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

TL;DR

This paper develops a computationally efficient method for risk-averse optimal control of PDE systems with uncertain parameters, using quadratic approximations and trace estimators to handle mean and variance objectives.

Contribution

It introduces a quadratic Taylor series approximation approach for PDE-constrained risk-averse control problems with explicit gradient and Hessian expressions, enabling scalable computations.

Findings

Method effectively computes risk-averse controls with PDE solves independent of parameter dimension.

Trace estimators reduce computational cost for Hessian trace evaluations.

Numerical experiments demonstrate the approach's efficiency and accuracy in uncertain PDE control problems.

Abstract

We present a method for optimal control of systems governed by partial differential equations (PDEs) with uncertain parameter fields. We consider an objective function that involves the mean and variance of the control objective, leading to a risk-averse optimal control problem. To make the problem tractable, we invoke a quadratic Taylor series approximation of the control objective with respect to the uncertain parameter. 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. The expressions for the mean and variance of the control objective under the quadratic approximation involve the trace of the (preconditioned) Hessian and are thus prohibitive to evaluate. To address this, we employ trace estimators that only require a modest number of Hessian-vector products. We illustrate our approach with two 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. For the latter problem, we derive adjoint-based expressions for efficient computation of the gradient of the risk-averse objective with respect to the controls. Our method ensures 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, and depends only on the number of random vectors employed in the trace estimation. Finally, we present a comprehensive numerical study of an optimal control problem for fluid flow in a porous medium with uncertain permeability field.