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.