# Welcome

### From DFG-SPP 1253

This Priority Program is financed by the DFG since July 2006.
For a brief sketch of the program's main features see below.

### Summary:

Solving optimization problems subject to constraints involving distributed parameter systems is one of the most challenging problems in the context of industrial, medical and economical applications. In particular, in the design of aircraft, ’moving bed’ processes in chemical engineering, crystal growth etc. the transition from model-based numerical simulations to model-based design and optimal control is crucial.
All of these applications involve modelling based on partial differential equations (PDEs), and can be classified as formand topology optimization problems or optimal control problems in an infinite-dimensional setting.
For the treatment of such problems the interaction of optimization techniques and numerical simulation is crucial. After proper discretization, the number of optimization variables varies between 10^{3} and 10^{10}.

It is only very recently that the enormous advances in computing power have made it possible to attack problems of this size. However, in order to accomplish this task it is crucial to utilize and further explore the

- specific mathematical structure of prototype applications
- and to develop new mathematical approaches concerning structure exploiting algorithms
- adaptivity in optimization and handling of control and state constraints.

The aim of this DFG-Priority Program (DFG-PP) is thus to combine numerical analysis of PDEs and optimization governed by prototype applications so that the most recent activities in the fields are merged and further explored, and new analytic and algorithmic paradigms will be developed, implemented, and, ultimately, validated in the context of real-world applications.

On the one hand, the proposed priority program requires basic research in single projects, while on the other hand, in order to foster a synthesis of ideas and methods, a synergetic networking of research groups is required. Clustering of research projects around prototype applications is particularly important in order to achieve landmark results with respect to solving the underlying industrial, economical, medical and environmental problems.

The proposed DFG-Priority Program is interdisciplinary, with a major focus on Applied Mathematics. It receives its motivation and validation platform from very recent grand scale applications. It becomes more and more apparent that such applications clearly define a class of mathematical problems in numerical analysis and optimization that need to be solved in order make a difference in future technology. It is, therefore, strongly believed that the research results will significantly benefit computational scientists from other disciplines and engineers in charge of grand-scale applications.

There is a critical potential of several senior and many young mathematicians that has evolved in Germany in recent years, driven by fascinating future applications and the challenging mathematical problems that are to be solved.

The proposed Priority Program would trigger a new and exciting development in computationally oriented Applied Mathematics and Engineering that would lead to ’cutting edge’ research in these fields. While the first funding period is mainly devoted to the development of new computational technologies, in the second period the engineering applications shall be given more room.