Multigrid preconditioners for optimal control problems with stochastic elliptic PDE constraints

ABSTRACT

In this work, we construct multigrid preconditioners to be used in the solution process of pathwise optimal control problems constrained by elliptic partial differential equations with random coefficients. We use a sparse-grid collocation method to discretize in the stochastic space and multigrid techniques in the physical space. Numerical results show that the proposed preconditioners lead to significant computational savings, with the number of preconditioned conjugate gradient iterations decreasing as the resolution in the physical space increases.

Keywords:

• PDE-constrained optimization
• stochastic collocation
• multigrid methods
• finite elements

2010 Mathematics Subject Classifications:

• 60H15
• 65N30
• 65F10

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported in part by the Office of Naval Research (ONR) [grant number N0001415WX01746].