The Second Israeli Mini-Workshop in Applied
and Computational Mathematics
David Kessler, Raz Kupferman, Vered Rom-Kedar, Jeremy Schiff, Edriss Titi
Following the success of
year's workshop at the Weizmann Institute,
we are pleased to invite the Israeli applied math community to participate
in the Second Israeli Mini-Workshop in Applied and Computational Mathematics,
to be held at Bar-Ilan University on Thursday December 23, 2004.
(For those who came last year and remember that the meeting was also on
December 23rd, please note this is purely coincidental, future meetings
may be held on different dates.)
The idea of the workshops is to create a forum for workers in applied
mathematics, especially younger faculty and students, to get to know other
members of the community, and promote discussion and collaboration.
This year's workshop will take place in the lecture hall on the first floor of
the Economics building (building 504) on Bar-Ilan's main campus. See
The Campus Map. For those arriving
by car, there are parking lots outside, but close to, the campus
(the big white things on the map next to building 1401). A large number of
bus routes serve the university.
The schedule of events is as follows:
Participation in the workshop is free, but you are asked
to register by
sending an email to Rivka Wolberger, so we can be adequately prepared
for the day. To get lunch you must register by
12:00 on Sunday December 19!
There will be an opportunity to display posters at the meeting. Poster boards are
1 meter wide and 2 meters tall. If you wish to reserve a board, please
send an email to Rivka Wolberger by 12:00 on December 19.
We thank the Department of Mathematics and the Faculty of Exact Sciences
at Bar-Ilan University for logistic and financial support
Visco-Elastic flow simulations and the High Weissenberg Number Problem
Raanan Fattal (Hebrew University)
For decades, the HWNP has been a major obstacle in the numerical
simulation of visco-elastic flows within meaningful conditions.
A large amount of work has been devoted in the development of
computational methods, yet all reported a limiting elasticity value
(the Weissenberg number) above which simulations break down.
In our work we present a simple analysis explaining the source of
this breakdown. Based on this observation, we propose a practical
solution that stabilizes most of the existing methods.
We also demonstrate our new approach by predicting the flow of an
Oldroyd B fluid at high Weissenberg numbers in a 4:1 planar abruplty
This is a joint work with Prof. Raz Kupferman
On a Lie-Poisson system and its Lie algebra
Arieh Iserles (Cambridge University)
In this talk, based on joint work with Tony Bloch, we consider
differential equations of the form X'=[N,X^2], where X(0) is a symmetric
matrix, while N is skew symmetric. Such flows can be considered as an
outcome of two distinct group actions: by similarity (hence they are
isospectral) and by congruence. We prove that they are endowed with a
Poisson structure. Hence, they correspond to a flow along an orbit of the
dual to the free Lie algebra generated by their structure constants. We
thus seek a faithful representation of the underlying Lie algebra and
attain it by using methods of matrix analysis and numerical linear algebra.
Numerical Methods and Nature
Eli Turkel (Department of Mathematics, Tel Aviv University)
In many numerical procedures one wishes to improve the initial
approach either to improve efficiency or else to improve accuracy.
Frequently this is based on an analysis of the properties of
the discrete system being solved. Using a linear algebra
approach one then improves the basic algorithm. We review
methods that instead use a continuous analysis and properties
of the differential equation rather than the algebraic system.
We shall see that frequently one wishes to develop methods
that destroy the physical significance of intermediate results.
We present cases where this procedure works and others where it
fails. Finally we present the opposite case where the physical
intuition can be used to develop improved algorithms.
Reproducibility of sequential dynamics in a model of neural circuits
Valentin Afraimovich (Universidad Autonoma de San Luis Potosi)
Reproducibility of sequential spatio-temporal responses is an essential
feature of many neural circuits in sensory and motor systems. Using the
framework of a generalized Lotka-Volterra model that describes the
dynamics of firing rates in an inhibitory network, we show that a stable
heteroclinic sequence is an appropriate mathematical image to describe
Abrikosov lattices in finite domains
Yaniv Almog (The Technion)
In 1957 Abrikosov published his work (for which he won a Nobel prize
in Physics a year ago) on periodic solutions to the linearized
Ginzburg-Landau equations, which serve as the basic model for
superconductivity. Abrikosov's analysis assumes periodic boundary conditions,
which are quite different than the natural boundary conditions the minimizer
of the Ginzburg-Landau energy functional should satisfy. In the present work
I demonstrate that the global minimizer of the fully non-linear functional is
close (for a certain range of applied magnetic field, and for a diminishing
small scale), in some sense to the minimizer of Abrikosov's classical
Graph theory applications in theoretical biology
Yoram Louzoun (Bar-Ilan University)
Graph theory tools are currently used in a wide domain of biological
applications. We here examplify the wide uses of graph theory in
biology, mainly in the context of birth death processes. The population
dynamics of most biological entities (creatures, cells, genes
organisms,...) can be described as a birth death process acompanied by
random changes. We define biological entities as nodes of a graph and
the resemblence between enitites as weighted links between those
entities. The statistical properties of the resulting weighted graph are
a function of its history. We use these properties to deduce the
mechanisms that lead to the creation of this graph. For simplified
processes, we can also quantitively estimate the different parameters of
the creation process (e.g. mutation and cell death rate).
A second application that we have studied is the translation of birth
death processes into a directed percolation problem. In this case we
show that a large varaiety of different birth death processes can be
translated into different directed percolation threholds.
Multiscale Asymptotic Analysis of Wave Propagating in Nonlinear Periodic Media
Ziad Musslimani (University of Central Florida)
New models describing wave propagation in transversely modulated
optically induced waveguide arrays are proposed. In the weakly guided
regime, a discrete nonlinear Schrodinger equation with the addition of
bulk diffraction term and an external ``optical trap'' is derived. In
the defocusing regime the optical trap induces a stable localized mode.
In the limit of strong transverse guidance, the dynamics is governed by
a model which represents the optical analogue of wave action.
Mathematical Challenges from Classical and Quantum Dynamics
David J. Tannor (Department of Chemical Physics,
Weizmann Institute of Science)
With the exception of a few instances where relativistic effects
are important, a single equation - the time-dependent Schrodinger
equation - describes all of atomic and molecular physics. The
problem is that formally, this wave equation has to be solved for
all the electrons and nuclei in the atom or molecule. For a small
molecule such as ozone (three oxygen atoms) there are already 24
electrons and three nuclei, for a total of 3 x 27 = 81 degrees of
freedom. For the smallest peptide molecule, glycine (a single
link in a DNA chain that typically extends for many thousands of
peptides) there are already 3 x 50 =150 degrees of freedom.
Solving a wave equation exactly in so many degrees of freedom
is out of the question. One of the main challenges for theoretical
chemistry is find accurate and efficient approximate solutions to
wave equations in so many degrees of freedom.
I belong to a group
of theoretical chemists who assume that the Schrodinger equation
for the electrons has already been solved. The remaining part of
the problem is the dynamics of the relative motion of the nuclei
in the presence of the electronic force field, so-called chemical
reaction dynamics. There are only 9 nuclear degrees of freedom
in ozone, and only 30 in glycine, but solving the wave equation
is still a formidable or impossible task.This talk will focus on
three aspects of the above problem. 1) Several classes of methods
used to solve the time dependent Schrodinger equation for the
nuclei exactly in up to 6 degrees of freedom will be described.
Recent work on pseudospectral methods that avoid an underlying
direct product basis will be presented (collaboration with Ilan
Degani and Jeremy Schiff). 2) A large literature devoted to
using classical mechanics reliably to approximate quantum
mechanics will be summarized. Recent ideas for using classical
mechanics to create pseudospectral representations that avoid
an underlying direct product basis will be described
(collaboration with Yair Goldfarb). I will also discuss
methods for using classical mechanics to propagate the time
dependent Schrodinger equation. 3) Finally, I will report
on ongoing efforts in my group to use optimal control theory
to design specially shaped laser pulses to control photochemical
reactions. Here, the wave character of the light and the
wave nature of the matter must be solved within a single coherent
framework. The mechanism for control is via constructive or
destructive interference, representing an interesting paradigm
for a control problem (collaboration with Shlomo Sklarz)