In this work, we apply the MG/OPT framework to a multilevel in-sample-space discretization of optimization problems governed by PDEs with uncertain coefficients. The MG/OPT algorithm is a template for the application of multigrid to deterministic PDE optimization problems. We employ MG/OPT to exploit the hierarchical structure of sparse grids in order to formulate a multilevel stochastic collocation algorithm. The algorithm is provably first-order convergent under standard assumptions on the hierarchy of discretized objective functions as well as on the optimization routines used as pre- and post-smoothers. We present explicit bounds on the total number of PDE solves and an upper bound on the error for one V-cycle of the MG/OPT algorithm applied to a linear quadratic control problem. We provide numerical results that confirm the theoretical bound on the number of PDE solves and show a dramatic reduction in the total number of PDE solves required to solve these optimization problems when compared with standard optimization routines applied to the same problem.

%B SIAM Conference on Computational Science & Engineering %8 02/2013 %G eng %1 http://www.mcs.anl.gov/papers/P4037-0213.pdf