Stochastic Galerkin Method and Hierarchical Preconditioning for PDE-constrained Optimization

TL;DR

This paper introduces hierarchical preconditioners for PDE-constrained optimization with uncertain coefficients, improving solver efficiency by exploiting stochastic Galerkin sparsity and balancing computational cost with preconditioning quality.

Contribution

It develops novel hierarchical preconditioners tailored for stochastic Galerkin discretizations in PDE-constrained optimization, enhancing scalability and robustness.

Findings

Preconditioners significantly accelerate iterative solver convergence.

Method effectively handles large, ill-conditioned systems in uncertainty quantification.

Numerical results confirm robustness for steady-state and time-dependent problems.

Abstract

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 projection, and advanced time-discretization schemes, the approach addresses challenges of scaling large and ill-conditioned linear systems arising in uncertainty quantification. By exploiting sparsity of linear systems in stochastic Galerkin method, we formulate 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 problems under uncertainty.