Reduced-Basis Methods for PDE-Constrained Elliptic Optimal Control Problems with Uncertain Coefficients

Abstract

This thesis covers a convex optimal control problem, which possesses an elliptic PDE subjected to uncertainty as constraint.It will be shown that with the help of the projected stochastic gradient method an optimal control over a discrete random space can be computed faster than with the projected gradient method. The determination of the necessary finite element (FE) solutions can be speeded-up by use of the reduced basis (RB) method. For the error between the FE and the reduced-order solution a-posteriori error estimates can be denoted for the state and the adjoint equation. The convergence behaviour between the FE and the reduced-order solution will be studied. The numerical results confirm the theoretical results of the convergence behaviour. Furthermore the difference between the optimal solution with and without the RB method can be bounded. The optimization with the projected stochastic gradient method and the RB method turns out to be a more efficient method.