The long-term scientific goal of the SFB is the design, verification, implementation, and analysis of numerical, symbolic, and graphical methods for solving large-scale direct and inverse problems with constraints and their synergetical use in scientific computing for real-life problems of high complexity. We have in mind so-called field problems (usually described by partial differential equations (PDEs)) and algebraic problems (e.g. involving constraints in algebraic formulation).

The particular emphasis of this SFB is put an the *integration*
of graphical, numerical, and symbolic methods on different levels:

*Numerical and symbolic methods* have been developed
so far by two fairly disjoint research communities. The University of
Linz is one of the few places with strong groups both in numerical and
symbolic computing. Thus, we intend to make the joint work on numerical
and symbolic methods one of the main focuses of the SFB.
For the first time, this will also encompass the phase in which new
mathematical results - as the basis of new numerical and symbolic
algorithms - are derived: The SFB also incorporates a subproject aiming
at computer-support for proving new mathematical theorems using advanced
symbolic techniques. The integration of numerical and symbolic methods
is also in the center of our study of the partial differential equations
that form the benchmark problems in the SFB: For the first time,
numerical methods will be brought together with algebraic methods (e.g.
generalized Gröbner bases and quantifier elimination) for handling
PDEs with complex constraints. And, more traditionally, symbolic methods
will be used in the preprocessing and postprocessing phases of the
numerical methods.

The *interaction of numerical and symbolic methods with graphical
methods* is another focus of the SFB. Of course, visualization by
advanced graphics and animation is indispensable both in understanding
and presenting the results of numerical and symbolic computations.
However, we will put an emphasis also on the reverse direction:
Refined numerical and, in particular, symbolic methods (e.g. algebraic
parametrization of curves and surfaces, topology analysis by algebraic
decomposition) must be developed for improving the topological
correctness and reliability of graphics for advanced applications.