The Eighth Israeli Mini-Workshop in Applied
and Computational Mathematics
Organizer: Jeremy Schiff.
We are pleased to invite the Israeli applied math community to participate
in the Eighth Israeli Mini-Workshop in Applied and Computational Mathematics,
to be held at Bar-Ilan University on Thursday December 27th, 2007.
These workshops have been held twice yearly for the last few years.
For information on the last meeting
click here.
The idea 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.
We thank the Department of Mathematics, the Faculty of Exact Sciences
and the Office of the Vice-President for Research
at Bar-Ilan University for logistic and financial support.
December 26th: Please note change in the workshop program!
Registration
Participation in the workshop is free, but you are asked
to register by
sending an email to Rivka Wolberger, mathoffice@macs.biu.ac.il,
so we can be adequately prepared
for the day. Please register by 12:00 on Monday December 24th.
Speakers and Schedule
Access and Parking
The workshop will take place in the Beck auditorium on Bar-Ilan's main campus.
The auditorium is accessed from the walkway between buildings 405 and 410.
(See the campus map.) Due to
the crowded conditions on campus we unfortunately cannot provide on-campus parking. Please use
the public parking lots outside (near building 1401 on the map, cost on the order of 101
sheqel). Bar-Ilan can be accessed by bus from Jerusalem (line 400), Rehovot (line 164) and Tel Aviv
(lines 45,64,68,70,87 and others). For access from Beer Sheva and Haifa we recommend taking the train
to the Tel Aviv central station and then a bus or a taxi.
Abstracts
Fast GL(n)-invariant framework for tensor regularization
Nir Sochen (Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv
University)
We propose a novel framework for regularization of symmetric
positive-definite (SPD) tensors (e.g., diffusion tensors). This
framework is based on a local differential geometric approach. The
manifold of symmetric positive-definite (SPD) matrices, Pn, is
parameterized in terms of the local coordinates via the Iwasawa
coordinate system. In this framework distances on Pn are measured
in terms of a natural GL(n)-invariant metric where this metric is
expressed in terms of the Iwasawa coordinates. Via the mathematical
concept of fiber bundles we describe the tensor-valued image as a
section where the metric over the section is induced then in terms
of the metric over Pn. Then, a functional over the sections
accompanied by a suitable data fitting term is defined. The
variation of this functional with respect to the Iwasawa coordinates
leads to a set of n(n+1)/2 coupled equations of motion.
By means of the gradient descent method, these equations of motion
define a flow over Pn. Thus, for an example,
regularization of Diffusion Tensor Imaging (DTI) datasets is done by
solving numerically a set of six coupled Beltrami equations with
respect to the Iwasawa coordinates. It turns to be that the local
coordinate approach via this coordinate system results in very
simple numerics that leads to fast convergence of the algorithm. We
demonstrate the efficiency of this framework on real volumetric DTI
datasets. Regularization results as
well as results of fibers tractography for DTI are presented.
(Joint work with Yaniv Gur and Ofer Pasternak.)
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Measuring and localizing homology
Daniel Freedman (Department of Computer Science, Rensselaer Polytechnic Institute and
Department of Computer Science and Applied Mathematics, Weizmann Institute)
A new field, combining ideas from algebraic topology and computer science,
has recently emerged. Some of the exciting developments in this young field
include fast algorithms for computing homology groups, topologically correct
reconstruction of manifolds from samples, and the concept of persistence.
In this talk, we introduce a geometrically natural definition of the size of
a homology class, as well as an efficient algorithm for computing both the
size and a smallest representative of the class. From the theoretical point
of view, this may be seen as an interesting way of combining geometry with
topology; from the applied side, the algorithms may find use in the
development of shape signatures. The talk will be self-contained.
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Inelastic soliton interactions in the perturbed KdV equation
Yair Zarmi (Jacob Blaustein Institute for Desert Research and Physics Department, Ben-Gurion University of the Negev)
Small-amplitude solutions of several complex dynamical systems are approximately governed by the KdV equation. The wave solutions of the equation are single- or multiple-soliton solutions. The multiple-soliton solution describes an elastic collision process: Asymptotically away from the origin, before and after the collision, each soliton is not affected by the existence of other solitons, except for a possible trivial phase shift. Re-instatement of the terms that have been neglected in the derivation of the KdV equation results in the perturbed KdV equation.
The solution of the perturbed equation is written as a sum of an elastic component and an inelastic one. The elastic component is described by the same perturbation series in the single- and multiple-soliton cases. In the multiple-soliton case, it preserves the elastic scattering character of the unperturbed solution.
The inelastic component is driven by that part in the perturbation, which represents the net effect of the difference between the single- and multiple-soliton solutions. This part corresponds to inelastic interactions amongst the solitons. It is highly localized in the x-t plane. The corrections it generates in the solution evolve along the characteristic lines of the original solitons. Hence, the soliton wave numbers and velocities are not affected. However, they do spoil the elastic character of the multiple-soliton solution.
The structure of the corrections included in the inelastic component depends on the initial data imposed on the solution. The common solution corresponds to solitary waves. Some initial data lead to the emergence of new types of corrections, including soliton-anti-soliton creation or annihilation, and soliton decay or amalgamation.
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Solution of the nonlinear Helmholtz equation
Guy Baruch
(Department of Applied Mathematics, School of Mathematical Sciences, Tel Aviv
University)
The nonlinear Helmholtz equation models the propagation of intense laser
beams in Kerr media such as water, silica and air.
It is a multidimensional semilinear elliptic equation, which requires
appropriate radiation boundary-conditions and remains unsolved in many
configurations.
Its commonly-used parabolic approximation, the nonlinear Schrodinger
equation (NLS), is known to possess singular solutions.
We therefore consider the question: do nonlinear Helmholtz solutions
exists, under conditions for which the NLS solution becomes singular ?
In other words, is the singularity removed in the elliptic model ?
In this work we develop a numerical method which produces such solutions
in some cases.
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Long time existence for the rapidly rotating shallow-water and Euler equations
Eitan Tadmor (University of Maryland)
We study the stabilizing effect of rotational forcing in the
nonlinear setting of two-dimensional shallow-water and more general models
of compressible Euler equations. The pressure-less version of these
equations admit global smooth solution for a large set of sub-critical
initial configurations. But what happens with more realistic models, in the
presence of pressure? It is shown that when rotational forcing dominates the
pressure, it prolongs the life-span of such sub-critical solutions, for a
time period dictated by the ratio δ=(Rossby number)/(Froude number)2.
Our study reveals a "nearby" periodic-in-time approximate solution in the
small δ regime, upon which hinges the long time existence of the exact
smooth solution. These results are in agreement with the close-to periodic
dynamics observed in the "near inertial oscillation" (NIO) regime which
follows oceanic storms.
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Incompressible, quasi-rigid deformations of 2-dimensional domains
Gershon Wolansky (Department of Mathematics, Technion)
Recently, there is an increasing interest in the analysis of near-rigid
deformation within the computer vision community, in particular pattern
recognition, image segmentation and face recognition.
A sensible definition of a deformation metric between 2-dimensional surfaces
obtained from each other by an area preserving (incompressible) mapping is
proposed. In addition, an algorithm for obtaining this metric, as well as the
optimal deformation is suggested.
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Chaos in the Pertubed Nonlinear Schrodinger Equation
Eli Shlizerman (Department of Computer Science and Applied Mathematics, Weizmann Institute)
We analyze the structure of the standing waves in the periodic non-linear
1d Schrodinger equation. Using a novel
framework by which bifurcations and stability of the standing waves are studied,
we are able to classify some of the chaotic solutions that emerge when the
equation is slightly perturbed. The route to temporal and spatiotemporal chaos
is nicely described within this framework.
(Joint work with Vered Rom-Kedar.)
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Statistical approximations of the Navier-Stokes equations
Fabio Ramos
(Department of Computer Science and Applied Mathematics, Weizmann Institute)
We present the Navier-Stokes-Voight (NSV) model as a smooth statistical
approximation for the Navier-Stokes equations. We show that the stationary
Reynolds equations hold for the NSV in the framework of stationary
statistical solutions (sss). Moreover, we show that sss of the NSV converge
to the ones of the Navier-Stokes when the relaxation parameter goes to
zero. Numerical support will also be presented at the end of the talk.
(Joint work with Edriss S. Titi and Boris Levant)
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Achieving convergence in mobile robot swarms
Reuven Cohen (Department of Mathematics, Bar-Ilan University)
Mobile robot swarms are the subject of considerable interest
recently, as they allow to complete tasks that are difficult or
impossible to complete by a single robot. The swarm is usually a system
of multiple simple robots with limited capabilities expected to jointly
complete a task in a robust and efficient way. The study of swarm
behavior combines concepts from geometry, analysis and distributed
computing. We discuss some of the difficulties of achieving even simple
tasks in a highly simplified model. We present some positive results and
discuss the convergence properties of simple operations and the effect
of errors.
(Joint work with David Peleg.)
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