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We present a method for optimal control of systems governed by partialdifferential equations (PDEs) with uncertain parameter fields. We consider anobjective function that involves the mean and variance of the controlobjective, leading to a risk-averse optimal control problem. To make theproblem tractable, we invoke a quadratic Taylor series approximation of thecontrol objective with respect to the uncertain parameter. This enablesderiving explicit expressions for the mean and variance of the controlobjective in terms of its gradients and Hessians with respect to the uncertainparameter. The risk-averse optimal control problem is then formulated as aPDE-constrained optimization problem with constraints given by the forward andadjoint PDEs defining these gradients and Hessians. The expressions for themean and variance of the control objective under the quadratic approximationinvolve the trace of the (preconditioned) Hessian and are thus prohibitive toevaluate. To address this, we employ trace estimators that only require amodest number of Hessian-vector products. We illustrate our approach with twoproblems: the control of a semilinear elliptic PDE with an uncertain boundarysource term, and the control of a linear elliptic PDE with an uncertaincoefficient field. For the latter problem, we derive adjoint-based expressionsfor efficient computation of the gradient of the risk-averse objective withrespect to the controls. Our method ensures that the cost of computing therisk-averse objective and its gradient with respect to the control, measured inthe number of PDE solves, is independent of the (discretized) parameter andcontrol dimensions, and depends only on the number of random vectors employedin the trace estimation. Finally, we present a comprehensive numerical study ofan optimal control problem for fluid flow in a porous medium with uncertainpermeability field.

Mean-variance risk-averse optimal control of systems governed by PDEs with random parameter fields using quadratic approximations