  Many problems in engineering and sciences require the solution of large scale
optimization constrained by partial differential equations (PDEs). Though
PDE-constrained optimization is itself challenging, most applications pose
additional complexity, namely, uncertain parameters in the PDEs. Uncertainty
quantification (UQ) is necessary to characterize, prioritize, and study the
influence of these uncertain parameters. Sensitivity analysis, a classical tool
in UQ, is frequently used to study the sensitivity of a model to uncertain
parameters. In this article, we introduce "hyper-differential sensitivity
analysis" which considers the sensitivity of the solution of a PDE-constrained
optimization problem to uncertain parameters. Our approach is a goal-oriented
analysis which may be viewed as a tool to complement other UQ methods in the
service of decision making and robust design. We formally define
hyper-differential sensitivity indices and highlight their relationship to the
existing optimization and sensitivity analysis literatures. Assuming the
presence of low rank structure in the parameter space, computational efficiency
is achieved by leveraging a generalized singular value decomposition in
conjunction with a randomized solver which converts the computational
bottleneck of the algorithm into an embarrassingly parallel loop. Two
multi-physics examples, consisting of nonlinear steady state control and
transient linear inversion, demonstrate efficient identification of the
uncertain parameters which have the greatest influence on the optimal solution.
