Author:

Abstract:

The optimal control of problems that are constrained by partial differential equations with uncertainties is addressed. The inclusion of the stochastic dimension provides additional freedom in the definition of cost functionals, for example by including statistics of the response in a cost functional. We formulate a one-shot approach to the stochastic optimal control problems and solve the resulting equations via a stochastic Galerkin or collocation finite element method. The stochastic collocation method is often preferred over the Galerkin approach as it converts a stochastic problem into a collection of decoupled deterministic problems. It is shown however that this so-called non-intrusivity property of the collocation method does not hold for a large class of stochastic PDE-constrained optimisation problems. The efficient solution of stochastic Galerkin finite element problems can hinge on the development and application of effective preconditioners. This aspect is addressed with two preconditioners that take the specific structure of the Galerkin one-shot systems into account. Numerical examples support the findings. The presented framework is sufficiently general to also consider a class of stochastic inverse problems and numerical examples of this type are also presented.