Risk-Averse PDE-Constrained Optimization: Analysis, Optimality, and Numerical Solution
Tuesday, March 7, 2017
1:15 pm - 2:15 pm
Gross Hall, 330 -- Ahmadieh Family Grand Hall
Dr. Thomas M. Surowiec, Philipps-University Marburg, Germany
Uncertainty arises in virtually all real-world applications, for instance,due to noisy measurements or a lack of observability. As many models inengineering and the natural sciences make use of partial differentialequations (PDEs), it is natural to consider systems of PDEs with randominputs. This presents a number of significant theoretical, algorithmic,and numerical challenges for optimization problems governed by systems ofPDEs with uncertain inputs. For example, since the solution of the PDE isa random field, the objective function becomes a random variable. Ifcoherent risk measures are used to handle this random variable objectivefunction, then the resulting optimization problem is non-smooth,non-convex, and infinite dimensional. In this talk, we motivate ourapproach via a model problem inspired by the mitigation of an airbornepollutant. We then present the necessary analytical framework and a briefoverview of risk measures. We then prove the existence of solutions andderive optimality conditions under mild assumptions. In order to exploitexisting numerical methods from PDE-constrained optimization, we suggestsmoothing schemes for the risk measures and discuss the consistency of theregularization schemes. Finally, we discuss solutions techniques andillustrate our results with numerical examples.