Preconditioners for robust optimal control problems under uncertainty

The discretization of robust quadratic optimal control problems under uncertainty using the finite element method and stochastic collocation leads to large saddle-point systems, which are fully coupled across random realizations. Despite its relevance for numerous engineering problems, solution such systems is notoriusly challenging. In this manuscript, we study efficient preconditioners all-at-once approaches both an algebraic operator preconditioning framework. We show in particular that values regularization parameter not too small, system can be efficiently solved by parallel all state adjoint equations. For small parameter, robustness recovered additional a linear system, however couples A mean approximation Chebyshev semi-iterative investigated solve reduced system. Our analysis considers elliptic partial differential equation whose diffusion coefficient $\kappa(x,\omega)$ modeled as almost surely continuous positive field, though necessarily uniformly bounded coercive. further provide estimates on dependence preconditioned variance field. Such involve either first or second moment variables $1/\min_{x\in \overline{D}} \kappa(x,\omega)$ $\max_{x\in \overline{D}}\kappa(x,\omega)$, where $D$ spatial domain. theoretical results confirmed numerical experiments, implementation details addressed.