Direct and Inverse Problems for PDEs with Random Coefficients - Abstract

Preusser, Tobias

Stochastic collocation for SPDE constrained optimal control

We discuss the use of stochastic collocation for the solution of optimal control problems which are constrained by stochastic partial differential equations (SPDE). Thereby the constraining SPDE depends on data which is not deterministic but random. Assuming a deterministic control, randomness within the states of the input data will propagate to the states of the system. For the solution of SPDEs there has recently been an increasing effort in the development of efficient numerical schemes based upon the mathematical concept of generalized Polynomial Chaos. Modal-based stochastic Galerkin and nodal-based stochastic collocation versions of this methodology exist, both of which rely on a certain level of smoothness of the solution in the random space to yield accelerated convergence rates. In the talk we show how to apply the stochastic collocation method in a gradient descent as well as a sequential quadratic program (SQP) for the minimization of objective functions constrained by an SPDE. The stochastic function involves several higher-order moments of the random states of the system as well as classical regularization of the control. In particular we discuss several objective functions of tracking-type. To demonstrate the performance of the method we show numerical test examples and we give an outlook to a real-world problem from medical treatment planning.