Block-Diagonal Preconditioning for Optimal Control Problems Constrained by PDEs with Uncertain Inputs

The goal of this paper is the efficient numerical simulation of optimization problems governed by either steady-state or unsteady partial differential equations involving random coefficients. This class of problems often leads to prohibitively high dimensional saddle-point systems with tensor product structure, especially when discretized with the stochastic Galerkin finite element method. Here, we derive and analyze robust Schur complement--based block-diagonal preconditioners for solving the resulting stochastic optimality systems with all-at-once low-rank iterative solvers. Moreover, we illustrate the effectiveness of our solvers with numerical experiments.