Scientific Program
The first three days of the symposium, January 10 to 12, will consist of
mini-courses for graduate students and post-doctoral associates. The second week, January 15 to 19,
will consist of plenary lectures given by invited speakers,
and parallel 20 minute talks intended to give young researches the opportunity
to lecture at an international conference.
Mini-Courses
Five mini-courses will be offered during the conference. They
will be held from January 10 to 12, 2007, and will consist of three lectures
of 90 minutes each.
The following mini-courses are planned:
Geometric Methods for Inverse Problems, by Matti Lassas (University of Technology, Finland)
We consider uniqueness results and counter examples for
inverse problems for the anisotropic
conductivity equation and
for hyperbolic equations in an anisotropic medium.
Typical inverse problems in an anisotropic medium
are not uniquely solvable. Indeed, it is well known that
a change of coordinates
changes the equation but does not change the boundary data.
This point of view makes it possible to consider both
uniqueness results and construct counter examples.
To prove uniqueness results, one may consider properties that
are invariant in smooth diffeomorphisms and try to reconstruct
them uniquely. Indeed,
there is often an underlying manifold structure
that can be uniquely determined.
Thus the inverse problem
in a subset of the Euclidean space can solved in two steps.
The first one is to reformulate the problem in terms of manifolds
and to reconstruct the
underlying manifold structure from the boundary data.
The second step is to find an embedding of the constructed
manifold to the Euclidean space.
The embedding is generally non-unique, but if we have
enough a priori knowledge about the form of equation,
we can determine the embedding uniquely, or
at least choose an optimal embedding.
To obtain counter examples for uniqueness of inverse problems,
we consider singular deformations
of a domain. This leads to degenerate Riemannian metrics, or
conductivity tensors that appear in boundary measurement
similar to a homogeneous domain.
To demonstrate these ideas in practice, we consider
the following particular problems:
1) Counter examples of inverse problems and
invisibility results
2) Inverse problems with an inaccurate model of the boundary.
The motivation for this kind of problem is that wrong
boundary modeling causes
severe errors for the reconstructions. We review recent methods for
solving inverse problems with inaccurately modeled boundary
3)
Inverse problems for hyperbolic equations. We review results
on reconstruction of a Riemannian manifold and metric
from the time domain data.
Coherent Interferometric Array Imaging in Random Media, by
Liliana Borcea (Rice University)
In important applications such as ultrasound medical imaging, foliage
or ground penetrating radar, land and shallow water mine detection,
etc., one seeks to detect and image small or extended scatterers
(reflectors) embedded in inhomogeneous, cluttered media. Such media
can be modeled as randomly inhomogeneous, with properties such as
acoustic impedance having a deterministic large scale variation,
assumed known, and an additional, small scale variation that is
unknown and is represented by a random function of space. The strong
scatterers embedded in such media are to be imaged with an array of
transducers which emit acoustic pulses and record the traces of the
backscattered echoes.
Traditional array imaging methods known as synthetic aperture sonar,
Kirchhoff Migration, etc., are well understood and work well in known
media. However, these methods fail in random media and new ideas must
be explored in order to obtain reliable images. I will describe a new,
coherent interferometric approach to imaging in clutter, which is
based on an asymptotic stochastic analysis of wave propagation in
random media, in regimes with strong multipath. To achieve stable
results, this method migrates (back propagates) cross-correlations of
the traces over appropriately chosen space-time windows, instead of
the traces themselves. The size of the space-time windows is critical
and it depends on two clutter dependent parameters. One is the
decoherence frequency, which is proportional to the reciprocal of the
delay spread in the traces produced by the clutter. The other
parameter quantifies the decoherence in the direction of arrival of
the echoes.
I will explain how coherent interferometry can be understood as a
statistically smoothed version of classic migration techniques. The
statistical smoothing is essential for achieving stable results (i.e.,
images that do not depend on the realization of the clutter).
However, stability comes at the cost of blurring the images and the
loss of resolution can be quantified explicitly, in terms of the
decoherence parameters. I will discuss an adaptive procedure for
getting the optimal trade-off between statistical stability and
blurring and therefore, for estimating the clutter dependent
decoherence parameters, without apriori knowledge of the clutter.
Finally, I will describe optimal illumination and waveform design
for improving the quality of the images in cluttered media.
Semiconductors and DtN Maps, by Antônio Leitão (UFSC, Brazil)
In this short course inverse problems related to stationary
drift-diffusion equations modeling semiconductor devices are investigated.
In this context we analyze some identification problems corresponding to
different types of measurements, where the parameter to be reconstructed
is an inhomogeneity in the PDE model (doping profile).
For a particular type of measurement, related to the voltage-current map,
we consider special cases of drift-diffusion equations, where the inverse
problems reduces to a classical {\em inverse conductivity problem}
Travel Time Tomography, by Plamen Stefanov (Purdue University)
This minicourse will cover topics in integral geometry of tensors and the
boundary rigidity problem. Let (M, g ) be a compact Riemannian manifold
with smooth boundary N. The first main problem we study is the following: given
a symmetric 2-tensor f_{ij}, is it possible to recover the tensor f from the associated
geodesic X-ray transform I_g f , i.e., from integrals along geodesics? It is known
that one can hope to recover the solenoidal part f^s of the tensor only, since all
potential tensors are in the kernel of that linear operator I_g. The second main
problem is the boundary rigidity problem: given the distance function \rho(x, y)
between any pair (x, y) on the boundary N, can we recover the metric g? If so,
this can be done only up to a pull-back of a diffeomorphism fixing the boundary
N. It turns out that the linearization of the boundary rigidity problem is
given by the X-ray transform I_g.
Those two problems are related in a natural way (through propagation of singularities) to other inverse problems, for example the problem of recovering of g
given the hyperbolic Dirichlet-to-Neumann map, the inverse scattering problem
related to a anisotropic medium, etc. They are motivated also by application
in geophysics, medical imaging, etc. On the other hand, they are problems of
independent interest in geometry.
A Survey of Inverse Problems and Techniques in the Biophysical Sciences, by
Jorge Zubelli (IMPA, Brazil)
Inverse problems impact the biophysical sciences in a multitude of shapes and
forms. To cite a few examples: Medical imaging, electron microscopy,
and parameter identification of ecosystems.
In the area of medical imaging, X-ray computerized tomography brought a
revolution in medicine. Such progress only happened because of the
contribution from many mathematical areas. In particular, Fourier analysis,
numerical analysis and operator theory.
During the early nineties the possibility of using infra-red
radiation for medical imaging was considered and its potential
in many fields has attracted the attention of several groups
both in academia and industry. Still, some of the mathematical
questions associated to it are formidable and mathematically
challenging.
In electron microscopy, an extremely powerful technique is the use of
X-ray transform techniques for imaging macromolecules. This leads to
deconvolution problems and ultimately to nonlinear problems associated
to the Radon transform without angular information.
As for the applications to ecosystems, important contributions
can be traced at least to the work of Bellmann in the sixties. Still
the challenges are enormous despite the fact that data collection techniques
and computer power have improved immensely.
In this mini-course we plan to address the issues above with a set of
theoretical lectures as well as some practical numerical experiments in
the computer labs.