Title:   On the calculation of gradients in certain PDE-constrained optimization problems

The presentation will discuss two PDE-constrained optimization problems in which the gradients of the objective function possess a non-standard structure. The first is an inverse problem concerning optimal reconstruction of constitutive relations in thermodynamics. The gradient in this problem represents the sensitivity of the objective function with respect to the structure of the governing equation. The second problem concerns shape optimization in which the gradients encode sensitivity with respect the shape of the domain on which the governing PDE system is defined. In addition to some analytical aspects of these problems, we will discuss efficient numerical techniques allowing one to solve such optimization problems in practice and will present a number of computational results.