Complexity Analysis of stochastic gradient methods for PDE-constrained optimal Control Problems with uncertain parameters

¹ Criteo AI Lab, 4 Rue des Méridiens, 38130 Échirolles, France.
² Department of Mathematics, RWTH Aachen, 52062 Aachen, Germany.
³ CSQI, Institute of Mathematics, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland.

^* Corresponding author: mat.martin@criteo.com

Received: 5 May 2020
Accepted: 22 May 2021

Abstract

We consider the numerical approximation of an optimal control problem for an elliptic Partial Differential Equation (PDE) with random coefficients. Specifically, the control function is a deterministic, distributed forcing term that minimizes the expected squared L² misfit between the state (i.e. solution to the PDE) and a target function, subject to a regularization for well posedness. For the numerical treatment of this risk-averse Optimal Control Problem (OCP) we consider a Finite Element discretization of the underlying PDE, a Monte Carlo sampling method, and gradient-type iterations to obtain the approximate optimal control. We provide full error and complexity analyses of the proposed numerical schemes. In particular we investigate the complexity of a conjugate gradient method applied to the fully discretized OCP (so called Sample Average Approximation), in which the Finite Element discretization and Monte Carlo sample are chosen in advance and kept fixed over the iterations. This is compared with a Stochastic Gradient method on a fixed or varying Finite Element discretization, in which the expectation in the computation of the steepest descent direction is approximated by Monte Carlo estimators, independent across iterations, with small sample sizes. We show in particular that the second strategy results in an improved computational complexity. The theoretical error estimates and complexity results are confirmed by numerical experiments.