GAMM Rundbrief 2002/Heft 2

84 Wissenschaftliche Tagungen
In the last twenty years, the field of inverse problems has undergone rapid
development: The enormous increase in computing power and the development of
powerful numerical methods made it possible to simulate real-world direct problems of
growing complexity. Since in many applications in science and engineering, the inverse
question of determining causes for desired or observed effects is really the final
question, this lead to a growing appetite in applications for posing and solving inverse
problems, which in turn stimulated mathematical research e.g., on uniqueness questions
and on developing stable and efficient numerical methods (regularization methods) for
solving inverse problems. This began mainly for linear problems, but more recently it
has also been done for nonlinear problems.
The Special Semester at IPAM will focus on new challenges that have appeared
recently in the field of inverse problems:
New application fields
Imaging Science including Image Processing, Computer Graphics and Computer
Vision. Many imaging problems are by their nature inverse problems, which suggests
the use of regularization methods for their solution. On the other hand, specific methods
developed in imaging like bounded-variation-regularization and diffusion filtering are
being applied to other inverse problems. From the applications side e.g. in medical
imaging, new techniques are emerging like elastographics and optical tomography,
which in turn pose new mathematical and computational questions.
Inverse problems in life sciences: Life sciences are a real growth field for mathematical
modeling. An important step in modeling is to determine parameters from
measurements. In life sciences, this usually leads to large-scale inverse problems, e.g.,
the simultaneous determination of hundreds of rate constants in very large reaction
diffusion systems. Other, already a bit more classical, inverse problems in life sciences
include inverse folding problems. Since many mathematical models in the life sciences
are just now being developed, this will be the right time to bring experts in life sciences
who develop such models and experts on inverse problems together in a kind of
exploratory workshop.
Inverse problems in industry: Knowledge about mathematical and numerical methods
for inverse problems has diffused faster into the scientific community than into
industry. On the other hand, many mathematical problems of interest to industry are in
essence inverse problems. At the IPAM Special Semester, such problems will be
studied and worked on in a "study group" format.
Inverse problems in physical sciences: Many measurements in the physical sciences are
indirect, thus their interpretation is an inverse problem whose ill-posedness is not
always appropriately addressed. An example are deconvolution problems for ground
based telescopes; deconvolution also appears in many other applications like in
spectroscopy or in the interpretation of time-resolved fluorescence data with a variety
of applications in medicine and biology. Also, modelling of epitaxial growth or other
growth processes involves various inverse problems like the determination of growth
rates from measured data or shape optimization problems; a methodological link to
inverse problems is the use of level set methods.