In March 2004 the SFB had successfully finished its second funding period: as a result of the peer-review organized by the FWF at the end of 2003, a third period (April 2004 to March 2008) had been granted.

# Overall Scientific Goal

The overall scientific goal of the SFB is the design, verification, implementation, and analysis of

- numerical,
- symbolic, and
- geometrical

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. This includes so-called field problems, usually described by partial differential equations (PDEs), and algebraic problems, e.g., involving constraints in algebraic formulation.

# General Scientific Concept

This overall goal in its essence is the same as defined in the original proposal written more than seven years ago. However, we note that during the evolution of the SFB the following minor changes have been made in its definition. In the proposal for the first funding period, instead of "geometrical" the term "graphical" was chosen. Since geometrical issues have started to play a stronger role, in the proposal for the second funding period, instead of "graphical" the term "geometrical and graphical" was chosen. Meanwhile scientific visualization tools are available (partly developed in F1301), and since different aspects of geometrical scientific computing are needed in almost all subprojects, for the last funding period the replacement of "geometrical and graphical" by "geometrical" was a natural decision.

As pointed out in the SFB Progress Report 2001-2003, concerning the fine structure of the Scientific Concept and of the Long Term Goals of the SFB, we permanently have made adaptations in order to focus more properly on our overall objective. These adjustments have been driven by the advice and the suggestions of the referees, by our experience made during the SFB work, but also by the changing requirements in the international research community.

To achieve the goal of a proper combination of numerical and symbolic scientific computing, also supplementary measures, like joint internal seminars between numerical and symbolic groups or a new target-oriented structure of the SFB status seminars, were introduced. As a result, the coherence between the numerical and symbolic groups has increased significantly. Instead of point-wise cooperations, a whole network of concrete links between numerics, symbolics, and geometry has emerged.

The scientific results obtained within the SFB by the participating institutes gave rise to various activities concerning knowledge and technology transfer to the industry, especially, in Upper Austria. The highlights are the foundation of the Software Competence Center Hagenberg and the Industrial Mathematics Competence Center in 1999. For more details see the section "Transfer of Knowledge and Technologies". On the academic level, the efforts of the institutes participating in the SFB to combine numericalsymbolic scientific computing with applied mathematics led to the foundation of the Johann Radon Institute for Computational and Applied Mathematics (RICAM) by the Austrian Academy of Sciences as a Center of Excellence in Applied Mathematics.

The following institutes of the University of Linz are currently involved in the subprojects of the SFB:

- Institute of Applied Geometry
- Institute of Computational Mathematics
- Institute of Industrial Mathematics
- Institute of Symbolic Computation

# Research Topics

Concerning more detailed information on the particular SFB research topics, please go to the individual project pages.

SpezialForschungsBereich SFB F013 | Special Research Program of the FWF - Austrian Science Fund