GAMM Rundbrief 2002/Heft 2

Wissenschaftliche Tagungen 85
Methodological challenges
In recent years, extremely powerful numerical methods have been developed for
solving complex direct problems, e.g., multi-field problems in three dimension, both
static and dynamic. Such methods include multigrid or, more general, multi-level
methods and domain decomposition. When solving inverse problems for such complex
problems, new questions arise also for the numerical treatment of the inverse problem,
which include the optimal coupling of regularization methods with direct solvers in
order to achieve overall optimal performance
A powerful numerical method whose main advantage is that it can easily handle
changes in the topology is the level set method. It has recently also been applied to
inverse problems.
Over the years, two major approaches have been followed in the inverse problems
community: statistical and functional-analysis based approaches. A full understanding
of the relations between these approaches is still lacking; this is also important for the
issue of "uncertainty".
During the proposed Special Semester, special emphasis will be laid on some of these
and other emerging challenges, although more classical topics will not be neglected.
The Special Semester is intended to bring together scientists and engineers with applied
and pure mathematicians interested in inverse problems.
The Special Semester will be structured as follows
Tutorials: In the second (and maybe also third) week of September 2003, a series of
tutorials will be held both on methodological and on applications issues of inverse
problems. These should also set the stage for research collaborations between
mathematicians and applications scientists that should go on throughout the semester,
and should prepare the participants for the subsequent events. The final list of topics
has not yet been decided, a tentative list is:
Methodology: regularization methods for inverse problems inverse spectral problems
statistical and wavelet methods for inverse problems level set methods.
Application fields inverse problems in the physical sciences, grouped according to
different application fields inverse problems in imaging science inverse problems in
biology inverse scattering and tomography.
Study group with industry: This format has probably the longest tradition in Oxford
and has been implemented also in other European countries, in Australia, and at RPI.
On the first day of such a study group, industrial researchers present problems for
which they want a mathematical model, solution, algorithm. In the following days, open
discussions in groups, which tend to be quite intensive, should lead to progress. The
outcome will generally not be a final solution, but a first mathematical model and a
clear plan for further work. Such a study group should focus on problems from West
Coast industries (but not exclusively) in order to make a follow-up by inverse problems
experts who visit IPAM for the semester possible. Topics where industrial contacts
have already been made include inverse problems and optimal design in photonics and
inverse problems in finance, especially identification of volatilities and interest rate
models.