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A very challenging problem in computational science is the solution of optimization problems whose constraints are given by the discretization of 3-dimensional partial differential equations. These problems are often so large that Hessian and Jacobian matrices cannot be formed or stored -- and this prevents us from applying contemporary optimization algorithms. To solve these kinds of problems, it is imperative to use multilevel techniques, matrix-free optimization methods and to compute steps inexactly so as to minimize the cost of the iteration. In this talk I will report on some recent contributions we have made in this area. Two case studies will be considered: long-range weather forecasting and a problem in cancer treatment planning

Note that this seminar is earlier than usual because of the Colloquium this week.