Optimization problems governed by partial differential equations (PDEs) arise in many engineering and science applications in the form of data assimilation, optimal control, optimal design, or parameter identification problems. The design, analysis, implementation, and application of algorithms for the efficient solution of such problems gives rise to many interesting mathematical questions.

In this talk I will use selected optimal control and optimal design problems to introduce some of the challenges that arise in the solution of PDE constrained optimization problems and to discuss approaches that have been used successfully for their solution.