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We present a method for optimal control of systems governed by partial differential equations (PDEs) with uncertain parameter fields. We will consider an objective function that involves the mean and variance of the control objective, leading to a risk-averse optimal control problem. Conventional numerical methods for optimization under uncertainty are prohibitive when applied to this problem. For example, sampling the (discretized infinite-dimensional) parameter space to approximate the mean and variance would require solution of an enormous number of PDEs. Thus, to make the optimal control problem tractable, we invoke a quadratic Taylor series approximation of the control objective with respect to the uncertain parameter field. This enables deriving explicit expressions for the mean and variance of the control objective in terms of its gradients and Hessians with respect to the uncertain parameter. The risk-averse optimal control problem is then formulated as a PDE-constrained optimization problem with constraints given by the forward and adjoint PDEs defining these gradients and Hessians. We illustrate our approach with two specific problems: the control of a semilinear elliptic PDE with an uncertain boundary source term, and the control of a linear elliptic PDE with an uncertain coefficient field. We will show that the cost of computing the risk-averse objective and its gradient with respect to the control-measured in the number of PDE solves-is independent of the (discretized) parameter and control dimensions, leading to an efficient quasi-Newton method for solving the optimal control problem. We will end the talk with a comprehensive numerical study of an optimal control problem for fluid flow in a porous medium with uncertain permeability field.

(Reference: https://epubs.siam.org/doi/abs/10.1137/16M106306X)