Topology and Dynamic Networks: Optimization with Application in Future Technologies

The optimal design and control of infrastructures, e.g. in traffic control, water-supply, sewer-systems and gas-pipelines, the optimization of structures, form and formation of materials, e.g. in lightweight structures, play a predominant role in modern fundamental and applied research. However, until very recently, simulation-based optimization has been employed in the sense that parameters are being adjusted in a forward simulation using either ‘trial-and-error’ or a few steps of a rudimentary unconstrained derivative-free and mostly stochastic optimization code. It has become clear by now that instead often a model-based and more systematic constrained optimization that exploits the structure of the problem under consideration may outperform the former more naive approaches. Thus, modern mathematical optimization methods respecting constraints in state and design variables can be seen as a catalyst for recent and future technologies. More and more success stories can be detected in the literature and even in the public press which underline the role of optimization as a key future technology. In particular, optimization with partial differential equations (PDEs) as constraints, or in other words ‘PDE-constrained optimization’ has become research topic of great influence. A DFG-Priority-Program (PP) has been established by the German Science Foundation (DFG) in 2006 in which well over 25 project are funded throughout Germany. The PP focuses on the interlocking of fundamental research in optimization, modern adaptive, hierarchical and structure exploiting algorithms, as well as visualization and validation. With similar goals in mind, a European network within the European Science Foundation (ESF) ‘PDE-constrained Optimization’ has been recently established that provides a European platform for this cutting edge technology. In this report the authors dwell on exemplary areas of their expertise within applications that are already important and will increasingly dominate future developments in mechanical and civil engineering. These applications are concerned with optimal material and design in material sciences and light-weight structures as well as real-time capable optimal control of flows in transportation systems such as gas-pipeline networks. ‘Advanced Materials’, ‘Energy-Efficiency’ and ‘Transport’ are key problems for the future society which definitely deserve public funding by national and international agencies.