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An algorithm is proposed to solve robust control problems constrained bypartial differential equations with uncertain coefficients, based on theso-called MG/OPT framework. The levels in this MG/OPT hierarchy correspond todiscretization levels of the PDE, as usual. For stochastic problems, therelevant quantities (such as the gradient) contain expected value operators oneach of these levels. They are estimated using a multilevel Monte Carlo method,the specifics of which depend on the MG/OPT level. Each of the optimizationlevels then contains multiple underlying multilevel Monte Carlo levels. TheMG/OPT hierarchy allows the algorithm to exploit the structure inherent in thePDE, speeding up the convergence to the optimum. In contrast, the multilevelMonte Carlo hierarchy exists to exploit structure present in the stochasticdimensions of the problem. A statement about the rate of convergence of thealgorithm is proven, and some additional properties are discussed. Theperformance of the algorithm is numerically investigated for three test cases.A reduction in the number of samples required on expensive levels and thereforein computational time can be observed.

MG/OPT and MLMC for Robust Optimization of PDEs