Stochastic Galerkin Method and Hierarchical Preconditioning for PDE-constrained Optimization

We develop efficient hierarchical preconditioners for optimal control problems governed by partial differential equations with uncertain coefficients. Adopting a discretize-then-optimize framework that integrates finite element discretization, stochastic Galerkin approximation, and advanced time-discretization schemes, the approach addresses the challenge of large-scale, ill-conditioned linear systems arising in uncertainty quantification. By exploiting the sparsity inherent in generalized polynomial chaos expansions, we derive hierarchical preconditioners based on truncated stochastic expansion that strike an effective balance between computational cost and preconditioning quality. Numerical experiments demonstrate that the proposed preconditioners significantly accelerate the convergence of iterative solvers compared to existing methods, providing robust and efficient solvers for both steady-state and time-dependent optimal control applications under uncertainty.