The Second Israeli Mini-Workshop in Applied and Computational Mathematics

Organizers: David Kessler, Raz Kupferman, Vered Rom-Kedar, Jeremy Schiff, Edriss Titi

Following the success of last 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:

9:30-10:00 registration and opening
10:00-10:30 Eli Turkel (Tel Aviv University) Numerical Methods and Nature
10:30-11:00 Valentin Afraimovich (Universidad Autonoma de San Luis Potosi) Reproducibility of sequential dynamics in a model of neural circuits
11:00-11:15 coffee break
11:15-11:45 Yoram Louzoun (Bar-Ilan University)Graph theory applications in theoretical biology
11:45-12:15 David Tannor (Weizmann Institute) Mathematical Challenges from Classical and Quantum Dynamics
12:15-14:00 lunch and discussions
14:00-14:30 Ziad Musslimani (University of Central Florida) Multiscale Asymptotic Analysis of Wave Propagating in Nonlinear Periodic Media
14:30-15:00 Raanan Fattal (Hebrew University) Visco-Elastic flow simulations and the High Weissenberg Number Problem(HWNP)
15:00-15:15 coffee break
15:15-15:45 Yaniv Almog (Technion) Abrikosov lattices in finite domains
15:45-16:30 Arieh Iserles (Cambridge University) On a Lie-Poisson system and its Lie algebra

Registration

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!

Posters

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 (HWNP)

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 contracting channel.

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 this phenomenon.


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 problem.


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)