Michael
Brenner (Department of Mathematics, Massachusetts Institute
of Technology) brenner@math.mit.edu
is one of the organizers of the IMA "Hot Topic" Workshop on Analysis
and Modeling of Industrial Jetting Processes held on January
1013, 2001. He provides the summary below.
The meeting was a success beyond my expectation. Not only were
the large majority of the talks of extremely high quality, but
they complemented themselves very well, and in ways we did not
anticipate when putting the program together. As an example,
at least four speakers dealt with the phenomenon of a spherical
drop of fluid splashing on a solid surface: David Wallace from
MicroFab, our first speaker, described how the details of this
question are crucial for depositing solder drops onto FlipChip
semiconductor devices; Our third speaker, Ely Sachs from MIT,
showed another beautiful example of this in the context of his
three dimensional printing processes: drops depositing on a
powder layer create craters, which limits the precision of their
engineering. Then we had two, more scientific speakers, who
touched on this subject: D. Poulikas from ETH showed his numerical
simulations which clearly established the phenomenon as due
to capillary waves. And then David Quere, from College de France,
showed his beautiful experiments of drops bouncing on hydrophobic
surfaces, which exposed the various regimes of bouncing drops.
One of David's drops lifted off the solid surface in a shape
close to a "baseball bat," which then oscillated through
a complex series of shapes. Rather remarkably, another of our
speakers, Osman Basaran from Purdue, had just presented a numerical
simulation of fluid started in a baseball bat shape, and found
a sequence of shapes which were very similar to those of Quere.
Basaran's "baseball bat shapes" was motivated by a
different experiment, the breaking of a pendant drop near a
nozzle, a problem discussed in detail by some of the other speakers.
This is just an example of the types of interactions that occurred
at the meeting. In a wide range of technologies, the same issues
kept coming up again and again, even though the background and
motivation of the speakers was very different. The combination
of mathematics, engineering and technology at the same meeting
allowed everyone to leave with a different perspective on the
problems they have been thinking about previously.
Vicent
Caselles (Dept. de Tecnologia, Universitat PompeuFabra)
vicent.caselles@tecn.upf.es
Summary of the work done at the IMA
During my stay at the IMA I wrote essentially two papers: one
on Qualitative Properties of the Total Variation Flow (joint
paper with F. Andreu, J.I Diaz and J.M. Mazon) pdf
(285KB) postscript
(314KB) and a paper for ICCV written jointly with C. Ballester,
J. Verdera, M. Bertalmio and G. Sapiro.
In the first of these papers we study the assymptotic behaviour
of the minimizing total variation flow, i.e., of the flow u_t
= div(Du/\vert Du\vert) under Dirichlet and Neumann boundary
conditions. Essentially, we prove that, if T denotes the extinction
time of the solution u(t,x) (corresponding to some initial condition
u_0) then v(t,x) = u(t,x)/(Tt) converges as t \to T to a function
V which is a solution of the eigenvalue problem  div(DV/\vert
DV\vert). Then we study the solutions of this eigenvalue problem
and give a complete description of them when the domain is a
ball and the solution is radial.
The second of these papers deals with the problem of inpainting
or desocclusion in images. We proposed a variational formulation
for inpainting which is based in an interpretation of the elastica.
Thus, our functional can be considered as a relaxed formulation
of the elastica and enables us to interpolate the level lines
in a missing part of the image. We display several experiments
showing the applications and limits of our model. This paper
has been submitted to ICCV and completes a previous paper of
the authors on this subject.
General description of my interests:
I am interested in image processing, both in its mathematical
and algorithmic aspects. Thus, in my work, those two aspects
are considered. Thus, in the last years, my interests focused
in
1) the study of the total variation model for restoration, proving
the well posedness and the geometric features of this flow.
As a consequence, I am working in on new models which try to
overcome its drawbacks.
2) The study of data structures for images which try to capture
the geometric shapes of the image. This is leading to some applications
which I am developping now.
3) The stuty of several interpolation problems in images: namely
operators which interpolate images when the dynamic range is
poor and variational models for singular interpolation (e.g.,
inpainting).
I will continue to develop these topics, at least, in the recent
future, insisting, in particular in the second topic. In all
of them, there is published work, but still there is much to
be done.
Marco Antonio Fontelos, Departamentos de Matematica
Aplicada Universidad Rey Juan Carlos fontelos@anacleto.escet.urjc.es
attended the IMA "Hot Topics" Workshop on Analysis
and Modeling of Industrial Jetting Processes held on January
10  13, 2001. Below is his report.
In
my last visit to the I.M.A. I have been working with two visitors
of the Institute and completed a personal project.
1) With Avner Friedman, I have been working in the modelling
of angiogenic processes. Our work originated the preprint "Mathematical
analysis of a model for the initiation of angiogenesis"
that has been submitted for publication in a mathematical journal.
The model consists of a system of 7 differential equations.
3 of them are nonlinear diffusion equations and describe the
densities of three types of cells and the other 4 equations
are just kinetic equations modelling the reactions between 4
different chemicals that take part in the process. We prove
that there exist global (in time) solutions to the system under
certain conditions.
2) With Santiago Betelú, a postdoctoral member of the
I.M.A., I have initiated work in a model for thin films of NonNewtonian
(powerlaw) fluids. Up to now, the work has originated a short
note intitled ''Capillarity driven spreading of powerlaw fluids''
that has been submitted. We hope to continue collaboration on
this or related topics since, as far as we know, this is a novel
approach to thin film equations.
3) I completed a personal project on the evolution of jets of
NonNewtonian (Oldroyd) fluids. The corresponding preprint is
entitled ''Breakup and no breakup in some models for the evolution
of NonNewtonian jets," submitted to a mathematical physics
journal.
Dirk
Horstmann, Mathematisches Institut der Universitaet
zu Koeln dhorst@MI.UniKoeln.DE
pdf
(16KB) postscript
(23KB)
W.
Christopher Lang,
Indiana University Southeast wclang@ius.edu
I was granted by Indiana University Southeast a onesemester
sabbatical leave for the Fall Semester of 2000. My goal for
this sabbatical was to become involved in research in interdisciplinary
or computational applied mathematics. My primary training and
experience as a mathematician is in pure mathematics (I am a
functional and abstract harmonic analyst), and it has long been
my desire to become involved in more applied work.
To
this end, I accepted an invitation for a longterm visitorship
to the Fall program on Mathematics in Multimedia scheduled for
the 2000/2001 academic year at the Institute for Mathematics
and its Applications at the University of Minnesota. I was at
the IMA for the months of September, October and November.
My
goal during my visit was to learn about the range of applications
represented by the workshops in the Fall program, particularly
applications of harmonic and wavelet analysis. I found the scientific
programs to be extremely interesting. As an outsider, I found
the talks to be generally clear and accessible, even those well
outside my original specialties. I also benefited from the generous
help of various workshop participants. I should thank in particular
Sanjeev Khudanpur, Alejandro Murua, JeanPhilippe Vert, Bradley
Love, Stu Geman, and Don Geman for questions answered and insights
shared.
I
now plan on focusing on one of the topics I learned about during
my visit, after some reflection to decide where to devote my
efforts. (Presently, I am most attracted to lowerlevel image
analysis and processing, but I'm planning on returning in May
to the IMA for the short course and workshop on computer graphics.)
I
believe that my visit to the IMA was well worth the effort and
expense; I learned an tremendous amount in a fast twoandahalf
months, and I am confident that I will be able to realize my
goal of doing productive research in an area I learned about
at the IMA. I would like to express my gratitude to the Director
of the IMA, Professor Willard Miller, and the staff of the IMA
(especially Fred Dulles, Judy Sweeny, Michelle Glubke and Steve
Bonneville), for making my visit to the IMA so valuable and
enjoyable.
20002001
Program: Mathematics in Multimedia
