
and collaboration was initiated by shortterm visitors. This
research is documented in many IMA technical reports and IMA
conference proceedings and other volumes to be published by
SpringerVerlag. A brief sample of the research is as
follows:
Assessment
by James Glimm: The spring
quarter emphasized applications of systems of hyperbolic
conservation laws in sev eral space dimensions. The goal was
to nurture strong and close interdisciplinary interactions among
diverse areas of modern applied mathematics including
 Large
(and small) scale computing
 Asymptotic
modeling
 Qualitative
modeling
 Rigorous
proofs for prototype problems
 Strong
interaction of theories from modern applied mathematics with
experimental data.
The
highlight of the spring quarter was the threeweek long workshop
during the month of April. One of the striking outgrowths of
this meeting was the fact that interdisciplinary interactions
using all of the tools of modern applied mathematics mentioned
in (1)(5) above are developing at such a rapid rate that new
predictions of phenomena for fluid dynamics, nonlinear elasticity,
the behavior of reacting materials, among other applications
can be made through a combination of theory and compu tations
in regimes which are not accessible readily to experimentalists.
Examples presented at the April workshop of this strong interdisciplinary
interaction leading to new predictions include: 1) the work
of Grove and Menikoff on anomalous shock diffra ction patterns;
2) the work of Glaz, Colella, and collaborators on shock reflections
from wedges; 3) the work of Woodward, Majda, and Artola on new
phenomena in nonlinear stability for supersonic vortex sheets
and jets; 4) the work of Krasny, Shelley, Di pernaMajda, and
Caflisch on striking new phenomena in the evolution of thin
layers of vorticity in turbulent incompressible flows; 5) the
discovery and analysis by T.P. Liu, Roytburd, Hunter, Rosales
and Majda among others of diverse types of new resona nt phenomena
leading to wave amplification in hyperbolic systems. Another
significant announcement concerned the results of PuiTah Kan's
thesis on existence for conservation laws: an example with an
isolated umbilic point.
A
special highlight of the workshops was the part of the program
held at the Minnesota Super ComputerCenter. Paul Woodward's
``state of the art'' flow visualization graphical techniques
revealed striking new vistas (for the first time for the va
st majority of participants) for processing data for documenting
and explaining new phenomena in applications.
Many
interdisciplinary interactions developed at I.M.A. in 19881989
during the spring quarter. These include the following: 1) collaboration
by Woodward and J. Glimm's group in merging fronttracking and
shockcapturing alogrithms for interface instabilities; 2) a
new approach by Rosales and Thomann to explain the experimental
phenomena in diffraction of weak shocks by caustics; 3) the
development by Embid, Hunter, and Majda of new simplified asymptotics
equations to explain the transition to detonation in granular
explosives; 4) a new theoretical conjecture by Majda for organizing
centers for waves bifurcations in shock diffraction was simulated
by computational results, reported by Glaz at the meeting. This
conjecture has stimulated Glaz to begin a new round of large
scale numerical simulations to test the conjecture.
The
Multiphase Flow Workshop produced research results
almost immediately. The kinetic theory ideas of Jenkins were
new to many people and the idea of ensemble averaging probably
was moved to the front during the workshop. The idea of diffusion
against a concentration gradient was an important new thought,
vital for stability of fluidized beds. Bud Homsy got its first
analytical result on this which he will publish in the IMA Worksh
op volume. He got this result using the symbol manipulation
program MAPLE, which he learned in his IMA office, in about
three all night sessions. The multiphase group is surely going
to meet again in August 1990 at the Stanford IUTAM conferences.
Jenkins and McTigue started a collaboration based on contacts
made at the IMA.
Martin
Golubitsky's
stay at the IMA was most productive as he collaborated on two
projects that he would not otherwise have become involved in
were it not for this visit. The two pro jects are:
a)
Particle sedimentation and timereversible symmetry, and
b)
Arrays of Josephson junctions and bifurcation of mappings with
permutation symmetry.
The
first project is joint with Martin Krupa and Chjan Lim while
the second is joint with Don Aronson and Martin Krupa.
In
(a) they study a model, called the Stokeslet model, for the
sedimentation of a finite number of particles. This model had
been studied by several groups, including one of Caflisch, Lim,
Luke & Sangani who showed that periodic solutions ex ist
in the three particle model. Golubitsky, Krupa and Lim proved
the existence of several families of periodic and quasiperiodic
solutions for the nparticle model using a combination of timereversibility
(due to an infinite viscosity limit in the mod el) and spatial
symmetries. Their work is based on previous work of Lim (as
indicated), Krupa (bifurcation from group orbits of equilibria)
and Golubitsky (Hopf bifurcation with dihedral group summetry).
Since Lim was a postdoc and Krupa was visiting th e Math department,
the IMA presented a unique opportunity to involve the three
in collaborative research.
In
the second project, Aronson, Krupa and Golubitsky studied period
doubling in the presence of permutation summetry classifying,
for the types of bifurcations occurring in arrays of identical
oscillators, all period two points emanating from this bi furcation.
They used the Apollo color graphics to check their abstract
results for a system of coupled Josephson junctions. Again the
facilities and synergism provided by the IMA was crucial to
their research.
There
were two main projects James Glimm
started as a result of the IMA meetings. Both involve interaction
with Paul Woodward and his group. The first (and simplest) is
to adapt and use Wo odward's graphics programs. The second is
to compare systematically results on KelvinHelmholtz instability.
Glimm and Woodward have each developed high quality computational
algorithms suitable for this problem. The two algorithms are
however different in some important ways. The problem is highly
unstable, and for this reason is difficult to compute correctly.
An ever present question is whether or at what level detail
the computations are correct. The presence of two very different
methods of comput ation is thus an important check on the validity
of the results in this area.
A SAMPLING OF RESEARCH ACCOMPLISHMENTS IN DYNAMICAL SYSTEMS
1989 90
The theory of dynamical systems was spawned in the attempt
to understand mathematical models of physical (and biological)
phenomena which change with time. While the early pioneers,
such as Poincare and Hill, were motivated by a desire to explain
properties of celestial mechanical systems, this theory today
has applications in many diverse areas. The program in Dynamical
Systems and Their Applications was planned to examine in depth
a number of these applications. The objectiv es of the yearlong
program were to (1) to seek new vistas for the theory of dynamical
systems and (2) to encourage researchers to look seriously at
areas of application which are ripe for major advances. Important
research results were obtained by many of the longterm participants
in the IMA program, and much significant research and collaboration
was initiated by shortterm visitors. This research is documented
in many IMA technical reports and IMA conference proceedings
and other volumes published b y SpringerVerlag. A brief
sample of the research is as follows:
Some
of the most important research was done by the postdocs.
For example, Chris Gole made progress on his analysis of invariant
sets for higher dimensional twist maps. Scott Dumas did excellent
work on the relationship between KAM theory and crystal structures.
Mohamed Elbialy established the C' regularization of simultaneous
binary collision in classical celestial mechanics.
One
of the major accomplishments of the IMA Year on Dynamical Systems
and Their Applications was a fundamental discovery of the existence
of a regular global attractor for the weak solution of the NavierSt
okes equation on threedimensional domains. While
this result has now been proven rigorously only for certain
this domain in R3, it gives strong evidence that the phenomenon
of the occurrence of global attractors may be more the ru le
than the exception. The recent history of this discovery is
noteworthy because it shows the importance of the IMA program
in this development. During the Fall of 1989 Genevi`eve Raugel
and George R. Sell were in residence at the IMA. Their proof
of th e global regularity of solutions in the Sobolev space
H^1 was completed at that time. In the Spring of 1990, while
reading a draft copy of the RaugelSell paper at the IMA, Ciprian
Foias suggested that the methods of Raugel and Sell may be strong
enough to prove the existence of global attractors. Foias was
correct, and the results were presented by Raugel at the IMA
workshop on Dynamical Theorem of Turbulence.
The
theory of Approximation Dynamics
is the branch of mathematics which attempts to answer the question
of how well is the longtime dynamics of a given equation reproduced
by the longtim e dynamics of a given equation reproduced by
the longtime dynamics of approximate equation. The key issue
here is the longtime dynamics, as opposed to the finitetime,
transient dynamics. This theory addresses fundamental issues
of science arising in m athematical modeling and in numerical
computations. Victor A. Pliss (Leningrad University), while
in residence at the IMA, collaborated with George R. Sell in
a study of the behavior of hyperbolic attractors under small
perturbations. In their IMA prepri nt on this subject they give
a proof of a basic theorem describing the dynamics on such an
attractor after a small perturbation.
The
IMA program on dynamics included major effort in the rapidly
evolving area of
approximate inertial manifolds.
Many of the researchers in this subject were in residence at
the IMA and widespread collaboration were begun. One of the
first fruits of this effort was the proof that, under a reasonable
assumption on the dimension, every approximate inertial manifold,
for any of a class of equations which include the 20 NavierStokes
equation, is an actual inertial manifold for an approximate
equation. This discovery should have a catalytic effect on future
research in Approximation Dynamics.
The IMA program on Mathematical Physiology and DifferentialDelay
Equation was organized as an attempt to introduce the researchers
in dynamical system to the new and challenging problems arising
in the biomedical areas. This was a short onem onth program,
and the main goal was to spawn new research efforts. One of
the promising new directions is the role of dynamics in the
area of high performance computation. This was initiated to
a great extent by the presentation by Raymond Winslow (Depar
tment of Physiology and Army High Performance Computing Research
Center) on his plans and those of his collaborators to model
a cellular network of heart cells on a massively parallel Convection
Machine. The study of the dynamics of large cellular networ
k is an emerging area of science. It is realistic to predict
that future collaboration between dynamical and physiologist
will soon lead to important insight into problems related to
the onset of and the control of heart diseases in humans.
Victor
Donnay was a visitor at the IMA from January to June
1990. While at the IMA, Donnay finished two papers. The first
was `` Hamiltonians on the TwoTorus for which the Hamiltoni
an flow is Ergodic'' in collaboration with Carlangelo Liverani.
The IMA invited Liverani for a short visit during which time
the authors where able to write the final section of the paper.
The second was ``Using Integrability to Produce Chaos: Billiards
with Positive Entropy''. In the first version of the paper,
Donnay had conjectured that a halfellipse was a focusing arc,
and hence could be used to build a stadium like billiard table
which would have positive entropy, providing the eccentricity
of the ellipse was less than \sqrt {2}. While at the IMA, Donnay
proved the conjecture. In trying to determine the best line
of attack for the problem, Donnay used the computer algebra
system Maple. He had just learned how to operate this system
at the IMA's w orkshop on Computer Algebra and Dynamical Systems.
Donnay solved another problem in billiards during his stay.
He found the first example of a Cinfinity smooth, strictly
convex billiard with chaotic behavior. More explicitly, he showed
that a small pert urbation of an ellipse produces a billiard
with positive topological entropy. The chaotic behavior in this
system happens on a set of measure zero; in his previous examples
the chaotic behavior happened on a set of full measure but the
billiard was only C1 smooth. A major advance in the field would
be to show that this new example actually has chaotic behavior
on a set of positive measure (positive measure entropy). This
problem arose out of a late night computer graphics session
with Mike Jolly and Cl int Scoval. Jolly had written a beautiful
interactive computer program on his Iris to model dynamical
systems. Donnay suggested applying the program to the elliptical
billiard, which is an integrable system, and hence would provide
a good test for the pr ogram. Unexpectedly, the system exhibited
zones of instability. Scoval pointed out that these zones where
due to the existence of a homoclinic connection and the error
due to roundoff error. A few days later, Donnay went to a meeting
at Northwestern. Ge rhard Kneiper, one of the participants,
mentioned a problem about geodesic flow on an ellipsoid. Looking
at the analogous question for billiards on an ellipse, Donnay
realized that this problem was connected to the phenomena that
Jolly, Scoval and he had noticed on the computer.
During
his stay at the IMA C. Foias
worked in collaboration with several other IMA visitors on several
topics concerning infinite dimensional dynamical systems mostly
centered on the concept of approximate inertial manifolds (AIM),
that is manifolds which rapidly epsilonattracts all trajectories.
Thus together with O.P. Manley and R. Temam he concluded research
on an approximate inertial manifold for the space periodic 2D
NavierSt okes equations which may describe the correct permanent
dynamic behavior of large eddies . Also together with E.S. Titi
he showed how some well chosen semidiscrete schemes for the
KuramotoSivashinsky equation are the compression of the partial
differen tial equation to appropriate AIMS . With M. Jolly and
E. Titi he continued by elaborating analytically and computationally
on the importance of the preservation of dissipativity in the
discretization of a dissipative partial differential equaiton.
With A . Eden, B. Nicolaenko and R. Temam he improved a theorem
by Man'c concerning the projections in Euclidian spaces which
are onetoone on compact sets of finite fractal dimension.
With A. Eden he found a very elementary proof of the 1D LiebThirrring
ineq ualities and of their GuidagliaMarionTemam generalization.
Finally with J.C. Saut, he completely clarified the algebra
of a canonical normal form for the NavierStokes equations (and
for other similar differential equations) which they discovered
in s tudying the asymptotic expansion of decaying solutions.
A very fruitful part of his stay at IMA was the scientific interaction
with many senior and junior IMA visitors and faculties of the
University of Minnesota.
BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS DURING PHASE TRANSITIONS
AND FREE BOUNDARIES PROGRAM 199091
The
scope of the program was to understand certain types of physical
behavior which occur in phase transitions and in phenomena which
involve free boundaries. The first half of the year was concerned
with phase transition s and concentrated on equilibrium and
dynamical problems involving two or more phases, with the transition
region a sharp interface or a transition layer. Among the physical
problems considered were solidification, dendritic growth, nucleation,
spinodal decomposition, solidsolid transitions, crystalline
states, shock induced phase transitions, and phase structures.
The
second half focused on freeboundary problems and on diffusion
problems involving a singular mechanism, such as a degeneracy
and free boundaries. Here the physical areas considered included
porous flow, jets, cavities, lubrication, combustio n, plasma,
coating flows, and the dispersal of biological populations.
Robert
V. Kohn
made the following extended observations about the fall program:
I can hardly contain my enthusiasm about the quality of your
fall,1990 concentration on Mathematical Modeli ng of phase transitions.
It was my privilege to visit IMA for a total of about 3 weeks
(September2428 and November821). I cannot imagine 3 weeks
of more intense or more productive mathematical activity. The
essence of IMA's success is its ability to bring people together
in a focused research environment. A list of specific projects
and publications cannot come close to describing IMA's true
impact. I hope the following summary captures some of the intangible
aspects of IMA's influence, as well as t he more tangible ones.
A)
New contacts.
A
visit to IMA is always an occasion to create new mathematical
connections. This time I was especially pleased to meet many
young people whom I had previously known only through correspondence,
and whose research interests are linked to my own. I hav e in
mind specifically K.Bhattacharya (now finishing Ph.D. work with
R.James), C.Collins (a recent student of M.Luskin), P.Pedregal
(a recent student of D.Kinderlehrer), G.Zanzotto (a recent student
of J.Ericksen), V.Sverak (a powerful young Czech mathem atician,
who had not visited the US before), L.Truskinovsky (a talented
young mechanician who recently immigrated from the USSR), and
L.Ma (a current student of M.Luskin). I also appreciated the
opportunity to meet several more senior individuals from th
e mechanics and phase transitions communities, including M.Glicksman
(whose work on mean field theories for Ostwald ripening makes
contact with my own on composite materials) and N.Goldenfeld
(whose work on intermediate asymptotics is strikingly and unex
pectedly related to mine on blowup of solutions of nonlinear
parabolic equations).
B)
Things learned.
It
would be impossible to list all the things I learned from other
visitors while at IMA. However, a few special things stand out
in my mind as unexpected and significant advances:
a)
P.Pedregal and V.Sverak had both made progress concerning the
socalled ``threewell problem.'' They gave two (different)
proofs that if a youngmeasure limit of gradient is supported
on three incompatible matrices, then the associated microstructu
re cannot be a ``laminate.'' N.Firoozye and I had previously
tried to prove this, without success.
b)
I.Fonseca presented a new framework for the variational modeling
of coherent phase transitions in the presence of crystalline
defects. Though the results are so far only preliminary in character,
this represents a significant advance for the theory .
c)
M.Chipot spoke about his work with Collins and Kinderlehrer
on the numerical minimization of nonconvex energies. I learned
from his talk that one can identify a (meshdependent) ``artificial
viscosity'' which behaves almost like a surface energy pe rturbation.
This provides an unexpected connection between the numerical
simulation of coherent phase microstructures and my work with
S.Muller (see below).
C)
Things taught.
I
spent many hours discussing points of common interest with other
IMA visitors. In some cases I think I was able to offer useful
insight or direction. Here are three specific examples:
a)
I had many discussions with N.Firoozye (my former student, now
an IMA postdoc) about modelling the evolution of phase boundaries
under the analogue of ``motion by mean curvature'' for a nonconvex
surface energy. Stimulated by the lectures of M.Gurt in, Nick
was formulating a research program based on ``regularization.''
We both began to understand some reasonable conjectures as a
result of those discussions.
b)
Collins and Luskin have been simulating the formation of microstructure
in coherent phase transitions via direct numerical minimization
of the nonconvex elastic energy functional. My recent work with
S.Muller (see below) strongly suggests that they are reaching
local rather than global minima in some cases. I suggested ways
to test this numerically; it remains to be seen whether they
will get numerical solutions similar to the ones Muller and
I found analytically.
c)
G.Zanzotto has been working with I.Muller on a model for shapememory
behavior based on a nonconvex ``free energy.'' I was able to
show him how the failure of convexity that Muller derives from
``surface energy'' can actually be obtained from bulk energy
considerations alone. I also introduce him to the work of Colli,
Fremond, and Visintin, which could represent a useful model
for the dynamics of shapememory alloys.
F)
A sense of community.
I
have emphasized aspects of my visit that were in some sense
specific to IMA. This has the effect of deemphasizing discussions
with other visitors I already knew well, such as J.Ball, R.Pego,
R.James, and Y.Giga. It would be wrong to ignore the impo rtance
of that aspect, However, Mathematical communities are formed
through interpersonal contact; to thrive they require the nourishment
of gatherings such as the one which took place at IMA this fall.
Forgive
me for going on such length. IMA's fall semester concentration
on phase transitions was by every measure a spectacular success.
The mix of interdisciplinary activity was just right, as was
the choice of a focus that was neither too broad nor too narrow.
The impact of this activity on the applied mathematical community
will been enormous  but the most important components of that
impact are the intangible ones. I hope this letter will serve
as a ``snapshot'' of this impact, including i ts less tangible
components specialized to my own individual experience. Please
feel free to share my comments with NSF, other funding agencies,
and with your participating institutions.
Steven
J. Altschuler
was a regular IMA postdoc. He reports as follows: Several years
ago, E. Calabi suggested a method for flowing through singularities
of plane curves evolving by the cur ve shortening flow. The
idea is to lift the plane curve to a family of periodic space
curves with lifing parameter. This family of curves has infinite
time existence and will converge in infinite time to straight
lines. One would like to approximate the plane curve evolution
with the parameter very small. One then attempts to get uniform
bounds on the family after the time of a singularity in order
to define the planar flow as a limit of the space curve flow.
In joint work with Matthew Grayson, this app roach was successfully
completed in October while the two authors were at the IMA.
For
the higher dimensional case of mean curvature flow, very little
is known about the formation of singularities. In joint work
with S. Angenent and Y. Giga, we were able to get a rather detailed
picture of singularitiy formation in the case of compa ct hypersurfaces
which are rotationally symmetric. In fact, using the language
of viscosity solutions and the level set approach, we were able
to extend the flow past the time of a singularity and flow until
the surface dissapears to a collection of poin ts. This work
was started and completed while all three authors were at the
IMA.
Finally,
in joint work with John Sullivan at the Minnesota Geometry Center
and the Supercomputer Institute, we made extensive use of computer
modeling in order to give a complete cataloge of all space curves
which can be self similar solutions to the curve shortening
flow. In a forthcomming paper, we will give asymptotic descriptions
of these curves and describe several of their most interesting
features.
David
C. Dobson
was a industrial postdoc during the 199091 year, working with
scientists from Honeywell, Inc. His research at the IMA has
centered on optical diffraction problems involvin g the scattering
of electromagnetic waves from various structures.
The
first problem studied involves the scattering of an electromagnetic
plane wave incident on an optical substrate into which a doubly
periodic (periodic in two orthogonal directions) relief profile
has been cut. The Maxwell equations are reduced to a coupled
pair of operator equations on the surface of discontinuity.
The equations are shown to be equivalent to a Fredholm system,
thus establishing existence and uniqueness of solutions for
all but a discrete set of parameters. This is joint work with
Avner Friedman.
The
work described above suggests an obvious numerical scheme to
solve the diffraction problem. The integral operator equations
on the surface of discontinuity are discretized and solved using
a boundary elementcollocation procedure. This metho d is now
partially implemented; initial test runs have been successful.
A brief outline of the method and some preliminary numerical
results will appear in an upcoming volume of Proceedings
of the SPIE with J. Allen Cox from Honeywell as coauthor
. A mathematical study of the method, convergence analysis,
and more complete numerical results has appeared as an IMA preprint.
Associated
with the forward diffraction problem discussed above is the
inverse problem of designing a substrate profile which diffracts
light in some specified way. In the Fraunhofer approximation,
the forward diffraction problem can be approximated b y an ordinary
Fourier transform. The inverse problem then reduces to finding
the phase of a complexvalued function from knowledge of its
magnitude and the magnitude of its Fourier transform. This nonlinear
problem can be efficiently solved with a quasi Newton method.
A future IMA preprint will describe this method in the infinitedimensional
setting, prove convergence, and give some numerical results.
A
code has already been implemented and has been applied to ``real
world'' optical design problems at Honeywel
BRIEF SELECTION OF RESEARCH RESULTS DURING APPLIED LINEAR ALGEBRA
PROGRAM 199192
The
academic year program was divided into three parts (corresponding
to the fall, winter and spring quarters), although there was
considerable fluidity between the various parts.
1.
Discrete Matrix Analysis with emphasis on the mathematical analysis
of sparse matrices and combinatorial structure;
2.
Matrix Computations with special emphasis on iterative methods
for solving systems of linear equations and computing the eigenvalues
of sparse, possibly structured matrices;
3.
Signal Processing, Systems and Control with emphasis on the
matrix analysis and computations that arise in this area of
application.
Richard
A. Brualdi prepared the following assessment of the
program: The IMA provides a very special environment which fosters
the cooperative interaction of mathematicians and other scientists.
This interaction has several effects. Research interests are
broadened as a result of one being exposed to new viewpoints
and new applications. Interest in an old research problem may
be rekindled because of contact with someone now consider ing
the problem or because one learns of a possible new application.
Unfinished
work may get finished because of the opportunity to work intensively
without the usual academic distractions. Problems, which might
otherwise go unsolved, get solved by cooperative work. The person
or persons with the right knowledge or rig ht technique becomes
newly aware of a problem and a successful attacked is made on
the problem. New problems are formulated, new techniques are
developed, new contacts are made of which some may have a lasting
and decisive effect on one's productivity fo r many years to
come. I believe that all of these things happened during the
Applied Linear Algebra Program. That so many people from applied
linear algebra were together for such a long time (one month,
one quarter, two quarters or more) was unprecedent ed. Such
an opportunity I expect will not occur again in the near future.
The effect and success of the program on Applied Linear Algebra,
like all IMA programs, cannot be properly assessed until several
years after the calendar says that the formal prog ram is over,
since the effects of the program go on for many more years after
that.
The
first quarter of the program concentrated on sparse matrix analysis,
and the application of Combinatorial analysis in matrix problems
and the application of linear algebra in combinatorial problems.
Here we had an unprecedented mix of some of the best sparse
matrix theorists (coming from computer science), like Joseph
Liu and John Gilbert, and core matrix theorists and discrete
mathematicians. Many of the latter group were exposed for the
first time to problems in sparse matrix analysis (of which many
require symbolic and therefore combinatorial analysis before
implementation). Some instances of problems successfully attacked
during the first quarter are: (i) Onequarter visitor M. Fielder
at the SIAM meeting on Applied Linear Algebra, conjectur ed
that the smallest number of nonzero entries in an orthogonal
(or unitary) matrix of order n which does not decompose into
two smaller orthogonal matrices is 4(n1) for all n\geq 2. This
problem was solved by LeRoy Beasley, postdoctorate Bryan L.
Shade r and Richard A. Brualdi by extracting the right combinatorial
property implied by orthogonality and then showing that lower
bound of 4(n1) is a consequence of this property. Shmuel Friedland
used powerful analytic techniques to obtain tight bounds on
t he real eigenvalues of almost skew symmetric matrices and
used these bounds to `almost' prove a conjecture of Brualdi
and Li(1983) concerning the maximum spectral radius of the class
of structured matrices known as tournament matrices. Wayne Barrett,
Cha rles R. Johnson and Raphael Loewy used the opportunity of
being together at the IMA to solve with Tamir Shalom an old
problem of theirs to determine the largest number of diagonal
entries of a matrix A of order n with rank A=k<n that have
to be pertur bed in order to increase the rank. In addition,
Shmuel Friedland, Alex Pothen and Richard A. Brualdi collaborated
on some analysis of the NPhard problem of finding a sparsest
basis of the row space of a matrix.
The
second and third quarters of the program were dominated by numerical
linear algebraists, most from computer science and most with
interests in numerical algorithms for problems that arise in
queuing theory and Markov processes, signal/image proces sing
and control theory, and linearizations of nonlinear problems
from applied science. During this time the twice weekly Applied
Linear Algebra Seminar was heavily attended, drawing people
from the computer sciences department of the University and
from the nearby Supercomputer Institute. The results here are
too technical to be reported in detail in this short report.
But there were significant advances, both theoretical (e.g problems
concerned with instability and robustness) and computational,
conce rning: computation of eigenvalues of large and structured
matrices (the kind that often come up in applications) by various
algorithms such as the QR method, symmetric and nonsymmetric
Lanczos method, subspace iteration, and the Arnoldi algorithm.
Some of the long term visitors that contributed to these investigations
were Gene Golub, G.W. Stewart, Anne Greenbaum, Robert Plemmons,
Roland Freund, Zdenek Strakos, Dianne O'Leary, Adam Bojanczyk)
and postdoctorates James Nagy, Walter Mascarenhas and Roy Ma
thias. Work on improving (and analyzing) algorithms for the
computation of the steady state eigenvector of large and structured
matrices that arise in Markov chains and queuing models was
done by G.W. Stewart, Dianne O'Leary, Robert Plemmons, Carl
Meyer, and Paul Schweitzer. Prompted by queries from two of
the developers of the linear algebra package LAPACK , namely
James Demmel and Z. Bai, Nicholas Higham obtained new error
estimates for the Sylvester equation, including stability analysis.
Many of his results will be included in the next release of
LAPACK. James Demmel and William Gragg characterized in a combinatorial
way (the bipartite graph must be acyclic) those matrices, like
bidiagonal matrices, which have the property that small relative
pert urbations of the entries result in small relative perturbations
of the singular values, independent of the values of the entries
of the matrix. G.W. Stewart developed a parallel implementation
for an algorithm for updating a URV decomposition (unitary t
imes upper triangular times unitary) of a matrix which reveals
its effective rank.
Victor
Klee
returned to the IMA for 1 month during the program. He writes:
While at IMA I made some progress on a longstanding joint effort
with an economist from Univ. of Alberta who vis ited Minneapolis
in order to spend some time with me. Of possibly greater interest
is the series of papers in highdimensional computational geometry
that Peter Gritzmann and I started while at IMA in 1987. Just
before coming to IMA, we had spent a year together in Seattle,
but our research projects started in Seattle were put on ``hold''
(and are still in that situation) when, at IMA we got some seemingly
good ideas for some important problems in computational geometry.
We have so far published five jo int papers (some with additional
coauthors, one of them being Laurent Habsieger (another IMAer))
that arose directly from our collaboration during the three
months at IMA. Two more such papers have been accepted for publication,
including the one towar d which several of the others were leading.
It will be 51 printed pages long, in Math. Programming early
in 1993. We have several other joint projects under way, and
I'll be spending the last three months of 1992 in Trier with
Peter in order to pursue th ese.
As
both Peter and I have often remarked, concerning that Fall of
1987 at IMA, ``That was a wonderful time!'' Impressively, our
wives feel the same way.
BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS DURING CONTROL
THEORY PROGRAM 199293
The
academic year program was divided into three parts (corresponding
to the fall, winter and spring quarters), although there was
considerable fluidity between the various parts.
1.
Linear and Distributed Parameter Systems
2.
Nonlinear Systems and Optimal Control
3.
Stochastic and Adaptive Systems
Hector
J. Sussmann prepared the following assessment of
the program: I spent the 199293 academic year at the I.M.A.,
participating in the organization and the activities of the
Cont rol Theory program. My own research benefitted tremendously
from the possibility of interacting with various longterm and
shortterm visitors. Naturally, when I look at the research
that I did at the I.M.A. and the one I am doing now, it is very
hard fo r me to draw a clear line between the work that I probably
would have done anyhow because it was a direct continuation
of my previous work, and the work that owes its existence to
the I.M.A. year. There are, however, a few examples where the
decisive rol e of the I.M.A. year is particularly clear, so
I shall start by describing these.
The
workshop on Nonsmooth Analysis and Differential Geometric Methods
in Optimal Control, held in February 1993, brought together
a number of researchers representing both approaches. Until
this workshop, there had been little contact between these tw
o subcultures with Optimal Control Theory. During the workshop,
many of us were able to engage in extensive discussions with
people representing the "other side," and this has led to the
birth of new directions of research where both approaches are
combi ned. I myself am now actively pursuing one of these directions.
Specifically, the visit of Prof. Martino Bardi, from Italy,
who gave a couple of lectures on viscosity solutions of firstorder
partial differential equations and the viscosity solution appr
oach to the problem of the characterization of the Value Function
in deterministic optimal control, made me renew my interest
in this issue, on which I had worked about four years earlier,
and about which I had taught a course at Rutgers. Prof. Bardi's
l ectures included a list of several important open problems
in the theory, such as the question of the characterization
of the value function as the unique positive viscosity solution
of the Bellman equation for linear quadratic optimal control.
It turned out that the techniques I had developed in my Rutgers
course made it possible to solve this and other problems. So
after extensive discussions with Bardi, I contacted a Rutgers
student who had the notes of the course and had them typed at
the I.M.A., an d I am now working on a number of papers, some
on my own, and some with Bardi, based on these notes. This could
only have happened in a setting such as that of the I.M.A.,
where one had plenty of time for discussion after the regular
lectures. In the spe cific case of my conversations with Bardi,
it took us several days until the precise correspondence between
my techniques and his formulation of the problems and techniques
became clear.
Another
important example of work that I am currently doing that owes
its existence to the I.M.A. year is a paper I am writing on
a new version of the Pontryagin Maximum Principle under weaker
hypotheses than all previous versions (including the Nonsmooth
Analysis version of "the Maximum Principle under Minimal Hypotheses"
due to F. Clarke) and with stronger conclusions (including highorder
necessary conditions for optimality ). I had been interested
in the Maximum Principle for a long time, a nd had used various
versions of it in my own work. However, most of my interest
arose from specific geometric control problems, such as local
controllability or the optimal control problems there was never
any difficulty arising from lack of smoothness o f the data.
I had thought quite a lot about nonsmooth versions of the Maximum
Principle, but I had never been sufficiently motivated to pursue
this activity. The I.M.A. year, and in particular the February
workshop, provided the motivation. Several exper ts on the Nonsmooth
approach in particular R.T. Rockafellar and B. Mordukhovitch
discussed versions of the Nonsmooth Maximum Principle, and this
made me aware of two things: (a) that there was a great lack
of awareness among the differentialgeomet rically oriented
practitioners of control theory, such as myself, of the importance
of extending our results to Nonsmooth settings, and (b) that
among the nonsmooth analysts there existed a misperception that
geometric methods were of more restricted app licability because
they required more smoothness. Based on my own thoughts on the
subject, I became convinced that it had to be possible to extend
the very best results of geometric optimal control which
included things such as highorder conditions t hat could not
be handled by Nonsmooth methods to general situations where
smoothness assumptions were not made. This eventually became
a true theorem, whose proof I found in August of 1993. I am
now completing a paper where a very general version of t he
Maximum Principle under minimal conditions is proved. Although
the actual writing of this work is taking place after the end
of the I.M.A. control year, the work owes its existence to the
I.M.A. year, in particular to the discussion with visitors that
took place during the February workshop.
The
above discussion of the effect of the I.M.A. control year on
my own work provides just one illustration of how much was achieved
during the year. Other visitors will tell about their own experiences,
and I probably should not speak for them, but from my own conversations
with several of these visitors I know that in many cases significant
new results were obtained and new directions of research were
born. For example, Jan Willems, form Groningen, in the Netherlands,
spent three months at th e I.M.A., where he met Karen Rudie,
who was there for the whole year as a postdoctoral fellow. Although
their areas of interest were in principle quite different, it
turned out that the set of concepts that Rudie had been developing
in order to formulate a general definition of "discrete event
system" was closely related to the ideas of Willems on dynamical
systems, and this led to a collaboration in which
a new version of Willems' general theory, incorporating discre
te events, was developed.
Besides
its direct effect on the research of participants such as myself,
the control theory year contributed in an important way to our
professional activity by helping us widen our knowledge and
become acquainted with the c urrent state of the arts in fields
of control theory other than our own special research area.
This is particularly significant for an area such as control
theory, which is mathematically very diverse, so that usually
a large investment of time and effor t is required even for
an experienced member of the control community to learn about
development in neighboring fields. Among the many events that
enabled us to acquire a new perspective of other fields within
control theory, I would particularly s ingle out the workshop
on Fuzzy Logic and its Applications. As is well known, the evaluation
of the applications of Fuzzy Logic a large number of which
are currently being carried out in Japan is a hotly contested
subject, on which opinions have be en expressed ranging all
the way from extremely enthusiastic to highly critical. The
format chosen for the Fuzzy Logic workshop, in which scientists
directly involved in specific applications presented their work,
and extensive discussions followed, made it possible for most
of us to become much better informed about the existing applications
and about the controversies surrounding them.
Summarizing,
I personally regard the I.M.A. control year as having had a
very significant impact on my own research, and I think that
its effect on the whole field of control theory has been felt
by most specialists in the area. The 199293 I.M.A. yea r will
be remembered for a long time as a major event in our field.
Scott
Hansen
was a regular IMA postdoc. He reports: Through interaction with
the other postdocs, senior researchers and workshop participants
the IMA has provided a rare opportunity to lea rn some of the
central research issues in a broad spectrum of control theory
applications. This broad exposure is particularly important
for several reasons. First, due to the highly interdisciplinary
nature of control theory it important to develop and maintain
lines of communication between the various disciplines involved.
In addition this breadth of exposure will be highly beneficial
in designing control theory courses and choosing research problems
for graduate students. This type of broad exposure to control
theory would have been completely impossible to obtain within
one university or through attending conferences.
More
directly, my own research conducted over the past year has greatly
benefited from interaction with other researchers in the area
of distributed parameter control. Mainly, I have been involved
in three projects: 1) control of thermoelastic s ystems, 2)
control of systems involving point masses and 3) modeling/control
of plates. I'll briefly describe each and how the program here
at the IMA has benefited the research.
Control
of thermoelastic systems: When I arrived at the IMA I was
writing a paper in which it is shown e.g., that by only controlling
the displacement at an end of a thermoelastic rod it is possible
to exactly control to zero both the tem perature and displacement.
Through discussions with Enrique Zuazua, John Lagnese and Vilmos
Komornik (who visited the IMA in the fall) I received several
valuable suggestions for improvement as well as suggestions
for further research in this area. Thus due to this interaction,
work on extending this result is ongoing in two different directions.
First, working with BingYu Zhang (IMA industrial postdoc),
we have been able to show that the same result holds for thermoelastic
beams. I expect to have a jo int paper on this sometime in the
next year. Secondly, Professor Zuazua has recently made some
progress extending the result to several dimensions. Still however,
many important questions remain unanswered and I expect to collaborate
with Prof. Zuazua on related problems in the future.
Exact
controllability of systems involving point masses: Working
with Prof. Zuazua, we considered the problem of boundary control
of a string (the one dimensional wave equation) having an interior
point mass. We were able to prove that a wave travelling along
the string is smoothed out one order as it crosses the point
mass. We then showed this result is sharp and consequently obtained
an exact controllability result. I was fortunate enough to be
invited to Universidad de Complutense in Madrid by Prof. Zuazua
for the month of July to work on extensions of this result.
Some progress was made showing that a one dimensional mass distribution
in a twodimensional membrane exhibits similar smoothing properties.
However this problem is much h arder than in the one dimensional
case and much work remains in connection with this problem.
Modelling/control
of plates: I was interested in modelling dissipation within
a plate or beam due to internal friction. Very roughly, the
idea is to ``glue'' two plates together (as a laminated plated)
in such a way that some amount of sl ip is possible at the interface,
where dissipation occurs through viscous friction (proportional
to the slip velocity). I was very fortunate to be able to discuss
this problem with Prof. Lagnese who is one on the leading experts
on plate modelling. I rec eived some very helpful suggestions
on the choice of notation and formulation of the problem. Later,
in proving existence and uniqueness results, I received some
highly appreciated help from Jiongmin Yong who has been with
the IMA all year.
Karen
Rudie
was a regular IMA postdoc. She reports: Probably the single
most important connection I made while at the IMA, resulting
in productive collaboration, was with Prof.\ Jan Willems. Our
first project was to determine the computational complexity
of a decentralized discreteevent systems problem. We found
that earlier work on centralized control by Tsitsiklis
could be generalized to the decentralized case, proving tha
t a particular decentralized supervisory control problem can
be solved in polynomial time. These results appear in the
Proc. of the 1993 European Control Conference, were presented
by me at the IMA workshop on discreteevent systems, and have
been submitted to a journal. Our second project focused on exploring
whether Jan's behavioral model for systems could be used to
describe discreteevent processes. This project has become more
of an ongoing discussion. Our collaboration led to Jan inviting
m e to spend a month working with him in Groningen (with travel
costs being supplied by the Systems and Control Theory Network
in the Netherlands).
During
my time there, we concentrated on trying to find an appropriate
newmodel for discreteevent systems problems. My area of research
is only about a decade old and there is as yet no standard model;
therefore, the modeling question is key in our f ield. My time
in Holland did not result in a paper but was extremely useful
in giving me guidance about how to direct my future research.
In this respect, I believe that working with Jan (who is experienced
and senior in the field) will prove invaluable for my career.
The
other person with whom I have worked at the IMA has been Dr.
Nahum Shimkin, a fellow postdoc. We have been focusing on modeling
a small industrial problem that had been mentioned to Nahum
by a friend who works on automating factories. This control
problem requires automating a system (using sensors, actuators
and a conveyor belt) that feeds cassettes into machines that
wind tape onto the cassettes. The problem specifications require
that certain timing constraints be met; consequently, we devoted
some time to reading literature on timed discreteevent systems
and realtime systems. We decided to formalize our industrial
problem using the model of Brandin and Wonham. Unfortunately,
like all other models of timed discreteevent syst ems,
this model suffers from computational statespace explosion.
We are currently trying to model the problem using a crude approximation,
in an effort to see whether we can get a reasonable solution
that is computationally tractable. Our efforts on thi s front
are ongoing. ......
While
at the IMA, I received many invitations to speak at conferences
and universities, some a very direct result of contacts I made
here. In particular, I was invited to give four talks while
I was in the Netherlands; was invited to give talks at Cal tech
(invited by Prof. John Doyle, who had been a longterm visitor
to the IMA in the fall) and UC, Santa Barbara (invited by Prof.
Roy Smith, who had visited the IMA for a workshop in the fall);
was invited (by Prof. P.R.~Kumar, a longterm visitor at t he
IMA) to give talks at the IMA tutorial and workshop on discreteevent
systems; ........
My
year at the IMA has been extremely productive in terms of making
professional contacts and exposing my work to others in the
field. I have, for instance, received an invitation to give
a talk at the University of Quebec at Montreal by Prof. O mar
Cherkaoui, whom I met at the IMA workshop on discreteevent
systems.
Bo
Bernhardsson
was a regular IMA postdoc. He reports: I followed the IMA program
in control from November 92 to June 93 as a post doc. This
time was very rewarding for me. The opportuni ties for interaction
with other researchers was great and the many workshops and
tutorials have given me a good chance to broaden my interests.
I followed most of the workshops and tutorials and also listened
to the industrial postdocs presentations.
My
own work during the year was concentrated on problems in the
areas of linear and robust control. During the spring time I
worked closely together with the two persons I shared room with,
Anders Rantzer and Li Qiu, both postdocs at IMA. This c ollaboration
resulted in a solution to the so called real
stability radius problem. This collaboration was
a direct consequence of our common stay at IMA.
The
problem is, after some work, given by the following linear algebra
problem: Given a complex matrix M, what is the smallest (induced
2) norm of a real matrix A such that rank (I_{n x n}  A M)
= nk . The crux is the condition of realness. I f this is relaxed
and complex A are allowed, the answer is given by the inverse
of the kth singular values of M. Our work, which was a continuation
of Li's work together with his supervisor Ted Davison, showed
that the mixed approximation problem above i s closely related
to some singular value like numbers. We worked out a formula
for the computation of these numbers and showed how this results
in the solution of the real stability radius problem. The solution
hints on interesting connections with other areas in linear
algebra, analysis and operator theory. The work was presented
at the IFAC conference in Sydney 1993, and has also been submitted
for publication. There are several ideas for generalizations
and further studies.
It
turned out that Minnesota was a good place also to meet swedish
researchers. This year the conference in acoustics and signal
processing was held in Minneapolis. During this week I meat
half a dozen of the swedish professors in control and signal
processing. It is probably hard to find such a concentration
of good swedish researchers even in Sweden at any given time.
During
the time at IMA I also prepared some work from my PhD thesis
for publication. By presenting the material at IMA I had the
chance to discuss and polish the material. I here had good help
with comments from other visitors. I especially want to mention
A. Rantzer, L. Qiu, P. Kumar and J. Doyle.
BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS DURING EMERGING
APPLICATIONS OF PROBABILITY PROGRAM 199394
The
academic year program was divided into three parts (corresponding
to the fall, winter and spring quarters), although there was
considerable fluidity between the various parts.
1.
Probability and Computer Science
2.
Genetics and Stochastics Network
3.
Stochastic Models
J.
Michael Steele chaired the Organizing Committee and
was in residence at the IMA for nine months during 199394.
Here is his Retrospective on the Special Year in Applied
Probabi lity: The bottom line is that one could not have
imagined the year to have been more successful. Some of the
results first promulgated during the special year seem destined
to become the focus of many years of future research and admiration.
The two such results that I have in mind are Talagrand's
Isoperimetric Inequality
and
Yuval Perez' new Capacity Theorem (and its applications to points
of multiplicity).< /P>
From
my perspective, these results are extraordinary, but the success
of the year should be measured more broadly. All of the workshops
really did ``work". I had a special appreciation of the Fall
workshops, because of having had a good hand in their organization,
and again because of this I found them quite inspiring. Not
too long ago there was no serious probabilistic theory of trees,
but now the subjectled substantially by Aldousis undeniably
rich. Similarly with the new theory of finite Markov chains,
where one now systematically exploits connections to differential
geometry and PDE. Even two years ago, almost no one could have
imagined these connections or their effectiveness on concrete,
discrete problems arising in c omputer science.
The
workshops of the Spring term are farther from my expertise,
but I have no trouble seeing that the workshop of Peter Donnelly
and Simon Tavare was an exceptional success. The witches' brew
that made this workshop so visibly effective was the blend of
theoretical geneticist and of individuals with real field experience.
The same combination was also afoot in the workshop on ``Hidden
Markov Models and their Speech Cousins" that was organized by
Steve Levinson and Larry Shepp. There were more c ore participants
of the mathematical genetics workshop who were in residence
at the IMA for a good stretch, and this, I think, added a lot
to the overall effectiveness of the genetics workshop.
Joel
Spencer
was an organizer for the September, 1993 workshop, and was active
at the IMA for three months. He writes:
Talagrand's Inequality
This
is really a nice story that shows what a positive effect IMA
can have. Let $\Omega=\prod\Omega_i$ be a product probability
space, $A\subset\Omega$. Talagrand defines, for $t>0$, a
``fattening'' $A_t$ containing $A$. To get a rough idea when
$\Omega=\{0,1\}^n$ is the Hamming cube then $y\in A_t$ implies
$y$ is within Hamming distance $t\sqrt{n}$ of $A$. Talagrand
proves $\Pr[A]\Pr[\ol A_t]<e^{t^2/4}$. Roughly, if $A$ is
moderate and $t$ is large then $A_t$ has most all the space.
Mike
Steele showed this to a whole group of us. He showed (as Talagrand
knew) how to use this to give sharp concentration results for
certain random variables. We (Eli Shamir, Svante Janson, Dominic
Welsh, myself, and others) started working on it and we (esp.
Svante) came up with a general application. Let $h:\Omega \rightarrow
R$ be a random variable. Suppose $h(x)$ is not too strongly
affected by changing one coordinate of $x=(x_1,\ldots,x_n)$.
Suppose further that if $h(x)\geq a$ then ther e is some ``small''
set of the coordinates $x_i$ that ``certify'' that $h(x)\geq
a$. Then one gets a strong concentration of $h$ around its mean.
Talagrand himself then came to a workshop and we had further
discussions. We've looked at concentration for a number of classical
problems. Consider the job assignment problem where the $i$th
person in the $j$th job gains a random $a_{ij}$ and $X$ is
the gain with the optimal assignment. Then $X$ is strongly concentrated.
Consider Minimal Spanning Tree with distances $\rho(x,y)$ being
random and $Y$ the size of the MST. Then $Y$ is strongly concentrated.
(Janson is working on much more precise results in this case.)
I have an IMA Jan '94 preprint that uses these ideas to give
an alternate (and I think clean er) proof of the Erd\H{o}sHanani
conjecture (first shown by R\"odl) on asymptotic packings. I
received a preprint by Alan Frieze and Bruce Reed on clique
coverings and using these ideas could greatly simplify the proof.
It seems clear to me (nothing wri tten yet) that studies of
extension statements for random graphs (e.g., every vertex lies
in a triangle) will be greatly aided by this idea. Overall,
it is a very exciting new concept.
What
role did IMA play? The original proof was due to one man (Talagrand)
thinking by himself. But the understanding of what the result
meant and the dissemination to the general community (which
is now fully underway) might not have taken place without the
fortuitous grouping of us at IMA and the time available there
to explore the potentials.
A
problem of Erdos
When
Uncle Paul comes to town there are always good problems. Paul
asked about the largest induced bipartite graph in a trianglefree
graph with $n$ vertices and $e$ edges. With Janson and Luczak
we were able pretty much to solve this and it will a ppear in
the Proceedings of the Random Structures workshop.
Large
Deviations
I
benefited tremendously by numerous discussions with Svante Janson
on large deviations. For example, let $X$ be the number of empty
bins when $m$ balls are thrown randomly into $n$ bins. We saw
how to estimate $E[e^{tX}]$ and how to use this to get g ood
bounds on the probability of $X$ being far from its mean. I
don't think a specific paper will come of this but it was a
case of ideas being clarified and distilled.
Ruth
J. Williams
was one of the organizers of the workshop on Stochastic Networks
at the beginning of March, 1994. Here is her report:
INTERACTIONS
WITH IMA VISITORS: During her three month stay at the IMA,
Professor Williams interacted with visitors in the three areas
of diffusion approximations to multiclass queueing networks,
passage time moments for reflected diffusions, a nd probabilistic
methods for solving nonlinear elliptic equations. These interactions
are described in more detail below.
 Diffusion
approximations to multiclass queueing networks. One of
the most intriguing open problems for multiclass queueing
networks is to determine conditions for the stability of these
networks and for approximating them in heavy traffic by r
eflected diffusion processes. A related problem is that of
determining conditions for positive recurrence of the reflected
diffusion processes. Paul Dupuis and Ruth Williams have shown
that stability of certain dynamical systems implies positive
recurren ce for these reflected diffusion processes. Motivated
by this result, Jim Dai has shown that stability of certain
fluid models implies positive recurrence of related queueing
networks. At the IMA, Professor Williams had discussions with
Jim Dai and Gideo n Weiss concerning their recent work on
using piecewise linear Lyapunov functions to determine sufficient
conditions for stability of fluid models. She also discussed
the use of Lyapunov functions to determine conditions for
positive recurrence of reflec ted random walks with Michael
Menshikov and Vadim Malyshev. Professor Williams is now investigating
whether piecewise linear Lyapunov functions might be used
to show stability of the dynamical systems of Dupuis and Williams
mentioned above. This is part of a larger project to determine
concrete conditions for stability of reflected diffusion processes
that arise as approximations to multiclass queueing networks.
 Passage
time moments for reflected diffusion processes. At the
IMA Professor Williams learned of the work of Michael Menshikov
and his coauthors on conditions for finiteness of passage
time moments of onedimensional discrete time semi martingales.
These authors applied their results, together with a martingale
functional of Varadhan and Williams, to determine conditions
for the finiteness of passage time moments of reflected random
walks in a quadrant. It is natural to try to extend t hese
semimartingale results to continuous time and to apply them
to reflected Brownian motions in a quadrant. Professor Williams
is collaborating with Michael Menshikov on such a project.
To date some progress has been made, but some difficulties
still r emain in generalizing the discrete time results to
continuous time. Assuming this can be done, it is intended
to seek other applications of the semimartingale results,
for example to other diffusion processes besides reflected
Brownian motions in a quadr ant.
 Probabilistic
methods for solving nonlinear elliptic partial differential
equations. Professor Williams is pursuing a mixed probabilistic
and analytic approach to solving nonlinear elliptic equations
with various boundary conditions in nonsmooth domains. Some
results that have already been obtained with Z. Q. Chen and
Z. Zhao for semilinear elliptic equations were presented at
the IMA during the period of concentration on probabilistic
methods for solving nonlinear partial differenti al equations.
Chen, Williams and Zhao are currently trying to extend these
results to include equations with a nonlinear first order
derivative term. One of the key ingredients needed for this
is a Green function estimate for a linear operator with possi
bly singular first order and zero order coefficients, in a
Lipschitz domain. Discussions, initiated at the IMA,are continuing
with Professor Eugene Fabes on how to derive such an estimate.
INTERACTIONS
OF OTHER VISITORS: Jim Dai and Bruce Hajek were good catalysts
for discussions on stochastic networks during the three month
Winter period. Informal discussions played at least as important
a role as formal talks. To give but one example, during a dinner
conversation, Bruce Hajek mentioned an open problem for ring
networks and the next day Jim Dai presented a solution based
on a technique he was developing for determining sufficient
conditions for stability of multiclass ne tworks. The four weeks
surrounding the stochastic networks workshop were especially
lively with the presence of a solid core of experts. Michael
Harrison and Frank Kelly were especially instrumental in creating
a friendly atmosphere in which they shared their expertise with
other participants. The area of stochastic networks is in rapid
development. Although I do not know that major open problems
were resolved during the IMA period devoted to this subject,
I do know that there was progress on understand ing their complexity
and subtlety and I expect that we will see some major breakthroughs
in this area in the next few years, fueled by the interactions
and connections that were made at the IMA.
STOCHASTIC
NETWORKS WORKSHOP: The Stochastic Networks workshop held
February 28March 4, 1994, and organized by Frank Kelly and
Ruth Williams, was the core for the period of activity on this
topic. Stochastic networks are currently used i n studying telecommunications,
computer networks, and manufacturing systems. The workshop featured
presentations on a variety of open problems for stochastic networks.
A notable feature of the workshop was the way in which experts
in different areas such as operations research, systems science,
and mathematics, converged to discuss problems of common interest
from different viewpoints. It is planned that the workshop proceedings
volume will convey the current state of knowledge in this rapidly
developin g area.
BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS DURING WAVES AND
SCATTERING PROGRAM 199495
The
academic year program was divided into three parts (corresponding
to the fall, winter and spring quarters), although there was
considerable fluidity between the various parts.
1.
Theory and Computation for Wave Propagation
2.
Inverse Problems in Wave Propagation
3.
Singularities, Oscillations, Quasiclassical and Multiparticle
Problems
Margaret
Cheney
spent the entire 199495 year in residence at the IMA. She reports:
Spending my sabbatical at the IMA, for its special year on waves
and scattering, was very constructive f or me. I learned a great
deal by attending most of the talks at most of the tutorials
and workshops. I had many useful conversations with visitors
to the IMA. I worked on the following problems.
Finishing the LayerStripping Paper. I finished up my
paper with David Isaacson on a layerstripping approach to reconstructing
a perturbed dissipative halfspace. This paper appeared in the
IMA preprint series, and was published in Inverse Pro blems.
While at the IMA, I gave a number of talks about this work.
Consulting
for Endocardial Solutions. The
presence of both myself and Eric Voth at the IMA led me to a
formal consulting arrangement with Endocardial Solutions, a
small startup company.
This company is attempting to develop an electrical system for
treating cardiac arrhythmias without openheart surgery. They
do this by passing into the heart a probe covered with electrodes.
The success of the system depends on using the probe to s ense
the location of conduction abnormalities. This involves the
following three main mathematical problems:
 Given
voltage measurements on the probe due to current sources on
the probe, determine the location of the endocardium;
 Given
the location of the endocardium, map the naturally occuring
electrical potentials on the endocardium;
 Use
the probe to determine the location of an additional ablation
catheter.
These
problems are all illposed inverse problems reminiscent of the
electrical impedance imaging problem I work on at Rensselaer.
Modeling
an Anesthesia Breathing System.
I worked with Paul Bigeleisen, a staff anesthesiologist at St.
PaulRamsey Medical Center, in developing a mathematical model
of the breathing system used to ad minister anesthesia.
This problem is very much in the spirit of the IMA Industrial
Mathematics program, in that it involved: 1) a practical, realworld
problem, 2) close collaboration between physician and mathematician,
3) modelling work, and 4) experiments. The problem is described
in the following excerpt from the introduction to our paper.
A
critical time in the administration of anesthesia is the
period called induction when the patient has received a
drug dose sufficient to paralyze the muscles of respiration,
but before tracheal intubation and mechanical ventilation
has commenced. So that the anesthetist will have the most
possible time to accomplish intubation before the patient
becomes hypoxic, the patient should start with the greatest
possible oxygen resevoir. In particular, the nitrogen normally
in the lungs should be replaced with oxygen before induction.
This process, called denitrogenation, is accomplished a
few minutes before the anesthetic drugs are given by having
the patient breathe pure oxygen through a mask connected
to the breathing circuit. This circuit is also con nected
to the anesthesia vapor machine and is used to administer
anesthetic gases after induction and intubation.
There
has been considerable debate within the anesthesia community
as to the most efficient method to accomplish denitrogenation.
In 1981 Gold, using blood gas samples, showed that four
deep breaths from a standard circle system produced an oxygen
con centration in the blood equivalent to that accomplished
by breathing normally for five minutes from the same circle
system. Thereafter, it became standard practice to have
the patient take four deep breaths of pure oxygen prior
to the induction of anest hesia. More recent studies, using
pulse oxymetry and in line mass spectrometry, have suggested
that four deep breathes are not as efficient at denitrogenation
as normal tidal breathing for three minutes. The latter
authors suggest, without evidence, that greater rebreathing
of nitrogen from the circle system when large breaths are
taken, is the cause for inefficient denitrogenation. With
rebreathing, the patient is actually recycling part of his/her
own breath, including nitrogen, through the circle sys tem.
This effect slows down the elimination of nitrogen from
the lungs.
In
order to determine the most efficient method of denitrogenation,
we developed a model of the circle system and compared it
to experimental data using the authors as subjects.
Our
mathematical model consisted of a system of 8 coupled differentialdelay
equations. We implemented this model in Matlab. We have nearly
completed a paper based on this work.
Processing
of Radar Data. I discussed with a number of visitors my
ideas for using a geometrical optics expansion to recover the
surface electrical properties of a material from the first reflection
of a radar pulse. Greg Beylkin suggested that a certain matrix
in my scheme might be illconditioned, which led me to a method
for overcoming this difficulty. Ingrid Daubechies gave me some
advice on windowing techniques for signal processing.
Wave
Propagation in Random Media. While listening to talks on
wave propagation in random media, I started wondering if the
method of multiple scales could be used to study the propagation
of electromagnetic waves in sea ice. This is a dif ficult wave
propagation problem, because sea ice contains a multitude of
pockets of brine and air, which are strong scatterers of electromagnetic
waves. I discussed this problem with a variety of visitors,
who uniformly found the problem very interesting but did not
know how to solve it. I am continuing to think about this problem
back here at RPI, and am discussing it with Julian Cole.
Other
Connections with Industry. Doug Huntley, one of the industrial
postdocs, arranged for me to give a talk at 3M. This eventually
led to an arrangement between 3M and my impedance imaging research
group at Rensselaer, in which 3M suppl ies us with electrodes.
I discussed with Keith Kastella, a staff scientist at Unisys,
the possibilities for using electrical impedance imaging to
determine the extent of ground contamination at waste disposal
sites. Our discussions led to an improvement in the algorithm
he was developing. I met with representatives of Renaissance
Technologies to discuss similarities between the Rensselaer
impedance imaging device and their IQ system for assessing cardiac
function. The IQ system uses the bulk electrical resistance
of the torso, together with sophisticated signal processing
techniques, to recover information about cardiac function.
Vladimir
Malkin of the Courant Institute visited IMA from
September 1 to December 31 in 1994. He reports: During this
period I published at two preprints: "On selffocusing of short
laser pulses" (IMA #1272, with George Papanicolaou) and "Kolmogorov
and nonstationary spectra of optical turbulence" (IMA #1283).
Apart from this, I gave a cycle of lectures on model making
in nonlinear mathematical physics, PDE seminar on singularities
for no nlinear Shroedinger equations and lecture on turbulence
theory at IMA, 2 seminars and 2 colloquiums at Mathematical
Departments of CWRU and University of Michigan, Ann Arbor. I
also gave lectures on instabilities of selfsimilar collapse
and on regulariz ation of singularities of nonlinear wave fields
at Mathematical Departments of Chicago University and University
of Illinois, Chicago, and a colloquium at Mathematical Department
of IUPUI. As a result of the IMA Letter with brief description
of my lectur es on model making in nonlinear mathematical physics,
I was offered to write a book of Lecture Notes on Nonlinear
Mathematical Physics for the World Scientific Publishing.
I
had many useful informal contacts at IMA and in participating
Universities. In particular, I met in IMA Prof. Tom Spencer
from IAS (Princeton) and have considered with him some problems
of weak turbulence theory. He suggested me to work out th ese
problems during the next academic year at IAS, and I accepted
his offer. I also was pleased to consider with Prof. Howard
Levine from Iowa State University possible extensions of his
famous theorem on blowup for nonlinear wave equations to a
more ge neral and very important for applications class of mixed
nonlinear Schroedinger and wave equations. (The possibilityof
so far reaching extensions follows from my recent results published
in Preprint IMA #1272). I appreciate considerations of selffocusin
g of ultrashort powerful laser pulses with Prof. Jeffrey Rauch
from University of Michigan, Ann Arbor. It was useful for me
to get some impression about industrial mathematics from the
"first hands".
BRIEF SELECTION OF RESEARCH ACCOMPLISHMENTS
DURING MATERIALS SCIENCE PROGRAM 199596
The
academic year program was divided into three parts (corresponding
to the fall, winter and spring quarters), although there was
considerable fluidity between the various parts.
1.
Phase Transitions, Optimal Microstructures and Disordered Materials
2.
Thin Films, Particulate Flows and Nonlinear Optical Materials
3.
Numerical Methods and Topological/Geometric Properties in Polymers
Richard
D. James
chaired the organizing committee. He reports on the IMA year
as follows: The goal of year on Mathematical Methods and Materials
Science was to actively encourage mathemati cal research on
materials and particularly an exchange of information between
mathematicians and materials scientists. This certainly occurred,
with particular success in bringing mathematicians in contact
with serious materials scientists.
The
breadth of coverage of the program can be partly judged based
on the variety of materials discussed. The following materials
were treated seriously from the point of view of both behavior
and mathematical modeling: composites, foams, sea ice, mart
ensitic materials, microelectronic materials, nonlinear optical
materials, shapememory materials, soft magnetic films, giant
magnetoresistive materials, giant magnetostrictive materials,
magnetic sensors, biomaterials, selfassembled composites, porous
rocks, disordered conductors, colloidal suspensions, rubber,
gels, granular materials, magnetic particles in bacteria, polymers
with various degrees of cross linking, optically active materials,
liquid crystals (nematic and smectic), piezocomposites, f erroelectrics,
magnetic fluids, materials that undergo diffusional phase transformations,
silicon, paper, high strength structural alloys, diamond films,
brittle ceramics, functionally graded materials, electrorheological
fluids and optical fibers! In m any cases there were several
talks on the class of material, and analysis of the appropriate
mathematical models.
Research
on mathematical problems in materials science is an area of
rapid growth in applied mathematics. Some of the workshops were
devoted to summarizing the evolving state of research. The workshop
on "Phase Transformations, Composite Materia ls and Microstructure"
was of this type. It became clear at that workshop that a great
many diverse approaches to the homogenization of composites
could be related, and streamlined, under the framework of weak
convergence methods, primarily compensated c ompactness. There
were also new methods revealed which do not fit this framework,
based on expansions (Bruno) and careful study of the differential
equations (Nesi). Similarly, the workshop on "Disordered Materials
and Percolation" summarized improvement s on the calculation
of critical exponents near percolation. But new directions emerged
on two fronts. First, there were a host of new disordered materials
 from sea ice to fractured ceramics  that infused this
field with new problems. Second, the mathematical implications
of conformal invariance of the equations were revealed. The
workshop on "Particulate flows: Processing and Rheology" offered
an uptodate summary of problems of the flow of fluids containing
particles. These problems are impor tant in industry and also
embody exceedingly challenging computational issues. These arise
from the sheer complexity of flows with many particles, and
from the fundamental difficulty  still far from resolved
 of computing nonNewtonian flows at hig h Deborah number.
In
addition to those workshops that summarized and energized an
existing area, new fields of applied mathematics were begun
during the year, some unanticipated to the organizers. These
are summarized below.
1.
The serious study by mathematicians of the schemes currently
used by physicists to
compute atomic forces and atomic configurations.
Physicists at the workshop, "The Mechanical Pro perties of Materials
from Angstroms to Meters" described their large scale computations
of atomic configurations at various levels of sophistication,
and indicated the hopelessness of computing the structure of
even the finest microstructural feature of interest in materials
science. Mathematicians summarized the success of weak convergence
methods and changeofscale in other areas. For the first time
it seems that there were suggested alternatives to the use of
conventional computing methods on larger and larger systems.
These are under development. As a byproduct of this discussion,
some mathematicians began a serious mathematical study of density
functional theory. A book by G. Friesecke based on lectures
(organized on the fly) and published throug h the IMA will summarize
these developments.
2.
New theories for the
growth and behavior of thin films.
During the program NSF/DARPA requested that the IMA organize
a workshop on the growth of thin films, with the goal of bring
ing new mathematical methods to bear on the growth of films
of interest in microelectronics. This preceded another workshop,
"Interfaces and Thin Films", that had been organized earlier
as part of the year long program. The NSF/DARPA workshop sparked
a f lurry of interest in the community, and this was translated
into serious mathematical modeling in the workshop on Interfaces
and Thin Films. This seems to be a good area for mathematics,
involving questions at the boundary between continuum and atomic
th eory. It appears that mathematical modeling can make a real
impact on the practice of thin film research in materials science,
and a variety of mathematicians are now involved.
3.
New approaches to the statistical mechanics
of polymers, via topological methods. The central
idea here is to account for entanglements of long chain polymer
molecules in statisti cal mechanical calculations, which were
previously not included by DeGennes' theory of reptation or
older treatments of rubber elasticity. This has led to discrepancies
with experiment on polymers of modern interest. Ideas in this
direction had previousl y been initiated by Whittington and
Sumners and their collaborators. The workshop exposed a much
larger variety of people to these ideas, and generated substantial
interest.
4.
The first interaction between people doing
largescale computations of turbulence and those doing largescale
computations of microstructure. Two groups of people,
similar difficulties with small scales, and severe limitations
of complexity which rules out many problems of practical interest.
The workshop on "The effect of Small Scales in Computations
of Microstructure and Turbulence" brought these two groups together.
Combined analyti cal/numerical methods involving weak convergence
methods to incorporate the effects of the small scales emerged
as a theme.
5.
Nonlinear optical materials.
While there is a relatively long history of mathematical research
on nonlinear optical phenomena, involving deep connections with
nonlinear dynamics, and al so on diffraction phenomena, the
organizers could not find any mathematician who was doing research
on the nonlinear optical material itself. Nevertheless, the
workshop attracted a variety of applied mathematicians knowledgeable
on optical phenomena, and research on optical materials was
begun.
On
a personal note, theories of microstructure for martensitic
materials have evolved from models that accurately
define the energy wells of the material. During the year I pondered
the form of the energy wells for a hypothetical material that
would combine two types of phenomena: a ferromagnetic transition
and a martensitic transformation. Thinking about the properties
of this energy well structure led to a strategy for seeking
thi s class of materials. The strategy has been put into practice
(Joint work with Manfred Wuttig, a materials scientist and IMA
visitor), and this class of materials has been realized (based
on Fe_3Pd). They currently exhibit the largest known magnetostrict
ive effect.
Stefan
Mueller (now a director of the MAXPLANCK INSTITUT
F. MATH IN THE NATURAL SCIENCES IN DRESDEN) was a three month
visitor at the IMA. He writes: One of the fundamental problems
in complex materials is to understand the interaction of different
scales (atomic, mesoscopic, macroscopic). Recent progress in
the understanding of fine microstructures in advanced materials
through continuum models raises important new questions about
the foundations of the continuum theory. I used my stay at the
IMA to learn about the hierachy of microscopic models (Nbody
quantum mechanics, density functional theory, embedded atom
method, pair potentials) and to continue joint efforts with
G. Alber ti, A. De Simone, G. Dolzmann, R.D. James, R. V. Kohn,
S. Spector and V. Sverak to develop mathematical tools for the
analysis of microstructures and the passage between different
scales. I also greatly benefited from discussion with many other
scientist s including K. Bhattacharya, D. Dahlberg, D. G. Pettifor,
R. Phillips, D. Schryvers, A. Sutton, L. Truskinovsky and J.
G. Zhu.
My
research mainly focused on elastic and on magnetic materials.
In collaboration with Scott Spector I obtained new results on
cavitation (spontaneous formation of voids) in elastomers. A
joint manuscript with J. Sivaloganathan is finished and w ill
be submitted for publication shortly. Jointly with R. D. James
I studied the relation between lattice and continuum models
of elasticity. We had previously obtained results relating lattice
and continuum models for magnetostatic interaction but the g
eneralization to elastic materials presents new difficulties.
With A. De Simone, G. Dolzmann, R. D. James and R. V. Kohn I
discussed the outline of a program for a rigorous derivation
of lowerdimensional theories for magnetic materials such as
thin film s. This program is at its beginnings. and only the
future will tell how successful it will be. An interaction of
formal and rigorous asymptotic analysis, analysis of experimental
data and numerical computation in this area seems, however,
very promising to me. The atmosphere at the IMA which encourages
exchange and discussion, as well as the opportunity to meet
experimentalists such as D. Dahlberg and researchers involved
in computation such as J. G. Zhu was an important boost for
our efforts.
In
joint work with G. Alberti I made progress in finding a suitable
mathematical description for microstructures that involve more
than one small scale. A paper which summarizes our results is
now almost finished. Applications are so far to some what academic
problems but I expect that similar tools will be important for
the understanding of crystal microstructure and the multiple
scales that arise in micromagnetics.
I
also finished a joint paper with G. Dolzmann and N. Hungerbuhler
on degenerate elliptic equations which has been accepted in
Math. Zeitschrift.
I
was also involved in the organization of the workshop `Mechanical
response of materials from Angstroms to meters' at the beginning
of the special year. Following after a week of excellent tutorials
by R. V. Kohn, A. Sutton and V. Sverak, it be came a very good
forum for exchange of ideas between people from very diverse
fields such as {\it ab initio} calculations, electronic structure
theory, continuum mechanics, microstructure and variational
methods. Many researchers met at this workshop for the first
time, and the groundwork was laid for a very fruitful interaction.
Of the many interesting contributions I would just like to mention
the approach by R. Phillips (and coworkers) which combines
the models of continuum mechanics and atomic theo ry in a single
numerical code to simulate dislocation nucleation.
As
another result of the `Angstroms to meters' workshop and the
surrounding workshops and tutorials G. Friesecke started to
work on a mathematically rigorous exposition of density functional
theory (DFT). This is the most common method for large scale
ab initio simulations. It is very successful, but a deeper
mathematical understanding of its success (and its occasional
failures) and thus a way of systematic improvement is almost
completely lacking. Friesecke taught a course during the fall
se mester at IMAand is now writing a book (to appear in the
IMA series) which will make DFT accessible to a large number
of researchers whose background is traditional in continuum
mechanics and partial differential equations. I think this is
one of many ex amples of new and fruitful interactions that
were initiated during the IMA year on `Mathematical methods
in materials science.'
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