The development, analysis and implementation of eﬃcient and robust numerical techniques for optimization problems associated with partial diﬀerential equations (PDEs) is of utmost importance for the optimal control of processes and the optimal design of structures and systems in modern technology. The successful realization of such techniques invokes a wide variety of challenging mathematical tasks and thus requires the application of adequate methodologies from various mathematical disciplines. During recent years, signiﬁcant progress has been made in PDE constrained optimization both concerning optimization in function space according to the paradigm ’Optimize ﬁrst, then discretize’ and with regard to the fast and reliable solution of the large-scale problems that typically arise from discretizations of the optimality conditions. The contributions at this Oberwolfach workshop impressively reﬂected the progress made in the ﬁeld. In particular, new insights have been gained in the analysis of optimal control problems for PDEs that have led to vastly improved numerical solution methods. Likewise, breakthroughs have been made in the optimal design of structures and systems, for instance, by the socalled ’all-at-once’ approach featuring simultaneous optimization and solution of the underlying PDEs. Finally, new methodologies have been developed for the design of innovative materials and the identiﬁcation of parameters in multi-scale physical and physiological processes.
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Ronald H.W. Hoppe, Matthias Heinkenschloss, Volker Schulz, Numerical Techniques for Optimization Problems with PDE Constraints. Oberwolfach Rep. 6 (2009), no. 1, pp. 191–284DOI 10.4171/OWR/2009/04